2.1. Fast connections

Taking the singular limit $\varepsilon \rightarrow 0$ in (2.2) yields the limiting fast system

(2.3) \begin{equation} \left \{ {\begin{array}{l} v^\prime =q,\\[5pt] q^{\prime }=G(u,v,0), \\[5pt] \end{array}} \right . \end{equation}

in which the slow variables $(u,p)$ serve as parameters.

By the assumption (A1), $G(u, 0, 0)=0$ holds for certain values of $u$ . It thus follows that the set

\begin{equation*} \mathcal{M}=\{(u, p, v, q)|v=q=0\} \end{equation*}

consists of the equilibria of system (2.3). Furthermore, $\mathcal{M}$ is normally hyperbolic since $G_v(u,0,\varepsilon )\gt 0$ , which ensures that each equilibrium in $\mathcal{M}$ is a saddle of system (2.3). Consequently, both the stable and unstable manifolds $\mathcal{W}^{s}(\mathcal{M})$ and $\mathcal{W}^{u}(\mathcal{M})$ are $3$ -dimensional in the $4$ -dimensional phase space.

System (2.3) is integrable with the Hamiltonian

(2.4) \begin{equation} H(v,q;\; u)=\frac {1}{2}q^{2}-\int ^v_0 G( u,s,0)\,\textrm {d}s. \end{equation}

The assumption (A4) ensures the existence of a homoclinic pulse in system (2.3). Moreover, the stable and unstable manifolds $\mathcal{W}^{s}(\mathcal{M})$ and $\mathcal{W}^{u}(\mathcal{M})$ coincide, forming a $3$ -dimensional homoclinic manifold, which is composed of a two-parameter family of homoclinic orbits (see Figure 1a).

Figure 1.

(a) The $3$ -dimensional homoclinic manifold of the fast limiting system (2.3) at $\varepsilon =0$ . (b) A transversal intersection between $\mathcal{W}^{s}(\mathcal{M}_\varepsilon )$ and $\mathcal{W}^{u}(\mathcal{M}_\varepsilon )$ for $0\lt \varepsilon \ll 1$ , which is homoclinic to the slow manifold $\mathcal{M}_\varepsilon$ .

Figure 1 Long description

Panel A: A three-dimensional plot shows the homoclinic manifold of the fast limiting system at epsilon equals zero. The plot features a cylindrical surface with a gradient from yellow to red, indicating varying values of q. The horizontal axis is labeled v, and the vertical axis is labeled u and p. The plane at epsilon equals zero is highlighted. Panel B: A two-dimensional plot illustrates a transversal intersection between the stable and unstable manifolds of the slow manifold for small epsilon values. The plot features two curved lines, one red and one blue, intersecting on a green plane. The horizontal axis is labeled u, and the vertical axis is labeled v. The slow manifold is labeled as M sub epsilon, and the trajectories T sub d and T sub o are marked.

When $\varepsilon \gt 0$ is sufficiently small, by GSPT, $\mathcal{M}$ perturbs to a $2$ -dimensional locally invariant manifold $\mathcal{M}_\varepsilon$ , which is $\mathcal{O}(\varepsilon )$ -close to its counterpart. Since $\mathcal{M}$ is invariant under the flow of (2.2) $_2$ , it follows $\mathcal{M}_\varepsilon =\mathcal{M}.$ Similarly, by GSPT again, $\mathcal{W}^{s/u}(\mathcal{M})$ perturbs to $3$ -dimensional locally invariant manifolds $\mathcal{W}^{s/u}(\mathcal{M}_\varepsilon )$ , which are also $\mathcal{O}(\varepsilon )$ -close to their counterparts. These manifolds consist of the fast stable and unstable fibres with their bases on the slow manifold $\mathcal{M}_\varepsilon$ . When $0\lt \varepsilon \ll 1$ , $\mathcal{W}^{s}(\mathcal{M}_\varepsilon )$ and $\mathcal{W}^{u}(\mathcal{M}_\varepsilon )$ no longer coincide. Under appropriate conditions, they can intersect transversely, giving rise to a $2$ -dimensional manifold $\mathcal{W}^{s}(T_d) \cap \mathcal{W}^{u}(T_o)$ (see Remark2 and Figure 1b).

