2.2.1. Proof of Theorem 2.2

1. (1) Using Fourier theory, we transform the Lax equation (2.7) into the integral equation:

   (2.22)\begin{equation} m_0(x_1,x_2,\lambda)= 1-\left[\frac{\widehat{u_0m_0}}{p_\lambda(\xi,\eta)}\right]^{\vee}=1-G_\lambda\ast u_0m_0, \end{equation}

   where the Green function is defined by:

   (2.23)\begin{equation} G_\lambda=\left[\frac{1}{p_\lambda}\right]^{\vee},\ \ p_\lambda(\xi,\eta)= (2\pi i\xi+\lambda)^2-(2\pi i\eta+\lambda^2). \end{equation}

   Hence, the unique solvability of the Lax equation and eigenfunction estimates follows from:

   (2.24)\begin{gather} \left|\frac 1{p_\lambda}\right|_{L^1(\Omega_\lambda, d\xi d\eta)}\le \frac C{(1+|\lambda_I|^2)^{1/2}},\ \left|\frac 1{p_\lambda}\right|_{L^2(\Omega_\lambda^c,d\xi d\eta)}\le \frac C{(1+|\lambda_I|^2)^{1/4}}, \end{gather}

   where $\Omega_\lambda=\{(\xi,\eta)\in{\mathbb R}^2\ :\ |p_\lambda(\xi,\eta)| \lt 1\}$.

2. (2) To define the scattering data, we compute the $ \partial_{\bar\lambda}$-data of the eigenfunction m ₀. This requires computing the $ \partial_{\bar\lambda}(1/ {p_\lambda})$ and establishing the commutative relation between p_λ and the exponential function:

   \begin{equation*} \begin{gathered} \partial_{\bar\lambda}\left[ \frac{1}{p_\lambda}\right] = -\frac{\operatorname{sgn}(\lambda_I)}{2\pi i}\delta{(\xi-\frac{\overline\lambda-\lambda}{2\pi i},\eta-\frac{\overline\lambda^2-\lambda^2}{2\pi i})},\\ p_\lambda(D)f= e^{(\overline\lambda-\lambda)x_1+(\overline\lambda^2-\lambda^2)x_2}p_{\overline \lambda}(D)e^{-[(\overline\lambda-\lambda)x_1+(\overline\lambda^2-\lambda^2)x_2 ]} f. \end{gathered} \end{equation*}

   From these two formulas, we obtain:

   \begin{gather*} \left[\partial_{\bar\lambda}G_\lambda\right]\ast u_0m_0= -s_c(\lambda)e^{(\overline\lambda-\lambda)x_1+(\overline\lambda^2-\lambda^2)x_2},\\ G_\lambda\, e^{(\overline\lambda-\lambda)x_1+(\overline\lambda^2-\lambda^2)x_2}= e^{(\overline\lambda-\lambda)x_1+(\overline\lambda^2-\lambda^2)x_2} \, G_{\overline\lambda}. \end{gather*}

   Consequently,

   (2.25)\begin{align} &\partial_{\bar\lambda}m_0(x_1,x_2, \lambda)= = s_c(\lambda)e^{(\overline\lambda-\lambda)x_1+(\overline\lambda^2-\lambda^2)x_2}m_0(x_1,x_2,\bar\lambda). \end{align}

   To prove the CIE for m ₀, applying Liouville’s theorem and the Lax equation, there exists $q(x_1,x_2)$ such that

   (2.26)\begin{gather} m_0(x_1,x_2,\lambda)= q(x_1,x_2)+\mathcal CT_0m_0(x_1,x_2,\lambda), \end{gather}

   (2.27)\begin{gather} u_0m_0= -2\lambda \partial_{x_1}q-\partial_{x_1}^2 q+ \partial_{x_2}q+\left(\partial_{x_2}-\partial_{x_1}^2 -2\lambda \partial_{x_1} \right)\mathcal CT_0m_0. \end{gather}

   Via a change of variables

   (2.28)\begin{equation} \begin{aligned} 2\pi i\xi= \overline\zeta-\zeta,&\quad 2\pi i\eta=\overline\zeta^2-\zeta^2,\\ \zeta=-i\pi\xi+ \frac \eta{2\xi}, &\quad d\overline\zeta\wedge d\zeta=\frac{i\pi}{|\xi|}d\xi d\eta, \end{aligned} \end{equation}

   and from (2.23), (2.24), we obtain,

   (2.29)\begin{align} &|\mathcal C T_0\phi| \le C| \iint \frac {{ s_c(\zeta)e^{(\overline\zeta-\zeta)x_1+(\overline\zeta^2-\zeta^2)x_2}\phi}}{\zeta-\lambda} d\overline\zeta\wedge d\zeta|\\ \le &\ C| \phi|_{L^\infty} \iint \frac {|{ s_c(\zeta(\xi,\eta))|}}{|(2\pi\xi)^ 2-4\pi i\xi\lambda+2\pi i \eta|}d\xi d\eta \nonumber\\ \le &\ C| \phi|_{L^\infty} \{| s_c(\zeta)| _{L^2(d\xi d\eta)}\left|\frac 1{p_\lambda}\right|_{L^2(\Omega^c_\lambda,d\xi d\eta)} +{| s_c(\zeta) |}_{L^\infty(d\xi d\eta)}\left|\frac 1{p_\lambda}\right|_{L^1(\Omega_\lambda, d\xi d\eta)}\}. \nonumber \end{align}

   Similarly, if $|\left(|\overline\lambda-\lambda|^{l_1} +| \overline\lambda^2-\lambda^2|^{l_2}\right) s_c (\lambda)| _{ L^2\cap L^\infty(d\xi d\eta)} \lt \infty$, $l_1\le 2,\, l_2\le 1$, from (2.24), one has $\left(\partial_{x_2}-\partial_{x_1}^2 -2\lambda \partial_{x_1} \right)\mathcal CT_0m_0\to o(|\lambda|)$ as $\lambda_I\to\infty$. Thus, from (2.27) and for $\lambda\gg 1$, we get $ q=q(x_2)$. By choosing $x_1\gg 1$ in (2.26), we find $q\equiv 1$, justifying the initial CIE (2.10).

2.2.2. Proof of Theorem 2.3

Note that

\begin{equation*} \partial_{\overline\lambda}\Phi(x, \lambda)=s_c(\lambda,x_3)\Phi(x, \overline\lambda). \end{equation*}

Denote

\begin{equation*} \mathcal M_\lambda=- \partial_{x_3}+ \partial_{x_1}^3+\frac 32u\partial_{x_1}+\frac 34u_{x_1}+\frac 34\partial_{x_1}^{-1}u_{x_2}+\rho(\lambda) ,\ \ \rho(\lambda)= -\lambda ^3, \end{equation*}

we have

(2.30)\begin{align} 0=\,&\partial_{\overline\lambda}\left[\mathcal M_\lambda\Phi(x, \lambda)\right]=\mathcal M_\lambda\left[ \partial_{\overline\lambda}\Phi(x, \lambda)\right]=\mathcal M_\lambda\left[ s_c(\lambda,x_3)\Phi(x, \overline\lambda)\right]\\ =\,&\Phi(x, \overline\lambda)\left[ {-\partial_{x_3}}+\rho(\lambda)\right] s_c(\lambda,{x_3})+s_c(\lambda,{x_3})\left[ {\mathcal M_\lambda}-\rho(\lambda)\right] \Phi(x, \overline\lambda)\nonumber\\ =\,&\Phi(x, \overline\lambda)\left[-\partial_{x_3}+\rho(\lambda)\right] s_c(\lambda,{x_3})+ s_c(\lambda,{x_3})\left[{ \mathcal M_{\overline \lambda}-\rho( \overline \lambda)}\right]\Phi(x, \overline\lambda)\nonumber\\ =\, &\Phi(x, \overline\lambda)\left[-\partial_{x_3}+\rho(\lambda)-\rho( \overline \lambda)\right] s_c(\lambda,{x_3}).\nonumber \end{align}

2.2.3. Proof of Theorem 2.4

1. (1) Unique solvability of the CIE (2.15) and the estimate (2.16) follow from

   (2.31)\begin{equation} \begin{aligned} \hskip.6 in|\mathcal C T\phi| \le \ C| \phi|_{L^\infty} \{| s_c(x_3,\zeta)| _{L^2(d\xi d\eta)}\left|\frac 1{p_\lambda}\right|_{L^2(\Omega^c_\lambda,d\xi d\eta)} \\ \hskip1 in +{| s_c(x_3,\zeta) |}_{L^\infty(d\xi d\eta)}\left|\frac 1{p_\lambda}\right|_{L^1(\Omega_\lambda, d\xi d\eta)}\} \hskip.2 in \end{aligned} \end{equation}

   which is proved by the same argument as (2.29).

2. (2) To prove the Lax equation (2.17), we introduce the shorthand notation for the heat operator

   \begin{equation*} \begin{gathered} -\partial_{x_2}+\partial_{x_1}^2+2 \lambda\partial_{x_1}=-\nabla_2+\nabla_1^2 ,\\ \nabla_1=\partial_{x_1}+\lambda,\ \nabla_2=\partial_{x_2}+\lambda^2 ,\ \left[\nabla_j,\ T\right]=0.\end{gathered} \end{equation*}

   Applying the heat operator to both sides of the CIE (2.15), formally,

   (2.32)\begin{equation} \begin{aligned} (-\nabla_2+\nabla_1^2 )m =& \left[-\nabla_2+\nabla_1^2, \mathcal CT \right] m +\mathcal CT(-\nabla_2+\nabla_1^2) m, \end{aligned} \end{equation}

   and

   (2.33)\begin{align} & \left[-\nabla_2+\nabla_1^2 ,\mathcal CT \right] m= \left[-\nabla_2+\nabla_1^2 ,\mathcal C \right] T m =2 \left[\lambda,\mathcal C \right] \partial_{x_1}\left(T m\right)\\ =& {\frac 1{\pi i}\partial_{x_1}\iint T m \ d\overline\zeta\wedge d\zeta\equiv-u(x)} , \nonumber \end{align}

   along with the unique solvability of the CIE, yields the Lax equation

   (2.34)\begin{align} \qquad\ \, &(-\nabla_2+\nabla_1^2 )m=-(1-\mathcal CT)^{-1}u(x)1=-u(x)(1-\mathcal CT)^{-1}1=-u(x)m(x,\lambda). \end{align}

   To rigorously justify the argument, we focus on a priori estimates for:

   \begin{equation*} \widehat u=-\widehat{m-1}\ast {\widehat u}-p_\lambda(\xi,\eta)\widehat{m-1}. \end{equation*}

   We will derive these estimates:

   (2.35)\begin{gather} |\widehat{m-1}|_{L^1(d\xi d\eta)}\le C|s_c|_{L^2\cap L^\infty(d\xi d\eta)}, \end{gather}

   (2.36)\begin{gather} |{(1+|\xi|^k+|\eta|^k)}p_\lambda(\xi,\eta)\widehat{m-1}|_{L^2\cap L^\infty(d\xi d\eta)}\le C | (1+|\xi|^{k+2}+|\eta|^{k+2}) s_c|_{L^2\cap L^\infty(d\xi d\eta)}, \end{gather}

   allowing us to apply Minkowski inequality to obtain:

   (2.37)\begin{gather} | (1+|\xi|^k+|\eta|^k) \widehat u(x) |_{L^2\cap L^\infty(d\xi d\eta)}\le C | (1+|\xi|^{k+2}+|\eta|^{k+2}) s_c|_{L^2\cap L^\infty(d\xi d\eta)}, \end{gather}

   (2.38)\begin{gather} (-\nabla_2+\nabla_1^2 )m\in L^\infty. \end{gather}

   Write the CIE (2.15) as

   (2.39)\begin{align} \widehat{m-1}=&\left[\mathcal CT(m-1) \right]^\wedge+ \left[\mathcal CT1 \right]^\wedge = \left[\mathcal CT(m-1) \right]^\wedge+\frac{2\pi is_c(\zeta(\xi,\eta))}{p_\lambda(\xi,\eta)}. \end{align}

   Using $\left|\frac{2\pi is_c(\zeta(\xi,\eta))}{p_\lambda(\xi,\eta)}\right|_{L^1}\le C|s_c|_{L^2\cap L^\infty(d\xi d\eta)}$ and

   \begin{align*} \left[\mathcal CTf \right]^\wedge=& -\frac 1{2\pi i}\iint_{\mathbb C}\frac{s_c(\zeta)\widehat f(\xi-\frac{\bar\zeta-\zeta} {2\pi i},\eta-\frac{\bar\zeta^2-\zeta ^2}{2\pi i},\bar\zeta)}{\zeta-\lambda}d\overline\zeta\wedge d\zeta\equiv R_{s_c}\widehat f, \end{align*}

   which is a contraction of $\widehat f\in L^1(d\xi d\eta)$, we prove (2.35).

   Next, we express (2.39) as:

   \begin{gather*} p_\lambda \widehat{m-1}= p_\lambda R_{s_c}\widehat{m-1}+ {2\pi is_c(\zeta(\xi,\eta))} \equiv M_{s_c}\left(p_\zeta \widehat{m-1} \right)+ {2\pi is_c(\zeta(\xi,\eta))}, \\ M_{s_c}f=\left(R_{s_c} f\right)(\xi,\eta,\lambda)-\left(R_{s_c}f\right)(\xi,\eta, \frac{\eta} {2\xi}-i\pi\xi ), \end{gather*}

   and we prove

   \begin{gather*} |(1+|\xi|^k+|\eta|^k) M_{s_c}f|_{L^2 \cap L^\infty} \le C|(1+|\xi|^{k+2}+|\eta|^{k+2})s_c|_{L^2\cap L^\infty} | (1+|\xi|^k+|\eta|^k)f|_{L^2 \cap L^\infty} . \end{gather*}

   This proves (2.36).

3. (3) To justify the KP equation (2.21), we verify the Lax pairs. If $| (|\overline\lambda-\lambda|^{l_1} +| \overline\lambda^2-\lambda^2|^{l_2} ) s_c (\lambda)| _{ L^2\cap L^\infty(d\xi d\eta)} \lt \infty$, $l_1\le 5,\, l_2\le 2$, using the representation formula (2.18), we define $\Phi(x,\lambda)= e^{\lambda x_1+ \lambda ^2x_2} m(x, \lambda)$ and the evolution operator:

   (2.40)\begin{equation} \mathcal M =- \partial_{x_3}+ \partial_{x_1}^3+\frac 32u\partial_{x_1}+\frac 34u_{x_1}+\frac 34\partial_{x_1}^{-1}u_{x_2} -\lambda ^3. \end{equation}

   Then

   \begin{equation*} \begin{aligned} \mathcal M \Phi(x,\lambda)=&e^{\lambda x_1+ \lambda ^2x_2}\left(\mathcal M+3\lambda\partial_{x_1}^2+3\lambda^2\partial_{x_1}+\lambda^3+\frac 32u\lambda\right)m(x,\lambda)\\ \equiv& e^{\lambda x_1+ \lambda ^2x_2}\mathfrak Mm. \end{aligned} \end{equation*}

   Reversing the procedure to prove (2.30), we obtain:

   (2.41)\begin{equation} \partial_{\overline\lambda}\left( \mathfrak Mm \right)(x,\lambda)=s_c(\lambda)e^{(\overline\lambda-\lambda)x_1+(\overline\lambda^2-\lambda^2)x_2 +(\overline\lambda^3-\lambda^3)x_3} \left( \mathfrak Mm \right)(x, \overline\lambda). \end{equation}

   As $ |\lambda|\to\infty$, $m( x,\lambda)\sim\sum_{j=0}^\infty\frac{M_j( x )}{\lambda^{j}}$. From the Lax equation (2.17), we obtain

   (2.42)\begin{gather} 2\partial_{x_1}M_{j+1} = (\partial_{x_2}-\partial_{x_1}^2-u)M_j,\\ M_0= 1,\ M_1= - \frac {1}2 \partial_{x_1}^{-1}u, \ M_2= -\frac 14 \partial_{x_2} \partial_{x_1}^{-2}u+ \frac 14 u+\frac 14 \partial_{x_1}^{-1}\left(u \partial_{x_1}^{-1} u\right),\ \cdots\nonumber \end{gather}

   As a result, as $\lambda\to\infty$,

   (2.43)\begin{align} &\mathfrak Mm\\ \to&\frac34 u_{x_1}+\frac 34 \partial_{x_1}^{-1}u_{x_2} +3\lambda\partial_{x_1}^2(1+\frac{M_1}{\lambda}) +3\lambda^2\partial_{x_1}(1+\frac{M_1}{\lambda}+\frac{M_2}{\lambda^2})+\frac 32u\lambda \nonumber\\ =&\frac34 u_{x_1}+\frac 34 \partial_{x_1}^{-1}u_{x_2}+\left( -\frac32u_{x_1}+3\partial_{x_1}[-\frac 14\partial_{x_2}\partial_{x_1}^{-2}u+\frac{u}{4}+\frac14\partial_{x_1}^{-1}(u\partial_{x_1}^{-1}u)] \right)\nonumber\\ +& \lambda\left( 3\partial_{x_1}M_1+\frac{3}{2}u\right) +\frac32u(-\frac 12\partial_{x_1}^{-1}u)\nonumber\\ =&0.\nonumber \end{align}

   Using the unique solvability of the CIE, we conclude that $\mathfrak M m(x,\lambda)=0$, $\mathcal M \Phi(x,\lambda)=0$, thus verifying the Lax pair and justifying the KPII equation.

3.2.1. Statement of results

Building upon Boiti et al.’s work [Reference Boiti, Pempinelli and Pogrebkov4–Reference Boiti, Pempinelli, Pogrebkov and Prinari10], rigorous direct scattering theory for perturbed $ {\mathrm{Gr}(1,2)_{ \gt 0}}$ KP solitons is carried out in [Reference Wu31, Reference Wu32].

Theorem 3.2. (Direct Scattering Theory)

[Reference Wu31, Reference Wu32] Given initial data

(3.10)\begin{equation} \begin{array}{c} u_0(x_1,x_2)=u_s(x_1,x_2,0)+v_0(x_1,x_2), \end{array} \end{equation}

where $ u_s(x)$ is a ${\mathrm{Gr}(1,2)_{ \gt 0}}$ KP soliton and $ \sum_{|l|\le {d+8}} |{{(1+|x_1|+|x_2|)}}\partial_x^lv _0|_ {L^1\cap L^\infty} \ll 1$, $d\ge 0$, we have:

1. (1) For $ \lambda\in{\mathbb C}\backslash\{0,\kappa_1,{\kappa_2}\}$, there exists a unique solution to the Lax equation:

   (3.11)\begin{align} (-\partial_{x_2}+\partial_{x_1}^2 +2\lambda\partial_{x_1} +u_0(x_1,x_2))m_0(x_1,x_2 ,\lambda)= 0, \end{align}

   (3.12)\begin{align} \lim_{|x|\to\infty}m_0(x_1,x_2,\lambda)= \chi(x_1,x_2,0,\lambda) . \end{align}

2. (2) The forward scattering transform is defined by

   (3.13)\begin{equation} \mathcal S(u_0 )=( 0, \kappa_1, \kappa_2,\mathcal D,s _c(\lambda)) \end{equation}

   where the scattering data satisfy the CIE and the $\mathcal D$-symmetry

   (3.14)\begin{gather} { m}_0(x_1,x_2, \lambda) =1+ \frac{ m_{0, \operatorname{res}} (x_1,x_2 )}{\lambda} +\mathcal C T_0 m_0(x_1,x_2, \lambda) , \end{gather}

   (3.15)\begin{gather} (e^{\kappa_1x_1+\kappa_1^2x_2} m_0(x_1,x_2,\kappa^+_1), e^{\kappa_2x_1+\kappa_2^2x_2} m_0(x_1,x_2,\kappa^+_2))\mathcal D=0. \end{gather}

   Here, $m_0\in W_0, \,\kappa_j^+=\kappa_j+0^+,\,\mathcal C$ is the Cauchy integral operator, T ₀ is the continuous scattering operator at $x_3=0$ defined by (2.13), and s_c is the continuous scattering data arising from the $\overline\partial$-characterization

   (3.16)\begin{equation} \begin{gathered} \partial_{\overline\lambda}m_0(x_1,x_2, \lambda) = s_c(\lambda) e^{(\overline\lambda-\lambda)x_1+(\overline\lambda^2-\lambda^2)x_2}m_0{(x_1,x_2, \overline\lambda)},\ \lambda\notin{\mathbb R},\\ s_c(\lambda) = \frac {\operatorname{sgn}(\lambda_I)}{2\pi i} \left[\xi(\cdot,0,\bar\lambda) v_0(\cdot) m_0 (\cdot,\lambda)\right]^\wedge(\frac{\overline\lambda-\lambda}{2\pi i},\frac{\overline\lambda^2-\lambda^2}{2\pi i}), \end{gathered} \end{equation}

   and $ \xi(x,\lambda) $ being the normalized Sato adjoint eigenfunction (see (3.34), (3.35) for definition). Moreover, $\mathcal D$ can be computed by

   (3.17)\begin{align} & \mathcal D = {\mathcal D}^\sharp \times \left({\mathcal D}_{11}^\sharp\right)^{-1} \times \kappa_1 = \left( \begin{array}{l} \kappa _1\\ \mathcal D_{21} \end{array} \right), \end{align}

   (3.18)\begin{align} &\mathcal D^\sharp=\left({\begin{array}{l}\mathcal D^\sharp_{1 1}\\ \mathcal D^\sharp_{2 1} \end{array}}\right)= {\left({\begin{array}{l}\mathcal D^\flat_{11}+\dfrac{c_{11}\mathcal D^\flat_{11}}{1-c_{11}}+\dfrac{c_{12}\mathcal D^\flat_{21}}{1-c_{11}}\\ \mathcal D^\flat_{21}+\dfrac{c_{22}\mathcal D^\flat_{21}}{1-c_{22}} \end{array}}\right)}, \end{align}

   (3.19)\begin{align} & \mathcal D^\flat= {diag}\,( \kappa _1 , \kappa _2 )\, A^T=\left( \begin{array}{l} \kappa _1\\ \kappa _2 a \end{array} \right), \end{align}

   with $c_{jl}=-\int\Psi _j(x_1,x_2,0 ) v_0(x_1,x_2)\varphi_l(x_1,x_2,0 )dx_1dx_2$, $\Psi_j(x)$, $\varphi_l(x)$ residues of the adjoint eigenfunction $\Psi(x,\lambda)$ at κ_j [Reference Wu32, (3.17))] and values of the Sato eigenfunction $\varphi(x,\lambda)$ at κ_l; ${W_0} =W_{(x_1,x_2,0)}$ is the eigenfunction space defined in Definition 3.3.

   Finally, the scattering data $\mathcal S(u_0)$ satisfies the algebraic and analytic constraints

   (3.20)\begin{align} & s_c(\lambda)= \left\{ {\begin{array}{ll} {\frac{\frac {i}{2} \operatorname{sgn}(\lambda_I)}{\overline\lambda-\kappa_j}\frac{\gamma_j}{1-\gamma _j|\alpha|}}+\operatorname{sgn}(\lambda_I) h_j(\lambda),&\lambda\in D^ \times_{\kappa_j} ,\\ \operatorname{sgn}(\lambda_I) { \hbar_0}(\lambda),&\lambda\in D^\times _{0}, \end{array}} \right.\\ & \mathcal D = \left( \kappa _1, \mathcal D_{21} \right)^T,\nonumber \end{align}

   and

   (3.21)\begin{align} & {\begin{array}{l} |(1-\sum_{j=1}^2\mathcal E_{{\kappa_j}} ) \sum_{|l|\le {d+8}}|\left(|\overline\lambda-\lambda|^{l_1} +| \overline\lambda^2-\lambda^2|^{l_2}\right) s_c (\lambda)| _{ L^\infty} \\ + \sum_{j=1}^2(|\gamma_j|+|h_j|_{L^\infty(D_{\kappa_j})})+ |\hbar_0|_{ C^1(D_{0})}+{|\mathcal D -\mathcal D^\flat|_{L^\infty}} \\ \le {C\sum_{|l|\le {d+8}} |{ (1+|x_1|+|x_2|)} \partial_{x} ^{l} v_0|_{L^1\cap L^\infty}} , \end{array}} \end{align}

   (3.22)\begin{align} &s_c(\lambda)= \overline{s_c( \overline\lambda)}, h_j(\lambda)=-\overline{h_j( \overline\lambda)}, \hbar_0(\lambda)=-\overline{\hbar_0( \overline\lambda)}. \end{align}

   Here $D_{z,a\delta}=\{ \lambda=z+re^{i\alpha}:0\le r\le {a \delta},|\alpha|\le\pi\}$, $D_{z,a\delta}^\times=D_{z,a\delta}\backslash\{z\} $, $1\ge \delta=\frac 12\inf \{|z-z'|:z, z' \in \{0,\, \kappa_1,\, \kappa_2 \},\ z\ne z'\}$, $ \mathcal E_{z,a\delta}(\lambda)\equiv 1$ on $D_{z,a\delta}$, $ \mathcal E_{z,a\delta} (\lambda)\equiv 0$ elsewhere. We suppress the $a\delta$-dependence for simplicity if a = 1.

Definition 3.3. The eigenfunction space ${W_0} =W_{x_1,x_2,0}$ consists of functions ϕ that satisfy the following conditions:

1. (a) $\phi (x_1,x_2, \lambda)=\overline{\phi (x_1,x_2, \overline\lambda)};$

2. (b) $(1- \mathcal E_{0} )\phi(x_1,x_2, \lambda)\in L^\infty;$

3. (c) For $\lambda \in D_{0}^\times$, $ \phi(x_1,x_2, \lambda)=\frac{{\phi_{0,\operatorname{res}}(x_1,x_2)}}{\lambda} +\phi_{0,r}(x_1,x_2, \lambda)$, $ \phi_{0,\operatorname{res}}$, $\phi_{0,r} \in L^\infty( D_{0})$;

4. (d) For $\lambda =\kappa_j+re^{i\alpha}\in D_{\kappa_j}^\times$, $\phi=\phi^\flat+\phi^\sharp $, $\phi^\flat =\sum_{l=0}^\infty \phi_l(X_1,X_2)(-\ln(1-\gamma_j|\alpha|))^l \in L^\infty(D_{\kappa_j}) $, $\phi^\sharp \in C_{\tilde\sigma}^{\mu} (D_{\kappa_j,\frac{1}{\tilde\sigma}})\cap L^\infty (D_{\kappa_j})$, and $\phi^\sharp(x_1,x_2,\kappa_j)=0$.

