2.1. Direct and inverse scattering for the NLS equation

The direct scattering transform is a mapping from a function $\psi(x)$ to the scattering data associated with the differential equation (1.2). We will give a very brief description here, and refer the reader to [Reference Borghese, Jenkins and McLaughlin7] for a more detailed description. One studies two sets of Jost solutions $\Phi^{\pm}(x,z)$ of this differential equation, which are uniquely determined by a normalisation condition

(2.1)\begin{equation} \Phi^{\pm}(x,z)e^{i z x \sigma_{3}} \to I \ \mbox{as } x \to \pm \infty \,, \quad \sigma_3=\begin{pmatrix} 1& 0 \\ 0&-1 \end{pmatrix}. \end{equation}

The matrix-valued function $\Phi^{+}(x,z)$ is a fundamental solution to (1.2), as is $\Phi^{-}(x,z)$. This fact yields the scattering relation between these two solutions,

(2.2)\begin{equation} \Phi^{-}(x,z) = \Phi^{+}(x,z) {\left( \begin{array}{cc}{a(z)}& {-\overline{b(z)}} \\ {b(z)} & {\overline{a(z)}}\end{array}\right)}, \ \ \ \ \mbox{det}{\left( \begin{array}{cc}{a(z)} & {-\overline{b(z)}}\\ {b(z)} & {\overline{a(z)}} \end{array}\right)}= |a(z)|^{2} + |b(z)|^{2} = 1 \ , \end{equation}

which in turn yields the reflection coefficient $r(z) = b(z)/a(z)$ and the transmission coefficient $T(z) = \frac{1}{a(z)}$.

It turns out that the Jost solutions have analyticity properties in the variable z, which follows from the usual existence theory. Specifically, the first column of $\Phi^{-}(x,z)$ (and the second column of $\Phi^{+}(x,z)$) has an analytic extension to $\mathbb{C}_{+}$, the upper-half of the complex z plane, where it exhibits exponential decay for $x \to - \infty$ (and the second column of $\Phi^{+}(x,z)$ decays exponentially for $x \to + \infty$). Similarly, the second column of $\Phi^{-}(x,z)$ and the first column of $\Phi^{+}(x,z)$ have analytic extensions to $\mathbb{C}_{-}$.

The transmission coefficient also has a meromorphic extension to $\mathbb{C}_{+}$, and each pole of T(z) in $\mathbb{C}_{+}$ corresponds to an L ² eigenfunction for the differential equation (1.2). For functions $\psi(x)$ in an open dense subset of $L^1({\mathbb R})$, there are at most a finite number of poles of T(z) in $\mathbb{C}_{+}$. At each pole λ_j, the first column of $\Phi^{-}(x,\lambda_{j})$ and the second column of $\Phi^{+}(x,\lambda_{j})$ are linearly dependent, and the proportionality constant c_j relating them is referred to as a normalisation constant, and is defined by

(2.3)\begin{eqnarray} \left(\Phi^{-}(x,\lambda_{j})\right)^{(1)} = c_{j} \left(\Phi^{+}(x,\lambda_{j})\right)^{(2)}. \end{eqnarray}

The scattering data (also referred to as the spectral data) corresponding to $\psi(x)$ consists of the reflection coefficient r(k), the eigenvalues $\{\lambda_{j} \}_{j=1}^{N}$, and the normalisation constants $\{c_{j}\}_{j=1}^{N}$. Following [Reference Borghese, Jenkins and McLaughlin7], we let $\mathcal{D}$ denote the scattering data:

(2.4)\begin{eqnarray} \mathcal{D} = \left\{r(z), \{\lambda_{j}, c_{j} \}_{j=1}^{N} \right\}. \end{eqnarray}

As $\psi(x,t)$ evolves in time according to the NLS equation (1.1), the scattering data also evolves in time, but that evolution is quite simple since the direct scattering transform linearises the flow: the eigenvalues do not change in time, and the reflection coefficient and normalisation constants evolve explicitly, so that

(2.5)\begin{eqnarray} \mathcal{D}(t) = \left\{r(z,t), \{\lambda_{j}, c_{j}(t) \}_{j=1}^{N} \right\} = \left\{r(z)e^{2 i t z^{2} }, \{\lambda_{j}, c_{j}e^{2 i t \lambda_{j}^{2} } \}_{j=1}^{N} \right\}. \end{eqnarray}

We note in passing that the direct scattering transform as discussed above is defined for generic initial data. However, for non-generic data, certain spectral singularities exist which require additional information and analysis. As we are interested in reflectionless potentials for which $r(z) \equiv 0$, spectral singularities are not an issue.

The solution $\psi(x,t)$ is uniquely determined by the scattering data, and the mapping from $\mathcal{D}(t)$ to $\psi(x,t)$ is referred to as the inverse scattering transform. The inverse scattering transform can be characterised in terms of a system of integral equations, referred to as Gelfand–Levitan–Marcenko equations. Equivalently, it can be characterised in terms of a Riemann–Hilbert problem, which is the approach we take here. Riemann–Hilbert formulations of the inverse problem have over the last 30 years yielded a remarkable number of different asymptotic results concerning the behaviour of solutions of integrable nonlinear partial differential equations, starting with the work of Deift and Zhou [Reference Deift and Zhou12] for the modified KdV equation.

For the NLS equation, the Riemann–Hilbert problem is to determine a $2 \times 2$ matrix $M = M(z) = M(z;x,t)$. We will usually suppress the dependence on x and t, and simply write M or M(z).

Existence and uniqueness for this Riemann–Hilbert problem follow from what are by now standard arguments: uniqueness follows from Liouville’s theorem, and existence follows from a careful application of a vanishing lemma as described in [Reference Zhou43]. The solution $\psi(x,t)$ to the NLS equation corresponding to the initial data $\psi(x)$, assuming that $\psi,\psi_x\in L^1({\mathbb R})$, and determined by the scattering data $\mathcal{D}(t)$ is obtained from $M(z;x,t)$ through the $z \to \infty$ normalisation:

(2.9)\begin{eqnarray} M(z;x,t) = I + \frac{1}{2 i z} {\left( \begin{array}{cc}{-\int_{x}^{\infty} |\psi(s,t)|^{2}{\rm d} s}& {\psi(x,t)} \\ {\overline{\psi(x,t)}} & {\int_{x}^{\infty} |\psi(s,t)|^{2}{\rm d} s}\end{array}\right)} + \mathcal{O} \left(\frac{1}{z^{2}} \right). \end{eqnarray}

We mention that the RHP approach to solve non-linear integrable PDEs is not just a relevant analytical tool, but it is also numerical one. Indeed, several authors [Reference Bilman, Nabelek and Trogdon5, Reference Bilman and Trogdon6, Reference Gkogkou, Prinari and Trogdon21, Reference Trogdon, Olver and Deconinck40] employed this method to numerically analyse various non-linear integrable PDEs.

Notation: We will use $\mathcal{D}$ to denote the scattering data at t = 0 (i.e. $\mathcal{D}= \mathcal{D}(0)$). In order to emphasise the dependence on the scattering data, we will occasionally write

(2.10)\begin{eqnarray} \psi(x,t;\mathcal{D}) \end{eqnarray}

to represent the solution to the NLS equation determined by the initial scattering data $\mathcal{D}$.

2.2. N-soliton solutions and soliton gases

In this paper, we are interested in N-soliton solutions, for which $r(z) \equiv 0$. Under this assumption, the above Riemann–Hilbert problem simplifies. Indeed, the jump relation becomes trivial: $M_{+}(z) = M_{-}(z)$ for $z \in \mathbb{R}$, and this implies that now M is a meromorphic function of z. The Riemann–Hilbert problem is referred to as a meromorphic Riemann–Hilbert problem, for which asymptotic analysis started in [Reference Kamvissis, Kenneth and Miller25] where the authors studied a semi-classical scaling of the NLS equation. (See also [Reference Baik, Kriecherbauer, Kenneth and Miller1], where the authors studied the asymptotic behaviour of discrete orthogonal polynomials, through the analysis of meromorphic Riemann–Hilbert problems).

The original set of kinetic equations in [Reference Zakharov42] was derived under the loose assumption of a dilute gas of solitons, described by an evolving density function $f(z,x,t)$. This density function yields (in some sense) the number of solitons in a small volume element around $(z,x,t)$. Of course, the notion of a soliton in a small space-time window, let alone the idea that there might be a large number of solitons with slightly different eigenvalue parameters in that same space-time window, while at the same time supposing there is enough space between solitons to consider them to be dilute, was quite challenging from the point of view of analysis.

One experimental prediction in [Reference Zakharov42] is that a ‘tracer soliton’, a soliton whose amplitude is much bigger than all the other solitons, launched into this gas, should interact with the gas in such a way that its velocity deforms as it propagates, and the explicit computation of this time-dependent velocity, as a function of density of the gas as well as the tracer’s initial eigenvalue parameter, yielded an experimental prediction which could in principle be tested. About 30 years later, El [Reference Gennady17] extended the kinetic theory to a dense gas, in which a coupled system of equations was derived, with the evolving density coupled to the evolving velocity of a tracer. Both [Reference Zakharov42] and [Reference Gennady17] focused on the KdV equation, but this was extended by El and Kamchatnov in [Reference El and Kamchatnov15] to other integrable nonlinear partial differential equations, including the NLS equation. Significant progress regarding the understanding of the kinetic equations arising in different settings, as well as more detailed analysis of the kinetic equations themselves, has continued, for example, in [Reference El and Tovbis13, Reference Kuijlaars and Tovbis27, Reference Tovbis and Wang38, Reference Tovbis and Wang39].

However, only recently has there been any validation of the kinetic theory, in the form of robust numerical [Reference Carbone, Dutykh and El8, Reference Congy, El and Roberti9, Reference Roberti, El, Tovbis, Copie, Suret and Randoux34] and experimental [Reference Maiden, Lowman, Anderson, Schubert and Hoefer30, Reference Mossman, Katsimiga, Mistakidis, Romero-Ros, Bersano, Schmelcher, Kevrekidis and Engels31] evidence (see the review manuscript [Reference Suret, Randoux, Gelash, Agafontsev, Doyon and El37]). There have also been a few limited analytical results, particularly [Reference Girotti, Grava, Jenkins and McLaughlin18–Reference Girotti, Grava, McLaughlin and Najnudel20, Reference Jenkins and Tovbis24].

In [Reference Girotti, Grava, Jenkins and McLaughlin18], the authors considered the $N\to\infty$ limit of an N-soliton solution for the KdV equation, with normalisation constants uniformly of size $\mathcal{O}\left(N^{-1} \right)$. The $N \to \infty$ analysis was carried out under the assumption that x and t were bounded. The fact that the normalisation constants were vanishingly small as $N \to \infty$ provided a necessary foothold for the rigorous analysis. This can be understood (intuitively) by considering the connection, in the case of a single soliton, between the normalisation constant and the initial location x ₀ of the soliton’s peak:

(2.11)\begin{eqnarray} x_{0} = \frac{1}{2 \eta}\log{\left| \frac{c}{2 \eta} \right| } \end{eqnarray}

(here c is the normalisation constant, and $\lambda = \xi + i \eta$ is the eigenvalue), so if $c \approx N^{-1}$, then x ₀ is shifted a distance $\log{N}$ to the left. One could then imagine that if one has an N-soliton solution with all the normalisation constants of size $\mathcal{O}\left(N^{-1}\right)$, all the solitons are shifted far to the left, and what remains for finite values of x is the accumulated tails of all these solitons. The results in [Reference Girotti, Grava, Jenkins and McLaughlin18] are consistent with this intuition. They prove convergence of the N-soliton solutions (as $N \to \infty$) to a ‘soliton gas solution’ of the KdV equation, but only for x and t in compact sets. They subsequently carry out an asymptotic analysis of the soliton gas solution for $|x| \to \infty$, and also for $t \to \infty$ with $\frac{|x|}{t}$ bounded. In the large-time analysis of the soliton gas solution, they established the existence of an evolving density function, depending on the spectral parameter, space and time, which describes the local, evolving solitonic content of the solution, providing a first analytical corroboration of the kinetic theory, and in the process showing that in the long-time regime, what emerges is a type of dense soliton gas, called a condensate. However, in [Reference Girotti, Grava, Jenkins and McLaughlin18] they did not consider the behaviour of the original N-soliton solution on space-time scales that would overlap with their asymptotic analysis of the soliton gas solution.

More recently, in [Reference Bertola, Grava and Orsatti2, Reference Bertola, Grava and Orsatti3, Reference Falqui, Grava and Puntini16] the authors considered the soliton gas and the breather gas for the NLS equation under the assumptions that the eigenvalues λ_j accumulate on a domain $D\subset {\mathbb C}_+$ (and its complex conjugate ${\overline{D}}$) and norming constants uniformly $\mathcal{O}(N^{-1})$.