Here, we want to remark that although the slow variables in system (2.2) are discontinuous due to the term $ W(u, \chi )$ , with each subregion, the system under consideration becomes autonomous with respect to $ \chi$ . Thus, within each subregion, the standard framework of GSPT can still be applied to the two autonomous subsystems. At the positions of interfaces $\chi = \pm L$ , we match the orbits of the two autonomous subsystems in a continuous, differentiable manner; that is, the orbits together with their derivatives are both connected. In this manner, we can still obtain the $C^1$ -smooth orbits of the full system (2.2) by using GSPT within each smooth region and smooth matching at the position of heterogeneities.

We now employ the adiabatic Melnikov method [Reference Palmer28, Reference Robinson30] to examine the persistence of the 3-dimensional homoclinic manifold, which is homoclinic to $\mathcal{M}_\varepsilon$ for $0 \lt \varepsilon \ll 1$ . The analysis reveals that the homoclinic orbit acts as a dynamical bridge that enables the flow originating from the slow unstable manifold of the saddle on $\mathcal{M}_\varepsilon$ to return to the slow stable manifold, thereby facilitating a dynamical transition.

To facilitate the analysis, we partition the entire spatial interval $(\!-\infty , +\infty )$ into three subregions, namely, two slow regions $I_s^- = (\!-\infty , -\frac {1}{\sqrt {\varepsilon }})$ and $I_s^+ = (\frac {1}{\sqrt {\varepsilon }}, \infty )$ and a fast region $I_f = [-\frac {1}{\sqrt {\varepsilon }}, \frac {1}{\sqrt {\varepsilon }}]$ . The boundaries of the fast region $I_f$ are placed in the transition zone, satisfying $|\xi | \gg 1$ and $|x| \ll 1$ . In other words, the exact location of $\partial I_f$ is not critical [Reference Doelman, Gardner and Kaper12, Reference Doelman and Veerman16]. Indeed, the width of $I_f$ should be chosen appropriately to ensure that $v_h(\xi )$ is exponentially small everywhere outside $I_f$ , while $u_h(\xi )$ remains approximately constant, to the leading order, within the fast region $I_f$ .

Remark 2. For convenience, we denote by $\mathcal{W}^{u}(T_o)$ the collection of unstable fibres with base points on the curve $T_o$ and by $\mathcal{W}^{s}(T_d)$ the collection of stable fibres with base points on the curve $T_d$ . These sets intersect forming a $2$ -dimensional manifold, denoted as $\mathcal{W}^{s}(T_d)\cap \mathcal{W}^{u}(T_o)$ . In the subsequent analysis, we will prove that

(2.5) \begin{equation} \mathcal{W}^{s}(\mathcal{M}_{\varepsilon })\cap \mathcal{W}^{u}(\mathcal{M}_{\varepsilon })=\mathcal{W}^{s}(T_d)\cap \mathcal{W}^{u}(T_o). \end{equation}

Remark 3. In our setup, the heterogeneity region $ [-L, L]$ is defined in terms of the super-slow spatial variable $ \chi = \varepsilon x=\varepsilon ^2 \xi$ , as introduced in equation (1.2). Thus, in terms of the original spatial variable $ \xi$ , the corresponding heterogeneous region is rescaled to $ \left [ -\frac {L}{\varepsilon ^2}, \frac {L}{\varepsilon ^2} \right ]$ , which becomes much wider as $ \varepsilon \to 0$ . On the other hand, the fast region is defined as $ I_f = \left [ -\frac {1}{\sqrt {\varepsilon }}, \frac {1}{\sqrt {\varepsilon }} \right ]$ . Therefore, we have

\begin{equation*} \left [ -\frac {1}{\sqrt {\varepsilon }}, \frac {1}{\sqrt {\varepsilon }} \right ] \subset \left [ -\frac {L}{\varepsilon ^2}, \frac {L}{\varepsilon ^2} \right ]. \end{equation*}

In the fast system (2.2), the Hamiltonian $H(v,q;\;u)$ becomes a slowly varying function. Its derivative with respect to the fast time variable $\xi$ is given by