Here, $ \tilde\sigma= \max\{1, |X_1|, \sqrt{|X_2|}\}$ is the rescaling parameter, and X_k is defined by the phase function coefficients:

(3.23)\begin{equation} \begin{aligned} \wp(x_1,x_2,\lambda)=& i[(\overline\lambda- \lambda){x_1}+(\overline\lambda^2-\lambda^2){x_2}] = X_1r\sin\alpha +X_2r^2 \sin2\alpha \equiv \wp(r,\alpha,X),\\ X_1 = &2(x_1+2x_2z ),\ X_2 = 2 x_2 , \ \lambda=z+re^{i\alpha} \in D_z. \end{aligned} \end{equation}

Finally, $C_{\tilde\sigma}^{\mu} (D_{\kappa_j,\frac{1}{\tilde\sigma}})=C (D_{\kappa_j,\frac{1}{\tilde\sigma}})\cap H_{\tilde\sigma}^{\mu} (D_{\kappa_j,\frac{1}{\tilde\sigma}})$ and $H_{\tilde\sigma}^{\mu} (D_{z,\frac{1}{\tilde\sigma}})$ is the rescaled H $\ddot{\mbox{o}}$lder space for $z\in{\mathbb R}$, where the norm is given by:

(3.24)\begin{equation} \begin{aligned} &|\phi|_{H_{\tilde\sigma}^\mu (D_{z,\frac{1}{\tilde\sigma}})} \equiv {\sup_{{\tiny {\begin{array}{c} \tilde r_1, \tilde r_2\le 1, |\alpha_1|,|\alpha_2|\le \pi\end{array}}}} \frac{|\phi(\frac{\tilde r_1}{\tilde\sigma}, \alpha_1,X_1,X_2)-\phi(\frac{\tilde r_2}{\tilde\sigma},\alpha_2,X_1,X_2)|}{|\tilde r_1e^{i\alpha_1}-\tilde r_2e^{i\alpha_2}|^\mu}} \lt \infty \end{aligned} \end{equation}

for $ \lambda_j=z+r_j e^{i\alpha_j}=z+\frac{\tilde r_j}{\tilde\sigma} e^{i\alpha_j}\in D_{z,\frac{1}{\tilde\sigma}}$ and $\phi(x,\lambda)\equiv \phi(r,\alpha,X_1,X_2)$.

Theorem 3.4. (Linearization Theorem) [Reference Wu31, Reference Wu32]

If $\Phi= e^{\lambda x_1+ \lambda ^2x_2} m(x, \lambda)$ satisfies the Lax pair (1.2) and

(3.25)\begin{gather} \partial_{\overline\lambda} m(x, \lambda)= { { s}_c(\lambda,x_3)}e^{(\overline\lambda-\lambda)x_1+(\overline\lambda^2-\lambda^2)x_2} m(x,\overline\lambda) , \end{gather}

(3.26)\begin{gather} (e^{\kappa_1x_1+\kappa_1^2x_2}m(x,\kappa^+_1),e^{\kappa_2x_1+\kappa_2^2x_2}m(x,\kappa^+_2)){\mathcal D(x_3)}=0, \end{gather}

with $\mathcal D(x_3)=(\kappa _1, \mathcal D_{21}(x_3) )^T$ then

(3.27)\begin{equation} \begin{gathered} { s}_c(\lambda, x_3)= {e^{(\overline\lambda^3-{\lambda}^3)x_3}}{ s}_c(\lambda ),\quad {\mathcal {\mathcal D}}_{mn}(x_3)= {e^{(\kappa_m^3-\kappa_n^3)x_3}} {\mathcal D}_{mn}. \end{gathered}\end{equation}

We make several remarks to conclude this subsection.

• • Comparing (3.16)–(3.19), (3.34), (2.8) and (3.7), we show that when the discrete or continuous scattering data vanish, the forward scattering transform for perturbed 1-line solitons reduces to transforms for rapidly decaying potentials or 1-line solitons.

• • For a perturbed 1-soliton, away from 0, κ_j, the continuous data s_c and eigenfunction m ₀ are regular, similar to the rapidly decaying potential case. But at 0 and κ_j, the Cauchy integral operator $\mathcal CT_0m_0$ is an oscillatory singular operator. Specifically, at κ_j, s_c has a ‘simple pole with a discontinuous residue’ and m ₀ is multi-valued. To address this, we introduce the rescaled H $\ddot{\mbox{o}}$lder structure $C_{\tilde\sigma}^{\mu} (D_{\kappa_j,\frac{1}{\tilde\sigma}})$ (see Section 3.3).

• • For the KPII equation, small ${L^1(R^2)\cap L^\infty}$ perturbations preserve the discrete scattering data κ_j, consistent with the fact that 1-line solitons form a discrete set in $L^p(R^2)$. In contrast, for the KdV equation, small ${L^1(R)\cap L^\infty}$ perturbations generically alter κ_j and even potentially increase the number of bound states.

3.2.2. The strategy of the proof of Theorem 3.2

Throughout Section 3.2.2, $x=(x_1,x_2,0),\,x'=(x'_1,x'_2,0)$ for simplicity.

1. (1) The Lax equation is proved by transforming the Lax equation into an integral equation with the Green function defined by:

   (3.28)\begin{align} m_0(x,\lambda)=& \chi(x , \lambda)-G \ast v_0m_0, \end{align}

   (3.29)\begin{align} G(x ,x' , \lambda) =&G_c(x,x',\lambda)+G_d(x,x',\lambda), \end{align}

   (3.30)\begin{align} G_c(x,x',\lambda)=&- \frac{\operatorname{sgn}(x_2-x_2')}{2\pi}e^{\lambda (x_1'-x_1)+\lambda^2 (x_2'-x_2)} \int_{{\mathbb R}} \theta((s^2-\lambda_I^2)(x_2 - x_2')) \, \end{align}

   (3.31)\begin{align} \times& \varphi(x ,\lambda_R+is) \psi(x' ,\lambda_R+is) \, ds,\nonumber\\ G_d(x,x',\lambda)=& -\theta(x_2' - x_2)e^{\lambda (x_1'-x_1)+\lambda^2 (x_2'-x_2)}\\ \times&( \theta(\lambda_R - \kappa_1)\varphi_1(x ) \psi_1(x' )+ \theta(\lambda_ R-\kappa_2) \varphi_2(x) \psi_2(x') ) ,\nonumber \end{align}

   and establishing the following estimates:

   (3.32)\begin{align} G\ast f(x,\lambda)\equiv&\iint G(x,x',\lambda)f(x' )dx',\nonumber\\ | G_c(x,x',\lambda)|\le& C (1+\frac1{\sqrt{|x_2-x_2'|}} ), \end{align}

   (3.33)\begin{align} | G_d(x,x',\lambda)|\le& C ,\\ \lim_{|x|\to\infty}G(x,x',\lambda)&\ast f(x') = 0.\nonumber \end{align}

   Here $\theta(s)$ is the Heaviside function, ψ, ξ are the Sato adjoint eigenfunction, normalized Sato adjoint eigenfunction [Reference Boiti, Pempinelli and Pogrebkov6, (2.12)], [Reference Dickey12, Theorem 6.3.8. (6.3.13)]

   (3.34)\begin{align} \psi(x_1,x_2,x_3,\lambda) = & e^{-(\lambda x_1+\lambda^2 x_2)}\frac{\frac {e^{\theta_1(x_1,x_2,x_3)} }{(1-\frac{\kappa_1}\lambda)}+ \frac{ae^{\theta_2(x_1,x_2,x_3)}}{(1-\frac{\kappa_2}\lambda)}}{e^{\theta_1(x_1,x_2,x_3)}+ a e^{\theta_2(x_1,x_2,x_3)} }\\ \equiv& e^{-[\lambda x_1+\lambda^2x_2]} \xi(x_1,x_2,x_3,\lambda),\nonumber \end{align}

   satisfying

   (3.35)\begin{equation} \begin{aligned} &\left(\partial_{x_2}+\partial^2_{x_1}+u_s(x_1,x_2,x_3)\right)\psi(x_1,x_2,x_3,\lambda)=0,\\ & \left(\partial_{x_2}+\partial^2_{x_1}{-}2\lambda\partial_{x_1}+u_s(x_1,x_2,x_3)\right)\xi(x_1,x_2,x_3,\lambda)=0. \end{aligned} \end{equation}

   Finally,

   \begin{equation*} \begin{aligned} \varphi_j(x)\equiv&\varphi(x,\kappa _j)=e^{\kappa_j x_1+\kappa_j^2x_2} \chi_j(x),\ \psi_j(x)\equiv \textrm{res}_{\lambda=\kappa_j}\psi(x,\lambda)=e^{-[\kappa_j x_1+\kappa_j^2x_2]}\xi_j(x). \end{aligned} \end{equation*}

   In the following, we will explain the construction of the Green function and provide estimates.

   • • $\blacktriangleright$ Construction of the Green function (3.29)–(3.31): [Reference Boiti, Pempinelli and Pogrebkov6, Reference Boiti, Pempinelli, Pogrebkov and Prinari10] Using Fourier inversion theorem, the residue theorem and the orthogonality

     (3.36)\begin{equation} \sum_{j=1}^2\varphi_j (x)\psi_j(x')=0, \end{equation}

     we first derive the orthogonality relation

     (3.37)\begin{equation} \begin{aligned} \hskip.8in \delta({x-x'})= \delta({x_2-x'_2}) \{\frac 1{2\pi} {\int_{\mathbb R}} \varphi (x,\lambda_R+is)\psi (x',\lambda_R+is) ds\\ - {\sum_{j=1}^2} \varphi_j(x)\psi_j(x')\theta( \lambda_R-\kappa_j)\}.\hskip.2in \end{aligned} \end{equation}

     Therefore, G defined by (3.29)–(3.31) satisfies

     \begin{equation*} \left(-\partial_{x_2}+\partial^2_{x_1}+2\lambda\partial_{x_1}+u_s(x)\right) G(x,x', \lambda)= \delta(x-x') \end{equation*}

     by applying (3.37) and

     \begin{equation*} \begin{aligned} \hskip.7in \operatorname{sgn} (x_2-x_2')\theta((s^2-\lambda_I^2)(x_2-x_2'))=\theta(x_2-x'_2)\chi_{-}(s)-\theta(x_2'-x_2)\chi_{+}(s), \end{aligned} \end{equation*}

     where $\chi_{\pm}(s)$ the characteristic function for $\{s|\,{Re}(\left[\lambda+is\right]^2-\lambda^2) \gt rless 0\}$.

   • • $\blacktriangleright$ Estimates of the Green function (3.32), (3.33): The proof for G_c only requires the totally non-negative (TNN) condition. For $\lambda\in D_{\kappa_1}^c\cap D_{\kappa_2}^c$, direct computation or properties of special functions give the estimate

     \begin{equation*} \begin{aligned} |G_c (x,x', \lambda)| \le &C (1+\frac 1{\sqrt{|x_2-x_2'|}} ). \end{aligned} \end{equation*}

     For $\lambda\in D_{\kappa_j}^\times$, we define

     (3.38)\begin{equation} G_c(x,x', \lambda)= -\frac{e^{i[\lambda_I(x_1'-x_1)+2\lambda_I\lambda_R(x'_2-x_2)]}}{2\pi}\left(I^{[1]}_j+I^{[2]}_j+I^{[3]}_j+I^{[4]}_j \right), \end{equation}

     where

     (3.39)\begin{align} I^{[1]}_j=:&\int_{- \delta}^ {\delta}\mbox{sgn}(x_2-x_2')\theta((s^2-\lambda_I^2)(x_2-x_2'))\chi(x, \lambda_R+is) \xi(x',\lambda_R+is)\\ &\times [e^{is[x_1-x_1'+2\lambda_R(x_2-x_2')]+(\lambda_I^2-s^2 )(x_2-x_2')}-1] ds,\nonumber\\ I^{[2]}_j=:&\int_{- \delta}^ {\delta}\mbox{sgn}(x_2-x_2')\theta((s^2-\lambda_I^2)(x_2-x_2')) \nonumber\\ &\times[ \chi(x, \lambda_R+is) \xi(x',\lambda_R+is) - \frac{\chi_j(x)\xi_j(x')} {\lambda_R+is-\kappa_j} ]ds, \nonumber\\ I^{[3]}_j=:&\int_{- \delta}^ {\delta}\mbox{sgn}(x_2-x_2')\theta((s^2-\lambda_I^2)(x_2-x_2'))\frac{\chi_j(x)\xi_j(x')} {\lambda_R+is-\kappa_j}ds,\nonumber\\ I^{[4]}_j=:& \left(\int_{-\infty}^{- \delta}+\int_\delta ^\infty\right)\mbox{sgn}(x_2-x_2')\theta((s^2-\lambda_I^2)(x_2-x_2')) \chi(x, \lambda_R+is)\nonumber\\ &\times \xi(x',\lambda_R+is)e^{(s^2-\lambda^2_I)(x_2'-x_2)- is[ (x_1'-x_1)+2 \lambda_R (x_2'-x_2)]} ds. \nonumber \end{align}

     We prove that

     \begin{equation*}|I^{[1]}_j|,\,|I^{[2]}_j|,\,|I^{[3]}_j| \lt C,\quad |I^{[4]}_j| \le C (1+\frac1{\sqrt{|x_2-x_2'|}} ) , \end{equation*}

     The uniform estimates for $|I^{[1]}_j|$ are derived using appropriate changes of variables and the residue theorem, while $|I^{[3]}_j|$ involves logarithmic functions, causing a discontinuity at κ_j.

     For G_d, note that $\varphi_j(x)$ and $\psi_j(x')$ have 2-cells and 0-cells in their nominators, respectively:

     \begin{equation*} \begin{aligned} \varphi_1(x)= \frac{1-a\frac{\kappa_2}{\kappa_1}e^{\theta_1(x)+\theta_2(x)}}{e^{\theta_1(x)}+a e^{\theta_2(x)}},&\quad \varphi_2(x)=\frac{1- \frac{\kappa_1}{\kappa_2}e^{\theta_1(x)+\theta_2(x)}}{e^{\theta_1(x)}+a e^{\theta_2(x)}}, \\ \psi_1(x')=\frac{\kappa_1}{e^{\theta_1(x')}+a e^{\theta_2(x')}},&\quad \psi_2(x')=\frac{a\kappa_2}{e^{\theta_1(x')}+a e^{\theta_2(x')}} . \end{aligned} \end{equation*}

     Following the argument in [Reference Boiti, Pempinelli and Pogrebkov5], [Reference Boiti, Pempinelli and Pogrebkov4] to permute and exchange cells, we obtain the decomposition

     (3.40)\begin{equation} G_d(x,x',\lambda)=G_{d}^1(x,x',\lambda)+G_{d}^2(x,x',\lambda), \end{equation}

     where

     (3.41)\begin{align} &G_{d}^1(x,x',\lambda)\nonumber\\ =\,&(\kappa_2- \kappa_1 ) a\,e^{i\lambda_I[x'_1-x_1 +2\lambda_R(x_2'-x_2 )]}\,\theta(k_{12}(x_2'-x_2))\,\theta((\lambda_R-\kappa_1)(z_{12}-z'_{12})) \nonumber\\ &\times {\operatorname{sgn}(z_{12}-z_{12}')}\dfrac{e^{-k_{12}(x'_2-x_2)+(\lambda_R-\kappa_1)(z'_{12}-z_{12})}\,e^{\theta_1(x')+\theta_2(x)}}{\tau(x)\tau(x')} , \end{align}

     and

     (3.42)\begin{align} &G_{d}^2(x,x',\lambda)\nonumber\\ = &(\kappa_1-\kappa_2 ) a\, e^{i\lambda_I[x'_1-x_1 +2\lambda_R(x_2'-x_2 )]}\theta(k_{12}(x_2'-x_2))\,\theta((\lambda_R-\kappa_2)(z_{12}-z'_{12})) \nonumber\\ &\times {\operatorname{sgn}(z_{12}-z_{12}')}\dfrac{e^{-k_{12}(x'_2-x_2)+(\lambda_R-\kappa_2)(z'_{12}-z_{12})}\,e^{\theta_1(x )+\theta_2(x')}}{\tau(x)\tau(x')} . \end{align}

     Here $z_{mn}=x_1+(\kappa_m+\kappa_n)x_2$, $ z'_{mn}=x'_1+(\kappa_m+\kappa_n)x'_2$, and $ k_{mn}=\lambda_I^2-(\lambda_R-\kappa_m)(\lambda_R-\kappa_n) $ for $m,\,n\in\{1,\,2\}$. Now all exponentials in the numerators are bounded or dominated by the tau functions in the denominators due to the TP condition, thus proving the result.

2. (2)

   • • $\blacktriangleright$The continuous scattering data s_c is derived in a similar way to that for rapidly decaying potentials. Specifically, we first compute $ \partial_{\bar\lambda} G(x,x' ,\lambda)$ and verify the commutative relation between the Green function and the exponential functions:

     (3.43)\begin{equation} \begin{gathered} \partial_{\bar\lambda} G(x,x' ,\lambda)= -\frac {\operatorname{sgn} (\lambda_I)}{2\pi i}e^{(\overline\lambda-\lambda)(x_1-x_1')+(\overline\lambda ^2-\lambda^2)(x_2-x_2')} \chi(x,\overline\lambda )\xi(x',\overline\lambda) ,\\ G_\lambda\, e^{(\overline\lambda-\lambda)x_1+(\overline\lambda^2-\lambda^2)x_2}= e^{(\overline\lambda-\lambda)x_1+(\overline\lambda^2-\lambda^2)x_2} \, G_{\overline\lambda}. \end{gathered}\end{equation}

     As a result,

     (3.44)\begin{align} &\partial_{\overline\lambda}m_0 {(x,\lambda)}= s_c(\lambda) e^{(\overline\lambda-\lambda)x_1+(\overline\lambda^2-\lambda^2)x_2}m_0{(x, \overline\lambda)}, \end{align}

     with $s_c(\lambda) = \frac {\operatorname{sgn}(\lambda_I)}{2\pi i} \left[\xi(\cdot,\bar\lambda) v_0(\cdot) m_0 (\cdot,\lambda)\right]^\wedge(\frac{\overline\lambda-\lambda}{2\pi i},\frac{\overline\lambda^2-\lambda^2}{2\pi i})$.

     We analyse the analytic properties of the continuous scattering data s_c at $\infty,\,\kappa_j$ and 0:

     $\ast$ Away from κ_j, the Fourier theory gives

     (3.45)\begin{align} \hskip.65in& |(1-\sum_{j=1}^2\mathcal E_{\kappa_j}(\lambda))(|\overline\lambda-\lambda|^{l_1}+|\overline\lambda^2-\lambda^2|^{l_2} ) s_c(\lambda)|_{L^\infty} \le C \sum_{h=0}^{l} |\partial_{x}^{h}v_0| _{L^1\cap L^\infty}. \end{align}

     $\ast$ Near κ_j, we derive the following asymptotics for the Green’s function:

     (3.46)\begin{gather} G(x,x', \lambda)={\mathfrak G}_j(x,x')+\frac 1\pi \chi_j(x)\xi_j(x')|\alpha|+\omega_j(x,x', \lambda), \end{gather}

     (3.47)\begin{gather} |{\mathfrak G}_j |_{C(D_{{\kappa}_j})} \le C (1+\frac1{\sqrt{|x_2-x_2'|}} ),\ \omega_j (x,x',\kappa_j)=0, \end{gather}

     (3.48)\begin{gather} |\omega_j |_{L^\infty(D_{ \kappa _j})\cap C^\mu_{\tilde \sigma}(D_{{\kappa}_j},\frac{1}{\tilde \sigma})}\le C(1+\frac {1+|x'|} {\sqrt{|x_2-x_2'|}} ). \end{gather}

     Here to derive (3.48), we have used, for $ \lambda_j=z+r_j e^{i\alpha_j}=z+\frac{\tilde r_j}{\tilde\sigma} e^{i\alpha_j}\in D_{z,\frac{1}{\tilde\sigma}}$, $z\in\{0,\kappa_1,\kappa_2\}$,

     \begin{align*} & |e^{(\overline\lambda-\lambda) ( x_1-x_1') +(\overline\lambda ^2-\lambda^2) (x_2-x_2')} -1|_{H_{\tilde\sigma}^\mu (D_{z,\frac{1}{\tilde\sigma}})}\\ =&\sup_{{\tiny {\begin{array}{c} \tilde r_1, \tilde r_2 \le 1, |\alpha_1|,|\alpha_2|\le \pi\end{array}}}} \\ &\frac{| e^{(\overline\lambda_1-\lambda_1) ( x_1-x_1') +(\overline\lambda _1^2-\lambda^2_1) ( x_2-x_2')} -e^{(\overline\lambda_2-\lambda_2) ( x_1-x_1') +(\overline\lambda _2^2-\lambda^2_2) ( x_2-x_2')}|}{|\tilde r_1e^{i\alpha_1}-\tilde r_2e^{i\alpha_2}|^\mu}\\ \le &\sup_{{\tiny {\begin{array}{c} \tilde r_1, \tilde r_2\le 1, |\alpha_1|,|\alpha_2|\le \pi\end{array}}}}\\ & \frac{|e^{i([X_1-X_1']\frac{\tilde r_1}{\tilde\sigma}\sin\alpha_1+[X_2-X_2'](\frac{\tilde r_1}{\tilde\sigma})^2\sin2\alpha_1)}-e^{i([X_1-X_1']\frac{\tilde r_2}{\tilde\sigma}\sin\alpha_2+[X_2-X_2'](\frac{\tilde r_2}{\tilde\sigma})^2\sin2\alpha_2)}|}{|\tilde r_1e^{i\alpha_1}-\tilde r_2e^{i\alpha_2}|} \\ \le &C (1+|x'|). \end{align*}

     Plugging (3.46) and (3.47) into (3.28), we obtain

     (3.49)\begin{equation} m_0 (x,\kappa_j+0^+e^{i\alpha})= \frac{\Theta_j(x)}{1-\gamma_j |\alpha|}, \end{equation}

     with

     (3.50)\begin{equation} \begin{aligned} \Theta_j(x)=& [1+{\mathfrak G}_j(x,x')\ast v_0(x')]^{-1}\chi_j(x'),\\ \gamma_j =& -\frac 1\pi\iint\xi_j(x )v_0(x )\Theta_j(x)dx, \end{aligned} \end{equation}

     and

     (3.51)\begin{gather} |m_0(x,\lambda)-m_0(x,\kappa_j+0^+e^{i\alpha})|_ {C_{\tilde\sigma}^{\mu} (D_{\kappa_j,\frac{1}{\tilde\sigma}})\cap L^\infty (D_{\kappa_j})} \lt C| { (1+|x|) v_0}|_{L^1\cap L^\infty}, \end{gather}

     (3.52)\begin{gather} |\frac{m_0(x,\lambda)-m_0(x,\kappa_j+0^+e^{i\alpha}))}{\lambda-\kappa_j}|_{ L^\infty(D_{\kappa_j})} \lt C(1+|x|)| { (1+|x|) v_0}|_{L^1\cap L^\infty}. \end{gather}

     Combining (3.34) and (3.49)–(3.52), we obtain:

     \begin{align*} &s_c(\lambda) = \frac {\operatorname{sgn}(\lambda_I)}{2\pi i} \iint e^{(\overline\lambda-\lambda)x_1+(\overline\lambda^2-\lambda^2)x_2} \left(\frac{\xi_j(x)}{\bar\lambda-\kappa_j}+h.o.t.\right) \\ \times& v_0(x) \left(\frac{\Theta_j(x)}{1-\gamma_j |\alpha|}+h.o.t.\right)dx=\dfrac{\frac {i}{2} {\operatorname{sgn}}(\lambda_I)}{\overline\lambda-\kappa_j}\dfrac {\gamma_j}{1-\gamma_j |\alpha|}+\operatorname{sgn}(\lambda_I)h_j(\lambda) \end{align*}

     with $|h_j|_{L^\infty(D_{\kappa_j})} \lt | { (1+|x|) v_0}|_{L^1\cap L^\infty}$.

     $\ast$ For $\lambda\in D_{0}^\times$, similarly, using $|m_{0,r} |_{C^1( D_0)} \lt C (1+|x|)\sum_{|k|=0}^1|\partial_x^k(1+|x|) v_0|_{L^1\cap L^\infty}$, we find:

     \begin{equation*} \begin{aligned} s_c(\lambda) =& \frac {\operatorname{sgn}(\lambda_I)}{2\pi i} \iint e^{(\overline\lambda-\lambda)x_1+(\overline\lambda^2-\lambda^2)x_2} \left( 0\cdot\lambda +h.o.t.\right) \\ \times& v_0(x) \left(\frac{m_{0,\operatorname{res}}(x)}{\lambda}+h.o.t.\right)dx= {\operatorname{sgn}}(\lambda_I)\hbar_0(\lambda) \end{aligned} \end{equation*}

     with $|\hbar_0|_{C^1(D_{0})} \lt |{(1+|x|) v_0}|_{L^1\cap L^\infty}$.

   • • $\blacktriangleright$ Building on the integral equation of m ₀ and the estimates regarding the Green function G, we can identify the following properties of m ₀:

     1. (a) $m_0 (x, \lambda)=\overline{m_0 (x, \overline\lambda)}$.