In [Reference Girotti, Grava, Jenkins, McLaughlin and Minakov19], the authors considered the mKdV equation and studied the interaction of a tracer soliton with a similar soliton gas with vanishingly small normalisation constants. In [Reference Girotti, Grava, McLaughlin and Najnudel20] the authors developed a probabilistic analysis of a soliton gas for the NLS equation, again under the assumption that the normalisation constants are uniformly $\mathcal{O}\left( N^{-1} \right)$. In other words, the ‘bulk’ asymptotic behaviour of N-soliton solutions, with normalisation constants chosen to be bounded (away from both 0 and $\infty$), has not been considered in the context of the kinetic theory of solitons. (However, the works [Reference Kamvissis, Kenneth and Miller25, Reference Robert and McLaughlin33] concerning the semi-classically scaled NLS equation, could be thought of as examples of this scaling.)

2.3. N-soliton solutions: condensate scaling

This manuscript considers the asymptotic behaviour of N-soliton solutions of the NLS equation, with eigenvalues accumulating on a horizontal line in $\mathbb{C}_{+}$ (and its complex conjugate contour in $\mathbb{C}_{-}$), with normalisation constants that are bounded, and bounded away from 0. Assuming the limit $N \to \infty$, such a configuration has been referred to (see, for example, [Reference Suret, Randoux, Gelash, Agafontsev, Doyon and El37]) as a soliton condensate, accompanied by the conjecture that soliton condensates achieve maximal soliton density.

We will refer to the sequence of such N-soliton solutions as an N-soliton condensate, and we will refer to the associated sequence of scattering data as an N-soliton condensate scattering data. Specifically, we consider the following assumptions.

Assumption 2.2. (N-soliton condensate scattering data)

For a positive integer N, we generate the scattering data as follows:

1. (1) There are two positive constants a and b, which define a line segment $ \eta_{1} = [- a + i b, a + i b]$, which we orient from right to left, and its complex conjugate $\eta_{2} = [-a-ib, a - ib]$, which we orient from left to right. We define

   (2.12)\begin{eqnarray} A = a + i b \end{eqnarray}

   to be the right-most endpoint of η ₁.

2. (2) There is a density function $\rho(z)$ which is analytic in a neighbourhood ${\mathcal U}$ of the domain enclosed by the line segment η ₁ and the arc of the circle of radius $|A|$ centred at 0 connecting A with $\overline{A}$ (ρ is also assumed to be analytic in the Schwarz reflection of ${\mathcal U}$). The density ρ satisfies

   (2.13)\begin{align} &\rho(z) \gt 0\ \mbox{for } z = t + ib, \ t\in(-a,a), \end{align}
   (2.14)\begin{align} & \int_{-a}^a \rho(t+ib){\rm d} t = 1, \end{align}
   (2.15)\begin{align} &\rho({\overline{z}}) = {\overline{\rho(z)}}. \end{align}

3. (3) The eigenvalues $\{\lambda_j\}_{j=1}^{N}$ on η ₁ are chosen to satisfy

   (2.16)\begin{eqnarray} \int_{-a}^{\Re(\lambda_j)} \rho(x + ib){\rm d} x = \frac{2j-1}{2N}, \quad \Im(\lambda_j) = b, \ j = 1, \ldots, N. \end{eqnarray}

   See Figure 1 for an example with uniform density.

4. (4) There is a ‘normalisation constant sampling function’ h(z) which is analytic in ${\mathcal U}$ and $\overline{{\mathcal U}}$ satisfying $ h({\overline{z}}) = {\overline{h(z)}}$, and $h(z)\neq0$ for $z\in \eta_1\cup\eta_2$.

5. (5) For each positive integer N, the normalisation constants $\{c_{j} \}_{j=1}^{N}$ are chosen to satisfy

   (2.17)\begin{eqnarray} c_j= h(\lambda_j), \ \ j = 1, \ldots, N \ , \end{eqnarray}

6. (6) There are two contours, $\Gamma_{1} \in \mathbb{C}_{+}$ and $\Gamma_{2} \in \mathbb{C}_{-}$, which are Schwarz reflections of each other (and so they are determined by Γ₁). The contour Γ₁ is a simple, closed, analytic curve that encircles the interval η ₁ (see Figure 2).

Figure 1.

Poles λ_j and their conjugates.

Figure 2.

Example of contour $\Gamma_1,\Gamma_2$.

We therefore have, for every positive integer N, our N-soliton scattering data

(2.18)\begin{eqnarray} \mathcal{D}_{N} = \left\{r \equiv 0, \left\{ \lambda_{j}, c_{j} \right\}_{j=1}^{N} \right\}, \end{eqnarray}

where λ_j is defined in (2.16) and the normalisation constants c_j are defined in (2.17). For each positive integer N, these scattering data correspond to an N-soliton solution to the NLS equation, which we will denote by

(2.19)\begin{eqnarray} \psi_{N}(x,t) = \psi(x,t;\mathcal{D}_{N}). \end{eqnarray}

In what follows, unless otherwise specified, we will assume that the scattering data under consideration is a sequence of N-soliton condensate scattering data, with N large and increasing towards $\infty$. We will refer to this collection of assumptions by saying, ‘assuming a sequence of N-soliton condensate scattering data’, or ‘under the N-soliton condensate scattering data assumption’.

2.4. Riemann–Hilbert problem for $\psi_{N}(x,t)$

Given the scattering data $\mathcal{D}_{N}$, the Riemann–Hilbert problem 2.1 simplifies significantly, since $r(z) \equiv 0$ for all real z. Indeed, since the jump relation across the real axis (2.6) becomes the trivial relation $M_{+}(z) = M_{-}(z)$ for all real z, Morera’s Theorem tells us that M(z) is then analytic across the real axis, and hence M(z) is meromorphic in $\mathbb{C}$. We will make one standard transformation, from M(z) to a new matrix A(z), following (for example) [Reference Girotti, Grava, Jenkins and McLaughlin18]. We use the contours Γ₁ and Γ₂ which are provided to us under the N-soliton condensate scattering assumption, Γ₁ encircling the interval η ₁ and Γ₂ its Schwarz reflection. Each of these contours is taken to be oriented clockwise, as shown in Fig. 2. We then define $A_N(z)$ as follows:

(2.20)\begin{equation} A_N(z) = \begin{cases} M(z)\begin{pmatrix} 1& 0 \\ - \sum_{j=1}^N \frac{c_j}{z-\lambda_j}e^{2i\theta(\lambda_j)} & 1 \end{pmatrix}, & z\ \textit{inside } \Gamma_1 \\ \\[2pt] M(z)\begin{pmatrix} 1 & \sum_{j=1}^N \frac{{\overline{c}}_j}{z-{\overline{\lambda}}_j}e^{-2i\theta({\overline{\lambda}}_j)} \\ 0& 1 \end{pmatrix}, & z\ \textit{inside } \Gamma_2 \\ \\[2pt] M(z), & \textit{otherwise}. \end{cases} \end{equation}

Now M(z) has simple poles at the eigenvalues λ_j and $\overline{\lambda_{j}}$, but the definition above is chosen so that $A_N(z)$ is analytic inside the contours Γ₁ and Γ₂. Indeed, the reader may check from the residue conditions (8.1) and (8.2) that (for example) the pole of $M_{11}(z)$ at λ_k is cancelled since

(2.21)\begin{eqnarray} A_{N;11}(z) = M_{11}(z) - \frac{c_{k} M_{12}(z)e^{2i\theta(\lambda_k)}}{z - \lambda_{k}} + \left \lt \mbox{analytic at}\ \lambda_{k} \right \gt , \end{eqnarray}

and the first two terms have the same exact residue at λ_k. Similar arguments show that the matrix $A_N(z)$ is analytic inside Γ₁ and inside Γ₂, and it is also obviously analytic for z outside the two contours. Finally, $A_N(z)$ clearly has boundary values as z approaches the contours Γ₁ and Γ₂ (with different boundary values as z approaches each contour from inside or outside).

Notation: boundary values for z on a contour: Following standard convention, for z in the contour Γ₁ (or Γ₂), we will let $A_{N,+}(z)$ denote the boundary value of $A_N(z)$ obtained by letting zʹ approach z from outside Γ₁ (or outside Γ₂):

(2.22)\begin{eqnarray} A_{N,+}(z) = \lim_{\substack{z' \to z \\ z' \in \mbox{exterior of } \Gamma_{j}}} A_N(z'), \ \ z \in \Gamma_{j}, \end{eqnarray}

and we will let $A_{N,-}(z)$ denote the boundary value from inside Γ₁ (or Γ₂). (Note that this is consistent with the usual convention that the + side of a contour is on the left of the contour as one traverses it according to its orientation).

The new matrix-valued function $A_N(z)$ satisfies a Riemann–Hilbert problem, which we will refer to as the N-soliton condensate Riemann–Hilbert problem.

For this Riemann–Hilbert problem, the poles are accumulating on the segments $\eta_1,\eta_2$ which we parametrise for future convenience:

(2.23)\begin{equation} \eta_1 = \left\{z\in{\mathbb C}\,\Big \vert \, z = a-2at+ib,\; t\in(0,1) \right\}\,,\quad \eta_2 = \left\{z\in{\mathbb C}\,\Big \vert \, z = -a +2ta - ib,\; t\in(0,1) \right\}\,. \end{equation}

In particular, η ₁ is the horizontal segment from A to $-{\overline{A}}$, and η ₂ is the one from −A to ${\overline{A}}$.

Under the N-soliton scattering assumption, we have the following Riemann–Hilbert problem.

Note that the above $z \to \infty$ asymptotic condition describes exactly how to extract the N-soliton condensate solution $\psi_{N}(x,t)$ from the solution A_N of the Riemann–Hilbert problem. Usually the asymptotic normalisation condition is described in the weak form $A_N = I + \mathcal{O} \left( \frac{1}{z} \right)$, but in this case, since the contours Γ₁ and Γ₂ are bounded, A possesses a complete asymptotic expansion in powers of $\frac{1}{z}$.

2.5. Riemann–Hilbert problem for the soliton gas

The N-soliton condensate Riemann–Hilbert problem behaves singularly as the number of eigenvalues grows to infinity, because the sums appearing in the jump relationships (2.24) do not converge as $N \to \infty$. However, their asymptotic behaviour can be determined:

(2.25)\begin{eqnarray} && \sum_{j=1}^{N} \frac{c_{j}e^{2 i \theta(\lambda_{j})} }{z-\lambda_{j}} = - N \int_{\eta_{1}} \frac{h(\lambda) \rho(\lambda) e^{2 i \theta(\lambda)} d\lambda}{z- \lambda} + \mathcal{O}\left( \frac{1}{N} \right) \ , \end{eqnarray}

(2.26)\begin{eqnarray} && \sum_{j=1}^N \frac{{\overline{c}}_j}{z-{\overline{\lambda}}_j}e^{-2i\theta({\overline{\lambda}}_j)} = N \int_{\eta_{2}} \frac{{h(\lambda)} \rho(\lambda) e^{- 2 i \theta(\lambda)}{\rm d} s}{z - \lambda} + \mathcal{O}\left( \frac{1}{N} \right)\,, \end{eqnarray}

where we recall that the contour η ₁ is oriented from right to left, while η ₂ is oriented from left to right, and in both cases the error term $\mathcal{O} \left( \frac{1}{N} \right)$ is uniform for all z in the respective contour Γ₁ or Γ₂. Indeed, under the N-soliton condensate assumption 3, a complete asymptotic expansion for this sum can be calculated, because the eigenvalues $\{\lambda_{j}\}_{j=1}^{N}$ have been selected to arrive at a uniform mid-point approximation, which is known to converge with an error ${\mathcal O}(N^{-2})$. The next term in the expansion can be computed explicitly, we perform such computation in Appendix A.

If we replace the finite sum by its leading order asymptotic behaviour, we arrive at a Riemann–Hilbert problem which still contains the large parameter N. It is this Riemann–Hilbert problem which we study as $N \to \infty$. The associated solution to the NLS equation, $\psi_{SG}=\psi_{SG}(x,t) = \psi_{SG}(x,t;N)$, is the soliton gas (or soliton condensate) solution.