(2.6) \begin{equation} H_{\xi }(v,q;\;u)=\big (G(u,v,\varepsilon )-G(u,v,0)\big )q-\varepsilon p\int ^v_0 G_u( u,s,0)\,\textrm {d}s +\mathcal{O}(\varepsilon ^2). \end{equation}

The unperturbed homoclinic orbit $(u^{(0)},p^{(0)},v_{0}(\xi ),q_{0}(\xi ))$ is perturbed to a solution of the full fast system (2.2), denoted by

\begin{equation*}\gamma _{h}(\xi )=\left (u_h(\xi ),\,p_h(\xi ),\,v_h(\xi ),\,q_h(\xi )\right ),\end{equation*}

which is homoclinic to $\mathcal{M}_{\varepsilon }$ , provided that the Melnikov integral

(2.7) \begin{equation} \begin{aligned} \triangle _{I_{f}}H(u,p) =& \int _{I_{f}} \big (G(u_{h}(\xi ),v_{h}(\xi ),\varepsilon )-G(u_{h}(\xi ),v_{h}(\xi ),0)\big )q_{h}(\xi )\,\textrm {d}\xi \\[5pt] & \quad -\varepsilon \int _{I_{f}} p_{h}(\xi ) \int ^{v_{h}(\xi )}_0 G_u( u_{h}(\xi ),s,0)\,\textrm {d}s \textrm {d} \xi +\mathcal{O}(\varepsilon ^2) \end{aligned} \end{equation}

has a simple zero. In that case, we have $\gamma _{h}(\xi )\subset \mathcal{W}^{s}(\mathcal{M}_\varepsilon )\cap \mathcal{W}^{u}(\mathcal{M}_\varepsilon )$ . The improper integral is well-defined because $\gamma _{h}(\xi )$ converges exponentially to $(0,0,0,0)$ as $\xi \rightarrow \pm \infty$ .

To leading order,

(2.8) \begin{equation} \begin{aligned} \triangle _{I_{f}}H(u^{(0)},p^{(0)}) =& \int _{I_{f}} \big (G(u^{(0)},v_{0}(\xi ),\varepsilon )-G(u^{(0)},v_{0}(\xi ),0)\big )q_{0}(\xi )\,\textrm {d}\xi \\[5pt] & \quad \quad \quad \quad -\varepsilon \int _{I_{f}} p^{(0)} \int ^{v_{0}(\xi )}_0 G_u( u^{(0)},s,0)\,\textrm {d}s\textrm {d}\xi . \end{aligned} \end{equation}

Since $v_{ 0}(\xi )$ is even and $q_{0}(\xi )$ is odd, so $\big (G(u^{(0)},v_{0}(\xi ),\varepsilon )-G(u^{(0)},v_{0}(\xi ),0)\big )$ and $\int ^{v_{0}(\xi )}_0 G_u( u^{(0)},s,0)\,\textrm { d}s$ are even functions. Note that $u^{(0)}$ is assumed to be positive; it then follows from (A1) that $G_u( u^{(0)},v,0)$ is also positive. Consequently, solving $\triangle _{I_{f}}H(u^{(0)},p^{(0)})=0$ gives $p^{(0)}=0$ .

To compute the correction, we expand $(u_{h}(\xi ), p_{h}(\xi ), v_{h}(\xi ), q_{h}(\xi ))$ in the power of $\varepsilon$ , namely,

(2.9) \begin{align} \begin{aligned} u_{h}(\xi )&=u^{(0)}+ \varepsilon u_{1}(\xi )+ \textrm {h.o.t.}, \\[5pt] p_{h}(\xi )&= \varepsilon p_{1}(\xi )+ \textrm {h.o.t.}, \\[5pt] v_{h}(\xi )&=v_{0}(\xi )+\textrm {h.o.t.},\\[5pt] q_{h}(\xi )&=q_{0}(\xi )+\textrm {h.o.t.}, \end{aligned} \end{align}

where the initial conditions are $u_{h}(0)=u^{(0)}$ and $u_{j}(0)=0$ for all $j\geq 1$ . Substituting (2.9) into (2.2) gives

\begin{equation*} u_{1}(\xi )\equiv 0,\,\,\, p_{1}(\xi )=\int _{0}^{\xi }F\big (u^{(0)},v_0(\omega )\big )\,\textrm {d}\omega + p_{1}(0). \end{equation*}