     2. (b) $(1- \mathcal E_{0} )m_0(x, \lambda)\in L^\infty$.

     3. (c) For $\lambda \in D_{0}^\times$, $ m_0(x_1,x_2, \lambda)=\frac{{m_{0,\operatorname{res}}(x_1,x_2)}}{\lambda} +m_{0,r}(x_1,x_2, \lambda)$, with $ m_{0,\operatorname{res}}$, $m_{0,r} \in L^\infty( D_{0})$.

     4. (d) For $\lambda =\kappa_j+re^{i\alpha}\in D_{\kappa_j}^\times$,

        (3.53)\begin{equation} \begin{gathered} \qquad\qquad\qquad m_0=m_0^\flat+m_0^\sharp ,\qquad m_0^\flat =\frac{\Theta_j(x)}{1-\gamma_j |\alpha|}. \end{gathered} \end{equation}

        Therefore, $m_0^\sharp \in C_{\tilde\sigma}^{\mu} (D_{\kappa_j,\frac{1}{\tilde\sigma}})\cap L^\infty (D_{\kappa_j})$, $ m_0^\sharp(x_1,x_2,\kappa_j)=0$, and

        \begin{equation*} \begin{gathered} m_0^\flat =\sum_{l=0}^\infty m_{0,l}(X_1,X_2)(-\ln(1-\gamma_j|\alpha|))^l \in L^\infty(D_{\kappa_j}) . \end{gathered} \end{equation*}

     These observations confirm that m ₀ is in the eigenfunction space W ₀.

   • • $\blacktriangleright$ To prove the CIE for m ₀, we follow the same method as used for rapidly decaying potentials. Using $T_0m_0\in L^1$, we can apply Liouville’s theorem to show that there exists g(x) such that

     (3.54)\begin{equation} m_0(x, \lambda)=g(x )+\frac{ m_{0,\operatorname{res}} (x )}{\lambda} +\mathcal CT _0 m_0(x ,\lambda). \end{equation}

     To prove $g\equiv 1$, we apply the Lax operator to both sides of (3.54) and utilize the Lax equation, yielding

     (3.55)\begin{equation} \begin{aligned} u(x ) m_0(x, \lambda)=\left(\partial_{x_2}-\partial_{x_1}^2-2 \lambda\partial_{x_1}\right) [g(x )+\frac {m_{0,\operatorname{res}} (x )}{\lambda} ]\\ +\left(\partial_{x_2}-\partial_{x_1}^2-2 \lambda\partial_{x_1}\right)\mathcal CT_0 m_0. \end{aligned} \end{equation}

     Then it reduces to demonstrate that these CI’s are uniformly $o(|\lambda|)$.

     To this aim, we decompose

     (3.56)\begin{align} \mathcal C T_0m_0 = &\iint_{D_0\cup D_{\kappa_1}\cup D_{\kappa_2}} \frac {{ s_c(\zeta)e^{(\overline\zeta-\zeta)x_1+(\overline\zeta^2-\zeta^2)x_2}m_0}}{\zeta-\lambda} d\overline\zeta\wedge d\zeta\\ +&\iint_{{\mathbb C}\backslash (D_0\cup D_{\kappa_1}\cup D_{\kappa_2})} \frac {{ s_c(\zeta)e^{(\overline\zeta-\zeta)x_1+(\overline\zeta^2-\zeta^2)x_2}m_0}}{\zeta-\lambda} d\overline\zeta\wedge d\zeta \nonumber\\ \equiv &P_1+ P_2 .\nonumber \end{align}

     The estimate for P ₁ is standard. The estimate for P ₂ corresponds to the CI near infinity, and this follows from the arguments presented by Wickerhauser.

   • • $\blacktriangleright$ [Reference Boiti, Pempinelli and Pogrebkov6] To prove the $\mathcal D$-symmetry (3.15), we introduce the total Green function $\mathcal K$, which is the fundamental solution of the full Lax operator:

     (3.57)\begin{equation} \begin{gathered} \overrightarrow{\mathcal L_{v_0}}\mathcal K =\mathcal K\overleftarrow{\mathcal L_{v_0}}=\delta(x-x') ,\\ \mathcal L_{v_0}=\mathcal L+v_0,\quad \mathcal L=-\partial_{x_2}+\partial^2_{x_1}+u_s(x) \end{gathered} \end{equation}

     with $\overrightarrow {\mathcal L}$ the operator $\mathcal L$ applying to the x variable of $\mathcal K$ and $\overleftarrow{\mathcal L}$ the operator applying to the xʹ variable of $\mathcal K$. The total Green function $\mathcal K$ can be solved using these integral equations.

     (3.58)\begin{gather} \mathcal K(x,x',\lambda)= \mathcal G-\mathcal G\ast v_0\,\mathcal K,\nonumber\\ \mathcal K(x,x',\lambda)= \mathcal G-\mathcal K\ast v_0\,\mathcal G,\\ \mathcal G(x,x', \lambda) = e^{\lambda (x_1-x_1')+\lambda^2 (x_2-x_2')} G(x,x', \lambda).\nonumber \end{gather}

     Therefore, the eigenfunction Φ and adjoint eigenfunction Ψ can be written as

     (3.59)\begin{equation} \begin{aligned} \Phi(x,\lambda)=& \mathcal K(x,x',\lambda)\ast_{x'}\overleftarrow{\mathcal L}\varphi (x',\lambda) \equiv \mathcal K \ast \overleftarrow{\mathcal L}\varphi ,\\ \Psi(x' ,\lambda)=&\psi(x ,\lambda)\ast_{x} \overrightarrow{\mathcal L}\mathcal K (x,x',\lambda) \equiv \psi\ast\overrightarrow{\mathcal L}\mathcal K , \end{aligned} \end{equation}

     with φ and ψ the Sato eigenfunction and the Sato adjoint eigenfunction (see (3.4), (3.5), (3.34), (3.35)). Furthermore, letting $\mathcal G _j=\lim_{\lambda\to \kappa_j^+}\mathcal G$, $\mathcal K _j=\lim_{\lambda\to \kappa_j^+}\mathcal K$, and successively using (3.59) and (3.58) [Reference Boiti, Pempinelli and Pogrebkov6, Reference Wu32], we can prove:

     (3.60)\begin{align} &(\varphi_1(x), \varphi_2 (x))\mathcal D^\flat=0, \end{align}

     (3.61)\begin{align} &\mathcal G _{j-1} =\mathcal G _j +\varphi_j(x)\psi_j(x'), \end{align}

     (3.62)\begin{align} &\mathcal K _{j-1} =\mathcal K _j +\frac{\Phi_j(x)\Psi_j(x')}{1-c_{jj}}. \end{align}

     Using these formulas, we establish:

     (3.63)\begin{gather} \sum_{j= 1}^{2}\frac{\Phi_j(x)\Psi_j(x')}{1-c_j}=0, \end{gather}

     (3.64)\begin{gather} \mathcal K_{l}=\mathcal K_i+\sum_{j=l+1}^{i+2}\frac{\Phi_j(x)\Psi_j(x')}{1-c_j}. \end{gather}

     Here $ c_j= c_{jj}$ and the mod 2-condition is adopted.

     Applying $ \overleftarrow{\mathcal L}\varphi_i$ to (3.64) from the right and using (3.59), we obtain

     (3.65)\begin{equation} \mathcal K_l \ast \overleftarrow{\mathcal L}\varphi_i=\Phi_i+\sum_{j=l+1}^{i+2}\frac{\Phi_j(x)c_{ji}}{1-c_j}. \end{equation}

     Multiplying (3.65) by $\mathcal D_{im}^\flat$, summing up and using the symmetry (3.60), we derive

     (3.66)\begin{equation} \sum_{i=1}^{2} \Phi_i\mathcal D^\flat_{im}+\sum_{i=1}^{2}\sum_{j=l+1}^{i+2}\frac{\Phi_j(x)c_{ji}\mathcal D^\flat_{im}}{1-c_j}=0. \end{equation}

     Taking l = 2 in (3.66) and using (3.63), we prove

     (3.67)\begin{equation} (e^{\kappa_1x_1+\kappa_1^2x_2} m_0(x_1,x_2,\kappa^+_1), e^{\kappa_2x_1+\kappa_2^2x_2} m_0(x_1,x_2,\kappa^+_2))\mathcal D^\sharp=0. \end{equation}

     Multiplying $\left(\mathcal D_{11}^\sharp\right)^{-1}\kappa_1$ from the right to both sides, we justify

     \begin{equation*} (e^{\kappa_1x_1+\kappa_1^2x_2} m_0(x_1,x_2,\kappa^+_1), e^{\kappa_2x_1+\kappa_2^2x_2} m_0(x_1,x_2,\kappa^+_2))\mathcal D =0. \end{equation*}

3.2.3. The strategy of the proof of Theorem 3.4

Note that [Reference Boiti, Pempinelli and Pogrebkov6, Reference Wu32]

\begin{equation*} \begin{aligned} &\partial_{\overline\lambda}\Phi(x, \lambda)=s_c(\lambda,x_3)\Phi(x, \overline\lambda),\\ &-\kappa_1\Phi(x, \kappa_1)= \mathcal D_{21}(x_3)\Phi(x, \kappa_2). \end{aligned} \end{equation*}

Denote

\begin{equation*} \mathcal M_\lambda=- \partial_{x_3}+ \partial_{x_1}^3+\frac 32u\partial_{x_1}+\frac 34u_{x_1}+\frac 34\partial_{x_1}^{-1}u_{x_2}+\rho(\lambda) ,\ \ \rho(\lambda)= -\lambda ^3. \end{equation*}

The linearity of the continuous scattering data s_c can be proved in the same way as that for rapidly decaying potentials. Similarly,

(3.68)\begin{align} 0= &-\mathcal M_{\kappa_1}\left[\kappa_1\Phi(x, \kappa_1)\right]= \mathcal M_{\kappa_1}\left[ \mathcal D_{21}(x_3)\Phi(x, \kappa_2)\right]\\ = &\Phi(x, {\kappa_2})\left[ {-\partial_{x_3}}+\rho(\kappa_1)\right] \mathcal D_{21}(x_3) + \mathcal D_{21}(x_3)\left[ {\mathcal M_{\kappa_2}}-\rho(\kappa_1)\right]\Phi(x, \kappa_2)\nonumber\\ =&\Phi(x, {\kappa_2})\left[ {-\partial_{x_3}}+\rho(\kappa_1)\right] \mathcal D_{21}(x_3) + \mathcal D_{21}(x_3)\left[ {\mathcal M_{\kappa_2}}-\rho(\kappa_2)\right]\Phi(x, \kappa_2)\nonumber\\ = &\Phi(x, {\kappa_2})\left[-\partial_{x_3}+\rho(\kappa_1)-\rho( \kappa_2)\right] \mathcal D_{21}(x_3).\nonumber \end{align}

3.3.1. Statement of results

We begin with the definition of admissible scattering data:

Definition 3.5. Let $0 \lt \epsilon_0\ll 1$, $d\ge 0$ and u_s be a $ {\mathrm{Gr}(1, 2)_{ \gt 0}}$ KP soliton defined by $\{\kappa_j\}, A=(1,a)$. A scattering data $ {\mathcal S} =(\{0\},\{\kappa_1,\kappa_2\}, \mathcal D ,s _c(\lambda))$ is called d-admissible if

(3.69)\begin{align} & s_c(\lambda)= \left\{ {\begin{array}{ll} {\frac{\frac {i}{2} \operatorname{sgn}(\lambda_I)}{\overline\lambda-\kappa_j}\frac{\gamma_j}{1-\gamma _j|\alpha|}}+\operatorname{sgn}(\lambda_I) h_j(\lambda),&\lambda\in D^ \times_{\kappa_j} ,\\ \operatorname{sgn}(\lambda_I) { \hbar_0}(\lambda),&\lambda\in D^\times _{0}, \end{array}} \right. \end{align}

(3.70)\begin{align} &\mathcal D={{(\kappa_1,\mathcal D _{21})^T}}, \end{align}

and

(3.71)\begin{equation} \begin{array}{l} \epsilon_0\ge (1-\sum_{j=1}^2\mathcal E_{{\kappa_j}} ) \sum_{|l|\le {d+8}}|\left(|\overline\lambda-\lambda|^{l_1} +| \overline\lambda^2-\lambda^2|^{l_2}\right) s_c (\lambda)| _{ L^\infty} \\ \qquad + \sum_{j=1}^2(|\gamma_j|+|h_j|_{L^\infty(D_{\kappa_j})})+ |\hbar_0|_{C^1(D_{0})}+ |\mathcal D - {\mathcal D}^\flat|_{L^\infty},\nonumber\\ s_c(\lambda)= \overline{s_c( \overline\lambda)},\, h_j(\lambda)=-\overline{h_j( \overline\lambda)}, \, \hbar_0(\lambda)=-\overline{\hbar_0( \overline\lambda)},\\ \mathcal D^\flat= {diag}\,( \kappa _1 , \kappa _2 )\, A^T= ( \kappa _1, \kappa _2 a )^T, \kappa_j\ne 0. \end{array} \end{equation}

Define T as the continuous scattering operator

(3.71)\begin{equation} T \phi (x ,\lambda) \equiv { s}_c(\lambda )e^{(\overline\lambda-\lambda)x_1+(\overline\lambda^2-\lambda^2)x_2 +(\overline\lambda^3-\lambda^3)x_3} \phi(x, \overline\lambda). \end{equation}

Definition 3.6. The eigenfunction space ${W} =W_x$ consists of ϕ satisfying

1. (a) $\phi (x, \lambda)=\overline{\phi (x, \overline\lambda)};$

2. (b) $(1- \mathcal E_{0} )\phi(x, \lambda)\in L^\infty;$

3. (c) for $\lambda \in D_{0}^\times$, $ \phi(x, \lambda)=\frac{{\phi_{0,\operatorname{res}}(x)}}{\lambda} +\phi_{0,r}(x, \lambda)$, $ \phi_{0,\operatorname{res}}$, $\phi_{0,r} \in L^\infty( D_{0})$;

4. (d) for $\lambda =\kappa_j+re^{i\alpha}\in D_{\kappa_j}^\times$, $\phi=\phi^\flat+\phi^\sharp $, $\phi^\flat =\sum_{l=0}^\infty \phi_l(X)(-\ln(1-\gamma_j|\alpha|))^l \in L^\infty(D_{\kappa_j}) $, $\phi^\sharp \in C_{\tilde\sigma}^{\mu} (D_{\kappa_j,\frac{1}{\tilde\sigma}})\cap L^\infty (D_{\kappa_j})$, $\phi^\sharp(x,\kappa_j)=0$.

Here

(3.72)\begin{equation} \tilde\sigma= \max\{1, |X_1|, {\sqrt{|X_2|}, \sqrt[3]{|X_3|}}\}, \end{equation}

is the rescaling parameter with X_k the phase function coefficients:

(3.73)\begin{equation} \begin{aligned} \wp( x,\lambda)=& i[(\overline\lambda- \lambda){x_1}+(\overline\lambda^2-\lambda^2){x_2}+(\overline\lambda^3-\lambda^3){x_3}]\qquad \lambda=z+re^{i\alpha} \in D_z \\ =& X_1r\sin\alpha +X_2r^2 \sin2\alpha +X_3r^3 \sin3\alpha \equiv \wp(r,\alpha,X),\\ X_1( x,z)= &2(x_1+2x_2z +3x_3z^2),\ X_2(x,z)= 2(x_2 +3x_3z),\ X_3(x,z)= 2x_3. \end{aligned} \end{equation}

Finally, for $\phi\in W$, define

(3.74)\begin{equation} \begin{aligned} |\phi|_W \equiv & |(1- \mathcal E_{0})\phi|_{L^\infty} + (| \phi_{0,\operatorname{res}}|_{L^\infty}+ | \phi_{0,r}|_{L^\infty(D_{0})}) \\ +& \sum_{j=1}^2 (| \phi^\flat |_{L^\infty(D_{\kappa_j})}+| \phi^\sharp |_{C_{\tilde\sigma}^{\mu} (D_{\kappa_j,\frac{1}{\tilde\sigma}})\cap L^\infty(D_{\kappa_j})}) . \end{aligned} \end{equation}

The theorem of the inverse problem is stated as follows:

Using the direct and inverse scattering theories (Theorems 3.2 and 3.7), we can solve the Cauchy problem for the KPII on 1-line soliton backgrounds (see Corollary 4.6).

3.3.2. The strategy of the proof of Theorem 3.7

1. (1) We will demonstrate the existence, uniqueness and estimates of this system by taking the limit of the iteration sequence

   (3.83)\begin{align} & \phi^{(k)}(x ,\lambda) = 1+ \frac{\phi^{(k)}_{0,\operatorname{res}} (x )}{\lambda} +\mathcal CT \phi^{(k-1)}(x, \lambda) ,\ \ k \gt 0, \end{align}

   (3.84)\begin{align} &(e^{\kappa_1x_1+\kappa_1^2x_2+\kappa_1^3x_3}\phi^{(k)}(x,\kappa^+_1), e^{\kappa_2x_1+\kappa_2^2x_2+\kappa_2^3x_3}\phi^{(k)}(x,\kappa^+_2))\mathcal D=0, \end{align}

   (3.85)\begin{align} &\phi^{(0)}(x ,\lambda)= \chi(x,\lambda) \end{align}

   in the eigenfunction space W.

   Evaluating the CIE (3.83) at $\kappa_1^+$, $\kappa_2^+$ and using the $\mathcal D$-symmetry (3.84), one obtains a linear system of $2+1$ variables $\phi^{(k)}(x,\kappa^+_1)$, $\phi^{(k)}(x,\kappa^+_2)$, and $\phi^{(k)}_{0,\operatorname{res}} (x )$. Hence the iteration turns into

   \begin{align*} \phi^{(k)}(x ,\lambda) = & 1+ \frac{\phi^{(k)}_{0,\operatorname{res}} (x )}{\lambda} +\mathcal CT \phi^{(k-1)}(x, \lambda) ,\ \ k \gt 0,\\ \phi^{(k)}_{0,\operatorname{res}} (x )=&-\dfrac{\mathcal D_{11} e^{\kappa_1x_1+\kappa_1^2x_2+\kappa_1^3x_3}+\mathcal D_{21} e^{\kappa_2x_1+\kappa_2^2x_2+\kappa_2^3x_3}}{\frac{\mathcal D_{11}}{\kappa_1} e^{\kappa_1x_1+\kappa_1^2x_2+\kappa_1^3x_3}+\frac{\mathcal D_{21}}{\kappa_2} e^{\kappa_2x_1+\kappa_2^2x_2+\kappa_2^3x_3}} \\ &-\dfrac{\mathcal D_{11} e^{\kappa_1x_1+\kappa_1^2x_2+\kappa_1^3x_3}} {\frac{\mathcal D_{11}}{\kappa_1} e^{\kappa_1x_1+\kappa_1^2x_2+\kappa_1^3x_3}+\frac{\mathcal D_{21}}{\kappa_2} e^{\kappa_2x_1+\kappa_2^2x_2+\kappa_2^3x_3}}\mathcal CT \phi^{(k-1)}(x, \kappa^+_1)\\ &-\dfrac{\mathcal D_{21}e^{\kappa_2x_1+\kappa_2^2x_2+\kappa_2^3x_3}}{\frac{\mathcal D_{11}}{\kappa_1} e^{\kappa_1x_1+\kappa_1^2x_2+\kappa_1^3x_3}+\frac{\mathcal D_{21}}{\kappa_2} e^{\kappa_2x_1+\kappa_2^2x_2+\kappa_2^3x_3}}\mathcal CT \phi^{(k-1)}(x, \kappa^+_2),\\ \phi^{(0)}(x ,\lambda)= & \chi(x,\lambda) . \end{align*}

By the TP condition, the convergence of the iteration sequence reduces to deriving uniform estimates of CIO near $ \infty$, κ_j and 0.

• • $\blacktriangleright$ Estimates on $D_\infty$: Away from 0, κ_j, the continuous data s_c and the eigenfunction m ₀ are regular, similar to the case of rapidly decaying potentials. Thus, we can derive estimates using Wickerhauser’s arguments.

• • $\blacktriangleright$ Estimates on $D_{\kappa_j}$: To derive estimates on $D_{\kappa_j}$, we take advantage of several factors:

1. (1) The boundedness of h_j and estimates of the Cauchy integral operator:

   (3.86)\begin{equation} \begin{aligned} &|h_j|_{L^\infty(D_{\kappa_j})} \lt | { (1+|x|) v_0}|_{L^1\cap L^\infty},\\ &|\mathcal C\mathcal E_z\phi|_{L^\infty} \le C|\phi|_{L^p(D_z)},\quad |\mathcal C\mathcal E_z\phi|_{H^\nu(D_z)} \le C|\phi|_{L^p(D_z)}, \end{aligned} \end{equation}

   allow us to replace CIO estimates with kernel s_c by those with kernel $\widetilde\gamma_j=\frac{\frac {i}{2} \operatorname{sgn}(\lambda_I)}{\overline\lambda-\kappa_j}\frac{\gamma_j}{1-\gamma _j|\alpha|}$, the leading singular term.

2. (2) The special form of $\widetilde\gamma_j $ allows us to apply Stokes’ theorem to integrate the leading singularity:

   (3.87)\begin{align} &\widetilde\gamma_j (\lambda)=\frac{\frac {i}{2} \operatorname{sgn}(\lambda_I)}{\overline\lambda-\kappa_j}\frac{\gamma_j}{1-\gamma _j|\alpha|}=-\partial_{\bar\lambda}\ln(1-\gamma_j|\alpha|)\\ &\mathcal C\widetilde\gamma_j\mathcal E_{\kappa_j}[-{\ln (1-\gamma_j|\beta|)}]^l =\frac{[- {\ln (1-\gamma_j|\alpha|)}]^{l+1}}{l+1}\nonumber\\ &\hskip.9in -\frac {1}{2\pi i}\oint_{|\zeta-\kappa_j|= \delta}\frac{\frac 1{l+1}[-\ln (1-\gamma_j|\beta|)]^{l+1}} {\zeta-\lambda} d\zeta .\nonumber \end{align}

3. (3) The scaling invariant properties:

   (3.88)\begin{align} &|\mathcal C_\lambda\widetilde\gamma_j e^{-i\wp(x,\zeta)} \mathcal E_{\kappa_j} f(\kappa_j+se^{i\beta}) |_{C_{\tilde\sigma}^{\mu} (D_{\kappa_j,\frac 1{\tilde\sigma}})\cap L^\infty(D_{\kappa_j} )}\\ &\hskip.9in= |\mathcal C_{\tilde\lambda}\widetilde\gamma_je^{-i\wp(\frac{\tilde s}{\tilde\sigma},\beta, X)}\mathcal E_{\kappa_j,\tilde\sigma\delta} f(\kappa_j+\frac{\tilde s}{\tilde\sigma} e^{i\beta})|_{C ^{\mu} (D_{\kappa_j,1})\cap L^\infty(D_{\kappa_j,\tilde\sigma\delta})},\nonumber \end{align}

   where the dilating polar coordinates near $z=\kappa_j$ is defined by

   \begin{equation*} \begin{aligned} \lambda=z+ re^{i\alpha}=z+ \frac{\tilde r}{{\tilde\sigma}}e^{i\alpha},&\quad \zeta=z+ se^{i\beta}=z+ \frac{\tilde s}{{\tilde\sigma}}e^{i\beta}, \\ \tilde\lambda=z+ \tilde r e^{i\alpha},&\quad \tilde\zeta=z+\tilde se^{i\beta}, \end{aligned} \end{equation*}

   and $r,\,s\le\delta$, $ \tilde r,\,{\tilde s} \le {\tilde\sigma}\delta$, $ |\alpha|,|\beta|\le \pi$.