This Riemann–Hilbert problem arises formally, by replacing the jump matrices (2.24) with (2.27). Since it is not related to the Riemann–Hilbert problem 2.3 by explicit transformations, the existence and uniqueness of a solution must be established independently. Note that this statement extends to the soliton gas condensate solution $\psi_{SG}(x,t;N)$. Uniqueness for the Riemann–Hilbert problem 2.4 (regardless of whether or not the solution is known to exist) is completely straightforward since $\mbox{det}J_{\tilde{A}} \equiv 1$. See, for example, the first paragraph of the proof of Theorem 3.1 on P. 1503 of [Reference Deift, Kriecherbauer, McLaughlin, Venakides and Zhou10]. Regarding existence, the crucial fact is that the jump matrices obey the symmetry

(2.28)\begin{eqnarray} J_{\tilde{A}}(z) = {\left( \begin{array}{cc}{0} & {1} \\ {1} & {0} \end{array}\right)} \overline{J_{\tilde{A}}(\overline{z})} {\left( \begin{array}{cc}{0} & {1} \\ {1} & {0} \end{array}\right)}. \end{eqnarray}

This permits the direct application of an existence theory via the connection between Riemann–Hilbert problems and Fredholm singular integral operators (see Theorems 9.1, 9.2 and 9.3 in [Reference Zhou43], or the proof of Theorem 5.3 in [Reference Deift, Kriecherbauer, McLaughlin, Venakides and Zhou11]). Regarding the asymptotic behaviour of the solution, using a standard dressing argument, see for example [Reference Bilman, Ling and Miller4, Reference Borghese, Jenkins and McLaughlin7], one can verify that the z ⁻¹ coefficient in the large z expansion for ${\widetilde{A}}(z)$ has the given form. Thus, we state without further discussion the following result:

2.6. Statement of the results

In Sections 3–6, we determine the asymptotic behaviour of the solution $\tilde{A}$ of the Riemann–Hilbert problem 2.4. The main application of this work is to determine the asymptotic behaviour of the soliton gas condensate $\psi_{SG}(x,t;N)$, which is described in the following theorem.

Theorem 2.6. Under the N-soliton condensate scattering data assumption, for all (x, t) in a compact set ${\frak{K}}$, there is N ₀ so that for all $N \gt N_{0}$, $\psi_{SG}(x,t;N)$ satisfies the asymptotic description (7.7). Specifically, for ψ_SG, we have

(2.29)\begin{equation} \begin{aligned} &\psi_{SG}(x,t;N) \\ & = -2 i \frac{\Re(A)\Im(A)}{|A|} \operatorname{sd}\left( -2|A|x + \frac{2K(\cos(\theta))}{\pi}\ln(N) - \frac{2|A|}{\pi}\Re\left(\int_{\eta_1} \frac{\log(2\pi h(s)\rho(s))}{R_+(s)}{\rm d} s\right); \sin(\theta)\right) e^{it(A^2+{\overline{A}}^2) + i \phi_0} \\ \unicode{x2205} & \quad +\mathcal{O} \left(\frac{1}{\log{N}}\right), \end{aligned} \end{equation}

where the error term $\mathcal{O}\left( \frac{1}{\log{N} } \right)$ is uniform for all (x, t) in the compact set ${\frak{K}}$. Here, $\operatorname{sd}$ is the Jacobi elliptic function $\operatorname{sd}$ [Reference Olver, Daalhuis, Lozier, Schneider, Boisvert, Clark, Miller, Saunders, Cohl and McClain32, Chap. 22], and R(z) and ϕ ₀ are given by

(2.30)\begin{align} &R(z) = (z-A)^{\frac{1}{2}}(z+A)^{\frac{1}{2}}(z-{\overline{A}})^{\frac{1}{2}}(z+{\overline{A}})^{\frac{1}{2}}\,, \end{align}

(2.31)\begin{align} &\phi_0 = \ln(N) \frac{K(\cos(\theta))}{K(\sin(\theta))} + \Re\left( \int_{\eta_1} \frac{s\ln(2\pi h(s)\rho(s))}{R_+(s)}{\rm d} s\right)\,, \quad \theta = \arg(A)\ \end{align}

where the function K(k) is the complete elliptic integral of the first kind [Reference Olver, Daalhuis, Lozier, Schneider, Boisvert, Clark, Miller, Saunders, Cohl and McClain32, Chap. 19], the branch-cuts are the canonical ones and the integration is performed on the straight line connecting the endpoints.

Remark 2.2. It is interesting to observe that the leading order asymptotic behaviour of the modulus of the soliton gas condensate solution $\psi_{SG}(x,t;N)$ (2.29) does not depend on time. The explanation for this fact is that our assumption about the scattering data, the poles are accumulating on symmetric contours. Essentially, the soliton gas condensate is in a sort of equilibrium, and can be interpreted as an elliptic wave with zero velocity. This could also be interpreted as another form of soliton-shielding, similar to [Reference Bertola, Grava and Orsatti2, Reference Bertola, Grava and Orsatti3]. In particular, we want to emphasise that the solution (2.29) and the solution [Reference Bertola, Grava and Orsatti3, eq. (4.44)] are similar, indeed they both arise as solutions of similar RHPs, the two main differences are that in our case the contours where the poles are accumulating are horizontal, and the norming constants c_j are bounded away from zero, while in [Reference Bertola, Grava and Orsatti3] the contours are vertical and the constants are uniformly $\mathcal{O}(N^{-1})$.

We notice that, in view of the Galilean invariance of the NLS equation, for $v\in {\mathbb R}$, $\psi(x-vt,t)e^{ivx-i\frac{v^2t}{2}}$ solves the NLS equations (1.1) whenever $\psi(x,t)$ does. Furthermore, if the scattering maps gives $\psi(x,0)\to\{r(z),\{\lambda_j,c_j\}_{j=1}^N\}$ then $\psi(x,0)e^{ivx}\to\{r\left(z+\frac{v}{2}\right),\{\lambda_j-\frac{v}{2},c_j\}_{j=1}^N\}$; therefore, in the setting where the two line segments η ₁ and η ₂ are shifted so that they are not symmetric with respect to the origin, one can always reduce to the symmetric situation up to a simple scaling. Finally, it is worth noticing that the solution depends just on the product of $h(z),\rho(z)$ and not on each function separately. Fig. 3 shows plots of the leading order asymptotic of the solution of the NLS equation for different values of N.

Figure 3.

Solution to the NLS equation (1.1) in assumptions 2.2. Here, $A=1+i$, and N is specified in the plots.

Proposition 2.7. Let $A_N(z),{\widetilde{A}}(z)$ be the solutions of RHP 2.3 and RHP 2.4 respectively. Under the N-soliton condensate scattering data assumption, there is N ₀ so that for all $N \gt N_{0}$, we have

(2.32)\begin{equation} A_N(z) = \left(I + \mathcal{O} \left( \frac{1}{N \left( 1 + |z| \right) } \right) \right) {\widetilde{A}}(z)\,. \end{equation}

where the error term, $\mathcal{O} \left( \frac{1}{N \left( 1 + |z| \right) } \right)$ is uniform in the entire complex plane, and (x, t) in a compact set.

The proof of this proposition is presented in Appendix C, and relies on the complete asymptotic description of the solution $\tilde{A}$ to the Riemann–Hilbert problem 2.4, which as mentioned above is developed in Sections 3–6.

It is an immediate consequence of this uniform approximation that the N-soliton solution is extremely well approximated by the soliton gas condensate solution.

Corollary 2.8. Under the N-soliton condensate scattering data assumption, for all (x, t) in a compact set, there is N ₀ and a constant c ₀ so that for all $N \gt N_{0}$, we have

(2.33)\begin{eqnarray} \left| \psi_{N}(x,t) - \psi_{SG}(x,t;N) \right| \leqslant \frac{c_{0}}{N}. \end{eqnarray}

In the next four sections, we develop the asymptotic analysis of the solution of the Riemann–Hilbert problem 2.4, to establish, in Section 7, the asymptotic behaviour of the soliton gas condensate solution $\psi_{SG}(x,t;N)$. In Section 8, we consider a more complex solution which contains one additional soliton, and using our analysis, we provide a rigorous validation of the kinetic theory of solitons in this setting. We prove Proposition 2.7, Corollary 2.8 and other technical results in Appendix A.

3.1. The g(z) function

To solve RHP 3.3, we proceed as follows. First, consider the RHP for $g'(z)$. Directly from the RHP for g(z), we get the following.

The previous RHP can be solved explicitly as

(3.11)\begin{equation} g'(z) = \frac{c_0}{R(z)}, \quad R(z) = (z-A)^{\frac{1}{2}}(z+{\overline{A}})^{\frac{1}{2}}(z+A)^{\frac{1}{2}}(z-{\overline{A}})^{\frac{1}{2}}\,, \end{equation}

for some $c_0 \in {\mathbb C}$, where for each factor $(\cdot )^{1/2}$ we use the principal branch, so that R(z) is analytic in $\mathbb{C} \setminus \{\eta_1 \cup \eta_2$}, and is normalised such that

\begin{equation*} R(z) \sim z^2\,, \quad z\to\infty \,.\end{equation*}

Define the function

(3.12)\begin{equation} g(z) = c_0 \int_A^z \frac{1}{R(s)}{\rm d} s + \frac{1}{2}\,, \end{equation}

then it is easy to check that g satisfies the following conditions:

(3.13)\begin{align} &g_{+}(z) + g_{-}(z) = 1, \quad z \in \eta_1 \end{align}

(3.14)\begin{align} &g_{+}(z) + g_{-}(z) = 2c_0 \int_{\gamma} \frac{1}{R(s)}{\rm d} s + 1 , \quad z \in \eta_2 \end{align}

(3.15)\begin{align} &g_{+}(z) - g_{-}(z) = 2c_0 \int_{\eta_1} \frac{1}{R_+(s)}{\rm d} s, \quad z \in \gamma \end{align}

Imposing the conditions

(3.16)\begin{equation} c_0= - \left(\int_\gamma \frac{1}{R(s)}{\rm d} s\right)^{-1}\, \mbox{and} \quad c_1 = 2c_0 \int_{\eta_1}\frac{1}{R_{+}(s)}{\rm d} s \, , \end{equation}

one may verify that g(z) in (3.12) solves RHP 3.3.

For later usage, we list a series of properties that the function R and its integrals satisfy. Since the proof is technical, we defer it to the appendix A.

Proposition 3.6. The function R defined as

(3.17)\begin{equation} R(z)=(z-A)^{\frac{1}{2}}(z+{\overline{A}})^{\frac{1}{2}}(z+A)^{\frac{1}{2}}(z-{\overline{A}})^{\frac{1}{2}}\,, \end{equation}

where for each factor $(\cdot )^{1/2}$ we use the principle branch, satisfies the following properties

1. (1) $R(z) = z^{2} - \frac{A^{2} + {\overline{A}}^{2}}{2} + \mathcal{O} \left( \frac{1}{z^{2}}\right)$, as $z \to \infty$,

2. (2) $R({\overline{z}}) = {\overline{R}}(z)$,

3. (3) $R(- z) = R(z)$,

Furthermore,

(3.18)\begin{equation} \int_{\eta_1} \frac{1}{R_+(z)}{\rm d} z = -\frac{K(\cos(\theta))}{|A|}\,,\quad \int_{-{\overline{A}}}^{-A}\frac{1}{R(s)}{\rm d} s=-i\frac{K(\sin(\theta))}{|A|}\,, \end{equation}

where $ \theta=\arg(A)$, and K(k) is the complete elliptic integral of the first kind [Reference Olver, Daalhuis, Lozier, Schneider, Boisvert, Clark, Miller, Saunders, Cohl and McClain32, Chap. 19].

Remark 3.1. For later usage, we define

(3.19)\begin{equation} c= \frac{1}{2\int_{\eta_1} \frac{1}{R_+(z)}{\rm d} z}=- \frac{|A|}{2K(\cos(\theta))}\,,\quad \tau = \frac{\int_{-{\overline{A}}}^{-A} \frac{1}{R(z)}{\rm d} z}{\int_{\eta_1} \frac{1}{R_+(z)}{\rm d} z}=2i\frac{K(\cos(\theta))}{K(\sin(\theta))}\,, \end{equation}

and we notice that $c\in {\mathbb R}_-$, $\tau\in i{\mathbb R}_+$, and $\frac{2c}{\tau}\in i {\mathbb R}_+$. In this notation

(3.20)\begin{equation} c_0 = - \frac{2c}{\tau}\,, \qquad c_1 = - \frac{2}{\tau}\,, \end{equation}

therefore, one can rewrite g(z) as

(3.21)\begin{equation} g(z) = -\frac{2c}{\tau}\int_A^z\frac{{\rm d} s}{R(s)} + \frac{1}{2} \end{equation}

Furthermore, as a corollary of Proposition 3.6 the following holds:

(3.22)\begin{equation} \frac{2c}{\tau}\int_{-{\overline{A}}}^{- A} \frac{1}{R(s)} {\rm d} s = 1, \quad \frac{2c}{\tau}\int_A^{-{\overline{A}}} \frac{1}{R_+(s)} {\rm d} s = \frac{1}{\tau}, \quad -\frac{2c}{\tau}\int_A^\infty \frac{1}{R(s)} {\rm d} s = \frac{1}{2\tau} - \frac{1}{2}\,. \end{equation}

3.2. The y(z) function

We now find the explicit solution to RHP 3.4.