Note that the integral term in $p_{1}(\xi )$ is an odd function since $v_{ 0}(\xi )$ is an even function; thus, (2.7) becomes

(2.10) \begin{equation} \begin{aligned} \int _{I_{f}}H_{\xi }\,\textrm {d}\xi =&\int ^{+\infty }_{-\infty } \big (G(u^{(0)},v_{0}(\xi ),\varepsilon )-G(u^{(0)},v_{0}(\xi ),0)\big )q_{0}(\xi )\,\textrm {d}\xi \\[5pt] &\quad -\varepsilon \int ^{+\infty }_{-\infty }(p^{(0)}+ \varepsilon p_{1}(\xi ))\int ^{v_{0}(\xi )}_0 G_u( u^{(0)},s,0)\,\textrm { d}s \textrm {d}\xi + \textrm {h.o.t.}\\[5pt] =&-2\varepsilon (p^{(0)}+\varepsilon p_1(0))\int ^{+\infty }_{0} \int ^{v_{0}(\xi )}_0 G_u( u^{(0)},s,0)\,\textrm {d}s \textrm { d}\xi +\textrm {h.o.t.} \end{aligned} \end{equation}

Since $u_0$ is positive, the integral in (2.10) is also positive by the assumption (A1). Therefore, we conclude that the condition for vanishing Melnikov integral up to $\mathcal{O}(\varepsilon ^2)$ is

\begin{equation*}p^{(0)}=0,\,\,\,\qquad \,\,\, p_1(0)=0.\end{equation*}

Remark 4. For any orbit

\begin{equation*} \gamma _{h}(\xi )=(u_{h}(\xi ), p_{h}(\xi ), v_{h}(\xi ), q_{h}(\xi )) \subset \mathcal{W}^{s}(\mathcal{M}_\varepsilon )\cap \mathcal{W}^{u}(\mathcal{M}_\varepsilon ),\end{equation*}

it follows from the reversibility symmetry that

(2.11) \begin{align} u_{h}(\xi )=u_{h}(\!-\xi ),\quad \quad v_{h}(\xi )=v_{h}(\!-\xi ). \end{align}

Therefore, $\xi = 0$ must be an extremum (specifically, a local maximum) of both $u_{h}(\xi )$ and $v_{h}(\xi )$ . By Fermat’s Lemma, this yields $p_{h}(0)=0$ and $q_{h}(0)=0$ .

Define the Take-off curve $T_{o}$ and the Touch-down curve $T_{d}$ on $\mathcal{M_{\varepsilon }}$ by

(2.12) \begin{equation} T_{o/d}=\left \{(u,p,v,q)\left |\,u=u_h\left (\mp \frac {1}{\sqrt {\varepsilon }}\right ),\,p=p_h\left (\mp \frac {1}{\sqrt {\varepsilon }}\right ),\right .v=q=0\right \} \end{equation}

with $p=\frac {1}{\varepsilon }\frac {d\,u}{d\,\xi }$ . By (2.2), we have

(2.13) \begin{equation} \begin{aligned} \Delta u=\int _{I_{f}}u_{\xi }\,\text{d}\xi =&\int _{I_{f}}\varepsilon ^2p_{1}(\xi )\,\text{d}\xi =\mathcal{O}(\varepsilon ^{3/2}),\\[5pt] \Delta p= \int _{I_{f}}p_{\xi }\,\text{d}\xi =&\int _{I_{f}}\varepsilon F(u,v(\xi )) + \mathcal{O}(\varepsilon ^{3})\,\text{d}\xi \\[5pt] =&\,2\varepsilon \int ^{0}_{-\infty } F(u,v_0(\xi )) \,\text{ d}\xi +\mathcal{O}(\varepsilon ^{2}), \end{aligned} \end{equation}

where we have used the facts that $u_{\xi }=\varepsilon p$ and $p=\mathcal{O}(\varepsilon )$ during $I_f$ in the first equation, and $v_0(\xi )$ is an even function with respect to $\xi$ in the second equation. So, to leading order, the explicit representations of the Take-off curve $T_{o}$ and the Touch-down curve $T_{d}$ on $\mathcal{M_{\varepsilon }}$ are, respectively

(2.14) \begin{equation} T_{o,d}(u)=\big \{(u,p,0,0)\in \mathcal{M_{\varepsilon }}\,|\,p= \mathcal{T}_{\mp }(u)+\mathcal{O}(\varepsilon ^{2}),\,u\gt 0\,\big \}, \end{equation}

where $\mathcal{T}_{\mp }(u)=\mp \varepsilon \int ^{0}_{-\infty } F(u,v_0(\xi ))\text{ d}\xi$ and $p=\frac {1}{\varepsilon }\frac {du}{d\xi }$ .