To achieve this, we decompose

(3.89)\begin{align} & \mathcal C_\lambda\widetilde\gamma_je^{-i\wp(x,\zeta)}\mathcal E_{\kappa_j} f\equiv I_1+I_2+I_3+I_4+I_5, \end{align}

with

(3.90)\begin{align} &\hskip.6inI_1= -\frac {\theta(1-\tilde r)}{2\pi i}\iint_{\tilde s \lt 2} \frac{ \widetilde \gamma_j(\tilde s, \beta) f^{\flat}(\frac {\tilde s}{{\tilde\sigma}} ,-\beta,X)} {\tilde\zeta-\tilde \lambda}d\overline{\tilde\zeta} \wedge d\tilde\zeta , \end{align}

(3.91)\begin{align} &\hskip.6inI_2= -\frac {\theta(1-\tilde r)}{2\pi i}\iint_{\tilde s \lt 2} \frac{ \widetilde \gamma_j(\tilde s, \beta)[ e^{-i\wp(\frac {\tilde s}{{\tilde\sigma}} ,\beta,X)}-1]f^{\flat}(\frac {\tilde s}{{\tilde\sigma}} ,-\beta,X)} {\tilde\zeta-\tilde \lambda}d\overline{\tilde\zeta} \wedge d\tilde\zeta , \end{align}

(3.92)\begin{align} &\hskip.6inI_3= -\frac {\theta(1-\tilde r)}{2\pi i}\iint_{\tilde s \lt 2} \frac{ \widetilde \gamma_j(\tilde s, \beta) e^{-i\wp(\frac {\tilde s}{{\tilde\sigma}} ,\beta,X)}f^{\sharp}(\frac {\tilde s}{{\tilde\sigma}} ,-\beta,X)} {{\tilde\zeta}-\tilde \lambda}d\overline{\tilde\zeta} \wedge d{\tilde\zeta} , \end{align}

(3.93)\begin{align} &\hskip.6inI_4= -\frac {\theta(1-\tilde r)}{2\pi i}\iint_{2 \lt \tilde s \lt {\tilde\sigma} \delta} \frac{\widetilde \gamma_j(\tilde s, \beta) e^{-i\wp(\frac {\tilde s}{{\tilde\sigma}},\beta,X)}f(\frac {\tilde s}{{\tilde\sigma}},-\beta,X)}{{\tilde\zeta}-\tilde \lambda}d\overline{\tilde\zeta} \wedge d{\tilde\zeta}, \end{align}

(3.94)\begin{align} &\hskip.6inI_5= -\frac {\theta(\tilde r-1)}{2\pi i}\iint_{ \tilde s \lt {\tilde\sigma} \delta} \frac{ \widetilde\gamma_j(\tilde s, \beta) e^{-i\wp(\frac {\tilde s}{{\tilde\sigma}},\beta,X)}f(\frac {\tilde s}{{\tilde\sigma}} ,-\beta,X)} {{\tilde\zeta}-\tilde\lambda}d\overline{\tilde\zeta} \wedge d{\tilde\zeta}. \end{align}

For I ₁, I ₂ and I ₃, integrals over uniformly compact domains, we will apply Stokes’ theorem (3.87) and Hölder interior estimates [Reference Gakhov13] to derive:

\begin{equation*} |I^\flat_1|_{L^\infty},\ |I^\sharp_1|_{C^\mu_{\tilde\sigma}(D_{\kappa_j,\frac{1}{\tilde\sigma}})},\ |I_2 |_{C^\mu_{\tilde\sigma}(D_{\kappa_j,\frac{1}{\tilde\sigma}})},\ |I_3 |_{C^\mu_{\tilde\sigma}(D_{\kappa_j,\frac{1}{\tilde\sigma}})}. \end{equation*}

For the estimates of

\begin{equation*} |I_4|_{C^\mu_{\tilde\sigma}(D_{\kappa_j,\frac{1}{\tilde\sigma}})} ,\qquad |I_5|_{L^\infty(D_{\kappa_j})}, \end{equation*}

slowly decaying integrals over non-uniformly compact domains, we first express them as iterated integrals in polar coordinates

\begin{align*} &\hskip.6in I_4= -\frac {\theta(1-\tilde r)}{2\pi i}\int_{-\pi}^\pi d\beta[\partial_ {\beta} \ln (1-\gamma_j |\beta| )] \int_{2 \lt \tilde s \lt {\tilde\sigma} \delta} \frac{ e^{-i\wp(\frac {\tilde s}{{\tilde\sigma}} ,\beta,X)}f(\frac {\tilde s}{{\tilde\sigma}},-\beta,X)}{\tilde s - \tilde r e^{i (\alpha-\beta)} } d\tilde s , \\ &\hskip.6in I_5= -\frac {\theta(\tilde r-1)}{2\pi i} \int_{-\pi}^\pi d\beta[\partial_ {\beta} \ln (1-\gamma_j |\beta| )] \int_{0} ^{\widetilde\sigma \delta} \frac{ e^{-i\wp(\frac {\tilde s}{{\tilde\sigma}} ,\beta,X)}f(\frac {\tilde s}{{\tilde\sigma}},-\beta,X)}{\tilde s - \tilde r e^{i (\alpha-\beta)} } d\tilde s. \end{align*}

Then, estimates will be derived by applying holomorphic extension properties in $\tilde s$, the deformation method and stationary point analysis of $\wp$. The main ideas for the deformation are as follows: near stationary points $\tilde s_\ast$, we deform

(3.95)\begin{equation} \tilde s\in{\mathbb R}\quad\longrightarrow\quad \tilde se^{i\tau}\in{\mathbb C} \end{equation}

such that the integral domain of $ \tilde s $ deforms into a union of line segments Γ’s and arcs S’s, satisfying the following conditions:

(3.96)\begin{align} (a)\ &\ {on}\ \Gamma{'s,}\ \mathfrak {Re} ({-i\wp(\frac {\tilde se^{i\tau}}{{\tilde\sigma}} ,\beta,X)})\le -\frac 1C |\sin (k\beta)(\tilde s-\tilde s_\ast)^k| \nonumber \\ (b)\ &\ {on}\ \Gamma{'s,}\ {|{\tilde\zeta}-\tilde\lambda}|\ge \frac 1C\max\{|\tilde s-\tilde s_\ast|,|\tilde r e^{i\alpha}-\tilde s_\ast e^{i\beta}|\},\\ (c)\ &\ {on}\ S{'s,}\ \mathfrak {Re} ({-i\wp(\frac {\tilde se^{i\tau}}{ \tilde\sigma},\beta,X)})\le 0. \nonumber \end{align}

Therefore,

• • $\ast$ using improper integrals, if $\tilde\sigma=\sqrt[k]{|X_k|}$, $k=3,2 $, estimates can be established when $\tilde s$ is away from the stationary point;

• • $\ast$ When both $\tilde s$ and $\tilde r$ are close to the stationary point, we will show that I ₅ near the stationary point is no longer a singular integral, allowing us to obtain estimates (see Proposition 3.14 for $\tilde\sigma=\sqrt[k]{|X_k|}$, $k=3,2 $).

The estimate for $\tilde\sigma=|X_1|$ is more complicated because $e^{\mathfrak {Re} ({-i\wp(\frac {\tilde se^{i\tau}}{{\tilde\sigma}} ,\beta,X)})} $ either decays non-uniformly in X or leads to a $\frac 1{|\sin\beta|}$ singularity which is not suitable for an improper integral. To address these difficulties, we will either utilize the scaling invariant properties of the Hilbert transform or look for a finer decomposition (see Proposition 3.17 and remarks before Definition 3.15).

More details regarding the key estimates for I ₃, I ₄ and I ₅ will be provided in Sections 3.3.3 and 3.3.4, as they represent significant additional analytic features of the inverse problem for perturbed multi-line solitons.

• • $\blacktriangleright$ Estimates on D ₀ : To leverage the methods for estimates on $D_{\kappa_j}$, we adopt a similar decomposition of the CI near 0, that is, a combination of uniformly compact domain integrals and non-uniformly compact domain integrals:

  \begin{align*} &\qquad \mathcal C T\mathcal E_{0}\phi \\ &\quad = -\frac {1}{2\pi i}\iint_{D_{0, {\tilde\sigma}\delta}} \frac{ \operatorname{sgn}(\beta) \hbar_0( \frac{\tilde s}{\tilde\sigma},\beta) e^{-i\wp(\frac {\tilde s}{{\tilde\sigma}} ,\beta,X)}\phi_{0,\operatorname{res}}(x)} {(\tilde\zeta-\tilde \lambda) \overline{\tilde\zeta}} d\overline{\tilde\zeta} \wedge d\tilde\zeta +\mathcal C_\lambda TE_{0}\phi_{0,r} \\ &\quad \equiv II_1+II_2+II_3+II_4+II_5, \end{align*}

  where

  \begin{align*} &\qquad II_1= -\frac {\theta(1-\tilde r)}{2\pi i}\iint_{\tilde s \lt 2} \frac{ \operatorname{sgn}(\beta) \hbar_0(\frac{\tilde s}{\tilde\sigma}, \beta) \phi_{0,\operatorname{res}}(x)} {(\tilde\zeta-\tilde \lambda) \overline{\tilde\zeta}} d\overline{\tilde\zeta} \wedge d\tilde\zeta ,\\ &\qquad II_2= -\frac {\theta(1-\tilde r)}{2\pi i}\iint_{\tilde s \lt 2} \frac{ \operatorname{sgn}(\beta) \hbar_0(\frac{\tilde s}{\tilde\sigma}, \beta)[ e^{-i\wp(\frac {\tilde s}{{\tilde\sigma}} ,\beta,X)}-1]\phi_{0,\operatorname{res}}( x)} {(\tilde\zeta-\tilde \lambda) \overline{\tilde\zeta}}d\overline{\tilde\zeta} \wedge d\tilde\zeta ,\\ &\qquad II_3= \ \mathcal C_\lambda E_{0}T\phi_{0,r} ,\\ &\qquad II_4= -\frac {\theta(1-\tilde r)}{2\pi i}\iint_{2 \lt \tilde s \lt {\tilde\sigma} \delta} \frac{ \operatorname{sgn}(\beta) \hbar_0(\frac{\tilde s}{\tilde\sigma}, \beta) e^{-i\wp(\frac {\tilde s}{{\tilde\sigma}} ,\beta,X)} \phi_{0,\operatorname{res}}(x)} {(\tilde\zeta-\tilde \lambda) \overline{\tilde\zeta}}d\overline{\tilde\zeta} \wedge d{\tilde\zeta},\\ &\qquad II_5= -\frac {\theta(\tilde r-1)}{2\pi i}\iint_{ \tilde s \lt {\tilde\sigma} \delta} \frac{ \operatorname{sgn}(\beta) \hbar_0(\frac{\tilde s}{\tilde\sigma}, \beta) e^{-i\wp(\frac {\tilde s}{{\tilde\sigma}} ,\beta,X)} \phi_{0,\operatorname{res}}(x)} {(\tilde\zeta-\tilde \lambda) \overline{\tilde\zeta}}d\overline{\tilde\zeta} \wedge d{\tilde\zeta}. \end{align*}

  The estimate for II ₁, the leading singular term, can be derived using $|\hbar_0|_{C^1(D_0)} \lt |v_0|_{L^1\cap L^\infty}$, the mean value theorem, Cauchy integral estimates and the Hilbert transform theory [Reference Gakhov13].

  II ₂ and II ₃ are estimated using (3.86).

  Writing $\hbar_0(\zeta)=\hbar_0(0)+ [\hbar_0(\zeta)-\hbar_0(0)]$ and decompose $II_4=II_{41}+II_{42}$, $II_5=II_{51}+II_{52}$ with

  \begin{align*} II_{41}=&-\frac {\theta(1-\tilde r)}{2\pi i}\iint_{2 \lt \tilde s \lt {\tilde\sigma} \delta} \frac{ \operatorname{sgn}(\beta) \hbar_0(0) e^{-i\wp(\frac {\tilde s}{{\tilde\sigma}} ,\beta,X)} \phi_{0,\operatorname{res}}(x)} {(\tilde\zeta-\tilde \lambda) \overline{\tilde\zeta}}d\overline{\tilde\zeta} \wedge d{\tilde\zeta},\\ II_{51}=&-\frac {\theta(\tilde r-1)}{2\pi i}\iint_{ \tilde s \lt {\tilde\sigma} \delta} \frac{ \operatorname{sgn}(\beta) \hbar_0(0) e^{-i\wp(\frac {\tilde s}{{\tilde\sigma}} ,\beta,X)} \phi_{0,\operatorname{res}}(x)} {(\tilde\zeta-\tilde \lambda) \overline{\tilde\zeta}}d\overline{\tilde\zeta} \wedge d{\tilde\zeta}. \end{align*}

  Thanks to $\tilde s$-meromorphic properties and adapting argument for estimating I ₄, I ₅ (see Section 3.3.3) for II ₄₁, II ₅₁, we obtain

  \begin{equation*} \begin{aligned} |II_{41}|_{L^\infty(D_{0})},\,|II_{51}|_{L^\infty(D_{0})}\le & C\epsilon_0|\phi_{0,\operatorname{res}}|_{L^\infty}. \end{aligned} \end{equation*}

  For the remaining terms, by (3.86) and $|\hbar_0|_{C^1(D_0)} \lt |v_0|_{L^1\cap L^\infty}$,

  \begin{equation*} \begin{gathered} |II_{42}|_{L^\infty(D_{0})},\,|II_{52}|_{L^\infty(D_{0})}\le C\epsilon_0|\phi_{0,\operatorname{res}}|_{L^\infty}. \end{gathered} \end{equation*}

1. (2) To derive the Lax equation, we introduce the shorthand notation:

   \begin{equation*} \begin{gathered} -\partial_{x_2}+\partial_{x_1}^2+2 \lambda\partial_{x_1}=-\nabla_2+\nabla_1^2 ,\\ \nabla_1=\partial_{x_1}+\lambda,\ \nabla_2=\partial_{x_2}+\lambda^2 ,\ \ J \, \phi = \frac{\phi _{0,\operatorname{res}}(x)}{\lambda} .\end{gathered} \end{equation*}

   By applying the heat operator to the system of the Cauchy Integral Equation (CIE) and the D-symmetry, formally,

   (3.97)\begin{gather} (-\nabla_2+\nabla_1^2 )m = \left[-\nabla_2+\nabla_1^2, J + \mathcal CT \right] m + (J +\mathcal CT)(-\nabla_2+\nabla_1^2) m, \end{gather}

   (3.98)\begin{gather} (e^{\kappa_1x_1+\kappa_1^2x_2+\kappa_1^3x_3}\left[(-\nabla_2+\nabla_1^2 )m\right](x,\kappa_1^+) , e^{\kappa_2x_1+\kappa_2^2x_2+\kappa_2^3x_3}\left[(-\nabla_2+\nabla_1^2 )m\right](x,\kappa_2^+) ){\mathcal D}=0, \end{gather}

   and

   \begin{equation*} \left[-\nabla_2+\nabla_1^2 , J +\mathcal CT\right] m = {+\frac 1{\pi i}\partial_{x_1}\iint T m \ d\overline\zeta\wedge d\zeta} +2 \partial_{x_1} m_{0,\operatorname{res}}(x)\equiv -u(x) .\end{equation*}

   Hence, formally, the unique of the CIE and the $\mathcal D$ symmetry constraint implies

   (3.99)\begin{align} (-\nabla_2+\nabla_1^2 )m= -(1-J-\mathcal CT)^{-1}u(x)1=&-u(x)(1-J-\mathcal CT)^{-1}1 = -u(x)m(x,\lambda). \end{align}

   Therefore, (3.78)–(3.80) are verified.

   A rigorous argument is carried out by introducing

   (3.100)\begin{equation} \begin{gathered} \phi^{(k)}= 1+J\phi^{(k)}+\mathcal CT\phi^{(k-1)}, \\ (e^{\kappa_1x_1+\kappa_1^2x_2+\kappa_1^3x_3}\left[(-\nabla_2+\nabla_1^2 )\phi^{(k)}\right](x,\kappa_1^+) , e^{\kappa_2x_1+\kappa_2^2x_2+\kappa_2^3x_3}\left[(-\nabla_2+\nabla_1^2 )\phi^{(k)}\right](x,\kappa^+_2))\mathcal D=0, \end{gathered} \end{equation}

   and using the estimates of the CIO’s and the iteration method to establish:

   \begin{align*} \left[-\nabla_2+\nabla_1^2 , J \right]\phi^{(k)}&\ {is independent of}\ \lambda,\\ \left[-\nabla_2+\nabla_1^2 , \mathcal CT \right]\phi^{(k-1)}&\ {is independent of}\ \lambda, \\ \left[-\nabla_2+\nabla_1^2 , J \right]\phi^{(k)}\to \ &\left[-\nabla_2+\nabla_1^2 , J \right]m=2\partial_{x_1} m_{0,\operatorname{res}}(x),\nonumber\\ \left[-\nabla_2+\nabla_1^2 , \mathcal CT \right]\phi^{(k-1)}\to &\ \left[-\nabla_2+\nabla_1^2 , \mathcal CT \right]m= \frac 1{\pi i}\partial_{x_1}\iint T m \, d\overline\zeta\wedge d\zeta , \nonumber\\ J (-\nabla_2+\nabla_1^2 ) \phi^{(k)}\to & J (-\nabla_2+\nabla_1^2 ) m\quad {in}\ W,\\ {\mathcal CT (-\nabla_2+\nabla_1^2 ) \phi^{(k)}\to} & {\mathcal CT (-\nabla_2+\nabla_1^2 ) m\quad {in}\ W,} \\ \sum_{0\le l_1+2l_2+3l_3\le d+4} |\partial_x^l ( [ - \nabla_2& +\nabla_1^2 , J ]m+u_s ) |_{L^\infty}\le C\epsilon_0, \\ |\left[-\nabla_2+\nabla_1^2 , \mathcal CT \right]\phi^{(k-1)}&-\left[-\nabla_2+\nabla_1^2 , \mathcal CT \right] \chi|_{L^\infty}\le(C\epsilon_0)^{k} . \end{align*}

2. (3) The KP equation: The KP equation will be derived by justifying the Lax pair. By the representation formula (3.80) and $\Phi(x, \lambda)= e^{\lambda x_1+ \lambda ^2x_2} m(x, \lambda)$, we define the evolution operators

   \begin{equation*}\begin{aligned} \mathcal M =&- \partial_{x_3}+ \partial_{x_1}^3+\frac 32u\partial_{x_1}+\frac 34u_{x_1}+\frac 34\partial_{x_1}^{-1}u_{x_2} -\lambda ^3, \\ \mathcal M \Phi(x,\lambda)=&e^{\lambda x_1+ \lambda ^2x_2}\left(\mathcal M+3\lambda\partial_{x_1}^2+3\lambda^2\partial_{x_1}+\lambda^3+\frac 32u\lambda\right)m(x,\lambda) \\ \equiv & e^{\lambda x_1+ \lambda ^2x_2}\left(\mathfrak Mm\right)(x,\lambda), \end{aligned} \end{equation*}

   We reverse the procedure in the linearization theorem Theorem 3.4 to prove

   (3.101)\begin{gather} \partial_{\overline\lambda}\left( \mathfrak Mm \right)(x,\lambda)=s_c(\lambda)e^{(\overline\lambda-\lambda)x_1+(\overline\lambda^2-\lambda^2)x_2 +(\overline\lambda^3-\lambda^3)x_3} \left( \mathfrak Mm \right)(x, \overline\lambda), \end{gather}

   (3.102)\begin{gather} (e^{\kappa_1x_1+\kappa_1^2x_2+\kappa_1^3x_3}\mathfrak Mm(x,\kappa _1^+) ,e^{\kappa_2x_1+\kappa_2^2x_2+\kappa_2^3x_3}\mathfrak Mm(x,\kappa _2^+)){\mathcal D}=0. \end{gather}

   As $ |\lambda|\to\infty$, letting $m \sim \sum_{j=0}^\infty\frac{M_j( x )}{\lambda^{j}}$, from the Lax equation (3.78),

   (3.103)\begin{gather} \qquad 2\partial_{x_1}M_{j+1} = (\partial_{x_2}-\partial_{x_1}^2-u)M_j,\nonumber\\ \qquad M_0= 1,\ M_1= - \frac {1}2 \partial_{x_1}^{-1}u,\ M_2= -\frac 14 \partial_{x_2} \partial_{x_1}^{-1}u+ \frac 14 u+\frac 14 \partial_{x_1}^{-1}\left(u \partial_{x_1}^{-1} u\right),\cdots \end{gather}

   As a result, as $\lambda\to\infty$,

   (3.104)\begin{align} &\mathfrak Mm\\ \to&\frac34 u_{x_1}+\frac 34 \partial_{x_1}^{-1}u_{x_2} +3\lambda\partial_{x_1}^2(1+\frac{M_1}{\lambda}) +3\lambda^2\partial_{x_1}(1+\frac{M_1}{\lambda}+\frac{M_2}{\lambda^2})+\frac 32u\lambda\nonumber \\ =&\frac34 u_{x_1}+\frac 34 \partial_{x_1}^{-1}u_{x_2}+\left( -\frac32u_{x_1}+3\partial_{x_1} (-\frac 14\partial_{x_2}\partial_{x_1}^{-2}u+\frac{u}{4}+\frac14\partial_{x_1}^{-1}[u\partial_{x_1}^{-1}u]) \right) \nonumber\\ +&\lambda\left( 3\partial_{x_1}M_1+\frac{3}{2}u\right)+\frac32u(-\frac 12\partial_{x_1}^{-1}u)=0.\nonumber \end{align}

   Therefore, $\mathfrak M m\in W$. Together with (3.101), (3.102), and unique solvability of the system of the CIE and $\mathcal D$ symmetry, yields $\mathfrak M m(x,\lambda)=0$ and the Lax pair.

3.3.3. Highlight: estimates for I ₃

The scaled H $\ddot{\mbox{o}}$lder estimate for

\begin{equation*} I_3= -\frac {\theta(1-\tilde r)}{2\pi i}\iint_{\tilde s \lt 2} \frac{ \widetilde \gamma_j(\tilde s, \beta) e^{-i\wp(\frac {\tilde s}{{\tilde\sigma}} ,\beta,X)}f^{\sharp}(\frac {\tilde s}{{\tilde\sigma}} ,-\beta,X)} {{\tilde\zeta}-\tilde \lambda}d\overline{\tilde\zeta} \wedge d{\tilde\zeta} \end{equation*}

is reminiscent of the H $\ddot{\mbox{o}}$lder estimate of the Beltrami’s equation [Reference Vekua25, Theorem 1.32], which involves mainly estimates of

\begin{equation*} I_3'g=\iint_{D} \frac{ g(\zeta)} {( \zeta - \lambda)^2}d\overline{\zeta} \wedge d{\zeta} ,\qquad |g|_{C^\mu} \lt \infty. \end{equation*}

Both leading singular terms of I ₃ and $I_3'$ can be integrated by Stokes’ theorem.

We only give the proof for the case $\kappa_j=\kappa_1$ since the proof for j = 2 is the same.

Proposition 3.8.

(3.105)\begin{equation} | I_3|_{C^\mu_{\tilde\sigma}(D_{\kappa_1,\frac{1}{\tilde\sigma}})}\le C\epsilon_0 | f^\sharp|_{C^\mu_{\tilde\sigma}(D_{\kappa_1,\frac{1}{\tilde\sigma}} )} . \end{equation}

Proof. From $ f^\sharp\in C^{\mu}_{\tilde\sigma}(D_{\kappa_1,\frac{1}{\tilde\sigma}})$ and $ f^\sharp(x,\kappa_1)=0$,

\begin{equation*} | \widetilde\gamma_1(\tilde s,\beta)f ^\sharp(\frac {\tilde s}{\tilde\sigma},\beta,X)|_{L^\infty(D_{\kappa_1} )}\le C \epsilon_0 |f^\sharp|_{H^\mu_{\tilde\sigma}(D_{\kappa_1,\frac{1}{\tilde\sigma}})}\tilde s^{\mu-1}. \end{equation*}

Therefore, an improper integral yields

(3.106)\begin{equation} | I_3|_{L^\infty(D_{\kappa_1})}\le C\epsilon_0 |f^\sharp|_{H^\mu_{\tilde\sigma}(D_{\kappa_1,\frac{1}{\tilde\sigma}})} . \end{equation}

To derive the $H^\mu_{\tilde\sigma}$-estimate of I ₃, let $\tilde \lambda_j=\kappa_1+\tilde r_je^{i\alpha_j}$, $ \tilde r_j \le 1$, $j=1,2$, define

(3.107)\begin{equation} \varphi_{f^\sharp}(x,\zeta)=e^{-i\wp(\frac {\tilde s}{\tilde\sigma} ,\beta,X)} {f^\sharp}(x, \overline\zeta), \end{equation}

and decompose

(3.108)\begin{align} & I_3(x,\lambda_1)-I_3(x, \lambda_2)\\ =& -\frac {\tilde\lambda_1-\tilde\lambda_2}{4\pi i} \iint_{\tilde s \lt 2}\widetilde\gamma_1({\tilde\zeta})\frac{\varphi_{f^\sharp}(\frac {\tilde s}{ \tilde\sigma},\beta,X) -\varphi_{f^\sharp}(\frac {\tilde r_1}{\tilde\sigma} ,\alpha_1,X) } {({\tilde\zeta}-\tilde\lambda_1)({\tilde\zeta}-\tilde\lambda_2)} d\bar{\tilde\zeta}\wedge d{\tilde\zeta}\nonumber\\ - & \frac {\tilde\lambda_1-\tilde\lambda_2}{4\pi i} \iint_{\tilde s \lt 2}\widetilde\gamma_1({\tilde\zeta})\frac{\varphi_{f^\sharp}(\frac {\tilde s}{ \tilde\sigma},\beta,X) -\varphi_{f^\sharp}(\frac {\tilde r_2}{\tilde\sigma},\alpha_2,X) } {({\tilde\zeta}-\tilde\lambda_1)({\tilde\zeta}-\tilde\lambda_2)} d\bar{\tilde\zeta}\wedge d{\tilde\zeta}\nonumber\\ +&\frac {\varphi_{f^\sharp}(\frac {\tilde r_1}{\tilde\sigma} ,\alpha_1,X)}{4\pi i} \iint_{\tilde s \lt 2}\widetilde\gamma_1({\tilde\zeta})[\frac{1 }{{\tilde\zeta}-\tilde\lambda_2}-\frac{1 }{{\tilde\zeta}-\tilde\lambda_1}] d\bar{\tilde\zeta}\wedge d{\tilde\zeta}\nonumber\\ +&\frac {\varphi_{f^\sharp}(\frac {\tilde r_2}{\tilde\sigma} ,\alpha_2,X)}{4\pi i} \iint_{\tilde s \lt 2}\widetilde\gamma_1({\tilde\zeta})[\frac{1 }{{\tilde\zeta}-\tilde\lambda_2}-\frac{1 }{{\tilde\zeta}-\tilde\lambda_1}] d\bar{\tilde\zeta}\wedge d{\tilde\zeta}.\nonumber \end{align}

In view of $ f^\sharp\in C^{\mu} _{\tilde\sigma}(D_{\kappa_1,\frac{1}{\tilde\sigma}})$ and $ f^\sharp(x,\kappa_1)=0$, we have

(3.109)\begin{equation} |\varphi_{f^\sharp}(\frac {\tilde r}{\tilde\sigma} ,\alpha,X)|_{L^\infty(D_{\kappa_1} )}\le C |f^\sharp|_{H^\mu_{\tilde\sigma}(D_{\kappa_1,\frac{1}{\tilde\sigma}} )}\tilde r^\mu . \end{equation}

Therefore, estimates for the last two terms can be derived using Stokes’ theorem to integrate the integrals.