Proposition 3.7. The solution $y \,:\, {\mathbb C}\to {\mathbb C}$ to the Riemann–Hilbert problem 3.4 is given by

(3.23)\begin{align} y(z) & = \frac{R(z)}{2 \pi i} \left( \int_{\eta_1} \frac{\log \left( 2 \pi h(s) \rho(s) \right) {{+ 2i\theta(s,x,t)}}}{R_{+}(s)(s-z)}{\rm d} s - \int_{\eta_2} \frac{\log \left( 2 \pi h(s) \rho(s) \right) {{- 2i\theta(s,x,t)}}}{R_{+}(s)(s-z)}{\rm d} s \right. \nonumber\\ & \quad \left. + \int_{\gamma} \frac{\Delta(x)}{R(s)(s-z)}{\rm d} s\right) \,, \end{align}

with $\Delta(x)$

(3.24)\begin{equation} \Delta(x) = -\frac{2c}{\tau} \left( 2\pi x +2\Re\left( \int_{\eta_1} \frac{\log \,( 2 \pi h(s) \rho(s))}{R_{+}(s)} {\rm d} s\right)\right)\,. \end{equation}

Lastly, $y_\infty = \lim_{z\to\infty}y(z)$ is finite.

Proof. By applying the Cauchy–Plemelj–Sokhotski formula, we obtain that the function y(z) has the following form

(3.25)\begin{align} y(z) & = \frac{R(z)}{2 \pi i} \left( \int_{\eta_1} \frac{\log \left( 2 \pi h(s) \rho(s) \right) {{+ 2i\theta(s,x,t)}}}{R_{+}(s)(s-z)}{\rm d} s - \int_{\eta_2} \frac{\log \left( 2 \pi h(s) \rho(s) \right) {{- 2i\theta(s,x,t)}}}{R_{+}(s)(s-z)}{\rm d} s \right. \nonumber\\ & \quad \left. + \int_{\gamma} \frac{{\widehat{\Delta}}(x,t)}{R(s)(s-z)}{\rm d} s\right) \,, \end{align}

with ${\widehat{\Delta}}(x,t)$ defined as

(3.26)\begin{gather} {\widehat{\Delta}}(x,t) = \frac{2c}{\tau} \left( - \int_{\eta_1} \frac{\log \, (2 \pi h(s) \rho(s)) + {{2i\theta(s,x,t)}}}{R_{+}(s)}{\rm d} s + \int_{\eta_2} \frac{\log (2 \pi h(s) \rho(s)) - {{2i\theta(s,x,t)}} }{R_{+}(s)}{\rm d} s \right)\,. \end{gather}

It is straightforward to verify, using the asymptotic behaviour $R(z) = z^{2} \left( 1 + \mathcal{O}\left( \frac{1}{z^{2}}\right) \right)$ as $z \to \infty$, that

(3.27)\begin{align} \begin{aligned} y(z) = \frac{z^2}{2\pi i} \Big( &-\frac{1}{z}\int_{\eta_1} \frac{\log \left( 2 \pi h(s) \rho(s) \right) + {{2i\theta(s,x,t)}} }{R_{+}(s)} {\rm d} s + \frac{1}{z} \int_{\eta_2} \frac{\log \left( 2 \pi h(s) \rho(s) \right) - 2i\theta(x,t,s)}{R_{+}(s)}{\rm d} s\\ & - \frac{1}{z} \int_{\gamma} \frac{{\widehat{\Delta}}(x,t)}{R(s)}{\rm d} s + \mathcal{O}(z^{-2})\Big) \end{aligned} \end{align}

and (3.26) guarantees that $y_{\infty}:=\lim_{z\to\infty}y(z)$ exists. To conclude, we must show that ${\widehat{\Delta}}(x,t) = \Delta(x)$ in (3.24). First, we notice that ${\widehat{\Delta}}(x,t)$ is independent of time. Indeed, the coefficient of t in ${\widehat{\Delta}}(x,t)$ is given by

(3.28)\begin{equation} - 2i \Bigg( \int_{\eta_1} \frac{s^2}{R_{+}(s)}{\rm d} s + \int_{\eta_2} \frac{s^2}{R_{+}(s)}{\rm d} s \Bigg) \end{equation}

which can be shown to be zero by residue calculation and by leveraging the symmetry of the endpoints of η ₁. Therefore, (3.26) reads

(3.29)\begin{equation} {\widehat{\Delta}}(x,t) = \left( \int_{\gamma} \frac{1}{R(s)}{\rm d} s \right)^{-1} \left( - \int_{\eta_1} \frac{\log \, (2 \pi h(s) \rho(s)) + 2ixs}{R_{+}(s)}{\rm d} s + \int_{\eta_2} \frac{\log (2 \pi h(s) \rho(s)) - 2ix s }{R_{+}(s)}{\rm d} s \right). \end{equation}

Moreover, by residue calculation, one can show that

(3.30)\begin{equation} -2ix \left( \int_{\eta_1} \frac{s}{R(s)}{\rm d} s + \int_{\eta_1} \frac{s}{R(s)}{\rm d} s \right) = -2\pi x\,. \end{equation}

Finally, applying Proposition 3.6, we know that $\int_\gamma \frac{1}{R(s)} {\rm d} s \in i {\mathbb R}$; so to conclude it remains to show that

(3.31)\begin{equation} \left( - \int_{\eta_1} \frac{\log \,( 2 \pi h(s) \rho(s)) }{R_{+}(s)} {\rm d} s + \int_{\eta_2} \frac{\log \left(2 \pi h(s) \rho(s)\right)}{R_{+}(s)} {\rm d} s \right) = -2\Re \left( \int_{\eta_1} \frac{\log \,( 2 \pi h(s) \rho(s)) }{R_{+}(s)} {\rm d} s \right)\,. \end{equation}

Setting $z={\widetilde{x}} -ib$, the second integral can be rewritten as

(3.32)\begin{equation} \begin{aligned} \int_{-a}^a \frac{\log \left(2 \pi h({\widetilde{x}}-ib) \rho({\widetilde{x}}-ib)\right) }{R_+({\widetilde{x}}-ib)}{\rm d} {\widetilde{x}} & ={\overline{\int_{-a}^a \frac{\log \left(2 \pi h({\widetilde{x}}+ib) \rho({\widetilde{x}}+ib)\right) }{R_+({\widetilde{x}}+ib)}{\rm d} {\widetilde{x}} }}\\ & \stackrel{s={\widetilde{x}} + ib}{=} {\overline{\int_{-{\overline{A}}}^A\frac{\log \left(2 \pi h(s) \rho(s)\right) }{R_+(s)}{\rm d} s}}\\ & = - {\overline{\int_{\eta_1}\frac{\log \left(2 \pi h(s) \rho(s)\right) }{R_+(s)}{\rm d} s}}\,, \end{aligned} \end{equation}

from which it follows that ${\widehat{\Delta}}(x,t)\equiv\Delta(x)$, which completes the proof.

4.1. Jump matrix factorisation

On η ₃, the jump matrix J_D factors as follows

(4.10)\begin{gather} J_{D}(z) = \begin{pmatrix} e^{-\ln(N)u(z) + \widetilde{y}_{-}(z) - \widetilde{y}_{+}(z)}& 0\\ \\[2pt] i & e^{\ln(N)u(z) + \widetilde{y}_{+}(z) - \widetilde{y}_{-}(z)} \end{pmatrix} \end{gather}

(4.11)\begin{gather} = \begin{pmatrix} 1 & -i e^{-\ln(N) u(z) + \widetilde{y}_{-}(z) - \widetilde{y}_{+}(z)} \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix} \begin{pmatrix} 1 & -i e^{\ln(N)u(z) + \widetilde{y}_{+}(z) - \widetilde{y}_{-}(z)} \\ 0 & 1 \end{pmatrix} \end{gather}

and on η ₄ as

\begin{gather*} J_{D}(z) = \begin{pmatrix} e^{-\ln(N)\left( \tilde{u}(z) \right) + \widetilde{y}_{-}(z) - \widetilde{y}_{+}(z)} & i \\ \\[2pt] 0 & e^{\ln(N)\left( \tilde{u}(z) \right) + \widetilde{y}_{+}(z) - \widetilde{y}_{-}(z)} \end{pmatrix}\\ = \begin{pmatrix} 1 & 0\\ -i e^{\ln(N)\tilde{u}(z) + \widetilde{y}_{+}(z) - \widetilde{y}_{-}(z)} & 1 \end{pmatrix} \begin{pmatrix} 0 & i\\ i & 0 \end{pmatrix} \begin{pmatrix} 1 & 0\\ -i e^{-\ln(N) \tilde{u}(z) + \widetilde{y}_{-}(z) - \widetilde{y}_{+}(z)} & 1 \end{pmatrix}. \end{gather*}

Thanks to the conditions that the function $\widetilde{y}$ satisfies on η ₃ and η ₄, we can rewrite the jump for D:

\begin{gather*} J_{D}(z)= \begin{cases} \begin{pmatrix} 1 & -i e^{-\ln(N) u(z)} \frac{e^{2 \widetilde{y}_{-}(z)}}{2 \pi h(z) \rho (z) e^{2 i \theta(z)}} \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 0 & i\\ i & 0 \end{pmatrix} \begin{pmatrix} 1 & -i e^{\ln(N) u(z)} \frac{e^{2 \widetilde{y}_{+}(z)}}{2 \pi h(z) \rho (z) e^{2 i \theta(z)}} \\ 0 & 1 \end{pmatrix}, \quad & z \in \eta_3 \\ \\[4pt] \begin{pmatrix} 1 & 0 \\ -i e^{\ln(N) \tilde{u}(z)} \frac{e^{-2 \widetilde{y}_{-}(z)}}{2 \pi h(z) \rho (z) e^{-2 i \theta(z)}} & 1 \end{pmatrix} \begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix}\begin{pmatrix} 1 & 0 \\ -i e^{-\ln(N) \tilde{u}(z)} \frac{e^{-2 \widetilde{y}_{+}(z)}}{2 \pi h(z) \rho (z) e^{-2 i \theta(z)}} & 1 \end{pmatrix}, \quad & z \in \eta_4\\ \\[4pt] \begin{pmatrix} e^{\frac{2}{\tau} \ln(N) -\Delta(x) } & 0 \\ 0 & e^{-\frac{2}{\tau} \ln(N) + \Delta(x) } \end{pmatrix}, \quad & z \in \gamma \end{cases} \end{gather*}

We define the matrices

(4.12)\begin{equation} \begin{aligned} &\mathfrak{M}^{-}(z) = \begin{pmatrix} 1 & -i e^{-\ln(N) u(z)} \frac{e^{2 \widetilde{y}(z)}}{2 \pi h(z) \rho(z) e^{2 i \theta (z)}} \\ 0 & 1 \end{pmatrix}, \quad \mathfrak{M}^{+}(z) =\begin{pmatrix} 1 & -i e^{\ln(N) u(z)} \frac{e^{2 \widetilde{y}(z)}}{2 \pi h(z) \rho(z) e^{2 i \theta (z)}} \\ 0 & 1 \end{pmatrix}\\ & \mathfrak{N}^{-}(z) = \begin{pmatrix} 1 & 0\\ -i e^{-\ln(N) \tilde{u}(z) } \frac{e^{-2 \widetilde{y}(z)}}{2 \pi h(z) \rho(z) e^{-2i \theta(z)}} & 1 \end{pmatrix}, \quad \mathfrak{N}^{+}(z) = \begin{pmatrix} 1 & 0\\ -i e^{\ln(N)\tilde{u}(z) } \frac{e^{-2 \widetilde{y}(z)}}{2 \pi h(z) \rho(z) e^{-2i \theta(z)}} & 1 \end{pmatrix}\,. \end{aligned} \end{equation}

The off-diagonal entries of the matrices $ \mathfrak{M}^{\pm}$ are piecewise analytic functions with jumps across the contours η ₁ and η ₃. Notice that because u changes signs across the contour η ₁, we have

(4.13)\begin{equation} \left(\mathfrak{M}^{+}\right)_{-}(z) = \left(\mathfrak{M}^{-}\right)_{+}(z), \ z \in \eta_{1}. \end{equation}

Analogously, the off-diagonal entries of $\mathfrak{N}^{\pm}$ are piecewise analytic, with jumps across the contours η ₂ and η ₄. Notice that because $\tilde{u}$ changes signs across η ₂, we have

(4.14)\begin{equation} \left(\mathfrak{N}^{-}\right)_{-}(z) = \left(\mathfrak{N}^{+}\right)_{+}(z), \ z \in \eta_{2}. \end{equation}

We can rewrite the jump matrix J_D for the matrix D as follows.

\begin{gather*} J_{D}(z) = \begin{cases} \left(\mathfrak{M}^{-}\right)_{-}(z) \begin{pmatrix} 0 & i\\ i & 0 \end{pmatrix} \left( \mathfrak{M}^{+}\right)_{+}(z), \quad & z \in \eta_3\\ \\[2pt] \left( \mathfrak{N}^{+}\right)_{-}(z) \begin{pmatrix} 0 & i\\ i & 0 \end{pmatrix} \left( \mathfrak{N}^{-}\right)_{+}(z), \quad & z \in \eta_4\\ \\[2pt] \begin{pmatrix} e^{\frac{2}{\tau} \ln(N) -\Delta(x) } & 0 \\ 0 & e^{-\frac{2}{\tau} \ln(N) + \Delta(x) } \end{pmatrix}, \quad & z \in \gamma \end{cases} \end{gather*}

The next transformation is from D(z) to E(z), the so-called ‘opening of lenses’ transformation. Our aim is to arrive at a new RHP for a function E(z) with constant jumps on the contours $\eta_3,\eta_4,\gamma$ and ‘small’ jumps on the other 4 contours $\omega_1,\omega_2,\omega_3,\omega_4$ that we construct according to the following proposition.