Remark 5. The homoclinic solution $(v_{0}(\xi;\;u), q_{0}(\xi;\;u))$ of system (2.3) provides the leading order solution of (2.2), given by

(2.15) \begin{equation} \phi _h(\xi )\;:\!=\;\left (u, \varepsilon \int ^{\xi }_{0} F(u,v_0(x;\;u))\,\textrm {d}x,v_{0}(\xi;\;u), q_{0}(\xi;\;u)\right ), \end{equation}

which is homoclinic to $\mathcal{M}_{\varepsilon }$ . Moreover, its asymptotic limits $\lim _{\xi \to \pm \infty } \phi _h(\xi )$ correspond precisely to the Touch-down and Take-off curves.

2.2. Slow dynamics on the slow manifold

A singular homoclinic orbit is constructed by concatenating trajectories of the fast and slow limiting systems. Though this singular homoclinic orbit is not an exact solution to system (2.2), GSPT ensures that a true homoclinic orbit exists within a small neighbourhood of this singular trajectory. In the previous subsection, the trajectories of the fast limiting system had been derived. In this section, we focus on the slow dynamics on the slow manifold and match them with the fast segments to complete the full orbit.

On the slow manifold

(2.16) \begin{equation} \mathcal{M}_{\varepsilon }=\{(u, p, v, q) | v=q=0,\,u\gt 0\}, \end{equation}

the dynamics of system (2.2) are governed by

(2.17) \begin{equation} u_{\xi \xi }=\varepsilon ^{4}\frac {\partial }{\partial u}W(u,\chi ). \end{equation}

In the super-slow coordinate $\chi =\varepsilon ^2 \xi$ , the slow limiting system can be transformed into

(2.18) \begin{equation} \left \{ {\begin{array}{l} u_{\chi }=p,\\[10pt] p_{\chi }=\dfrac {\partial }{\partial u}W(u,\chi ),\\[5pt] \end{array}} \right . \end{equation}

which is a non-autonomous Hamiltonian system with

(2.19) \begin{equation} H_s(u,p)=\frac {1}{2}p^2-W(u,\chi ). \end{equation}

It follows from (A3) that system (2.18) admits a saddle $(0,0)$ . The slow trajectories can be constructed via the phase plane analysis on two autonomous Hamiltonian systems,

(2.20) \begin{equation} \left \{ {\begin{array}{l} u_{\chi }=p,\\[5pt] p_{\chi }=W^{\prime }_i(u), \end{array}} \right .\qquad i=1,2, \end{equation}

where $W^{\prime }_i(u)$ are the potentials outside and inside the heterogeneous region, respectively. Appropriate boundary conditions must be imposed to ensure that the matching between the trajectories and their derivatives at the endpoints of each region can be performed. These trajectories decay exponentially to $(0,0)$ as $\chi \rightarrow \pm \infty$ . In this manner, the solutions to system (2.18) are $\textrm{C}^1$ -smooth.

In the following, we will demonstrate that the value of the Hamiltonian $h$ inside the heterogeneity with the length $2L$ serves as an important parameter to characterise the pinned pulses. A pinned pulse is generally determined by four points in the phase plane, namely:

• • The points where the pinned pulse enters and exits the heterogeneous region, respectively, denoted by $(u_{in}, p_{in})$ and $(u_{out}, p_{out})$ .

• • The points where the pinned pulse enters and exits the fast field, respectively, denoted by $(u_0, p_0)$ and $(u_d, p_d)$ .