We focus on the proof of the first term on the RHS of (3.108), as the proof of the second right-hand term is identical. Applying (3.86), it suffices to derive the estimate for all λ ₁, λ ₂ with $\tilde D\subset \{\tilde s\le 2\} $ being a disk centred at $\tilde\lambda_1$ with radius l and $l=2|\tilde\lambda_2-\tilde\lambda_1|$. Write

(3.110)\begin{align} &-\frac {\tilde\lambda_1-\tilde\lambda_2}{4\pi i}\iint_{\tilde s\le 2}\widetilde\gamma_1({\tilde\zeta})\frac{\varphi _{f^\sharp}(\frac {\tilde s}{ \tilde\sigma},\beta,X) -\varphi _{f^\sharp}(\frac {\tilde r_1}{\tilde\sigma},\alpha_1,X) } {({\tilde\zeta}-\tilde\lambda_1)({\tilde\zeta}-\tilde\lambda_2)} d\bar{\tilde\zeta}\wedge d{\tilde\zeta}\\ =&-\frac {\tilde\lambda_1-\tilde\lambda_2}{4\pi i} \iint_{\tilde D}\widetilde\gamma_1({\tilde\zeta})\frac{\varphi _{f^\sharp}(\frac {\tilde s}{ \tilde\sigma} ,\beta,X) -\varphi _{f^\sharp}(\frac {\tilde r_1}{\tilde\sigma},\alpha_1,X) } {({\tilde\zeta}-\tilde\lambda_1)({\tilde\zeta}-\tilde\lambda_2)} d\bar{\tilde\zeta}\wedge d{\tilde\zeta}\nonumber\\ -& \frac {\tilde\lambda_1-\tilde\lambda_2}{4\pi i}\iint_{\{\tilde s\le 2\}/\tilde D}\widetilde\gamma_1({\tilde\zeta})\frac{\varphi _{f^\sharp}(\frac {\tilde s}{\tilde\sigma},\beta,X) -\varphi _{f^\sharp}(\frac {\tilde r_1}{\tilde\sigma} ,\alpha_1,X) } {({\tilde\zeta}-\tilde\lambda_1)({\tilde\zeta}-\tilde\lambda_2)} d\bar{\tilde\zeta}\wedge d{\tilde\zeta}.\nonumber \end{align}

Let $\tilde D_0=\{\zeta: |\zeta-\tilde \lambda_1| \lt \frac{3l}2\} $.

• • $\blacktriangleright$ If ${\tilde\zeta}\in \{\tilde s\le 2\}/\tilde D$ and $\kappa_1\in \tilde D_0$, then

  \begin{equation*} \frac 1C\le|\frac{{\tilde\zeta}-\tilde\lambda_1}{{\tilde\zeta}-\tilde\lambda_2}|,|\frac{{\tilde\zeta}-\kappa_1}{{\tilde\zeta}-\tilde\lambda_1}|,|\frac{{\tilde\zeta}-\kappa_1}{{\tilde\zeta}-\tilde\lambda_2}|\le C. \end{equation*}

  In this case, using $ f^\sharp\in C^{\mu} _{\tilde\sigma}(D_{\kappa_1,\frac{1}{\tilde\sigma}})$ and [Reference Vekua25, Chapter 1,§ 6.1],

  (3.111)\begin{align} &|-\frac {\tilde\lambda_1-\tilde\lambda_2}{4\pi i}\iint_{\{\tilde s\le 2\}/\tilde D}\widetilde\gamma_1({\tilde\zeta})\frac{\varphi _{f^\sharp}(\frac {\tilde s}{ \tilde\sigma} ,\beta,X) -\varphi _{f^\sharp}(\frac {\tilde r_1}{ \tilde\sigma} ,\alpha_1,X) } {({\tilde\zeta}-\tilde\lambda_1)({\tilde\zeta}-\tilde\lambda_2)} d\bar{\tilde\zeta}\wedge d{\tilde\zeta}|\\ \le &C\epsilon_0|f^\sharp|_{C^{\mu} _{\tilde\sigma}(D_{\kappa_1,\frac{1}{\tilde\sigma}})}|\tilde\lambda_1-\tilde\lambda_2|\iint_{\{\tilde s\le 2\}/\tilde D} \frac{1}{|\tilde\zeta-\lambda_2| |{\tilde\zeta}-\tilde\lambda_1|^{2-\mu}} d\bar{\tilde\zeta}\wedge d{\tilde\zeta} \nonumber\\ \le & C\epsilon_0|f^\sharp|_{C^{\mu} _{\tilde\sigma}(D_{\kappa_1,\frac{1}{\tilde\sigma}})}|\tilde\lambda_1-\tilde\lambda_2|^\mu.\nonumber \end{align}

• • $\blacktriangleright$ If ${\tilde\zeta}\in \{\tilde s\le 2\}/\tilde D$ and $\kappa_1\notin \tilde D_0$ then

  \begin{equation*} \begin{gathered} \frac 1C\le|\frac{{\tilde\zeta}-\tilde\lambda_1}{{\tilde\zeta}-\tilde\lambda_2}|\le C,\quad |\tilde\lambda_1 -\tilde\lambda_2 |\le \frac 1C\min\{| {\tilde\lambda_1}-\kappa_1 |,|{\tilde\lambda_2}-\kappa_1 |\}. \end{gathered} \end{equation*}

  In this case, using $ f^\sharp\in C^{\mu} _{\tilde\sigma}(D_{\kappa_1,\frac{1}{\tilde\sigma}})$ and [Reference Vekua25, Chapter 1,§ 6.1],

  (3.112)\begin{align} &|-\frac {\tilde\lambda_1-\tilde\lambda_2}{4\pi i}\iint_{\{\tilde s\le 2\}/\tilde D}\widetilde\gamma_1({\tilde\zeta})\frac{\varphi _{f^\sharp}(\frac {\tilde s}{ \tilde\sigma} ,\beta,X) -\varphi _{f^\sharp}(\frac {\tilde r_1}{ \tilde\sigma} ,\alpha_1,X) } {({\tilde\zeta}-\tilde\lambda_1)({\tilde\zeta}-\tilde\lambda_2)} d\bar{\tilde\zeta}\wedge d{\tilde\zeta}|\\ \le & C\epsilon_0|f^\sharp|_{C^{\mu} _{\tilde\sigma}(D_{\kappa_1,\frac{1}{\tilde\sigma}})}|\tilde\lambda_1-\tilde\lambda_2|\iint_{\{\tilde s\le 2\}/\tilde D} \frac{1 } {|\tilde\zeta-\kappa_1| |{\tilde\zeta}-\tilde\lambda_1|^{2-\mu}} d\bar{\tilde\zeta}\wedge d{\tilde\zeta} \nonumber\\ \le & C\epsilon_0|f^\sharp|_{C^{\mu} _{\tilde\sigma}(D_{\kappa_1,\frac{1}{\tilde\sigma}})}|\tilde\lambda_1-\tilde\lambda_2|^\mu.\nonumber \end{align}

Therefore, the second term on the RHS of (3.110) is done.

Let $\tilde L(\zeta)=0$ be the line perpendicular to $\overline{\lambda_1\lambda_2}$ passing through $\frac 12(\lambda_1+\lambda_2)$. Set

\begin{equation*} \begin{aligned} \tilde D_{\tilde\lambda_1,\pm}=\tilde D\cap\{\zeta: { \tilde L(\zeta) \tilde L(\lambda_1)} \gt rless 0\}. \end{aligned} \end{equation*}

Therefore, thanks to $ f^\sharp \in C^\mu_{\tilde\sigma}(D_{\kappa_1,\frac{1}{\tilde\sigma}})$, setting $\eta=\frac{\tilde\zeta-\tilde\lambda_1}{|\tilde\lambda_1-\tilde\lambda_2|}$, $\frac{\tilde\zeta-\kappa_1}{|\tilde\lambda_1-\tilde\lambda_2|}=\eta-r_0e^{i\alpha_0}$, and using [Reference Vekua25, Chapter 1,§ 6.1],

(3.113)\begin{align} &|\frac {\tilde\lambda_1-\tilde\lambda_2}{4\pi i} \iint_{\tilde D_{\tilde\lambda_1,+}}\widetilde\gamma_1({\tilde\zeta})\frac{\varphi _{f^\sharp}(\frac {\tilde s}{ \tilde\sigma},\beta,X) -\varphi _{f^\sharp}(\frac {\tilde r_1}{\tilde\sigma},\alpha_1,X) } {({\tilde\zeta}-\tilde\lambda_1)({\tilde\zeta}-\tilde\lambda_2)} d\bar{\tilde\zeta}\wedge d{\tilde\zeta}|\\ \le &C\epsilon_0|\tilde\lambda_1-\tilde\lambda_2||f^\sharp|_{C^\mu_{\tilde\sigma}(D_{\kappa_1,\frac{1}{\tilde\sigma}})} |\iint_{\tilde D_{\tilde\lambda_1,+}} \frac 1{|\tilde\zeta-\kappa_1||{\tilde\zeta}-\tilde\lambda_1|^{1-\mu}|{\tilde\zeta}-\tilde\lambda_2|}d\bar{\tilde\zeta}\wedge d{\tilde\zeta}|\nonumber\\ \le &C\epsilon_0|\tilde\lambda_1-\tilde\lambda_2|^\mu|f^\sharp|_{C^\mu_{\tilde\sigma}(D_{\kappa_1,\frac{1}{\tilde\sigma}})} |\iint_{\{|\eta|\le 2\}\cap \tilde D_{\tilde\lambda_1,+}} \frac 1{|\eta-r_0e^{i\alpha_0}||\eta|^{1-\mu} |\eta-e^{i\alpha'}|} d \eta_Rd\eta_I|\nonumber\\ \le &C\epsilon_0|\tilde\lambda_1-\tilde\lambda_2|^\mu|f^\sharp|_{C^\mu_{\tilde\sigma}(D_{\kappa_1,\frac{1}{\tilde\sigma}})} |\iint_{\{|\eta|\le 2\}\cap \tilde D_{\tilde\lambda_1,+}} \frac 1{|\eta-r_0e^{i\alpha_0}||\eta|^{1-\mu}} d \eta_Rd\eta_I|\nonumber\\ \le &C\epsilon_0|f^\sharp|_{C^\mu_{\tilde\sigma}(D_{\kappa_1,\frac{1}{\tilde\sigma}})}|\tilde\lambda_1-\tilde\lambda_2|^\mu. \nonumber \end{align}

By analogy,

\begin{align*} &|\frac {\tilde\lambda_1-\tilde\lambda_2}{4\pi i} \iint_{\tilde D_{\tilde\lambda_1,-}}\widetilde\gamma_1({\tilde\zeta})\frac{\varphi _{f^\sharp}(\frac {\tilde s}{ \tilde\sigma},\beta,X) -\varphi _{f^\sharp}(\frac {\tilde r_1}{\tilde\sigma},\alpha_1,X) } {({\tilde\zeta}-\tilde\lambda_1)({\tilde\zeta}-\tilde\lambda_2)} d\bar{\tilde\zeta}\wedge d{\tilde\zeta}|\\ \le &|\frac {\tilde\lambda_1-\tilde\lambda_2}{4\pi i} \iint_{\tilde D_{\tilde\lambda_1,-}}\widetilde\gamma_1({\tilde\zeta})\frac{\varphi _{f^\sharp}(\frac {\tilde s}{ \tilde\sigma},\beta,X) -\varphi _{f^\sharp}(\frac {\tilde r_2}{\tilde\sigma},\alpha_2,X) } {({\tilde\zeta}-\tilde\lambda_1)({\tilde\zeta}-\tilde\lambda_2)} d\bar{\tilde\zeta}\wedge d{\tilde\zeta}|\\ +&|\frac {\tilde\lambda_1-\tilde\lambda_2}{4\pi i} \iint_{\tilde D_{\tilde\lambda_1,-}}\widetilde\gamma_1({\tilde\zeta})\frac{\varphi _{f^\sharp}(\frac {\tilde r_2}{ \tilde\sigma},\alpha_2,X) -\varphi _{f^\sharp}(\frac {\tilde r_1}{\tilde\sigma},\alpha_1,X) } {({\tilde\zeta}-\tilde\lambda_1)({\tilde\zeta}-\tilde\lambda_2)} d\bar{\tilde\zeta}\wedge d{\tilde\zeta}|\\ \le &C\epsilon_0|f^\sharp|_{C^\mu_{\tilde\sigma}(D_{\kappa_1,\frac{1}{\tilde\sigma}})}|\tilde\lambda_1-\tilde\lambda_2|^\mu\\ +&|\frac {\tilde\lambda_1-\tilde\lambda_2}{4\pi i} \iint_{\tilde D_{\tilde\lambda_1,-}}\widetilde\gamma_1({\tilde\zeta})\frac{\varphi _{f^\sharp}(\frac {\tilde r_2}{ \tilde\sigma},\alpha_2,X) -\varphi _{f^\sharp}(\frac {\tilde r_1}{\tilde\sigma},\alpha_1,X) } {({\tilde\zeta}-\tilde\lambda_1)({\tilde\zeta}-\tilde\lambda_2)} d\bar{\tilde\zeta}\wedge d{\tilde\zeta}|. \end{align*}

Applying $ f^\sharp \in C^\mu_{\tilde\sigma}(D_{\kappa_1,\frac{1}{\tilde\sigma}})$, Stokes’ theorem and $| {\tilde\zeta-\tilde\lambda_1}| $, $ | {\tilde\zeta-\tilde\lambda_2}|\sim| {\tilde\lambda_1}-\tilde\lambda_2|$ on the boundary of $\tilde D_{\tilde\lambda_1,-}$ (assured by $|\tilde\lambda|\le 1$, $\tilde D\subset\{\tilde s \lt 2\}$),

\begin{align*} &|\frac {\tilde\lambda_1-\tilde\lambda_2}{4\pi i} \iint_{\tilde D_{\tilde\lambda_1,-}}\widetilde\gamma_1({\tilde\zeta})\frac{\varphi _{f^\sharp}(\frac {\tilde r_2}{ \tilde\sigma},\alpha_2,X) -\varphi _{f^\sharp}(\frac {\tilde r_1}{\tilde\sigma},\alpha_1,X) } {({\tilde\zeta}-\tilde\lambda_1)({\tilde\zeta}-\tilde\lambda_2)} d\bar{\tilde\zeta}\wedge d{\tilde\zeta}|\\ \le &C|f^\sharp|_{C^\mu_{\tilde\sigma}(D_{\kappa_1,\frac{1}{\tilde\sigma}})}|\tilde\lambda_1-\tilde\lambda_2|^{1+\mu} \iint_{\tilde D_{\tilde\lambda_1,-}}\widetilde\gamma_1({\tilde\zeta})\frac{1} {({\tilde\zeta}-\tilde\lambda_1)({\tilde\zeta}-\tilde\lambda_2)} d\bar{\tilde\zeta}\wedge d{\tilde\zeta}|\\ =&C|f^\sharp|_{C^\mu_{\tilde\sigma}(D_{\kappa_1,\frac{1}{\tilde\sigma}})}|\tilde\lambda_1-\tilde\lambda_2|^{1+\mu} \iint_{\tilde D_{\tilde\lambda_1,-}} \frac{\partial_{\overline\zeta}\left[\ln(1-\gamma|\beta|) \frac 1{{\tilde\zeta}-\tilde\lambda_1}\right]}{{\tilde\zeta}-\tilde\lambda_2} d\bar{\tilde\zeta}\wedge d{\tilde\zeta}|\\ \le &C\epsilon_0|f^\sharp|_{C^\mu_{\tilde\sigma}(D_{\kappa_1,\frac{1}{\tilde\sigma}})}|\tilde\lambda_1-\tilde\lambda_2|^\mu. \end{align*}

Therefore the first term on the RHS of (3.110) is done. Thus

(3.114)\begin{equation} \begin{aligned} \qquad\qquad|\frac {\tilde\lambda_1-\tilde\lambda_2}{4\pi i}\iint_{\tilde s\le 2}\widetilde\gamma_1({\tilde\zeta})\frac{\varphi _{f^\sharp}(\frac {\tilde s}{ \tilde\sigma} ,\beta,X) -\varphi _{f^\sharp}(\frac {\tilde r_1}{\tilde\sigma} ,\alpha_1,X) } {({\tilde\zeta}-\tilde\lambda_1)({\tilde\zeta}-\tilde\lambda_2)} d\bar{\tilde\zeta}\wedge d{\tilde\zeta}|\\ \le C\epsilon_0|f^\sharp|_{C^\mu_{\tilde\sigma}(D_{\kappa_1,\frac{1}{\tilde\sigma}})}|\tilde\lambda_1-\tilde\lambda_2|^\mu,\qquad \end{aligned} \end{equation}

for $|\tilde\lambda_j-\kappa_1|\le 1,\ j=1,2$. Hence (3.105) can be justified.

3.3.4. Highlight: estimates for $I_4,I_5$

Without loss of generality and for simplicity, we assume

(3.115)\begin{equation} \begin{gathered} \kappa_j=\kappa_1,\quad |\lambda-\kappa_1|\le \frac \delta 2,\quad X_3 \gt 0,\quad X_1,X_2\ge 0 , \end{gathered} \end{equation}

in this section. For $ \widehat \sigma\in\{X_1,\sqrt{X_2},\sqrt[3]{X_3}\}$, consider the $\widehat\sigma$-scaled coordinates

\begin{equation*} \zeta =\kappa_1+se^{i\beta}=\kappa_1+\frac{\widehat s} {\widehat \sigma}e^{i\beta}\quad\in D_{\kappa_1}, \end{equation*}

and the deformation

\begin{equation*} \widehat s\in{\mathbb R} \longrightarrow \widehat se^{i\tau}\in{\mathbb C}. \end{equation*}

Observe that

\begin{align*} \mathfrak {Re} ({-i\wp(\frac {\widehat se^{i\tau}}{\widehat \sigma},\beta,X)})=&\frac{X_3}{\widehat \sigma ^3} \sin3\tau\sin3\beta\widehat s^3+ \frac{X_2}{\widehat \sigma^2} \sin2\tau\sin2\beta\widehat s^2 +\frac{X_1}{\widehat \sigma} \sin \tau\sin \beta\widehat s \\ \equiv &\frac{X_3}{\widehat \sigma ^3} \sin3\tau\sin3\beta\widehat s(\widehat s- \rho_{+} )(\widehat s- \rho_{-}),\\ \partial_{\widehat s}\wp(\frac{\widehat s}{\widehat \sigma},\beta,X)=&3\frac{X_3}{\widehat \sigma^3} \sin3\beta\widehat s^2+ 2 \frac{X_2}{\widehat \sigma^2} \sin2\beta\widehat s +\frac{X_1}{\widehat \sigma} \sin \beta\\ \equiv& 3\frac{X_3}{\widehat \sigma^3} \sin3\beta( \widehat s -\widehat s _+)(\widehat s -\widehat s _-). \end{align*}

Hence as $|\tau|\ll 1$, $\rho_\pm\sim\widehat s_\pm $. This is the primary motivation behind our definition of stationary points.

Definition 3.9. The stationary points are defined to be

(3.116)\begin{equation} \widehat s_\pm = \frac{ - 1\pm\sqrt{1-\Delta}\,} {3\frac{X_3} {\widehat\sigma X_2}\frac{\sin3\beta}{\sin2\beta}},\quad \Delta= 3\frac{{X_1}{X_ 3}}{X_2^2}\frac{\sin\beta\sin3\beta}{\sin^22\beta}, \end{equation}

which satisfy

\begin{equation*} \partial_{\widehat s}\wp(\frac{\widehat s_\pm}{\widehat \sigma},\beta,X)=3\frac{X_3}{\widehat \sigma^2} \sin3\beta \widehat s^2_\pm+2\frac{X_2}{\widehat \sigma^{2} }\sin2\beta\widehat s_\pm+\frac{X_1}{\widehat \sigma} \sin\beta=0. \end{equation*}

Denote

(3.117)\begin{equation} \begin{array}{lll} \Omega_1=\{0\le|\beta|\le\frac\pi 3 \},& \Omega_2=\{\frac {\pi} 3\le|\beta|\le\frac {\pi} 2\},& \Omega_3=\{\frac {\pi} 2\le|\beta|\le\frac {2\pi} 3\},\\ \Omega_4=\{\frac {2\pi} 3\le|\beta|\le\pi\},& & \end{array} \end{equation}

one has Figure 1 for signatures of $sin (k\beta)$ on $\Omega_j$. Moreover, according to the determinant Δ, we have Table 1 to classify points in $(\beta,\,X)$ into 5 Types and their subtypes. We outline the properties of stationary points for each type or subtype. Note that

(3.118)\begin{equation} |\widehat s_ +-\widehat s_ - |= |\frac{\sqrt{1-\Delta}}{3{\frac{X_3}{\widehat\sigma X_2} \frac{\sin3\beta}{ \sin2\beta}} }|. \end{equation}

Figure 1. Signatures of $ ( \sin\beta,\sin2\beta,\sin3\beta)$ for $X_1,X_2,X_3 \gt 0$.

Table 1. Properties of $\widehat s_\pm$ and Δ for Type $\mathfrak A,\cdots,\mathfrak E$ when $X_1 \gt 0,X_2,X_3\ge 0$

In the following, we will define essential stationary points $\widehat s_{j,\ast}$ and decompose $[0,\widehat\sigma]$ into intervals $\mho_j$ around $\widehat s_{j,\ast}$. The deformation will be defined on $\mho_j $.

Definition 3.10. We define the essential stationary points $\widehat s_{j,\ast}=\widehat s_{j,\ast}(\widehat\sigma,\beta, X)$ by

(3.119)\begin{align} \widehat s_{0,\ast} = & 0, \nonumber\\ \widehat s_{1,\ast}= & \left\{ {\begin{array}{ll} {\frac{\widehat s_{+} +\widehat s_{-} }2 \gt rless 0} , & Type\,\mathfrak A' \wedge(\widehat{\sigma}\in\{\sqrt{X_2},\sqrt[3]{X_3}\}), \,\mathfrak A^{\prime\prime} , \\ { \inf\widehat s_{\pm} } \gt 0 , & Type\, \mathfrak B^{\prime\prime},\mathfrak C^{\prime\prime},\\ \sup\widehat s_{\pm} \gt 0 ,& Type\, \mathfrak D, \mathfrak E, \\ - ,& Type\, \mathfrak A '\wedge(\widehat{\sigma}=X_1),\, \mathfrak B' ,\mathfrak C' , \end{array}}\right. \ &\nonumber\\ \widehat s_{2,\ast}= & \left\{ {\begin{array}{ll} {\sup\widehat s_{\pm} }, & \hskip.3in Type\, \mathfrak B^{\prime\prime},\mathfrak C^{\prime\prime} ,\\ - ,& \hskip.3in {others,} \end{array}}\right. \end{align}

where − means no definition. Given $0 \lt \epsilon_1 \lt \frac{\pi}{2k}\ll 1$, define neighborhood $ \mho_j(\widehat\sigma,\beta,X )$ of essential critical points $\widehat s_{j,\ast}$ by

(3.120)\begin{align} \mho_0 =&\left\{ {\begin{array}{l} { [0, \frac 12]}, \hskip2.9 in Type\, \mathfrak A'\wedge(\widehat{\sigma}\in\{\sqrt{X_2},\sqrt[3]{X_3}\}),\mathfrak A^{\prime\prime}, \\ { [0, \frac 1{2\cos\epsilon_1} \widehat s _{1,\ast} ],} \hskip2.45 in Type\, \mathfrak B^{\prime\prime},\mathfrak C^{\prime\prime}, \mathfrak D , \mathfrak E,\\ {[0, \widehat\sigma\delta]=[0,\frac 12]\cup[\frac 12,\widehat\sigma\delta]=\mho_{0, \lt }\cup\mho_{0, \gt }} , \hskip.9 in Type\,\mathfrak A '\wedge(\widehat{\sigma}=X_1), \, \mathfrak B',\mathfrak C', \end{array}}\right.\nonumber\\ \mho_1 =& \left\{ {\begin{array}{l} { [\frac 12,\widehat s_{1,\ast}]\cup[\widehat s_{1,\ast}, \widehat\sigma\delta]} \equiv\mho_{1, \lt }\cup\mho_{1, \gt }, \hskip .1in Type\, \{[\mathfrak A'\wedge(\widehat{\sigma}\in \{\sqrt{X_2},\sqrt[3]{X_3}\})]\vee\mathfrak A^{\prime\prime}\}\wedge (\widehat{s}_{1,\ast} \gt 0),\\ { [\frac 12, \widehat\sigma\delta]} \equiv \mho_{1, \gt }, \hskip 1.4in Type\, \{[\mathfrak A'\wedge(\widehat{\sigma}\in\{\sqrt{X_2}, \sqrt[3]{X_3}\})]\vee \mathfrak A^{\prime\prime}\}\wedge(\widehat{s}_{1,\ast} \lt 0),\\ {[ (1- \frac 1{2\cos\epsilon_1} ) \widehat s_{1,\ast}, \widehat s_{1,\ast} ]\cup [ \widehat s_{1,\ast}, \widehat s_{1,\ast}+ \frac {\widehat s_{2,\ast}-\widehat s_{1,\ast}}{2\cos\epsilon_1} ]\equiv\mho_{1, \lt }\cup\mho_{1, \gt },} \hskip 1in Type\, \mathfrak B^{\prime\prime},\mathfrak C^{\prime\prime},\\ {[ (1- \frac 1{2\cos\epsilon_1} )\widehat s_{1,\ast}, \widehat s_{1,\ast} ]\cup [ \widehat s_{1,\ast},\widehat\sigma\delta ] \equiv\mho_{1, \lt }\cup\mho_{1, \gt },} \hskip 1.56in Type\ \, \mathfrak D ,\ \mathfrak E,\\ {\phi,} \hskip 3. in Type\,\mathfrak A '\wedge(\widehat{\sigma}=X_1),\, \mathfrak B',\mathfrak C', \end{array}}\right.\nonumber\\ \mho_2 =& \left\{ {\begin{array}{l} {[ \widehat s_{2,\ast}-\frac {\widehat s_{2,\ast}-\widehat s_{2,\ast}}{2 \cos\epsilon_1} , \widehat s_{2,\ast} ]\cup [ \widehat s_{2,\ast},\widehat\sigma\delta ] \equiv\mho_{2, \lt }\cup\mho_{2, \gt },} \hskip.4 in Type\, \mathfrak B^{\prime\prime},\mathfrak C^{\prime\prime},\\ \phi \hskip3. in {otherwise}, \end{array}}\right. \end{align}