Proposition 4.3. Fix $\min(\Re(A),\Im(A)) \gt 3\varepsilon \gt 0$, let ${\mathcal U}_A,{\mathcal U}_{{\overline{A}}},{\mathcal U}_{-A},{\mathcal U}_{-{\overline{A}}}$ be 4 disks of radius ɛ centred at $A,{\overline{A}}, -A, -{\overline{A}}$ respectively and define ${\mathcal V} = {\mathbb C} \setminus \{{\mathcal U}_A\cup {\mathcal U}_{{\overline{A}}}\cup{\mathcal U}_{-A}\cup{\mathcal U}_{-{\overline{A}}}\}$. Consider the matrices $\mathfrak{M}^{-},\mathfrak{N}^{+}$ (4.12), then there exist four contours $\omega_1,\omega_2,\omega_3,\omega_4$, a κ > 0 and a N ₀ such that for all $N \gt N_0$

(4.15)\begin{align} \mathfrak{N}^{+}(z) & = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} + O(N^{-\kappa}), \quad z\in \left(\omega_{1} \cup\omega_2\right) \cap {\mathcal V};\nonumber\\ \mathfrak{M}^{-}(z) & = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} + O(N^{-\kappa}), \quad z\in \left(\omega_3\cup\omega_4 \right)\cap {\mathcal V}\,. \end{align}

In particular, the contours $\omega_{1},\omega_2\subset {\mathbb C}_-\setminus\overline{\Omega}_2^+$ lie on opposing sides of η ₄, and $\omega_{3},\omega_4\subset {\mathbb C}_+\setminus\overline{\Omega}_3^+$ lie on opposing sides of η ₃. An example of these contours is shown in Fig. 7.

Figure 7.

Contour for RHP 4.4.

Proof. We focus on $\mathfrak{M}^-$, the proof in the other situation is analogous. First, we notice that in view of the definition (4.12), we must show that there exist two contours $\omega_3,\omega_4$ connecting A with $-{\overline{A}}$, a κ > 0 and a N ₀ such that for all $N \gt N_0$

(4.16)\begin{equation} -i e^{-\ln(N) u(z)} \frac{e^{2 \widetilde{y}(z)}}{2 \pi h(z) \rho(z) e^{2 i \theta (z)}} = O(N^{-\kappa})\,, \quad z\in \left(\omega_3\cup \omega_4\right)\cap {\mathcal V}\,. \end{equation}

Consider the set ${\mathbb C}_+\setminus(\overline{\Omega}_3^+ \cup \gamma)$. We notice that in these regions the quantities h(z), $\rho(z)$, $\theta(z)$ and ${\widetilde{y}}(z)$ are all analytic and bounded. Moreover, (4.5) holds. (See Assumptions 2.2 and Proposition 3.7, and Lemma 4.1 to verify these properties.) Therefore, we can choose $\omega_3,\omega_4\subset {\mathbb C}_+\setminus(\overline{\Omega}_3^+ \cup \gamma)$ (it is convenient to pick $\omega_{3},\omega_4$ to be analytic arcs). Finally, since $\Re(u(z))$ is continuous on $(\omega_3\cup\omega_4)\cap {\mathcal V}$, there exists κ > 0 such that

\begin{equation*} \min_{z\in\omega_3\cap{\mathcal V}} \Re(u(z)) = \kappa\,,\end{equation*}

so there exists N ₀ such that for all $N \gt N_0$

(4.17)\begin{equation} -i e^{-\ln(N) u(z)} \frac{e^{2 \widetilde{y}(z)}}{2 \pi h(z) \rho(z) e^{2 i \theta (z)}} = O(N^{-\kappa})\,, \quad z\in \omega_3\cap {\mathcal V}\,, \end{equation}

proving the result.

Given these 4 contours, we can define a new function E(z) as

(4.18)\begin{gather} E(z) = \begin{cases} D(z) \, \mathfrak{N}^{+}(z), \quad & z \in \Omega_1\\ D(z) \, \left( \mathfrak{N}^{-}(z)\right)^{-1}, \quad & z \in \Omega_2^{+}\\ D(z) \, \left(\mathfrak{N}^{+}(z)\right)^{-1}, \quad & z \in \Omega_2^{-}\\ D(z) \, \left(\mathfrak{M}^{+}(z)\right)^{-1}, \quad & z \in \Omega_3^{+}\\ D(z) \, \left(\mathfrak{M}^{-}(z)\right)^{-1}, \quad & z \in \Omega_3^{-}\\ D(z) \, \mathfrak{M}^{-}(z), \quad & z \in \Omega_4\\ D(z), \quad & \text{elsewhere} \end{cases} \end{gather}

where the sets $\Omega_2^{+},\Omega_2^{-},\Omega_3^{+},\Omega_3^{-}$ are as in Figure 7 and $\mathfrak{M}^{+},\mathfrak{M}^{-},\mathfrak{N}^{+},\mathfrak{N}^{-}$ are defined in (4.12). The matrix E(z) solves the following RHP.

5.1. Solution on the Riemann surface

Now we can explicitly find W(p) satisfying the conditions (5.12). To this end, we introduce the Abel map related to ${\mathcal R}$ as follows

(5.31)\begin{equation} \mathcal{A}(p) = \int_{A}^{p} \frac{c}{\rho (s)} \, {\rm d} s. \end{equation}

This is typically multi-valued, and if one wants to create a single-valued function, one may restrict the path of integration to avoid $P_{I}(\gamma) \cup P_{II}(\gamma)\cup P_{I}([{\overline{A}},\infty]) \cup P_{II}([{\overline{A}},\infty])$.

Next, we consider the Riemann Theta function [Reference Olver, Daalhuis, Lozier, Schneider, Boisvert, Clark, Miller, Saunders, Cohl and McClain32, 21]

(5.32)\begin{equation} \Theta(z;\tau) = \sum_{n \in \mathbb{Z}} e^{2 \pi i n z + \pi n^2 i \tau}, \quad z \in \mathbb{C} \end{equation}

which is an even function of z satisfying the following properties

(5.33)\begin{align} &\Theta(z+1;\tau) = \Theta(z;\tau) \end{align}

(5.34)\begin{align} &\Theta(z+\tau;\tau) = e^{-\pi i \tau - 2 \pi i z} \Theta(z;\tau) \end{align}

(5.35)\begin{align} &\Theta(z + n + m \tau;\tau) = e^{- \pi i m^2 \tau - 2 \pi i m z} \Theta(z;\tau), \quad n, m \in \mathbb{Z} \end{align}

and has a simple zero at $z=\frac{\tau + 1}{2}$. We notice that the Riemann Theta function is related to the Jacobi θ ₃ as $\Theta(z;\tau)=\theta_3(\pi z|\tau)$. We use the Θ-function to construct W(p) for $p \in {\mathcal R}$, and then through (5.11) we find $\Xi(z)$ solving RHP 5.4. Recall that the function W(p) has simple poles at the points ${\overline{A}},-{\overline{A}}$ and that two entries of W(p) vanish at $\infty_1$ and $\infty_2$.

Define the functions

(5.36)\begin{align} & w_1(p) = \frac{1}{Z_1}\frac{\Theta \left( \mathcal{A}(p) - \mathcal{A}(\infty_2) + \frac{\tau + 1}{2}; \tau \right)\Theta \left( \mathcal{A}(p) - \mathcal{A}(Q_1) + \frac{\tau + 1}{2}; \tau \right)}{\Theta \left( \mathcal{A}(p) - \mathcal{A}({\overline{A}}) + \frac{\tau + 1}{2}; \tau \right)\Theta \left( \mathcal{A}(p) - \mathcal{A}(-{\overline{A}}) + \frac{\tau + 1}{2}; \tau \right)} e^{-i \delta(p)}\,, \end{align}

(5.37)\begin{align} & w_2(p) = \frac{1}{Z_2} \frac{\Theta \left( \mathcal{A}(p) - \mathcal{A}(\infty_1) + \frac{\tau + 1}{2}; \tau \right)\Theta \left( \mathcal{A}(p) - \mathcal{A}(Q_2) + \frac{\tau + 1}{2}; \tau \right)}{\Theta \left( \mathcal{A}(p) - \mathcal{A}({\overline{A}}) + \frac{\tau + 1}{2}; \tau \right)\Theta \left( \mathcal{A}(p) - \mathcal{A}(-{\overline{A}}) + \frac{\tau + 1}{2}; \tau \right)} e^{-i \delta(p)}\,, \end{align}

where

(5.38)\begin{equation} \begin{aligned} &Z_1 = \frac{\Theta \left( \mathcal{A}(\infty_1) - \mathcal{A}(\infty_2) + \frac{\tau + 1}{2}; \tau \right)\Theta \left( \mathcal{A}(\infty_1)) - \mathcal{A}(Q_1) + \frac{\tau + 1}{2}; \tau \right)}{\Theta \left( \mathcal{A}(\infty_1)) - \mathcal{A}({\overline{A}}) + \frac{\tau + 1}{2}; \tau \right)\Theta \left( \mathcal{A}(\infty_1)) - \mathcal{A}(-{\overline{A}}) + \frac{\tau + 1}{2}; \tau \right)}\\ &Z_2 = \frac{\Theta \left( \mathcal{A}(\infty_2) - \mathcal{A}(\infty_1) + \frac{\tau + 1}{2}; \tau \right)\Theta \left( \mathcal{A}(\infty_2) - \mathcal{A}(Q_2) + \frac{\tau + 1}{2}; \tau \right)}{\Theta \left( \mathcal{A}(\infty_2) - \mathcal{A}({\overline{A}}) + \frac{\tau + 1}{2}; \tau \right)\Theta \left( \mathcal{A}(\infty_2) - \mathcal{A}(-{\overline{A}}) + \frac{\tau + 1}{2}; \tau \right)}\,. \end{aligned} \end{equation}

We notice that defining $W(p) = (w_1(p), w_2(p))^\intercal$, it has the right asymptotic behaviour for $p \to \infty$, but, in general, it is not single-valued on the Riemann surface. Indeed, with the notational convention that $p + \beta$ means adding a β-cycle, we have

(5.39)\begin{equation} \begin{aligned} & w_1(p+\beta) = \exp\left( 2\pi i \left( {\mathcal A}(\infty_2) + {\mathcal A}(Q_1) - {\mathcal A}({\overline{A}}) - {\mathcal A}(-{\overline{A}}) - \frac{\zeta(x,N)}{2\pi} \right) \right)w_1(p)\,, \\ & w_2(p+\beta) = \exp\left( 2\pi i \left( {\mathcal A}(\infty_1) + {\mathcal A}(Q_2) - {\mathcal A}({\overline{A}}) - {\mathcal A}(-{\overline{A}}) - \frac{\zeta(x,N)}{2\pi} \right) \right)w_1(p)\,. \end{aligned} \end{equation}

Therefore, to have W(p) single-valued on the Riemann surface, we must set

(5.40)\begin{equation} \begin{aligned} &{\mathcal A}(\infty_2) + {\mathcal A}(Q_1) - {\mathcal A}({\overline{A}}) - {\mathcal A}(-{\overline{A}}) - \frac{\zeta(x,N)}{2\pi} = 0 \,,\\ & {\mathcal A}(\infty_1) + {\mathcal A}(Q_2) - {\mathcal A}({\overline{A}}) - {\mathcal A}(-{\overline{A}}) - \frac{\zeta(x,N)}{2\pi} = 0\,, \end{aligned} \end{equation}

which implies that

(5.41)\begin{align} & {\mathcal A}(Q_1) = {\mathcal A}({\overline{A}}) + {\mathcal A}(-{\overline{A}}) + \frac{\zeta(x,N)}{2\pi} - {\mathcal A}(\infty_2)\,, \end{align}

(5.42)\begin{align} & {\mathcal A}(Q_2) = {\mathcal A}({\overline{A}}) + {\mathcal A}(-{\overline{A}}) + \frac{\zeta(x,N)}{2\pi} - {\mathcal A}(\infty_1)\,. \end{align}

Note that this determines the zeros $Q_1, Q_2$ of $w_1(p;\tau)$ and $w_2(p;\tau)$, respectively, although the path of integration from A to Q ₁ or Q ₂ will necessarily involve many traverses of the cycle β, since $\zeta(x,N)$ is growing in N. We can further simplify the expression for W(p) by applying the following proposition, which is a Corollary of (3.22).