Since system (2.2) is symmetric under $\chi \rightarrow -\chi$ , we have

\begin{align*} \begin{aligned} u_{in} &= u_{out}, \qquad \qquad p_{in} = -p_{out},\\[5pt] u_{0}\, &= u_{d},\qquad \qquad \,\,\, p_{0}\, = -p_{d}. \end{aligned} \end{align*}

Therefore, only $(u_{in}, p_{in})$ and $(u_0, p_0)$ need to be determined. More precisely, $(u_{in}, p_{in})$ is the intersection between the unstable manifold of the saddle $(0, 0)$ outside the heterogeneity (the purple solid curve in Figure 2) and the level set $H_s(u, p) = h$ within the heterogeneous region (the black curve). They satisfy

(2.21) \begin{equation} \begin{aligned} 0&=p_{in}^2-2W_1(u_{in}),\\[5pt] 2h&=p_{in}^2-2W_2(u_{in}), \end{aligned} \end{equation}

where

\begin{equation*} W^{\prime }_1(u_{in})-W^{\prime }_2(u_{in})\neq 0 \end{equation*}

is required to ensure that the intersection between such two slow trajectories is transverse, and $h$ represents the Hamiltonian value within the heterogeneity. By solving system (2.21), we can obtain $(u_{in}, p_{in}) = (u_{in}(h), p_{in}(h))$ , where $h$ acts as a free parameter.

Figure 2.

Singular homoclinic orbit of system (2.2) with $1$ fast and $4$ slow segments. Left panel: two slow segments outside the heterogeneity are marked by the solid purple curves, indicating the stable and unstable manifolds of the saddle $(0, 0)$ ; two slow segments inside the heterogeneity (the black curves), representing the orbits defined by $H_s(u,p)=h$ ; and a fast segment is indicated by the red dashing line. In the figure, the yellow dot represents $(u_{in}, p_{in})$ , and the green one stands for $(u_{out}, p_{out})$ . Right panel: the thick red curve indicates the whole singular homoclinic orbit.

Based on the curves $T_{o,d}$ on $\mathcal{M}_\varepsilon$ defined in (2.14), a homoclinic orbit $\phi _h$ of the limiting fast system (2.3) can be connected with the slow orbits on the slow manifold $\mathcal{M}_\varepsilon$ to construct a singular homoclinic orbit. To guarantee that this singular orbit persists under small perturbations, it is essential that the Take-off curve $T_o$ intersects transversely with the orbit of the 2nd system (2.20). The corresponding intersection point $(u_0, p_0)$ is governed by

(2.22) \begin{equation} \begin{aligned} p_0 & = \mathcal{T}_-(u_0),\\[5pt] 2h & = p_{0}^2 - 2W_2(u_{0}), \end{aligned} \end{equation}

where it is also required that

\begin{equation*} \mathcal{T}_-(u_0)\mathcal{T}_-^{\prime }(u_0) - W^{\prime }_2(u_{0})\neq 0 \end{equation*}

such that the matching is transverse.

If both equations (2.21) and (2.22) admit non-degenerate solutions $(u_{in}(h), p_{in}(h))$ and $(u_0(h), p_0(h))$ , this implies that

• • The outer and inner slow trajectories intersect transversely at the position of heterogeneity.

• • The slow trajectory intersects with the fast jump trajectory transversely.

However, the Hamiltonian value $h$ remains undetermined now. To determine it, we impose the condition that the ‘time’ for the slow trajectory to evolve from $(u_{in}(h), p_{in}(h))$ to $(u_0(h), p_0(h))$ is exactly $L$ (the half-width of the heterogeneous region). Once this condition is satisfied, we get a singular homoclinic orbit from the saddle to itself with transversality. By GSPT, there are true homoclinic orbits in a small neighbourhood of this singular configuration.

3.1. Case 1: $g(\chi )\equiv 0$ .

Through this simplest case, we can illustrate the key idea of the geometric approach clearly.