Write

(3.121)\begin{equation} \begin{gathered} \lambda=\kappa_1+\frac{\widehat re^{i\alpha}} {\widehat \sigma}=\kappa_1+\frac{\widehat s_{j,\ast}e^{i\beta}+\widehat r_je^{i\alpha_j}}{\widehat \sigma} ,\\ \widehat r_j=\widehat r_j(\widehat\sigma,\beta,X,\lambda),\ \alpha_j=\alpha_j(\widehat\sigma,\beta,X,\lambda),\,j=0,1,2. \end{gathered} \end{equation}

We define the deformation defined by

(3.122)\begin{equation} \begin{gathered} \zeta=\kappa_1+se^{i\beta}=\kappa_1+\frac{\widehat s e^{i\beta}}{\widehat\sigma}\\ \widehat s \mapsto \xi_j\equiv \widehat s_{j,\ast} +\widehat s_je^{i\tau_j} , \ \widehat s\equiv\widehat s_{j,\ast}\pm\widehat s_j\in \mho_j ,\, |\tau_j|\lessgtr\frac\pi 2,\,\widehat s_j\ge 0,\, j=0,1,2, \end{gathered} \end{equation}

with

\begin{equation*}\begin{aligned} &\left\{\begin{array}{lll} { \mp\pi\lessgtr\tau_1\lessgtr \mp\pi\pm\epsilon_1 ,} & for\, \sin 3\beta \gt rless 0,\ \ ||\alpha_1-\beta|- \pi|\le \frac{\epsilon_1}2,& \widehat s\in\mho_{1 \lt }\ne\phi,\, Type\, \mathfrak A , \\ {\mp\epsilon_1\lessgtr \tau_1\lessgtr 0 ,} &for\, \sin 3\beta \gt rless0,\ \ |\alpha_1-\beta|\le \frac{\epsilon_1}2,&\widehat s\in\mho_{1 \gt }\ne\phi, \, Type\, \mathfrak A ,\\ { \mp\pi\lessgtr \tau_1\lessgtr \mp\pi\pm\epsilon_1 ,} &for\, \sin 3\beta \gt rless0,\ \ ||\alpha_1-\beta|- \pi|\le \frac{\epsilon_1}2,&\widehat s\in\mho_{1 \lt }, Type\, \mathfrak B^{\prime\prime},\mathfrak C^{\prime\prime},\\ { \pm\epsilon_1 \gt rless\tau_1 \gt rless 0 ,} & for\, \sin 3\beta \gt rless0,\ \ |\alpha_1-\beta|\le \frac{\epsilon_1}2,& \widehat s\in\mho_{1 \gt } , Type\, \mathfrak B^{\prime\prime},\mathfrak C^{\prime\prime},\\ { \pm\pi \gt rless\tau_1 \gt rless\pm\pi\mp\epsilon_1,} &for\, \sin 3\beta \gt rless0,\ \ ||\alpha_1-\beta|- \pi|\le \frac{\epsilon_1}2,& \widehat s\in\mho_{1 \lt } , Type\, \mathfrak D,\mathfrak E,\\ { \mp\epsilon_1\lessgtr\tau_1\lessgtr 0,} &for\, \sin 3\beta \gt rless 0,\ \ |\alpha_1-\beta|\le \frac{\epsilon_1}2,& \widehat s\in\mho_{1 \gt } , Type\, \mathfrak D,\mathfrak E,\mathfrak B',\mathfrak C',\\ {\mp\pi\lessgtr\tau_1\lessgtr \mp\pi\pm\frac{\epsilon_1} 4 ,}& for\, \sin 3\beta \gt rless 0,\ \ ||\alpha_1-\beta|- \pi|\ge \frac{\epsilon_1}2,& \widehat s\in\mho_{1 \lt }\ne\phi,\, Type\, \mathfrak A , \\ { \mp\frac{\epsilon_1}4\lessgtr\tau_1\lessgtr 0 ,} & for\, \sin 3\beta \gt rless 0,\ \ |\alpha_1-\beta|\ge \frac{\epsilon_1}2,&\widehat s\in\mho_{1 \gt }\ne\phi,\, Type\, \mathfrak A ,\\ { \mp\pi\lessgtr \tau_1\lessgtr \mp\pi\pm\frac{\epsilon_1} 4,} &for\, \sin 3\beta \gt rless 0,\ \ ||\alpha_1-\beta|- \pi|\ge \frac{\epsilon_1}2,& \widehat s\in\mho_{1 \lt }, Type\, \mathfrak B^{\prime\prime},\mathfrak C^{\prime\prime},\\ { \pm\frac{\epsilon_1} 4 \gt rless\tau_1 \gt rless 0 ,} &for\, \sin 3\beta \gt rless 0,\ \ |\alpha_1-\beta|\ge \frac{\epsilon_1}2,&\widehat s\in\mho_{1 \gt }, Type\, \mathfrak B^{\prime\prime},\mathfrak C^{\prime\prime},\\ {\pm\pi \gt rless\tau_1 \gt rless \pm\pi\mp\frac{\epsilon_1}4,} &for\, \sin 3\beta \gt rless 0,\ \ ||\alpha_1-\beta|- \pi|\ge \frac{\epsilon_1}2,& \widehat s\in\mho_{1 \lt } , Type\, \mathfrak D,\mathfrak E,\\ {\mp\frac{\epsilon_1}4\lessgtr\tau_1\lessgtr 0,}&for\, \sin 3\beta \gt rless 0,\ \ |\alpha_1-\beta|\ge \frac{\epsilon_1}2,& \widehat s\in\mho_{1 \gt } , Type\, \mathfrak D, \mathfrak E,\mathfrak B',\mathfrak C', \end{array}\right.\\ &\left\{ {\begin{array}{lll} { \pm\pi \gt rless\tau_2 \gt rless \pm\pi\mp\epsilon_1,} & for\, \sin 3\beta \gt rless 0,\ \ ||\alpha_2-\beta|- \pi|\le \frac{\epsilon_1}2,& \widehat s\in\mho_{2 \lt }, Type\, \mathfrak B^{\prime\prime}, \mathfrak C^{\prime\prime},\\ { \mp\epsilon_1\lessgtr\tau_2\lessgtr 0 ,} & for\, \sin 3\beta \gt rless 0,\ \ |\alpha_2-\beta|\le \frac{\epsilon_1}2,& \widehat s\in\mho_{2 \gt }, Type\, \mathfrak B^{\prime\prime}, \mathfrak C^{\prime\prime},\\ { \pm\pi \gt rless\tau_2 \gt rless \pm\pi\mp\frac{\epsilon_1} 4 ,} &for\, \sin 3\beta \gt rless 0,\ \ ||\alpha_2-\beta|- \pi|\ge \frac{\epsilon_1}2,& \widehat s\in\mho_{2 \lt } , Type\, \mathfrak B^{\prime\prime}, \mathfrak C^{\prime\prime},\\ {\mp\frac{\epsilon_1} 4\lessgtr\tau_2\lessgtr 0 ,} &for\, \sin 3\beta \gt rless 0,\ \ |\alpha_2-\beta|\ge \frac{\epsilon_1}2,&\widehat s\in\mho_{2 \gt } , Type\, \mathfrak B^{\prime\prime}, \mathfrak C^{\prime\prime}, \end{array}}\right.\\ & \left\{ {\begin{array}{l@{\qquad\quad\ \,}l@{\qquad\quad\,}l} {\tau_0\equiv 0 ,} &for\, \sin 3\beta \gt rless 0,\ \ |\alpha_0-\beta|\le \frac{\epsilon_1}2,&\widehat s\in [0,\frac 12]\subset \mho_0, Type\ \mathfrak A,\mathfrak B',\mathfrak C',\\ {\tau_0\equiv 0 ,} & for\, \sin 3\beta \gt rless 0,\ \ |\alpha_0-\beta|\ge \frac{\epsilon_1}2,&\widehat s\in [0,\frac 12]\subset \mho_0, Type\ \mathfrak A,\mathfrak B',\mathfrak C',\\ {\pm\epsilon_1 \gt rless\tau_0 \gt rless 0 ,}&for\, \sin 3\beta \gt rless 0,\ \ |\alpha_0-\beta|\le \frac{\epsilon_1}2,& \widehat s\in\mho_0, Type\ \mathfrak D,\mathfrak E, \\ {\pm\frac{\epsilon_1} 4 \gt rless\tau_0 \gt rless 0 ,}& for\, \sin 3\beta \gt rless 0,\ \ |\alpha_0-\beta|\ge \frac{\epsilon_1}2,&\widehat s\in\mho_0 , Type\ \mathfrak D, \mathfrak E, \\ {\mp\epsilon_1\lessgtr\tau_0\lessgtr 0 ,} &for\, \sin 3\beta \gt rless 0,\ \ |\alpha_0-\beta|\le \frac{\epsilon_1}2,&\widehat s\in \mho_{0, \gt }, Type\ \mathfrak A',\mathfrak B',\mathfrak C',\\ {\mp\frac{\epsilon_1} 4\lessgtr\tau_0\lessgtr 0 ,}& for\, \sin 3\beta \gt rless 0,\ \ |\alpha_0-\beta|\ge \frac{\epsilon_1}2,&\widehat s\in \mho_{0, \gt }, Type\ \mathfrak A',\mathfrak B',\mathfrak C',\\ {\mp\epsilon_1\lessgtr\tau_0\lessgtr 0 ,} &for\, \sin 3\beta \gt rless 0,\ \ |\alpha_0-\beta|\le \frac{\epsilon_1}2,& \widehat s\in\mho_0, Type\, \mathfrak B^{\prime\prime},\mathfrak C^{\prime\prime},\\ {\mp\frac{\epsilon_1} 4\lessgtr\tau_0\lessgtr 0 ,} & for\, \sin 3\beta \gt rless 0,\ \ |\alpha_0-\beta|\ge \frac{\epsilon_1}2,&\widehat s\in\mho_0, Type\, \mathfrak B^{\prime\prime},\mathfrak C^{\prime\prime}, \end{array}}\right. \end{aligned} \end{equation*}

and

\begin{equation*} \begin{aligned}\tau_{0,\dagger}&= \left\{ \begin{array}{l@{\qquad\ \ \ }l@{\qquad\quad\,}l} {0 ,}& for\, \sin 3\beta \gt rless 0,\ \ |\alpha_0-\beta|\le \frac{\epsilon_1}2,&\widehat s\in [0,\frac 12]\subset \mho_{0} , Type\ \mathfrak A ,\mathfrak B',\mathfrak C', \\ {0 ,} & for\, \sin 3\beta \gt rless 0,\ \ |\alpha_0-\beta|\ge \frac{\epsilon_1}2,&\widehat s\in [0,\frac 12]\subset \mho_{0} , Type\ \mathfrak A ,\mathfrak B',\mathfrak C',\\ {\pm\epsilon_1 ,} &for\, \sin 3\beta \gt rless 0,\ \ |\alpha_0-\beta|\le \frac{\epsilon_1}2,& \widehat s\in\mho_0,\,Type\, \mathfrak D, \mathfrak E ,\\ {\pm\frac{\epsilon_1} 4 ,}& for\, \sin 3\beta \gt rless 0,\ \ |\alpha_0-\beta|\ge \frac{\epsilon_1}2,&\widehat s\in\mho_0 ,\,Type\, \mathfrak D, \mathfrak E,\\ {\mp\epsilon_1 ,}&for\, \sin 3\beta \gt rless 0,\ \ |\alpha_0-\beta|\le \frac{\epsilon_1}2,&\widehat s\in \mho_{0, \gt } , Type\ \mathfrak A' ,\mathfrak B',\mathfrak C', \\ {\mp\frac{\epsilon_1} 4 ,}& for\, \sin 3\beta \gt rless 0,\ \ |\alpha_0-\beta|\ge \frac{\epsilon_1}2,&\widehat s\in \mho_{0, \gt } , Type\ \mathfrak A' ,\mathfrak B',\mathfrak C',\\ {\mp\epsilon_1 ,}&for\, \sin 3\beta \gt rless 0,\ \ |\alpha_0-\beta|\le \frac{\epsilon_1}2,& \widehat s\in\mho_0 , Type\, \mathfrak B^{\prime\prime},\mathfrak C^{\prime\prime}, \\ {\mp\frac{\epsilon_1} 4 ,} & for\, \sin 3\beta \gt rless 0,\ \ |\alpha_0-\beta|\ge \frac{\epsilon_1}2,&\widehat s\in\mho_0 , Type\, \mathfrak B^{\prime\prime},\mathfrak C^{\prime\prime}, \end{array}\right. \end{aligned}\end{equation*}

\begin{equation*} \begin{aligned} \tau_{1,\dagger}&= \left\{ \begin{array}{lll} { \mp\pi\pm\epsilon_1 ,}&for\, \sin 3\beta \gt rless 0,\ \ ||\alpha_1-\beta|- \pi|\le \frac{\epsilon_1}2,& \widehat s\in\mho_{1 \lt }\ne\phi,\, Type\, \mathfrak A , \\ {\mp\epsilon_1 ,} &for\, \sin 3\beta \gt rless0,\ \ |\alpha_1-\beta|\le \frac{\epsilon_1}2,&\widehat s\in\mho_{1 \gt }\ne\phi, \, Type\, \mathfrak A ,\\ { \mp\pi\pm\epsilon_1 ,} &for\, \sin 3\beta \gt rless0,\ \ ||\alpha_1-\beta|- \pi|\le \frac{\epsilon_1}2,&\widehat s\in\mho_{1 \lt }, Type\, \mathfrak B^{\prime\prime}, \mathfrak C^{\prime\prime},\\ { \pm\epsilon_1 ,} &for\, \sin 3\beta \gt rless0,\ \ |\alpha_1-\beta|\le \frac{\epsilon_1}2,& \widehat s\in\mho_{1 \gt } , Type\, \mathfrak B^{\prime\prime}, \mathfrak C^{\prime\prime},\\ {\pm\pi\mp\epsilon_1 ,}&for\, \sin 3\beta \gt rless0,\ \ ||\alpha_1-\beta|- \pi|\le \frac{\epsilon_1}2,& \widehat s\in\mho_{1 \lt } , Type\, \mathfrak D, \mathfrak E,\\ { \mp\epsilon_1 ,}&for\, \sin 3\beta \gt rless 0,\ \ |\alpha_1-\beta|\le \frac{\epsilon_1}2,& \widehat s\in\mho_{1 \gt } , Type\, \mathfrak D, \mathfrak E,\\ { \mp\pi\pm\frac{\epsilon_1} 4 ,}& for\, \sin 3\beta \gt rless 0,\ \ ||\alpha_1-\beta|- \pi|\ge \frac{\epsilon_1}2,& \widehat s\in\mho_{1 \lt },\ Type\, \mathfrak A , \\ { \mp\frac{\epsilon_1}4 ,} & for\, \sin 3\beta \gt rless 0,\ \ |\alpha_1-\beta|\ge \frac{\epsilon_1}2,&\widehat s\in\mho_{1 \gt },\ Type\, \mathfrak A ,\\ { \mp\pi\pm\frac{\epsilon_1} 4,}&for\, \sin 3\beta \gt rless 0,\ \ ||\alpha_1-\beta|- \pi|\ge \frac{\epsilon_1}2,& \widehat s\in\mho_{1 \lt }, Type\, \mathfrak B^{\prime\prime}, \mathfrak C^{\prime\prime}, \\ {\pm\frac{\epsilon_1} 4 ,} &for\, \sin 3\beta \gt rless 0,\ \ |\alpha_1-\beta|\ge \frac{\epsilon_1}2,&\widehat s\in\mho_{1 \gt }, Type\, \mathfrak B^{\prime\prime}, \mathfrak C^{\prime\prime},\\ {\pm\pi\mp\frac{\epsilon_1}4 ,}&for\, \sin 3\beta \gt rless 0,\ \ ||\alpha_1-\beta|- \pi|\ge \frac{\epsilon_1}2,& \widehat s\in\mho_{1 \lt } , Type\, \mathfrak D, \mathfrak E,\\ {\mp\frac{\epsilon_1}4 ,} & for\, \sin 3\beta \gt rless 0,\ \ |\alpha_1-\beta|\ge \frac{\epsilon_1}2,& \widehat s\in\mho_{1 \gt } , Type\, \mathfrak D, \mathfrak E, \end{array}\right. \\ \tau_{2,\dagger}&= \left\{ \begin{array}{lll} {\pm\pi\mp\epsilon_1 ,} &for\, \sin 3\beta \gt rless 0,\ \ ||\alpha_2-\beta|- \pi|\le \frac{\epsilon_1}2,& \widehat s\in\mho_{2 \lt }, Type\, \mathfrak B^{\prime\prime}, \mathfrak C^{\prime\prime},\\ {\mp\epsilon_1 ,} &for\, \sin 3\beta \gt rless 0,\ \ |\alpha_2-\beta|\le \frac{\epsilon_1}2,& \widehat s\in\mho_{2 \gt }, Type\, \mathfrak B^{\prime\prime}, \mathfrak C^{\prime\prime},\\ {\pm\pi\mp\frac{\epsilon_1} 4 ,} &for\, \sin 3\beta \gt rless 0,\ \ ||\alpha_2-\beta|- \pi|\ge \frac{\epsilon_1}2,& \widehat s\in\mho_{2 \lt }, Type\, \mathfrak B^{\prime\prime}, \mathfrak C^{\prime\prime},\\ {\mp\frac{\epsilon_1} 4 ,} &for\, \sin 3\beta \gt rless 0,\ \ |\alpha_2-\beta|\ge \frac{\epsilon_1}2,&\widehat s\in\mho_{2 \gt }, Type\, \mathfrak B^{\prime\prime}, \mathfrak C^{\prime\prime}, \end{array}\right. \end{aligned} \end{equation*}

We confirm Goals (b) and (c) in the following lemma:

Lemma 3.11. Define the deformation $\widehat s\mapsto \xi_j=\widehat s_je^{i\tau_{j}}+\widehat s_{j,\ast}$ on $\mho_j$ by Definition 3.10. We have, for $j=0,1,2$,

(3.125)\begin{align} {|\widehat s_j e^{i\tau_{j,\dagger}}-\widehat r_je^{i(\alpha_j-\beta)}|} \ge & \frac 1C \max\{\widehat r_j,\widehat s_j\},\quad\ {if}\ \,\tau_{j,\dagger}\ne 0, \end{align}

and

(3.126)\begin{equation} \mathfrak{Re}(-i\wp(\frac{\xi_j}{\widehat\sigma},\beta,X))\le 0. \end{equation}

Definition 3.13. Let $\tilde\sigma= \max\{1, X_1 , \sqrt{X_2} , \sqrt[3]{X_3} \}$ with X_j defined by (3.73) and scaled coordinates

\begin{equation*} \zeta =\kappa_1+\frac{\tilde s} {\tilde \sigma}e^{i\beta}\quad\in D_{\kappa_1} \end{equation*}

by replacing $\widehat \sigma$, $\widehat s$, $\widehat s_j$, $\widehat s_{j,\ast}$, $\widehat \lambda$, $\widehat \lambda_{j,\ast}$ by $\tilde \sigma$, $\tilde s$, $\tilde s_j$, $\tilde s_{j,\ast}$, $\tilde \lambda$, $\tilde \lambda_{j,\ast}$ in Definition 3.10. We decompose $X_1,X_2\ge 0,X_3 \gt 0$ into following three cases

(3.145)\begin{equation} (F1) \ \tilde\sigma= X_1 , \qquad\quad (F2) \ \tilde\sigma= \sqrt{X_2} , \qquad\quad (F3) \ \tilde\sigma= \sqrt[3]{X_3} . \end{equation}

Thus, we have achieved Goals (a), (b) and (c) for Cases $(F3)$ and $(F2)$. We will now demonstrate the estimates for I ₄ and I ₅.

Proposition 3.14. For Case $(F3)$, $(F2)$ and $f\in L^\infty (D_{\kappa_j})$ is $\tilde s$-holomorphic,

(3.146)\begin{align} |I_4| _{C^\mu_{\tilde\sigma}(D_{\kappa_j,\frac 1{\tilde\sigma}})}\le & C\epsilon_0 | f|_{L^\infty(D_{\kappa_j})}, \end{align}

(3.147)\begin{align} |I_5| _{L^\infty(D_{\kappa_j )}}\le & C\epsilon_0 | f|_{L^\infty(D_{\kappa_j})}. \end{align}

Estimates for Case $(F1)$ are complex. Below, we outline difficulties and our approach:

• • For $\mho_j$ ( $j \geq 1$): Lemma 3.12 shows no uniform estimates exist for $\mathfrak{Re}(-i\wp(\frac{\tilde s_je^{i\tau_{j,\dagger}}+ \tilde s_{j,\ast}}{X_1}$, $\beta, X))$. We utilize the scaling invariance of the Hilbert transform and estimates (3.137) and (3.138), where $\sqrt{X_2}$ and $\sqrt[3]{X_3}$ are not equal to $\tilde\sigma=X_1$, to derive estimates for $\Gamma_{4j}$ or $\Gamma_{5j}$, $j \geq 1$. Additionally, the Cauchy integral near renormalized critical points, as outlined in Lemma 3.16, needs to avoid singularities for the proper application of (3.86).

• • For $\mho_0$: The scaling argument does not work here since 0 is a singular point. We have to take advantage of estimate (3.140)! If we directly apply the arguments from Case $(F3)$ or $(F2)$ in Proposition 3.14, the difficulty lies in the Jacobian $\frac{1}{|\sin\beta|} d\beta$. Since it is not suitable for improper integrals. To address this for small $|\beta|$, we use a finer decomposition with the $\mathfrak J_1$– $\mathfrak J_5$ approach in (3.169)–(3.173) to extract extra $|\sin\beta|$ decay on Γ₄₀ or Γ₅₀.

Definition 3.15. For Case $(F1)$, introduce new scaled σ_j-parameters on $\mho_j(\beta,X)$, $j=0,1,2,$

(3.158)\begin{equation} \begin{aligned} &\left\{ {\begin{array}{ll} {\sigma_0={\tilde\sigma},\ { \sigma_1}= \sigma_2= \sqrt[3]{|X_3|}},& for\, \ Type\ \mathfrak A, \mathfrak B,\mathfrak E,\\ {\sigma_0={\tilde\sigma},\ { \sigma_1}= \sigma_2= \sqrt[2]{|X_2|}},& for\, \ Type\ \mathfrak C,\mathfrak D, \end{array}}\right. \end{aligned} \end{equation}

and scaled σ_j-coordinates on $\mho_j(\tilde\sigma, \beta,X)$

(3.159)\begin{gather} \lambda=\kappa_1+\frac{\tilde r} {\tilde\sigma}e^{i\alpha}=\kappa_1+\frac{ s_{j,\ast}e^{i\beta}+ r_je^{i\alpha_j}}{ \sigma_j} ,\\ s_{j,\ast}\equiv \tilde s_{j,\ast}\frac{\sigma_j}{\tilde\sigma},\ r_j=\tilde r_j \frac{ \sigma_j}{{\tilde \sigma} } \ge 0,\ \alpha_j = \alpha_j(\beta,X,\lambda),\nonumber \\ \frac{\tilde s}{\tilde \sigma} \mapsto \frac{\vartheta_j}{\sigma_j}\equiv \frac{s_{j,\ast}+s_je^{i\tau_j}}{\sigma_j} , \ \ \tilde s\equiv (s_{j,\ast}\pm s_j)\frac{\tilde\sigma}{\sigma_j} \in \mho_j(\tilde\sigma, \beta,X) \nonumber \end{gather}

where $\mho_j(\tilde\sigma, \beta,X)$, $\tilde s_{j,\ast}$, and $\tilde r_j$ are defined by Definition 3.13.