Proposition 5.6. Consider the Riemann surface ${\mathcal R}$, and define the Abel map ${\mathcal A}(p)$ (5.31); then

(5.43)\begin{equation} {\mathcal A}({\overline{A}}) = \frac{\tau}{2}\,, \quad {\mathcal A}(-{\overline{A}})= \frac{1}{2}\,,\quad {\mathcal A}(\infty_1)= - {\mathcal A}(\infty_2) = \frac{\tau - 1}{4}\,. \end{equation}

Therefore, we can rewrite the solution W(p) as

(5.44)\begin{equation} W(p) = \begin{pmatrix} w_1(p) \\ w_2(p) \end{pmatrix} = \begin{pmatrix} \frac{1}{Z_1}\frac{\Theta \left( \mathcal{A}(p) + \frac{1+3\tau}{4} ; \tau \right)\Theta \left( \mathcal{A}(p) - \frac{\zeta(x,N)}{2\pi} + \frac{1-\tau}{4}; \tau \right)}{\Theta \left( \mathcal{A}(p) + \frac{1}{2}; \tau \right)\Theta \left( \mathcal{A}(p) + \frac{\tau}{2}; \tau \right)} e^{-i \delta(p)}\\ \frac{1}{Z_2} \frac{\Theta \left( \mathcal{A}(p) + \frac{3+\tau}{4}; \tau \right)\Theta \left( \mathcal{A}(p) - \frac{\zeta(x,N)}{2\pi} + \frac{\tau - 1}{4}; \tau \right)}{\Theta \left( \mathcal{A}(p) + \frac{1}{2}; \tau \right)\Theta \left( \mathcal{A}(p) + \frac{\tau}{2}; \tau \right)} e^{-i \delta(p)} \end{pmatrix}\,. \end{equation}

5.2. Solution on the complex plane

We now have to project the solution from the Riemann surface ${\mathcal R}$ to the complex plane. Consider the two projections $P_I(z),P_{II}(z)$ defined from the complex plane to the first and the second sheet of ${\mathcal R}$ respectively, see Fig. 8. Then, one can rewrite the solution of the RHP 5.4 as

(5.45)\begin{equation} \Xi(z) = \begin{pmatrix} w_1(P_I(z)) & w_1(P_{II}(z)) \\ w_2(P_I(z)) & w_2(P_{II}(z)) \\ \end{pmatrix}=: \begin{pmatrix} s_{11}(z;x,t) & s_{12}(z;x,t) \\ s_{21}(z;x,t) & s_{22}(z;x,t) \end{pmatrix}\,, \end{equation}

from the definition of $w_1(p),w_2(p)$ (5.39), it is easy to see that the previously defined function solves RHP 5.4.

For future use, we write the previous solution explicitly, making use of our previously forewarned notational abuse in which ${\mathcal A}(z)$ and $\delta(z)$ are considered to be evaluated on the complex plane, or, equivalently, on the first Riemann sheet.

(5.46)\begin{equation} \begin{aligned} &s_{11}(z;x,t) = \frac{1}{Z_1}\frac{\Theta \left( \mathcal{A}(z) + \frac{1+3\tau}{4} ; \tau \right)\Theta \left( \mathcal{A}(z) - \frac{\zeta(x,N)}{2\pi} + \frac{1-\tau}{4}; \tau \right)}{\Theta \left( \mathcal{A}(z) + \frac{1}{2}; \tau \right)\Theta \left( \mathcal{A}(z) + \frac{\tau}{2}; \tau \right)} e^{-i \delta(z)} \\ &s_{12}(z;x,t) = \frac{1}{Z_1}\frac{\Theta \left( -\mathcal{A}(z) + \frac{1+3\tau}{4} ; \tau \right)\Theta \left( -\mathcal{A}(z) - \frac{\zeta(x,N)}{2\pi} + \frac{1-\tau}{4}; \tau \right)}{\Theta \left( -\mathcal{A}(z) + \frac{1}{2}; \tau \right)\Theta \left( -\mathcal{A}(z) + \frac{\tau}{2}; \tau \right)} e^{i \delta(z)} \\ & s_{21}(z;x,t) = \frac{1}{Z_2} \frac{\Theta \left( \mathcal{A}(z) + \frac{3+\tau}{4}; \tau \right)\Theta \left( \mathcal{A}(z) - \frac{\zeta(x,N)}{2\pi} + \frac{\tau - 1}{4}; \tau \right)}{\Theta \left( \mathcal{A}(z) + \frac{1}{2}; \tau \right)\Theta \left( \mathcal{A}(z) + \frac{\tau}{2}; \tau \right)} e^{-i \delta(z)}\\ & s_{22}(z;x,t) = \frac{1}{Z_2} \frac{\Theta \left( -\mathcal{A}(z) + \frac{3+\tau}{4}; \tau \right)\Theta \left( -\mathcal{A}(z) - \frac{\zeta(x,N)}{2\pi} + \frac{\tau - 1}{4}; \tau \right)}{\Theta \left( -\mathcal{A}(z) + \frac{1}{2}; \tau \right)\Theta \left( -\mathcal{A}(z) + \frac{\tau}{2}; \tau \right)} e^{i \delta(z)}. \end{aligned} \end{equation}

6.1. Local parametrix near $z= - A$

Recall from proposition 4.3 that $\mathcal{U}_{-A}$ is a small neighbourhood of $z=-A$. Let us denote by $E_{-A}(z)$ a local parametrix near $z=-A$: $E_{-A}(z)$ should satisfy all the Riemann–Hilbert jump relations satisfied by E(z) inside the neighbourhood $\mathcal{U}_{-A}$, and it should also behave on the boundary of $\mathcal{U}_{-A}$ so as to asymptotically match the global parametrix.

Inside $\mathcal{U}_{-A}$, we seek $E_{-A}(z)$ satisfying the Riemann–Hilbert jump relations:

(6.1)\begin{gather} E_{-A +}(z) = E_{-A -}(z) J_{E_{-A}}(z), \quad J_{E_{-A}}(z) = \begin{cases} \begin{pmatrix} 0 & i\\ i & 0 \end{pmatrix}, & \quad z \in \eta_4 \\\\[2pt] e^{\left( -\Delta(x) + \frac{2}{\tau} \ln(N) \right) \sigma_3}, & \quad z \in \gamma\\\\[2pt] \mathfrak{N}^{+}(z), & \quad z \in \omega_1 \cup \omega_2 \end{cases} \end{gather}

Following [Reference Kuijlaars, McLaughlin, Van Assche and Vanlessen28], we construct E _−A in terms of special functions. For this purpose, we define the function $\Psi(\xi)$ as follows:

(6.2)\begin{gather} \Psi(z) = \begin{cases} \begin{pmatrix} I_{0}(2 \xi^{1/2}) & \frac{i}{\pi} K_{0}(2 \xi^{1/2}) \\\\[1pt] 2 \pi i \xi^{1/2} I_{0}'(2 \xi^{1/2}) & -2 \xi^{1/2} K_{0}'(2 \xi^{1/2}) \end{pmatrix}, \quad & \arg(\xi) \in (-2\pi/3,2\pi/3) \\\\[1pt] \begin{pmatrix} \frac{1}{2} H_{0}^{(1)}(2(-\xi)^{1/2}) & \frac{1}{2} H_{0}^{(2)}(2(-\xi)^{1/2}) \\\\[1pt] \pi \xi^{1/2} (H_{0}^{(1)})'(2(-\xi)^{1/2}) & \pi \xi^{1/2} (H_{0}^{(2)})'(2(-\xi)^{1/2}) \end{pmatrix}, \quad & \arg(\xi) \in (2\pi/3,\pi) \\\\[1pt] \begin{pmatrix} \frac{1}{2} H_{0}^{(2)}(2(-\xi)^{1/2}) & -\frac{1}{2} H_{0}^{(1)}(2(-\xi)^{1/2}) \\\\[1pt] -\pi \xi^{1/2} (H_{0}^{(2)})'(2(-\xi)^{1/2}) & \pi \xi^{1/2} (H_{0}^{(1)})'(2(-\xi)^{1/2}) \end{pmatrix}, \quad & \arg(\xi) \in (-\pi,-2\pi/3) \end{cases} \end{gather}

where $H_{0}^{(1)}$ and $H_{0}^{(2)}$ are the Hankel functions and K ₀ and I ₀ are the modified Bessel functions, see [Reference Olver, Daalhuis, Lozier, Schneider, Boisvert, Clark, Miller, Saunders, Cohl and McClain32, Chapter 10]. In [Reference Kuijlaars, McLaughlin, Van Assche and Vanlessen28], it is shown that Ψ satisfies the jumps given in Fig. 9. For brevity, we introduce the notation $\mathcal{Y}(z) = 2 \pi h(z) \rho(z) e^{-2i \theta(z)}$.

Figure 9.

Jumps for $\Psi(\xi)$.

We will arrive at precisely this collection of jump relations through explicit transformations, starting with $E_{-A}(z)$.

We introduce the function m:

(6.3)\begin{gather} m(z) = \begin{cases} E_{-A}(z) \, e^{\left( \widetilde{y}(z) - \frac{1}{\tau} \ln(N) \right) \sigma_3}, \quad & z \in \Omega\\\\[2pt] E_{-A}(z) \, e^{\left( \widetilde{y}(z) + \frac{1}{\tau} \ln(N) \right) \sigma_3}, \quad & z \in \widetilde{\Omega}. \end{cases} \end{gather}

Using the jump of E _−A and $\widetilde{y}$ across γ, we can check that m is analytic across γ. Moreover, it satisfies the following jump relations across $\eta_4 \cup \omega_1 \cup \omega_2$

(6.4)\begin{gather} m_{+}(z) = m_{-}(z) J_{m}(z), \end{gather}

where the jump matrix $J_{m}(z)$ can be factored as follows

(6.5)\begin{gather} J_{m}(z) = \Bigg( \frac{1}{\mathcal{Y}(z)} \Bigg)^{-\sigma_3/2} \left\{\begin{array}{lr} \begin{pmatrix} 0 & i\\ i & 0 \end{pmatrix}, \quad & z \in \eta_4\\\\[2pt] \begin{pmatrix} 1 & 0\\ -i e^{\ln(N) \frac{4c}{\tau} \int_{-A}^{z} \frac{{\rm d} s}{S(s)}} & 1 \end{pmatrix}, \quad & z \in \omega_1 \cup \omega_2 \end{array}\right\} \Bigg( \frac{1}{\mathcal{Y}(z)} \Bigg)^{\sigma_3/2}. \end{gather}

Note that, in relation (6.5), we have replaced $\int_{-A}^{z} \frac{{\rm d} s}{R(s)}$ with $\int_{-A}^{z} \frac{{\rm d} s}{S(s)}$ since the two agree on the contours ω ₁ and ω ₂. It is more convenient to use the function S(s) instead of R(s), because it is branched along η ₃ and η ₄, which are the relevant contours of non-analyticity in this section.

Next, we define the function

(6.6)\begin{gather} P(z) = m(z) \, \Bigg( \frac{1}{\mathcal{Y}(z)} \Bigg)^{-\sigma_3/2}\,, \end{gather}

which satisfies the jump relations

\begin{gather*} P_{+}(z) = P_{-}(z) \begin{cases} \begin{pmatrix} 0 & i\\ i & 0 \end{pmatrix}, \quad & z \in \eta_4\\\\[2pt] \begin{pmatrix} 1 & 0\\ -i e^{\ln(N) \frac{4c}{\tau} \int_{-A}^{z} \frac{{\rm d} s}{S(s)}} & 1 \end{pmatrix}, \quad & z \in \omega_1 \cup \omega_2. \end{cases} \end{gather*}

Analysis of the local behaviour of the function $\frac{4c}{\tau}\int_{-A}^{z} \frac{{\rm d} s}{S(s)}$ near $z=-A$, prompts us to introduce the conformal mapping $\xi = \varphi_{-A}(z)$, with the analytic function φ _−A defined implicitly via the relation

(6.7)\begin{gather} \left( \varphi_{-A}(z) \right)^{1/2} \equiv - \ln(N) \frac{c}{\tau} \int_{-A}^{z} \frac{{\rm d} s}{S(s)}\,. \end{gather}

This function maps the point $z=-A$ to ξ = 0, and the contour η ₄ emanating from −A is mapped to $\mathbb{R}_{-}$. We choose the contours ω ₁ and ω ₂ (within $\mathcal{U}_{-A}$) so that in the ξ plane, they map to precisely the rays $\mbox{arg}(\xi) = \pm 2 \pi /3 $ as shown in Figure 9.