Lemma 3.16. For Case $(F1)$, introduce the new scaled σ_j-coordinates defined by Definition 3.15,

(3.160)\begin{equation} \inf_\beta s_{1,\ast} = c_0,\quad 0 \lt c_0 \lt 1. \end{equation}

Proposition 3.17. For Case $(F1)$, and $f\in L^\infty (D_{\kappa_j})$ is $\tilde s$-holomorphic,

(3.161)\begin{align} |I_4| _{C^\mu_{\tilde\sigma}(D_{\kappa_j,\frac 1{\tilde\sigma}})}\le & C\epsilon_0 | f|_{L^\infty(D_{\kappa_j})}, \end{align}

(3.162)\begin{align} |I_5| _{L^\infty(D_{\kappa_j )}}\le & C\epsilon_0 | f|_{L^\infty(D_{\kappa_j})}. \end{align}

Proof. Thanks to Lemma 3.16, by following the same argument (3.148)–(3.150), (3.152)–(3.154), we obtain

(3.163)\begin{align} |I_{4}|_{C^\mu_{\tilde\sigma}(D_{\kappa_1,\frac 1{\tilde\sigma}})} \le &C\epsilon_0 |f|_{L^\infty(D_{\kappa_1})} + \sum_{j=0}^2|I_{4j}|_{L^\infty (D_{\kappa_1})} , \end{align}

(3.164)\begin{align} |I_5|_{L^\infty(D_{\kappa_1})} \le & C\epsilon_0|f|_{L^\infty(D_{\kappa_1})}+\sum_{j=0}^2|I_{5j}|_{L^\infty (D_{\kappa_1})}, \end{align}

with

\begin{align*} I_{4j}=& \sum_{h= 1,2} \frac {\theta(1-\tilde r)}{2\pi i} \int_{-\pi}^\pi d\beta[\partial_ {\beta} \ln (1-\gamma |\beta| )]e^{-i(n-1)\beta} \int_{\Gamma _{4j}} \frac{e^{-i \wp(\frac{\tilde s_j}{\tilde\sigma}e^{i\tau_j},\beta,X)}f(\frac{\tilde s_j}{\tilde\sigma}e^{i\tau_j},-\beta,X)} {( \tilde s_je^{i\tau_j}- \tilde r_j e^{i (\alpha_j-\beta)} )^h}d\tilde s_j e^{i\tau_j}, \nonumber \\ I_{5j}=&\frac {\theta( \tilde r-1)}{2\pi i} \int_{-\pi}^\pi d\beta[\partial_ {\beta} \ln (1-\gamma |\beta| )] \int_{\Gamma _{5j,out}} \frac{e^{-i \wp(\frac{\tilde s_j}{\tilde \sigma}e^{i\tau_j},\beta,X)}f(\frac{\tilde s_j}{\tilde \sigma}e^{i\tau_j},-\beta,X)} { \tilde s_je^{i\tau_j}- \tilde r_j e^{i (\alpha_j-\beta)} } d \tilde s_j e^{i\tau_j}.\nonumber \end{align*}

• • $\blacktriangleright$ Proof for $I_{41}, I_{42},I_{51},I_{52}$: For $j\ge 1$, using the scaling invariant of the Hilbert transform and

• - for $\Gamma _{4j}\cap \{s_j \lt 1\}$: applying Lemma 3.16, (3.86);

• - for $\Gamma _{4j}\cap \{s_j \gt 1\}$: thanks to the scaling invariant of the Hilbert transform, applying (3.138) on $\mho_1,\,\mho_2 $ for $Type\,\mathfrak A^{\prime\prime},\,\mathfrak B^{\prime\prime}\,\mathfrak E$, and (3.139) on $\mho_1,\,\mho_2 $ for $Type\,\mathfrak C^{\prime\prime},\,\mathfrak D$,

we have

(3.165)\begin{align} &|I_{4j}|_{C^1(D_{\kappa_1,\frac{1}{\tilde\sigma}})}\\ \le & \sum_{h=0}^1\{|\frac {\theta(1-\tilde r)}{2\pi i} \int_{-\pi}^\pi d\beta[\partial_ {\beta} \ln (1-\gamma |\beta| )] \nonumber\\\times& \int_{\Gamma _{4j}\cap \{s_j \lt 1\}} \frac{e^{-i \wp(\frac{ s_j}{\sigma}e^{i\tau_j},\beta,X)}\widetilde \chi(\frac{ s_j}{\sigma}e^{i\tau_j},-\beta,X)} { (\tilde s_je^{i\tau_j}- \tilde r_j e^{i (\alpha_j-\beta)} )^h( s_je^{i\tau_j}- r_j e^{i (\alpha_j-\beta)} )} d s_j e^{i\tau_j}|_{L^\infty (D_{\kappa_1})}\nonumber \\ +& \sum_{h=0}^1\{|\frac {\theta(1-\tilde r)}{2\pi i} \int_{-\pi}^\pi d\beta[\partial_ {\beta} \ln (1-\gamma |\beta| )] \nonumber\\ \times&\int_{\Gamma _{4j}\cap \{s_j \gt 1\}} \frac{e^{-i \wp(\frac{ s_j}{\sigma}e^{i\tau_j},\beta,X)}\widetilde \chi(\frac{ s_j}{\sigma}e^{i\tau_j},-\beta,X)} { (\tilde s_je^{i\tau_j}- \tilde r_j e^{i (\alpha_j-\beta)} )^h( s_je^{i\tau_j}- r_j e^{i (\alpha_j-\beta)} )} d s_j e^{i\tau_j}|_{L^\infty(D_{\kappa_1})} \}\nonumber\\ \le &C\epsilon_0|\widetilde \chi|_{L^\infty(D_{\kappa_1})}.\nonumber \end{align}

In an entirely similar way, namely, for $j\ge 1$,

• • for $\Gamma _{5j}\cap \{s_j \lt 1\}$: applying Lemma 3.16, (3.86);

• • for $\Gamma _{5j}\cap \{s_j \gt 1\}$: thanks to the scaling invariant of the Hilbert transform, applying (3.138) on $\mho_1,\,\mho_2 $ for $Type\,\mathfrak A^{\prime\prime},\,\mathfrak B^{\prime\prime}\,\mathfrak E$, and (3.139) on $\mho_1,\,\mho_2 $ for $Type\,\mathfrak C^{\prime\prime},\,\mathfrak D$,

we have

(3.166)\begin{equation} \begin{aligned} &|I_{5j}|_{L^\infty(D_{\kappa_1})}\le C\epsilon_0|f|_{L^\infty(D_{\kappa_1})}. \end{aligned} \end{equation}

As a result, the proof for $I_{4j}$, $I_{5j}$ for $j\ge 1$ is done.

• • $\blacktriangleright$ Proof for $I_{40} ,I_{50}$: Decompose

  (3.167)\begin{align} I_{40}=&-\frac {\theta(1-\tilde r)}{2\pi i} \int_{0}^\pi d\beta[\partial_ {\beta} \ln (1-\gamma |\beta| )] \int_{\Gamma_{40}} \mathfrak J_{ \lt } , \end{align}

  (3.168)\begin{align} \mathfrak J_ \lt =& \left\{ {\begin{array}{ll} \mathfrak J_{1}+\mathfrak J_{2}+\mathfrak J_{3}+\mathfrak J_{4}+\mathfrak J_{5} , &\textit{if} 0 \lt \beta \lt \epsilon_1/8,\\ 0,&\textit{if} -\epsilon_1/8 \lt \beta \lt 0,\\ \frac{ e^{-i \wp(\frac{\vartheta_0} {{\sigma_0}},\beta,X)}f(\frac{\vartheta_0} {\sigma_0} ,-\beta,X)}{s_0 e^{i\tau_0} - r_0 e^{i (\alpha_0 -\beta)} } d{\vartheta_0} , &\textit{if} |\beta| \gt \epsilon_1/8; \end{array}}\right. \end{align}

  and $\mathfrak J_1,\cdots,\mathfrak J_5$ defined by

  (3.169)\begin{align} \mathfrak J_1= & \theta(\frac 1{|\sin\beta|}-|\tilde s-\tilde r|) \frac{{[e^{-i\wp(\frac{\tilde se^{i\tau} }{X_1} ,\beta,X)}-1] f(\frac{\tilde se^{i\tau} }{X_1} ,-\beta,X)} }{\tilde s e^{i\tau}- \tilde r e^{i (\alpha-\beta )} }d\tilde se^{i\tau} , \end{align}

  (3.170)\begin{align} \mathfrak J_2= & \theta(\frac 1{|\sin\beta|}-|\tilde s-\tilde r|) \frac{{[ 1-e^{-i\wp(\frac{\tilde se^{-i\tau} }{X_1} ,-\beta,X)} ] f(\frac{\tilde se^{-i\tau} }{X_1} ,-\beta,X )} }{\tilde se^{-i\tau} - \tilde r e^{i (\alpha-\beta )} }d\tilde se^{-i\tau} , \end{align}

  (3.171)\begin{align} \mathfrak J_3=& \theta(\frac 1{|\sin\beta|}-|\tilde s-\tilde r|) e^{-i\wp(\frac{\tilde se^{-i\tau} }{X_1} ,-\beta,X)} f(\frac{\tilde se^{-i\tau} }{X_1} ,-\beta,X) \end{align}

  (3.172)\begin{align} \times& [\frac{1}{\tilde se^{-i\tau} - \tilde r e^{i (\alpha-\beta )} }-\frac{1}{\tilde se^{-i\tau} - \tilde r e^{i (\alpha+\beta )} }]d\tilde se^{-i\tau} ,\nonumber\\ \mathfrak J_4= &\theta(\frac 1{|\sin\beta|}-|\tilde s-\tilde r|) e^{-i\wp(\frac{\tilde se^{-i\tau} }{X_1} ,-\beta,X)}\frac{f(\frac{\tilde se^{-i\tau} }{X_1} ,-\beta,X)- f(\frac{\tilde se^{-i\tau} }{X_1} ,+\beta,X)} {\tilde se^{-i\tau} - \tilde r e^{i (\alpha+\beta )} } d\tilde se^{-i\tau} , \end{align}

  (3.173)\begin{align} \mathfrak J_5=& \theta(|\tilde s-\tilde r|-\frac 1{|\sin\beta|}) ( e^{-i\wp(\frac{\tilde se^{i\tau} }{X_1} ,\beta,X)} \frac{{f(\frac{\tilde se^{i\tau} } {X_1} ,-\beta,X)} }{\tilde se^{i\tau} -\tilde r e^{i (\alpha-\beta )} }d\tilde se^{i\tau} \\ - &e^{-i\wp(\frac{\tilde se^{-i\tau} }{X_1} ,-\beta,X)} \frac{{f(\frac{\tilde se^{-i\tau} } {X_1} ,+\beta,X)} }{\tilde se^{-i\tau} - \tilde r e^{i (\alpha+\beta )} }d\tilde se^{-i\tau} ),\nonumber \end{align}

  with τ defined by Definition 3.10 for $\beta\in[0,\pi]$.

  Let’s explain our strategy before providing detailed estimates for $\mathfrak J_1,\cdots,\mathfrak J_5$.

• - For $\mathfrak J_1$- $\mathfrak J_4$, where $|\tilde s-\tilde r| \lt \frac 1{|\sin\beta|}$, we can extract additional $|\sin\beta|$-decay from the numerators of the integrands by utilizing the difference terms. Moreover, through a change of variables,

  (3.174)\begin{equation} \tilde s\mapsto t=\tilde s|\sin\beta|, \end{equation}

  the t-domain is compact. This allows us to derive effective estimates.

• - For $\mathfrak J_5$, where $|\tilde s-\tilde r| \gt \frac 1{|\sin\beta|}$. We work on the same change of variables, leverage the scaling invariance of the Hilbert transform to cancel the Jacobian. The cut-off function ensures that the kernel of the t-Hilbert transform remains bounded, while the numerator exhibits an exponential decay in t. Consequently, we can derive estimates.

Precisely,

• - $\mathfrak J_1,\mathfrak J_2$: From the mean value theorem and (3.140),

  (3.175)\begin{align} &|e^{-i\wp(\frac{\tilde se^{\pm i\tau} }{X_1} ,\pm \beta,X)} -1|\le C | \sin\beta|,\quad\ for\ \tilde s\in \mho_0(\beta,X),\ |\beta| \lt \frac{\epsilon_1}8 . \end{align}

• - $\mathfrak J_3$: From $|\beta| \lt \epsilon_1/8$,

  (3.176)\begin{align} & |\frac1{\tilde se^{-i\tau_\dagger} - \tilde r e^{i (\alpha-\beta)} } -\frac1{\tilde se^{-i\tau_\dagger} - \tilde r e^{i (\alpha+\beta)} }|\\ =& |\frac{\tilde r e^{i\alpha}2\sin\beta}{(\tilde se^{-i\tau_\dagger} - \tilde r e^{i (\alpha-\beta)} )(\tilde se^{-i\tau_\dagger} - \tilde r e^{i (\alpha+\beta)} )}| \le C|\sin\beta| .\nonumber \end{align}

• - $\mathfrak J_4$: From $\tilde s$-holomorphic properties of f,

  (3.177)\begin{align} &|\frac{ f(\frac{\tilde s}{\tilde \sigma}e^{-i\tau_\dagger} ,-\beta,X)-f(\frac{\tilde s}{\tilde \sigma}e^{-i\tau_\dagger} ,+\beta,X)} {\tilde se^{-i\tau_\dagger} - \tilde r e^{i (\alpha+\beta)} }|\\ \le & C|f|_{L^\infty(D_{\kappa_1})}\frac{\tilde s| \sin\beta|} {|\tilde se^{-i\tau_\dagger} - \tilde r e^{i (\alpha+\beta)} |} \le C|f|_{L^\infty(D_{\kappa_1})}| \sin\beta|. \nonumber \end{align}

Applying (3.169)–(3.172), (3.175)–(3.177) and the change of variables (3.174), for $j=1,\ldots, 4$,

(3.178)\begin{align} & |-\frac {\theta(1-\tilde r)}{2\pi i} \int_{0}^\pi d\beta[\partial_ {\beta} \ln (1-\gamma |\beta| )] + \int_{ \Gamma _{40}} \mathfrak J_j |_{L^\infty(D_{\kappa_1} )}\\ \le & C|f|_{L^\infty(D_{\kappa_1})}|\int_{0}^\pi d\beta[\partial_ {\beta} \ln (1-\gamma |\beta| )] \int_{2|sin\beta|} ^{X_1\delta|\sin\beta|} \theta(1- \left|t-(\tilde r|\sin\beta|)\, \right|) dt |_{L^\infty(D_{\kappa_1})}\nonumber\\ \le &C\epsilon_0 |f|_{L^\infty(D_{\kappa_1})}.\nonumber \end{align}

On the other hand, using (3.173), $\epsilon_1 \gt 0$, the rescaling (3.174), the scaling invariant property of the Hilbert transform, one obtains

(3.179)\begin{align} &|-\frac {\theta(1-\tilde r)}{2\pi i} \int_{0}^\pi d\beta[\partial_ {\beta} \ln (1-\gamma |\beta| )] \int_{ \Gamma _{40}} \mathfrak J_5|_{L^\infty(D_{\kappa_1})}\\ \le & C\epsilon_0|f|_{L^\infty(D_{\kappa_1})}\int_{2|sin\beta|} ^{X_1\delta|\sin\beta|} \theta(| t-\tilde r|\sin\beta|\,|-1) \frac{ e^{-t\sin\frac{\epsilon_1}4} } {|t-\tilde r|\sin\beta|\,|} dt|_{L^\infty(D_{\kappa_1})} \nonumber\\ \le & C\epsilon_0 |f|_{L^\infty(D_{\kappa_1})}. \nonumber \end{align}

From (3.178) and (3.179), $ |I_{40}|_{L^\infty(D_{\kappa_1})}\le C\epsilon_0 |f|_{L^\infty(D_{\kappa_1})}$. In an entirely similar way, $|I_{50}|_{L^\infty(D_{\kappa_1})}\le C\epsilon_0 |f|_{L^\infty(D_{\kappa_1})}$. Combining with (3.163)–(3.166), the proof for (3.161), (3.162) is completed.

4.2.1. The Lax–Sato formulation of the KP equation

Our approach to establish an IST of perturbed $\textrm{Gr}(N,M)_{ \gt 0}$ KP solitons is based on (3.1), (3.2), (4.9)–(4.11). To verify these formulas, we summarize the Lax–Sato formulation of the KP equation [Reference Kodama17, § 2.1–2.4] in this subsection.

• • $\blacktriangleright$(The KP hierarchy, the KP equation, and the Lax pair): Suppose the operator $\mathfrak L$ can be gauge transformed into the trivial operator $\partial=\partial_{x_1}$, i.e.

  (4.41)\begin{equation} \partial={{\mathcal W}}^{-1}\mathfrak L{{\mathcal W}}, \end{equation}

  where

  (4.42)\begin{equation} \begin{aligned} {{\mathcal W}}=&1-w_1\partial^{-1}-w_2\partial^{-2}-w_3\partial^{-3}+\cdots,\\ w_j=&w_j(x_1,x_2,x_3,\ldots){,\quad x=(x_1,x_2,x_3,\ldots).} \end{aligned} \end{equation}

  If the Sato equation

  (4.43)\begin{equation} \partial_{x_n}{{\mathcal W}}=B_n{{\mathcal W}}-{{\mathcal W}}\partial^n\quad for\,n=1,2,\ldots, \end{equation}

  holds where $B_n=\left({{\mathcal W}}\partial^n{{\mathcal W}}^{-1}\right)_{\ge 0}$, the polynomial part of $ \mathfrak L^n$ in $\partial$, then

  (4.44)\begin{gather} \partial_{x_n}\mathfrak L=[B_n, \mathfrak L], \end{gather}

  (4.45)\begin{gather} \partial_{x_m} B_n-\partial_{x_n} B_m+[ B_n, B_m]=0, \end{gather}

  and the gauge transform (4.41) transforms the linear system of the vacuum wave function

  (4.46)\begin{equation} \left\{ {\begin{array}{ll} {\partial \phi_0}=\lambda\phi_0,&\\ \partial_{x_n}\phi_0=\partial^n\phi_0=k^n\phi_0,&n=1,2,\cdots, \end{array}} \right.\qquad \phi_0(x,\lambda)=\exp(\sum_{n=1}^\infty \lambda^nx_n), \end{equation}

  to the KP linear system

  (4.47)\begin{equation} \left\{ {\begin{array}{ll} {\mathfrak L \phi} =\lambda\phi ,&\\ \partial_{x_n}\phi =B_n\phi ,&n=1,2,\cdots, \end{array}} \right.\qquad \phi (x,\lambda)={{\mathcal W}}\phi_0. \end{equation}

  Note that given a pair (n, m) with n > m, from (4.45) (called the Zakharov–Shabat equations), we obtain a system of n − 1 equations for $u_2,u_3,\ldots, u_n$,

  (4.48)\begin{equation} \begin{gathered} \mathfrak L=\partial+u_2\partial^{-1}+u_3\partial^{-2}+\cdots,\\ u_2=w_{1,x_1},\ u_3=w_{2,x_1}+w_1w_{1,x_1}, \ldots, \end{gathered} \end{equation}

  in the variables $x_1,x_m,x_n$. For $(n,m)=(3,2)$, the Zakharov–Shabat equations yield the Kadomtsev–Petviashvili equation (1.1) for $u=2u_2=2w_{1,x_1}$, and, from the second equation of (4.47), we derive the Lax pair (1.2).

• • $\blacktriangleright$(The tau function and wave eigenfunctions): To derive the τ-function rep of solutions and wave eigenfunctions, let

  \begin{equation*} \begin{aligned} {{\mathcal W}}=& 1-w_1\partial^{-1}-w_2\partial^{-2}-\cdots-w_N\partial^{-N},\\ {{\mathcal W}}_N\equiv &{{\mathcal W}}\partial^N= \partial^{N}-w_1\partial^{N-1}-w_2\partial^{N-2}-\cdots-w_N. \end{aligned} \end{equation*}

  Hence the Sato equation (4.43) turns into

  (4.49)\begin{equation} \partial_{x_n}{{\mathcal W}}_N=B_n{{\mathcal W}}_N-{{\mathcal W}}_N\partial^n\quad\,for\,n=1,2,\ldots \end{equation}

  which yield

  \begin{equation*} \partial_{x_n}({{\mathcal W}}_Nf)=B_n({{\mathcal W}}_Nf)+{{\mathcal W}}_N(\partial_{x_n}f-\partial_{x_1}^nf). \end{equation*}

  We conclude that if the Sato equation holds for $\mathcal W_N$, then any solution of the N-th-order ODE $\mathcal W_Nf=0$ also satisfies the linear heat hierarchy, i.e., $\partial_{x_n}f=\partial_{x_1}^nf$ for $n=1,2,\ldots$.

  Conversely, if f_j for j = 1 up to N satisfy the N-th order ODE and the heat hierarchy, then the Sato equation (4.49) holds. Hence, an explicit KP solution can be found via the τ-function

  (4.50)\begin{equation} u(x)=2w_{1,x_1}= 2\partial^2_{x_1}\ln\tau(x)\equiv 2\partial^2_{x_1}\ln\textrm{Wr}(f_1,\cdots,f_N) \end{equation}

  by writing ${{\mathcal W}}_Nf_j=0$ as

  \begin{equation*} \left[ \begin{array}{cccc} f_1& f_1^{(1)} &\cdots & f_1^{(N-1)}\\ \vdots&\vdots&\ddots&\vdots\\ f_N& f_N^{(1)} &\cdots & f_N^{(N-1)} \end{array} \right] \left[ \begin{array}{c} w_N\\ \vdots\\ w_1 \end{array} \right] = \left[ \begin{array}{c} f_1^{(N)}\\ \vdots\\ f_N^{(N)} \end{array} \right]. \end{equation*}

  For the wave function $\phi={{\mathcal W}}_N\phi_0$ of the KP linear system (4.47), we begin by expressing it in the determinant form [Reference Kodama17, Proposition 2.2]:

  \begin{align*} \phi=&W_N\phi_0=(1-\frac{w_1}{\lambda}-\frac{w_2}{\lambda^2}-\cdots-\frac{w_N}{\lambda^N})\phi_0=\frac{1}{\tau}\left| \begin{array}{cccc} f_1 &f_1^{(1)}&\cdots &f_1^{(N)}\\ \vdots&\vdots&\ddots &\vdots\\ f_N &f_N^{(1)}&\cdots &f_N^{(N)}\\ \lambda^{-N} &\lambda^{-N+1} &\cdots &1 \end{array} \right|\phi_0. \end{align*}

  Using elementary column operations, the determinant of the above expression can be rewritten as

  (4.51)\begin{equation} \frac{(-1)^N}{\lambda^N}\left| \left(f_i^{(j)}-\lambda f_i^{(j-1)} \right)_{1\le i,j\le N} \right|. \end{equation}

  Besides, express $f_i(x)$ in the integral form $ f_i(x)=\int_Ce^{\sum\zeta^nx_n}\rho_i(\zeta)d\zeta$, which is satisfied by the ${\mathrm{Gr}(N,M)_{ \gt 0}}$ KP solitons. Consequently, the numerator takes on this specific form:

  \begin{align*} &f_i^{(j)}(x)-\lambda f_i^{(j-1)}(x)\nonumber\\ =&-\lambda\int_C\zeta^{j-1}(1-\frac{\zeta}{\lambda})e^{\sum\zeta^nx_n}\rho_i(\zeta)d\zeta\nonumber\\ =&-\lambda\int_C\zeta^{j-1}e^{-\sum\frac{\zeta^n}{n\lambda^n}} e^{\sum\zeta^nx_n}\rho_i(\zeta)d\zeta\nonumber\\ =&-\lambda f_i^{(j-1)}(x_1-\frac{1}{\lambda},x_2-\frac{1}{2\lambda^2},x_3-\frac{1}{3\lambda^3},\ldots)\nonumber. \end{align*}

  As a result,

  (4.52)\begin{equation} \phi(x,\lambda)=\frac{\tau(x_1-\frac{1}{\lambda},x_2-\frac{1}{2\lambda^2},x_3-\frac{1}{3\lambda^3},\ldots)}{\tau(x)}\phi_0(x). \end{equation}

  Moreover, a corresponding formula for the adjoint wave function $\phi^\dagger$ can also be derived [Reference Kodama17, § 2.4]:

  (4.53)\begin{equation} \phi^\dagger(x,\lambda)=\frac{\tau(x_1+\frac{1}{\lambda},x_2+\frac{1}{2\lambda^2},x_3+\frac{1}{3\lambda^3},\ldots)}{\tau(x)}\phi^{-1}_0(x). \end{equation}

• • $\blacktriangleright$ (The multi-line solitons and Sato eigenfunctions): The multi-line solitons (3.1) is defined by setting $1\le n\le 3$, letting

  (4.54)\begin{equation} \begin{gathered} \left( \begin{array}{c} f_1(x)\\ \vdots\\ f_N(x) \end{array} \right)= \left( \begin{array}{cccc} a_{11} &a_{12} & \cdots & a_{1M}\\ \vdots & \vdots &\ddots &\vdots\\ a_{N1} &a_{N2} & \cdots & a_{NM} \end{array} \right) \left( \begin{array}{c} E_{1} \\ E_{2} \\ \vdots \\ E_{M} \\ \end{array} \right),\\ A= (a_{ij})\in {\mathrm{Gr}(N, M)_{\ge 0}},\ \ \kappa_1 \lt \cdots \lt \kappa_M,\\ E_j(x)= \exp\theta_j(x)=\exp( \kappa_j x_1+\kappa_j^2 x_2+\kappa_j^3 x_3)\\ = \int_Ce^{\zeta x_1+\zeta^2 x_2+\zeta^3 x_3} \rho_j(\zeta)d\zeta,\ \ \rho_j(\zeta) is the point measure at \kappa_j, \end{gathered} \end{equation}

  in (4.50); and formula (4.9) of $ \varphi(x,\lambda)$ and (4.10) of $ \psi(x,\lambda)$) are derived by (4.50), (4.52)–(4.54), and

  \begin{align*} &e^{\theta_i(x_1-\frac{1}{\lambda},x_2-\frac{1}{2\lambda^2},x_3-\frac{1}{3\lambda^3},\cdots)}=e^{\left(\sum_{n=1}^\infty \kappa_i^n x_n\right)-\ln (1-\frac{\kappa_i}{\lambda} )}=\frac{\lambda-\kappa_i}{\lambda} e^{\theta_i(x)} . \end{align*}

4.2.2. The Lax equation

The Lax equation can be proved by replacing the Sato eigenfunction and Sato adjoint eigenfunction by (4.9) and adapting the procedure (3.28)–(3.42) for the proof of perturbed 1-solitons. Major difficulties and differences occur in proving the orthogonality relation (for the construction of Green’s function G) and boundedness of G_d. More precisely,

• • $\blacktriangleright$ (The orthogonality relation) : [Reference Boiti, Pempinelli and Pogrebkov5, Reference Boiti, Pempinelli, Pogrebkov and Prinari9, Reference Boiti, Pempinelli, Pogrebkov and Prinari10] Let

  (4.55)\begin{equation} \begin{aligned} \varphi_j(x)=\varphi(x,\kappa _j),&\quad \psi_j(x)=\textit{res}_{\lambda=\kappa_j}\psi(x,\lambda),\\ \varphi(x,\kappa)= (\varphi_1(x), \ldots, \varphi_M (x)),&\quad \psi(x,\kappa)= (\psi_1(x), \ldots, \psi_M (x)). \end{aligned} \end{equation}