Now we consider the jumps in the ξ plane satisfied by $P(\varphi_{-A}^{-1}(\xi))$:

(6.8)\begin{gather} \left(P ( \varphi_{-A}^{-1}(\xi)) \right)_{+} = \left(P ( \varphi_{-A}^{-1}(\xi)) \right)_{-} \begin{cases} \begin{pmatrix} 0 & -i\\ -i & 0 \end{pmatrix}, \quad & \xi \in (-\infty,0)\\ \\[4pt] \begin{pmatrix} 1 & 0\\ i e^{- 4 \xi^{1/2}} & 1 \end{pmatrix}, \quad & \mbox{Arg}(\xi) = \pm 2 \pi / 3 \end{cases} \end{gather}

It is then straightforward to verify that the quantity

(6.9)\begin{eqnarray} P(\varphi_{-A}^{-1}(\xi)) e^{2 \xi^{1/2} \, \sigma_3} e^{\frac{- i \pi \sigma_{3}}{4}} \end{eqnarray}

satisfies the jump relationships

(6.10)\begin{gather} \left( P(\varphi_{-A}^{-1}(\xi)) e^{2 \xi^{1/2} \, \sigma_3} e^{\frac{-i \pi \sigma_{3}}{4}} \right)_{+}= \left( P(\varphi_{-A}^{-1}(\xi)) e^{2 \xi^{1/2} \, \sigma_3} e^{\frac{-i \pi \sigma_{3}}{4}}\right)_{-} \begin{cases} \begin{pmatrix} 0 & 1\\ -1 & 0 \end{pmatrix}, \quad & \xi \in (-\infty,0)\\ \\[4pt] \begin{pmatrix} 1 & 0\\ 1 & 1 \end{pmatrix}, \quad & \mbox{Arg}(\xi) = \pm \frac{2 \pi }{3}, \end{cases} \end{gather}

which are precisely the jump relationships satisfied by Ψ, as shown in Figure 9. We therefore set

(6.11)\begin{eqnarray} P(\varphi_{-A}^{-1}(\xi)) e^{2 \xi^{1/2} \, \sigma_3} e^{\frac{-i \pi \sigma_{3}}{4}} = \Psi(\xi). \end{eqnarray}

Then, using the definition (6.2) of $\Psi(\xi) $ and the (conformal and hence invertible) transformation $\varphi_{-A}(z)$, (6.11) defines P(z) back in the z plane. Unravelling the prior transformations from E _−A to m and from m to P, we have established the following proposition.

Proposition 6.1. The matrix valued function E _−A given by the expression

(6.12)\begin{gather} E_{-A}(z) = Q_{-A}(z) \, e^{-i \pi \sigma_3/4} \, \Psi(\varphi_{-A}(z)) \, e^{i \pi \sigma_3/4}e^{-2 \varphi_{-A}(z)^{1/2} \sigma_3}\Bigg( \frac{1}{\mathcal{Y}(z)}\Bigg)^{\sigma_3/2} \begin{cases} e^{-(\widetilde{y}(z) - \frac{1}{\tau} \ln(N)) \sigma_3}, \quad z \in \Omega\\ e^{-(\widetilde{y}(z) + \frac{1}{\tau} \ln(N)) \sigma_3}, \quad z \in \widetilde{\Omega} \end{cases} \end{gather}

satisfies the jump relation (6.1). Here, $\Psi(\xi)$ is as in (6.2), Q _−A is any analytic and invertible matrix-valued function inside $\mathcal{U}_{-A}$ (to be determined later), the map φ _−A is defined as in equation (6.7), and the domains $\Omega,\, \widetilde{\Omega}$ are as in Figure 10.

Figure 10.

Contour for RHP 6.1.

6.2. Local parametrix near $z={\overline{A}}$

Recall that $\mathcal{U}_{{\overline{A}}}$ is a small neighbourhood of $z={\overline{A}}$. We seek $E_{{\overline{A}}}(z)$, a parametrix which satisfies the same jump relations as E(z) within $\mathcal{U}_{{\overline{A}}}$, and matches the global parametrix on the boundary of $\mathcal{U}_{{\overline{A}}}$. The jump relations for E are:

(6.13)\begin{equation} E_{+}(z) = E_{-}(z) \, \begin{cases} \begin{pmatrix} 0 & i\\ i & 0 \end{pmatrix}, \quad & z \in \eta_4\\ \\[4pt] \mathfrak{N}^{+}(z), \quad & z \in \omega_1 \cup \omega_2 \end{cases} \end{equation}

Proposition 6.2. The matrix valued function $E_{{\overline{A}}}$ given by the expression

(6.14)\begin{gather} E_{{\overline{A}}}(z) = Q_{{\overline{A}}}(z) e^{i \pi \sigma_3/4} \, \Psi \left(\varphi_{{\overline{A}}}(z)\right) \, e^{-i \pi \sigma_3/4} e^{-2 \varphi_{{\overline{A}}}(z)^{1/2} \sigma_3} \Big( e^{2 \widetilde{y}(z)} \mathcal{Y}(z) \Big)^{-\sigma_3/2} \end{gather}

satisfies the jump relations (6.13). Here Ψ is given by equation (6.2), $Q_{{\overline{A}}}$ is an analytic function inside $\mathcal{U}_{{\overline{A}}}$ and the map $\varphi_{{\overline{A}}}$ is defined as

(6.15)\begin{equation} \varphi_{{\overline{A}}}(z)^{1/2} = -\ln(N)\frac{c}{\tau}\int_{{\overline{A}}}^z \frac{{\rm d} s}{S(s)}\,. \end{equation}

Proof. It suffices to show that $E_{{\overline{A}}}(z)$ as in equation (6.14) satisfies the jump relation (6.13). We therefore compute the jump of $E_{{\overline{A}}}(z)$, first along η ₄. Note that the non-analytic functions in the definition of $E_{{\overline{A}}}(z)$ along η ₄ are: $\Psi \left( \varphi_{{\overline{A}}}(z) \right)$, $\varphi_{{\overline{A}}}(z)^{1/2}$ and $\widetilde{y}(z)$. We then have:

(6.16)\begin{align} &J_{E_{{\overline{A}}}}(z) =\left[\left( E_{{\overline{A}}} \right)_{-}(z)\right]^{-1} \left( E_{{\overline{A}}} \right)_{+}(z)\nonumber\\ &= \left( e^{2 \widetilde{y}_{-}(z)} \mathcal{Y}(z) \right)^{\sigma_3/2} e^{2 \varphi_{{\overline{A}}-}(z)^{1/2} \sigma_3} e^{i \pi \sigma_3/4}J_{\Psi} \left( \varphi_{{\overline{A}}}(z) \right) e^{-i \pi \sigma_3/4} e^{-2 \varphi_{{\overline{A}}+}(z)^{1/2} \sigma_3}\left( e^{2 \widetilde{y}_{+}(z)} \mathcal{Y}(z) \right)^{\sigma_3/2}. \end{align}

From the properties of Ψ, for $z \in \eta_4$, we have:

(6.17)\begin{gather} e^{i \pi \sigma_3/4} J_{\Psi}\left( \varphi_{{\overline{A}}}(z) \right) e^{-i \pi \sigma_3/4} = \begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix}. \end{gather}

In addition, $\varphi_{{\overline{A}}}(z)^{1/2}$ has jump across η ₄, specifically:

\begin{equation*}\varphi_{{\overline{A}}-}(z)^{1/2} = - \varphi_{{\overline{A}}+}(z)^{1/2}, \quad z \in \eta_4\ .\end{equation*}

Finally, recall that the function $\widetilde{y}$ satisfies the jump relation (4.8a) along η ₄. Using all of these relations, straightforward multiplication of the right-hand side of equation (6.16) yields the desired jump of $E_{{\overline{A}}}(z)$ along η ₄. Next, we compute the jump of $E_{{\overline{A}}}(z)$ along $\omega_1 \cup \omega_2$. Note that the only non-analytic piece in the definition of $E_{{\overline{A}}}(z)$ along these two rays is the function $\Psi \left( \varphi_{{\overline{A}}}(z) \right)$. We then have:

\begin{align*} J_{E_{{\overline{A}}}}(z) & = \left( e^{2 \widetilde{y}(z)} \mathcal{Y}(z) \right)^{\sigma_3/2} e^{2 \phi_{{\overline{A}}}(z)^{1/2} \sigma_3} e^{i \pi \sigma_3/4} J_{\Psi}\left( \varphi_{{\overline{A}}}(z) \right) e^{-i \pi \sigma_3/4} e^{-2 \varphi_{{\overline{A}}}(z)^{1/2} \sigma_3}\left( e^{2 \widetilde{y}(z)} \mathcal{Y}(z) \right)^{\sigma_3/2}. \end{align*}

From the properties of Ψ, for $z \in \omega_1 \cup \omega_2$, we have:

\begin{equation*}e^{i \pi \sigma_3/4} J_{\Psi} \left( \varphi_{{\overline{A}}}(z) \right) e^{-i \pi \sigma_3/4} = \begin{pmatrix} 1 & 0 \\ -i & 1 \end{pmatrix}.\end{equation*}

Inserting this into the equation for $J_{E_{{\overline{A}}}}(z)$, a straightforward calculation yields the desired jump.

6.3. Local parametrix near z = A

Recall that $\mathcal{U}_{A}$ is a small neighbourhood of z = A, and we denote by E_A the solution to the local problem near z = A. Inside $\mathcal{U}_{A}$, E_A satisfies the jump relation

(6.18)\begin{gather} E_{A +}(z) = E_{A +}(z) J_{E_{A}}(z), \quad J_{E_{A}}(z) = \begin{cases} \begin{pmatrix} 0 & i\\ i & 0 \end{pmatrix}, \quad & z \in \eta_3\\\\[2pt] \mathfrak{M}^{-}(z), \quad & z \in \omega_3 \cup \omega_4. \end{cases} \end{gather}

Proposition 6.3. The matrix valued function E_A given by the expression

(6.19)\begin{gather} E_{A}(z) = - Q_{A}(z) \sigma_2 e^{i \pi \sigma_3/4} \Psi \big( \varphi_{A}(z) \big) e^{-i \pi \sigma_3/4} \sigma_2 e^{-2 \varphi_{A}(z)^{1/2} \sigma_3} \left( \frac{e^{2 \widetilde{y}(z)}}{\mathcal{Y}(z) e^{4i \theta(z)}} \right)^{-\sigma_3/2}\,, \end{gather}

satisfies the jump relation (6.18). Here, Q_A is an analytic function inside $\mathcal{U}_{A}$ (to be determined later), and the map φ_A is defined as

(6.20)\begin{gather} \varphi_{A} (z)^{1/2} = -\ln(N) \frac{c}{\tau} \int_{A}^{z} \frac{{\rm d} s}{S(s)}\,. \end{gather}

Proof. First, define

(6.21)\begin{gather} \Psi_{A} (\xi) = - \sigma_2 e^{i \pi \sigma_3/4}\Psi (\xi)e^{-i \pi \sigma_3/4} \sigma_2. \end{gather}

The reader may verify that $\Psi_{A}$ satisfies the jump relation

(6.22)\begin{eqnarray} \left(\Psi_{A}\right)_{+} (\xi)=\left(\Psi_{A}\right)_{-} (\xi) \begin{cases} \begin{pmatrix} 0 & -i\\ -i & 0 \end{pmatrix}, \quad & \xi \in (-\infty,0)\\ \\[4pt] \begin{pmatrix} 1 & -i\\ 0 & 1 \end{pmatrix}, \quad & \mbox{Arg}(\xi) = \pm 2 \pi / 3 . \end{cases} \end{eqnarray}

The composition $\Psi_{A}(\varphi_{A}(z))$ possesses jumps across the contours η ₃, ω ₃ and ω ₄. However, these contours are oriented in the opposite way as their images in the ξ plane. The reader may verify that the jumps in the z plane are the inverses of those in the ξ plane:

(6.23)\begin{eqnarray} \left(\Psi_{A}(\varphi_{A}(z))\right)_{+}=\left(\Psi_{A}(\varphi_{A}(z))\right)_{-} \begin{cases} \begin{pmatrix} 0 & i\\ i & 0 \end{pmatrix}, \quad & z \in \eta_{3}\\ \\[4pt] \begin{pmatrix} 1 & i\\ 0 & 1 \end{pmatrix}, \quad & z \in \omega_{3} \cup \omega_{4} . \end{cases} \end{eqnarray}

For example, as z approaches $\hat{z}$ from the + side of η ₃, $\xi = \varphi_{A}(z)$ approaches $\varphi_{A}(\hat{z})$ from below, i.e. from the − side of $\mathbb{R}_{-}$. Similarly, if z approaches $\hat{z}$ from the − side of η ₃, then $\xi = \varphi_{A}(z)$ approaches $\varphi_{A}(\hat{z})$ from the + side of $\mathbb{R}_{-}$. This means that

(6.24)\begin{align} & \lim_{\substack{z \to \hat{z} \\ z \in + \mbox{side of } \eta_3}} \Psi_{A}(\varphi_{A}(z) )= \left(\Psi_{A} \right)_{-}(\varphi_{A}(\hat{z})) = \left(\Psi_{A} \right)_{+}(\varphi_{A}(\hat{z})) {\left( \begin{array}{cc}{0}&{i}\\{i}&{0}\end{array}\right)} \end{align}

(6.25)\begin{align} & = \lim_{\substack{z \to \hat{z} \\ z \in - \mbox{side of } \eta_3}} \Psi_{A}(\varphi_{A}(z) ) {\left( \begin{array}{cc}{0}&{i}\\{i}&{0}\end{array}\right)}, \end{align}

which explains (6.23).

Now the definition (6.19) can be rewritten as

\begin{eqnarray*} E_{A}(z) = Q_{A}(z) \Psi_{A}(\varphi_{A}(z)) e^{-2 \varphi_{A}(z)^{1/2} \sigma_3} \left( \frac{e^{2 \widetilde{y}(z)}}{\mathcal{Y}(z) e^{4i \theta(z)}} \right)^{-\sigma_3/2}, \end{eqnarray*}

and the known jump relations satisfied by the last two factors appearing on the right-hand side can be used to verify that $E_{A}(z)$ satisfies the jump relations in (6.18).

6.4. Error function and global parametrix

In this subsection, we define the global approximation to E, and introduce the error function ${\mathcal E}$. The global approximation is defined separately within each of the four disks and in the region exterior to all the disks.

(6.26)\begin{gather} \quad E_{{\rm approx}}(z) = \begin{cases} E_{A}(z), \quad & z\in \mathcal{U}_{A}\\ E_{{\overline{A}}}(z), \quad & z\in \mathcal{U}_{{\overline{A}}} \\ E_{-{\overline{A}}}(z), \quad & z\in\mathcal{U}_{-{\overline{A}}} \\ E_{-A}(z), \quad & z\in\mathcal{U}_{-A} \\ \Lambda(z), \quad & \text{otherwise}. \end{cases} \end{gather}

The error matrix, ${\mathcal E} (z)$ is defined as follows

(6.27)\begin{eqnarray} {\mathcal E} (z) = E(z) \, \left( E_{{\rm approx}}(z) \right)^{-1}. \end{eqnarray}

Here, recall that E is given by (4.18) and Λ is the outer approximate solution given in (5.2). We choose the functions Q_A, $Q_{{\overline{A}}}$, Q _−A, $Q_{-{\overline{A}}}$ appearing in the definitions of the local solutions so that the quantity ${\mathcal E}(z)$ is in the small norm setting. Specifically, we compute the jump of ${\mathcal E}(z)$ across the disk boundaries, and then choose these quantities to ensure that the jump matrices are of the form $I + \mbox{small}$.

We start with the jump relationship satisfied by ${\mathcal E}(z)$ for $z\in \partial\mathcal{U}_{{\overline{A}}}$ where $\partial\mathcal{U}_{{\overline{A}}}$ (as all the four neighbourhoods) has been chosen with counterclockwise orientation:

(6.28)\begin{align} J_{{\mathcal E}}(z) &= {\mathcal E}_{-}^{-1}(z) \, {\mathcal E}_{+}(z) \end{align}

(6.29)\begin{align} & = \Big( E(z) \Lambda^{-1}(z) \Big)^{-1} \, E(z) E_{{\overline{A}}}^{-1}(z) \end{align}

(6.30)\begin{align} &= \Lambda(z) \Big( e^{2 \widetilde{y}(z)} \mathcal{Y}(z) \Big)^{\sigma_3/2} e^{2 \varphi_{{\overline{A}}}(z)^{1/2} \sigma_3} e^{i \pi \sigma_3/4} \, \Psi^{-1} \left(\varphi_{{\overline{A}}}(z)\right) \, e^{-i \pi \sigma_3/4} Q_{{\overline{A}}}^{-1}(z). \end{align}

Next, we utilise the asymptotic behaviour of the function Ψ defined in (6.2), as presented in [Reference Kuijlaars, McLaughlin, Van Assche and Vanlessen28, p. 369]

(6.31)\begin{gather} \Psi(\xi) = \left( 2 \pi \xi^{1/2} \right)^{-\sigma_3/2} \, \frac{1}{\sqrt{2}} \begin{pmatrix} 1 + \mathcal{O}(\xi^{-1/2}) & i + \mathcal{O}(\xi^{-1/2})\\ i + \mathcal{O}(\xi^{-1/2}) & 1 + \mathcal{O}(\xi^{-1/2}) \end{pmatrix} e^{2 \xi^{1/2} \sigma_3} \end{gather}

uniformly as $\xi \to \infty$. Using this, we find

(6.32)\begin{align} J_{{\mathcal E}}(z) & = \frac{\sqrt{2}}{2} \, \Lambda(z) \Big( e^{2 \widetilde{y}(z)} \mathcal{Y}(z) \Big)^{\sigma_3/2} e^{2 \varphi_{{\overline{A}}}(z)^{1/2} \sigma_3} e^{i \pi \sigma_3/4} e^{-2 \varphi_{{\overline{A}}}(z)^{1/2} \sigma_3} \qquad \qquad \qquad \qquad \qquad \quad \end{align}

\begin{align*} & \qquad \left( I - \frac{1}{2} \begin{pmatrix} 1 & -i\\ -i & 1 \end{pmatrix} \begin{pmatrix} \mathcal{O}\left( \varphi_{{\overline{A}}}^{-1/2}(z) \right) & \mathcal{O}\left( \varphi_{{\overline{A}}}^{-1/2}(z) \right)\\\\[0.5pt] \mathcal{O}\left( \varphi_{{\overline{A}}}^{-1/2}(z) \right) & \mathcal{O}\left( \varphi_{{\overline{A}}}^{-1/2}(z) \right) \end{pmatrix} + \ldots \right) \begin{pmatrix} 1 & -i\\ -i & 1 \end{pmatrix} \left( 2 \pi \varphi_{{\overline{A}}}^{1/2}(z) \right)^{\sigma_3/2} e^{- i \pi \sigma_3/4} Q_{{\overline{A}}}^{-1}(z) \,. \end{align*}

The last one simplifies to:

(6.33)\begin{align} J_{\mathcal E}(z)& = \frac{\sqrt{2}}{2} \, \Lambda(z) \Big( e^{2 \widetilde{y}(z)} \mathcal{Y}(z) \Big)^{\sigma_3/2} e^{i \pi \sigma_3/4} \end{align}

(6.34)\begin{align} & \left( I - \frac{1}{2} \begin{pmatrix} 1 & -i\\ -i & 1 \end{pmatrix} \begin{pmatrix} \mathcal{O}\left( \varphi_{{\overline{A}}}^{-1/2}(z) \right) & \mathcal{O}\left( \varphi_{{\overline{A}}}^{-1/2}(z) \right)\\\\[0.5pt] \mathcal{O}\left( \varphi_{{\overline{A}}}^{-1/2}(z) \right) & \mathcal{O}\left( \varphi_{{\overline{A}}}^{-1/2}(z) \right) \end{pmatrix} + \dots \right) \begin{pmatrix} 1 & -i\\ -i & 1 \end{pmatrix} \left( 2 \pi \varphi_{{\overline{A}}}^{1/2}(z) \right)^{\sigma_3/2} e^{- i \pi \sigma_3/4} Q_{{\overline{A}}}^{-1}(z)\,. \end{align}

At this point, we require that $J_{{\mathcal E}}$ behaves like the identity matrix plus small corrections across $\partial\mathcal{U}_{{\overline{A}}}$. Therefore, matching the leading order terms on both sides of the last relation, we must define $Q_{{\overline{A}}}$ as follows:

(6.35)\begin{gather} Q_{{\overline{A}}}(z) = \frac{\sqrt{2}}{2} \Lambda(z) \Big( e^{2 \widetilde{y}(z)} \mathcal{Y}(z) \Big)^{\sigma_3/2} e^{i \pi \sigma_3/4} \begin{pmatrix} 1 & -i\\ -i & 1 \end{pmatrix} \left( 2 \pi \varphi_{{\overline{A}}}(z)^{1/2} \right)^{\sigma_3/2} e^{- i \pi \sigma_3/4}. \end{gather}

Proposition 6.4. The function $Q_{{\overline{A}}}$ defined in (6.35) is analytic for $z \in \mathcal{U}_{{\overline{A}}}$. Moreover, using this function in (6.14), we have

(6.36)\begin{eqnarray} J_{{\mathcal E}} = I + \mathcal{O} \left( \frac{1}{\ln{(N)}}\right), \ \ z \in \partial \mathcal{U}_{{\overline{A}}}. \end{eqnarray}

Remark 6.3. Note that we can apply a similar approach to that used in proposition 6.4 to identify the analytic functions Q present in all local approximate solutions. By utilising these functions and employing an argument analogous to that in proposition 6.4, we can establish that

(6.37)\begin{equation} J_{{\mathcal E}}(z) = \mathbb{I} + \mathcal{O} \left(\frac{1}{\log{N}}\right), \quad z\in \partial\left(\mathcal{U}_{A} \cup \mathcal{U}_{{\overline{A}}} \cup \mathcal{U}_{-{\overline{A}}} \cup \mathcal{U}_{-A} \right)\,. \end{equation}

Since the determination of each of Q_A, Q _−A and $Q_{-{\overline{A}}}$ is achieved by similar considerations, we omit these calculations for the sake of brevity. Next, we are interested in the jump of the function ${\mathcal E}$ across the remaining contours. As observed previously, ${\mathcal E}$ has no jumps within the four disks. From its definition, we can check that ${\mathcal E}$ is also analytic across $\eta_3 \cup \eta_4 \cup \gamma$. There remains the jumps across the contours $\omega_1 \cup \omega_2 \cup \omega_3 \cup \omega_4$:

(6.38)\begin{align} J_{{\mathcal E}}(z) = \begin{cases} \Lambda(z) \, \mathfrak{M}^{-}(z) \, \Lambda^{-1}(z), & \quad z \in \omega_3 \cup \omega_4\\ \Lambda(z) \, \mathfrak{N}^{+}(z) \, \Lambda^{-1}(z), & \quad z \in \omega_1 \cup \omega_2. \end{cases} \end{align}

From proposition (4.3) we can conclude that $J_{{\mathcal E}}$ behaves like identity plus small across the contour $\omega_1 \cup \omega_2 \cup \omega_3 \cup \omega_4$. We have established all the necessary ingredients to conclude that ${\mathcal E}$ defined in (6.26) satisfies a small norm Riemann–Hilbert problem. The contour for this Riemann–Hilbert problem consists of the four disk boundaries, and the contours $\{\omega_{j} \cap {\mathcal V}\}_{j=1}^{4}$, where we recall that ${\mathcal V} = {\mathbb C} \setminus \{{\mathcal U}_A\cup {\mathcal U}_{{\overline{A}}}\cup{\mathcal U}_{-A}\cup{\mathcal U}_{-{\overline{A}}}\}$, as shown in Fig. 11. On each of the contours, the jump matrix for ${\mathcal E}$ is uniformly close to I. Specifically, we have (6.37), and on each of the contours ω_j, we have the uniform bound

(6.39)\begin{equation} J_{{\mathcal E}}(z) = \mathbb{I} + \mathcal{O} \left(e^{- c \log{N} }\right), \quad z \in \omega_{j}\cap {\mathcal V}, \ j = 1, \ldots, 4. \end{equation}

Figure 11.

Jumps for the matrix $J_{\mathcal E}$.

Then, following (for example) the statement and proof of Theorem 7.10 on P. 1532 of [Reference Deift, Kriecherbauer, McLaughlin, Venakides and Zhou10], we may conclude that ${\mathcal E}(z)$ not only exists, it also satisfies

(6.40)\begin{equation} {\mathcal E} (z) = \mathbb{I} + \mathcal{O} \left(\frac{1}{\log{N} \left( 1 + |z| \right)}\right) \end{equation}

for all $z \in \mathbb{C}$. Therefore, the global solution E exists and is given by the expression

(6.41)\begin{equation} E(z) = \left( \mathbb{I} + \mathcal{O} \left(\frac{1}{\log{N}\left( 1 + |z| \right)}\right) \right) \, E_{{\rm approx}}(z), \quad z \in \mathbb{C}. \end{equation}

By unravelling all the transformations we have introduced in Sections 3 and 4, we have established a complete asymptotic description of the solution to RHP 2.4. In the next section, we will carry out this unravelling in order to determine the asymptotic behaviour of the solution to the NLS equation.