  Define

  (4.56)\begin{equation} \begin{aligned} &\mathcal D^\flat= \textrm{diag}\,( \kappa^N_1 ,\ldots,\kappa^N_M )\, A^T ,\\ &\mathcal D^{\flat,\dagger} = \left(\begin{array}{c}-d^T,\ I_{M-N}\end{array}\right)\,\pi\,\textrm{diag}\,( \kappa^{-N}_1 ,\ldots,\kappa^{-N}_M ), \end{aligned} \end{equation}

  where π is an M × M permutation matrix and d is an $N\times (M-N)$ matrix satisfying

  (4.57)\begin{equation} A = \left(\begin{array}{c} I_N,\ d \end{array} \right)\pi. \end{equation}

  We can implement various Pl $\ddot{\textrm u}$cker relations of $\varphi(x,\kappa)$ and $\psi(x,\kappa)$ to prove

  \begin{equation*} \mathcal D^{\flat,\dagger}\mathcal D^\flat=0,\ \ \varphi(x,\kappa)\mathcal D^\flat=0,\ \ \mathcal D^{\flat,\dagger}\psi(x,\kappa)^T=0 \end{equation*}
  (see [Reference Wu32, Lemma 2.1, 2.2] for detailed proofs). Together with setting
  \begin{equation*}P=\mathcal D(\mathcal D^T\mathcal D)^{-1}\mathcal D^T,\, P'=(\mathcal D')^T(\mathcal D'{\mathcal D'}^T)^{-1}\mathcal D',\, P\oplus P'=I_{M\times M},\end{equation*}

  we justify the orthogonality relation

  (4.58)\begin{equation} \sum_{j=1}^M\varphi_j (x)\psi_j(x')=0 . \end{equation}

• • $\blacktriangleright$ (Boundedness of G_d) :

  • - following argument to permute and exchange cells, one obtains the decomposition [Reference Boiti, Pempinelli and Pogrebkov5, (3.16),(3.17)], [Reference Boiti, Pempinelli and Pogrebkov4],

    (4.59)\begin{equation} G_d(x,x',\lambda)=G_{d}^1(x,x',\lambda)+G_{d}^2(x,x',\lambda), \end{equation}

    with

    (4.60)\begin{align} &G_{d}^1(x,x',\lambda)\\ = &-\frac{\theta(x_2-x_2')}{[(N-1)!]^2(N+1)}\sum_{\{m_i\},\{n_i\}} \operatorname{sgn}(z_{m_Nn_N}-z'_{m_Nn_N})\nonumber\\ \times& e^{-k_{m_Nn_N}(x_2-x_2')} V(\{m_i\},n_N) V(n_1,\ldots,n_{N-1}) \nonumber\\ \times &\theta((\lambda_R-\kappa_{m_N})(z_{m_Nn_N}-z'_{m_Nn_N})) e^{-(\lambda_R-\kappa_{m_N})(z_{m_Nn_N}-z'_{m_Nn_N})}\nonumber\\ \times&\dfrac{\mathcal D^\flat(\{m_i\}) \exp(\sum_{l=1}^{N-1} E_{m_l}(x)+E_{n_N}(x))}{\tau(x)} \nonumber\\ \times&\dfrac{\mathcal D^\flat(\{n_i\}) \exp(\sum_{l=1}^{N-1} E_{n_l}( x')+E_{m_N}(x'))}{\tau(x')} ,\nonumber \end{align}

    and

    (4.61)\begin{align} &G_{d}^2(x,x',\lambda)\\ = &+\frac{\theta(x_2-x_2')}{[(N-1)!]^2(N+1)}\sum_{\{m_i\},\{n_i\}} \operatorname{sgn}(z_{m_Nn_N}-z'_{m_Nn_N})\nonumber\\ \times& e^{-k_{m_Nn_N}(x_2-x_2')} V(\{m_i\},n_N) V(n_1,\ldots,n_{N-1}) \nonumber\\ \times &\theta((\lambda_R-\kappa_{n_N})(z_{m_Nn_N}-z'_{m_Nn_N})) e^{-(\lambda_R-\kappa_{n_N})(z_{m_Nn_N}-z'_{m_Nn_N})}\nonumber\\ \times&\dfrac{\mathcal D^\flat(\{m_i\}) \exp(\sum_{l=1}^{N} E_{m_l}(x) )}{\tau(x)} \dfrac{\mathcal D^\flat(\{n_i\}) \exp(\sum_{l=1}^{N} E_{n_l}(x '))}{\tau(x')} ,\nonumber \end{align}

    where $z_{mn}=x_1+(\kappa_m+\kappa_n)x_2$, $z'_{mn}=x'_1+(\kappa_m+\kappa_n)x'_2$, $k_{mn}=\lambda_I^2-(\lambda_R-\kappa_m)(\lambda_R-\kappa_n)$, and

    \begin{gather*} V(\{n_i\})= \det\left( \begin{array}{cccc} 1&1&\cdots &1\\ \kappa_{n_1}&\kappa_{n_2}&\cdots &\kappa_{n_N}\\ \vdots&\vdots&\ddots&\vdots\\ \kappa_{n_1}^{N -1}&\kappa_{n_2}^{N -1}&\cdots &\kappa_{n_N}^{N -1} \end{array} \right),\\ \mathcal D^\flat(\{n_i\})= \det\left( \begin{array}{ccc} \mathcal D^\flat_{n_1,1}&\cdots & \mathcal D^\flat_{n_1,N}\\ \vdots&\ddots&\vdots\\ \mathcal D^\flat_{n_N,1}&\cdots & \mathcal D^\flat_{n_N,N} \end{array} \right), \end{gather*}

    for $m,\,n\in\{1,\ldots,M\}$ and $\{m_i\}=\{m_1,\ldots,m_N\},\{n_1,\ldots, n_N\}$ denote unordered set of N indices from $\{1,\ldots,M\}$.

    Therefore, $G_d^2$ is uniformly bounded because, after permutation, the cells E_J that appear in the numerators are also present in the denominator of the tau function. Conversely, $G_d^1$ is uniformly bounded in x due to the Vandermonde matrix $V(\{m_i\},n_N)$ and the TP condition, though it is not necessarily uniformly bounded in xʹ. Since m_N may not be distinct from $n_1,\ldots,n_{N-1}$. Consequently,

    (4.62)\begin{align} |G_d(x,x',\lambda)| \lt &C (C_{x'}+1 ), \end{align}

    and, combining estimates for the continuous Green function G_c,

    (4.63)\begin{equation} |G(x,x',\lambda)| \lt C (C_{x'}+\frac1{\sqrt{|x_2-x_2'|}} ). \end{equation}

    To enhance the above estimate, we will utilize the duality between $\mathrm{Gr}(N,M)_{ \gt 0}$ and $\mathrm{Gr}(M-N, M)_{ \gt 0}$ [Reference Kodama18, Section 4.4]:

    (4.64)\begin{equation} \begin{aligned} &\textit{If}\ u_s(x)\ \textit{is a}\ {\mathrm{Gr}(N,M)_{ \gt 0}}\ \textit{KP soliton,} \\ &\textit{then}\ u_s(-x)\ \textit{is a}\ {\mathrm{Gr}(M-N,M)_{ \gt 0}}\ \textit{KP soliton.} \end{aligned} \end{equation}

    Consider the Lax operators

    (4.65)\begin{align} \mathcal L_\pm =&-\partial_{x_2}+\partial_{x_1}^2+u_s(\pm x_1,\pm x_2,0), \end{align}

    (4.66)\begin{align} \mathcal L_1=& +\partial_{x_2}+\partial_{x_1}^2+u_s(+ x_1,+ x_2,0), \end{align}

    and the associated Green functions by $\mathcal G_\pm(x,x', \lambda)$, $\mathcal G_1(x,x', \lambda)$ respectively, namely, $ \mathcal L_\pm\mathcal G_\pm(x,x', \lambda) = \mathcal L_1\mathcal G_1 (x,x', \lambda) =\delta(x-x') $, and define

    (4.67)\begin{align} \mathcal G_\pm(x,x', \lambda) =&e^{\lambda (x_1-x_1')+\lambda^2 (x_2-x_2')} G_\pm (x,x', \lambda), \end{align}

    (4.68)\begin{align} \mathcal G_1(x,x', \lambda) =&e^{\lambda (x'_1-x_1 )+\lambda^2 (x'_2-x_2 )} G_1 (x,x', \lambda). \end{align}

    Applying the duality theorem (4.64), (4.64), and (4.63), one has

    (4.69)\begin{align} |G_\pm(x,x',\lambda)| \lt &C (C_{x'}+\frac1{\sqrt{|x_2-x_2'|}} ). \end{align}

    Thanks to $\mathcal L_1=\mathcal L^d_+$,

    (4.70)\begin{equation} G_1(x,x',\lambda)= G_+(x',x,\lambda) \end{equation}
    [Reference Boiti, Pempinelli and Pogrebkov4, (1.6)]. From $ \mathcal L_1(x_1,x_2)=\mathcal L _-(-x_1,-x_2)$,
    (4.71)\begin{equation} \begin{aligned} G_1(x,x',\lambda)= G_-(-x,-x',\lambda). \end{aligned} \end{equation}

    Combining (4.69)–(4.71), we conclude

    (4.72)\begin{align} |G_\pm(x,x',\lambda)| \lt &C (1+\frac1{\sqrt{|x_2-x_2'|}} ). \end{align}

4.2.3. The inverse problem

To address the inverse problem, we will examine the limit of the iteration sequence within the eigenfunction space W:

(4.73)\begin{align} & \phi^{(k)}(x ,\lambda) = 1+\sum_{n=1}^N \frac{\phi^{(k)}_{z_n,\operatorname{res}} (x )}{\lambda -z_n} +\mathcal CT \phi^{(k-1)}(x, \lambda) ,\ \ k \gt 0, \end{align}

(4.74)\begin{align} &(e^{\kappa_1x_1+\kappa_1^2x_2+\kappa_1^3x_3}\phi^{(k)}(x,\kappa^+_1),\ldots, e^{\kappa_Mx_1+\kappa_M^2x_2+\kappa_M^3x_3}\phi^{(k)}(x,\kappa^+_M))\mathcal D=0, \end{align}

(4.75)\begin{align} &\phi^{(0)}(x ,\lambda)= \widetilde \chi(x,\lambda) . \end{align}

The previous arguments can be adapted with some modifications. However, we must clarify how to implement the $\mathcal D$-symmetry to reduce estimates of the residues to those of the Cauchy integral operator at $\kappa_j^+$ throughout the iteration process. The lemma is stated as follows.

Proposition 4.7. Suppose $ {\mathcal S}=(\{z_n\},\{\kappa_j\}, \mathcal D,s _c)$ is d-admissible and $\phi^{(k)}$, $\phi^{(k)}_{z_n,\operatorname{res}}$ satisfy (4.73), (4.74). Then for k > 0,

(4.76)\begin{equation} {\left( \begin{array}{c} \phi^{(k)}_{z_1,\operatorname{res}} \\ \vdots\\ \phi^{(k)}_{z_N,\operatorname{res}} \end{array} \right)= -B^{-1} \widetilde A \left( \begin{array}{c} 1+\mathcal C_{{\kappa_1^+}}T\phi^{(k-1)}\\ \vdots\\ \vdots\\ \vdots\\ \vdots\\ 1+\mathcal C_{{\kappa_M^+}}T\phi^{(k-1)} \end{array} \right),} \end{equation}

where

(4.77)\begin{align} \widetilde A= & \left( \begin{array}{cccccc} \kappa_1^Ne^{\theta_1}&\cdots&0&\mathcal D_{N+1,1}e^{\theta_{N+1}}&\cdots&\mathcal D_{M,1}e^{\theta_M} \\ \vdots&\ddots&\vdots&\vdots&\ddots&\vdots \\ 0&\cdots&\kappa_N^Ne^{\theta_N}&\mathcal D_{N+1,N}e^{\theta_{N+1}}&\cdots&\mathcal D_{M,N}e^{\theta_M} \end{array} \right) ,\ \ B= \widetilde A \left( \begin{array}{ccc} \frac 1{\kappa_1-z_1}&\cdots&\frac 1{\kappa_1-z_N}\\ \vdots&\ddots&\vdots\\ \vdots&\ddots&\vdots\\ \vdots&\ddots&\vdots\\ \vdots&\ddots&\vdots\\ \frac 1{\kappa_M-z_1}&\cdots&\frac 1{\kappa_M-z_N} \end{array} \right), \nonumber\\ \end{align}

and $ e^{\theta_j}= e^{\kappa_j x_1+\kappa_j^2 x_2+\kappa_j^3 x_3}$. Moreover, for k > 0,

(4.78)\begin{align} \sum_{0\le l_1+2l_2+3l_3\le d+5}\left|\partial_x^l\phi^{(k)}_{z_n,\operatorname{res}}\right|_{L^\infty}\le C(1+\epsilon_0\sum_{0\le l_1+2l_2+3l_3\le d+5}&\left|\partial_x^l\phi^{(k-1)} \right|_W), \end{align}

(4.79)\begin{align} \sum_{0\le l_1+2l_2+3l_3\le d+5}\left|\partial_x^l\left[\phi^{(k)}_{z_n,\operatorname{res}}-\phi^{(k-1)}_{z_n,\operatorname{res}}\right]\right|_{L^\infty}\le & (C\epsilon_0)^{k} , \end{align}

(4.80)\begin{align} \sum_{0\le l_1+2l_2+3l_3\le d+5}\left|\partial_x^l\left[\phi^{(k)}_{z_n,\operatorname{res}}-\widetilde\chi_{z_n,\operatorname{res}}\right]\right|_{L^\infty}\le& C\epsilon_0 . \end{align}

Proof. Write the $\mathcal D$-symmetry and the evaluation at $ \kappa_j ^+ $ of $\phi^{(k)}$ as a linear system for M + N variables $\{\phi^{(k)}(x,\kappa_j^+),\phi^{(k)}_{z_n,\operatorname{res}}(x)\}$,

(4.81)\begin{align} &\left( \begin{array}{ccccccccc} \kappa_1^Ne^{\theta_1}&\cdots&0&\mathcal D_{N+1,1}e^{\theta_{N+1}}&\cdots&\mathcal D_{M,1}e^{\theta_M}&0&\cdots&0\\ \vdots&\ddots&\vdots&\vdots&\ddots&\vdots&0&\ddots&0\\ 0&\cdots&\kappa_N^Ne^{\theta_N}&\mathcal D_{N+1,N}e^{\theta_{N+1}}&\cdots&\mathcal D_{M,N}e^{\theta_M}&0&\cdots&0\\ -1&\cdots&0&0&\cdots&0&\frac 1{\kappa_1-z_1}&\cdots&\frac 1{\kappa_1-z_N}\\ \vdots&\ddots&\vdots&\vdots&\ddots&\vdots&\vdots&\ddots&\vdots\\ \vdots&\ddots&\vdots&\vdots&\ddots&\vdots&\vdots&\ddots&\vdots\\ \vdots&\ddots&\vdots&\vdots&\ddots&\vdots&\vdots&\ddots&\vdots\\ \vdots&\ddots&\vdots&\vdots&\ddots&\vdots&\vdots&\ddots&\vdots\\ 0&\cdots&0&0&\cdots&-1&\frac 1{\kappa_M-z_1}&\cdots&\frac 1{\kappa_M-z_N} \end{array} \right) \left( \begin{array}{c} \phi^{(k)}(x,\kappa_1^+)\\ \vdots\\ \vdots\\ \vdots\\ \vdots\\ \phi^{(k)}(x,\kappa_M^+)\\ \phi^{(k)}_{z_1,\operatorname{res}} (x)\\ \vdots\\ \phi^{(k)}_{z_N,\operatorname{res}} (x) \end{array} \right)\nonumber\\ &= \left( \begin{array}{c} 0\\ \vdots\\ 0\\ -1-\mathcal C_{\kappa_1^+}T\phi^{(k-1)}\\ \vdots\\ \vdots\\ \vdots\\ \vdots\\ -1-\mathcal C_{\kappa_M^+}T\phi^{(k-1)} \end{array} \right) . \end{align}

Solving $\phi^{(k)}(x,\kappa_j^+)$ in terms of $\phi^{(k)}_{z_n,\operatorname{res}}(x)$ and plugging the outcomes into (4.81) yields

(4.82)\begin{equation} \begin{aligned} &B\left( \begin{array}{c} \phi^{(k)}_{z_1,\operatorname{res}}(x)\\ \vdots\\ \phi^{(k)}_{z_N,\operatorname{res}}(x) \end{array} \right)=-\widetilde A \left( \begin{array}{c} 1+\mathcal C_{\kappa_1^+}T\phi^{(k-1)}\\ \vdots\\ \vdots\\ \vdots\\ \vdots\\ 1+\mathcal C_{\kappa_M^+}T\phi^{(k-1)} \end{array} \right) , \end{aligned} \end{equation}

with B and $\widetilde A $ defined by (4.77). By the d-admissible condition, the system (4.81) is just determined and is equivalent to (4.76).

Next, the d-admissible condition implies that defining $\mathfrak A$ through:

\begin{equation*} \begin{gathered} {\mathfrak A}^T=\textit{diag}\,(\kappa_1^N, \ldots, \kappa_M^N)^{-1}\times {\mathfrak D}^\sharp , \end{gathered} \end{equation*}

then ${\mathfrak A}\in \mathrm {Gr}(N,M)_{ \gt 0}$. Let $\widetilde \chi'(x,\lambda) $ be the normalized Sato eigenfunction with data $(\{z_n\},\{\kappa_j\},{{\mathfrak A},}0)$, we have:

(4.83)\begin{gather} { { \widetilde\chi'}(x, \lambda) =1+\sum_{n=1}^N\frac{ \widetilde\chi'_{z_n, \operatorname{res}} (x )}{\lambda -z_n} ,} \end{gather}

(4.84)\begin{gather} (e^{\kappa_1x_1+\kappa_1^2x_2+\kappa_1^3x_3}\widetilde \chi'(x,\kappa _1),\ldots,e^{\kappa_Mx_1+\kappa_M^2x_2+\kappa_M^3x_3}\widetilde \chi'(x,\kappa _M))\mathcal D=0, \end{gather}

and, from the d-admissible condition, $\forall k$,

(4.85)\begin{equation} |\widetilde\chi'_{z_n, \operatorname{res}} (x )-\widetilde\chi _{z_n, \operatorname{res}} (x )|_{C^k} \le C_k\epsilon_0. \end{equation}

Moreover, using previous argument,

(4.86)\begin{equation} \quad{\left( \begin{array}{c} \widetilde\chi'_{z_1,\operatorname{res}} (x) \\ \vdots\\ \widetilde\chi'_{z_N,\operatorname{res}}(x) \end{array} \right)= - B^{-1} \widetilde { A} \left( \begin{array}{c} 1 \\ \vdots\\ \vdots\\ \vdots\\ \vdots\\ 1 \end{array} \right),} \end{equation}

with B and $\widetilde A $ defined by (4.77). Let $E_j =e^{\theta_j} = e^{\kappa_j x_1+\kappa_j^2 x_2+\kappa_j^3 x_3} $ and write

(4.87)\begin{align} \widetilde { A}= & \mathcal D^T \textit{diag}\,( E_1, \ldots, E_M),\\ B =& \mathcal D^T \textit{diag}\,( E_1, \ldots, E_M)\left( \begin{array}{ccc} \frac 1{\kappa_1-z_1} &\cdots&\frac{1}{\kappa_1-z_N} \\ \vdots&\ddots&\vdots\\ \vdots&\ddots&\vdots\\ \frac 1{\kappa_M-z_1} &\cdots&\frac{1}{\kappa_M-z_N} \end{array} \right).\nonumber \end{align}

From Sato’s theory, (4.9), (4.3), (4.86), (4.87), elementary row and column operations and matching the coefficients of $E_1\times\cdots\times E_N$,

(4.88)\begin{align} &B^{-1} = \frac{1}{\tau'(x)} \left( \begin{array}{ccc} b_{11}&\cdots & b_{1N}\\ \vdots&\ddots&\vdots\\ b_{N1}&\cdots & b_{NN} \end{array} \right),\nonumber \\ & {b_{kl}= \sum_{J(kl) =( j _{(kl),1},\cdots,j _{(kl),N-1})}\Lambda_{J(kl)} E_{J(kl)}(x),\ \ 1\le j _{(kl),1} \lt \cdots \lt j _{(kl),N-1}\le M,}\\ & \tau'(x)\ \textit{is the tau function with data}\ \kappa_j,\ {\mathfrak A},\nonumber\\ & |\Lambda _{J(kl)}|=|\Lambda _{J(kl)} (\{z_n\},\{\kappa_j\},{{\mathfrak A}})| \lt C. \nonumber \end{align}

As a consequence,

(4.89)\begin{align} &\tau'(x) \widetilde\chi'_{z_h,\operatorname{res}}(x)\\ =&\textit{the}\,h^{\mathrm{th}}\textit{-row of}\ { \left( \begin{array}{ccc} b_{11}&\cdots & b_{1N}\\ \vdots&\ddots&\vdots\\ b_{N1}&\cdots & b_{NN} \end{array} \right) \left( \begin{array}{c} \kappa_1^NE_1+\cdots+\mathcal D_{N+1,1}E_{N+1}+\cdots+\mathcal D_{M,1}E_M\\ \vdots\\ \vdots\\ \vdots\\ \vdots\\ \kappa_N^NE_{ 1}+\cdots+\mathcal D_{N+1,N}E_{N+1}+\cdots+\mathcal D_{M,N}E_M \end{array} \right)} \nonumber \\ =&(\kappa_1^NE_1+\cdots+\mathcal D_{N+1,1}E_{N+1}+\cdots+\mathcal D_{M,1}E_M)\sum_{|J(h1)|=N-1}\Lambda _{J(h1)} E_{J(h1)}(x)\nonumber\\ +&\cdots +(\kappa_N^NE_{ 1}+\cdots+\mathcal D_{N+1,N}E_{N+1}+\cdots+\mathcal D_{M,N}E_M) \sum_{|J(hN)|=N-1}\Lambda_{J(hN)} E_{J(hN)}(x) \nonumber\\ \equiv&(\widetilde{ a}_{11}E_1+\cdots+\widetilde{ a}_{1M}E_M)\sum_{|J(h1)|=N-1}\Lambda_{J(h1)} E_{J(h1)}(x)\nonumber\\ +&\cdots +(\widetilde{a}_{N1}E_1+\cdots+\widetilde{ a}_{NM}E_M) \sum_{|J(hN)|=N-1}\Lambda_{J(hN)} E_{J(hN)}(x). \nonumber \end{align}

Since $E_J(hl)$ are N − 1 cells. According to the formula of the Sato eigenfunction,

(4.90)\begin{equation} \begin{aligned} 0=& \widetilde { a}_{1k}E_k \ \sum_{k\in J(h1),\ |J(h1)|=N-1}\Lambda_{J(h1)} E_{J(h1)}(x) + \cdots \\ +&\widetilde { a}_{Nk}E_k \sum_{k\in J(hN), |J(hN)|=N-1}\Lambda_{J(hN)} E_{J(hN)}(x) . \end{aligned} \end{equation}

Using (4.76), (4.87)–(4.90), multi-linearity, and estimates of the CIO’s,

\begin{align*} &\tau'(x) \phi^{(k)}_{z_h,\operatorname{res}}(x) = \tau'(x) \widetilde\chi'_{z_h,\operatorname{res}}(x)\\ +&\textit{the}\ h\textit{-row of} {\left( \begin{array}{ccc} b_{11}&\cdots & b_{1N}\\ \vdots&\ddots&\vdots\\ b_{N1}&\cdots & b_{NN} \end{array} \right) \left( \begin{array}{c} \widetilde a_{11}E_1\mathcal C_{\kappa_1^+}T\phi^{(k-1)}+\cdots+ \widetilde a_{1M}E_M\mathcal C_{\kappa_M^+}T\phi^{(k-1)}\\ \vdots\\ \vdots\\ \vdots\\ \vdots\\ \widetilde a_{N1}E_1\mathcal C_{\kappa_1^+}T\phi^{(k-1)}+\cdots+ \widetilde a_{NM}E_M \mathcal C_{\kappa_M^+}T\phi^{(k-1)} \end{array} \right)}\\ =&{\tau'(x) \widetilde\chi'_{z_h,\operatorname{res}}(x)}\\ +&{(\widetilde{ a}_{11}E_1\mathcal C_{\kappa_1^+}T\phi^{(k-1)}+\cdots+\widetilde{ a}_{1M}E_M\mathcal C_{\kappa_M^+}T\phi^{(k-1)})\sum_{|J(h1)|=N-1}\Lambda_{J(h1)} E_{J(h1)}(x)}\\ +&{\cdots +(\widetilde{a}_{N1}E_1\mathcal C_{\kappa_1^+}T\phi^{(k-1)}+\cdots+\widetilde{ a}_{NM}E_M\mathcal C_{\kappa_M^+}T\phi^{(k-1)}) \sum_{|J(hN)|=N-1}\Lambda_{J(hN)} E_{J(hN)}(x)} \\ =&\sum_{|J(h)|=N}{\widetilde{\Lambda} _{J(h)}} E_{J(h)}(x),\qquad \end{align*}

with

\begin{gather*} \sum_{0\le l_1+2l_2+3l_3\le d+5}|\partial_x^l {\widetilde{\Lambda} _{J(h)}}| \lt C(1+\sum_{j=1}^M \sum_{0\le l_1+2l_2+3l_3\le d+5}|\partial_x^l\mathcal C_{\kappa_j^+}T\phi^{(k-1)}|). \end{gather*}

Along with the TP condition of $\mathfrak A $, yield

\begin{equation*} \sum_{0\le l_1+2l_2+3l_3\le d+5}|\partial_x^l \phi^{(k)}_{z_n,\operatorname{res}}(x)|\le C(1+\sum_{j=1}^M \sum_{0\le l_1+2l_2+3l_3\le d+5}|\partial_x^l\mathcal C_{\kappa_j^+}T\phi^{(k-1)}|). \end{equation*}

Combining with (4.85), we prove (4.78)–(4.80).