Under $g(\chi )\equiv 0$ , the matching conditions in equation (3.8) reduce to

(3.10) \begin{equation} \begin{aligned} 0&=p_{in}^2-\alpha _1 u_{in}^2,\\[5pt] 2h&=p_{in}^2-\alpha _2 u_{in}^2. \end{aligned} \end{equation}

Solving these two equations yields the explicit expressions for $u_{in}$ and $p_{in}$ in terms of $h$ , namely,

(3.11) \begin{equation} u_{in}(h)=\sqrt {\frac {2h}{\alpha _1-\alpha _2}},\qquad \quad p_{in}(h)=\sqrt {\frac {2h \alpha _1}{\alpha _1-\alpha _2}}, \end{equation}

where $\textrm {sign}(h)=\textrm {sign}(\alpha _1-\alpha _2)$ , which ensures that $u_{in}(h),\, p_{in}(h)\gt 0$ . By direct calculations, when $\sigma \gt 0$ and $\textrm {sign}(h)=\textrm {sign}(\alpha _1-\alpha _2)$ , (3.9) can be simplified as

(3.12) \begin{align} \begin{aligned} u_{0}(h)=u^\pm _{0}(h) =\frac {\left (\alpha _2\pm \sqrt {\alpha _2^2+72\sigma ^2h}\,\right )^\frac {1}{2}}{3\sqrt {2}\sigma }, \end{aligned} \end{align}

where we only consider $u_0(h)\in \mathbb{R}_+$ ( $u_{0}(h)=u^-_{0}(h)$ is valid only when $\alpha _1\lt \alpha _2$ , which is discarded). Thus, the condition $u_0(h)\gt u_{in}(h)$ is equivalent to

(3.13) \begin{equation} \frac {\left (\alpha _2\pm \sqrt {\alpha _2^2-72\sigma ^2h}\,\right )^\frac {1}{2}}{3\sqrt {2}\sigma } \gt \sqrt {\frac {2h}{\alpha _1-\alpha _2}}. \end{equation}

Within the heterogeneous region, the slow orbit satisfies

\begin{equation*}2h=p^2-\alpha _2 u^2\end{equation*}

with $p=u_{\chi }$ . Therefore, the relationship between the length $L$ and the Hamiltonian $h$ is

(3.14) \begin{equation} \begin{aligned} L&=\int ^0_{-L} \textrm{d}x=\int ^{u_0(h)}_{u_{in(h)}}\frac {\textrm{d}u}{p} =\int ^{u_0(h)}_{u_{in(h)}}\frac {\textrm{d}u}{\sqrt {2h+\alpha _2 u^2}}=I(u_0(h))-I(u_{in}(h)), \end{aligned} \end{equation}

where

(3.15) \begin{equation} I(u)= \begin{cases} \dfrac {1}{\sqrt {\alpha _2}} \log \, (2 \sqrt {\alpha _2 (2h+\alpha _2 u^2)}+2\alpha _2 u ), \quad \quad \quad & [\alpha _2 u\gt \sqrt {-\Delta },\, \quad \Delta \lt 0],\\[10pt] \dfrac {1}{\sqrt {\alpha _2}} \log \, ( \sqrt {\alpha _2 (2h+\alpha _2 u^2)}-2\alpha _2 u ), \quad & [2\alpha _2 u\lt -\sqrt {-\Delta }, \quad \Delta \lt 0],\\[10pt] \dfrac {1}{\sqrt {\alpha _2}} \operatorname {arcsinh}\left (\dfrac { \alpha _2 u}{\sqrt {\Delta }}\right ), \quad &[ \Delta \gt 0], \\[10pt] \dfrac {1}{\sqrt {\alpha _2}} \log \, (2\alpha _2 u), \quad &[\Delta =0], \end{cases} \end{equation}

and $\Delta =\alpha _2h$ . By combining (3.11), (3.12) and (3.14), the value of $h$ can be implicitly determined. This determines the exact slow orbit used in the construction of the pinned pulse solution.

The unstable and stable manifolds of the saddle $(0,0)$ are given by

(3.16) \begin{equation} u_{h,\mp }(\chi )=e^{\pm \sqrt {\alpha _1}\,(\chi \mp x_*)}, \end{equation}

where $x_*\gt 0$ is introduced such that $u_{h,-}(\!-L )=u_{in}$ . By the reversible symmetry, it follows that $u_{h,+}(L )=u_{out}$ . Within the heterogeneous region ( $|x|\lt L$ ), the expression of the orbit is given explicitly by

(3.17) \begin{equation} u_{i,\mp }(\chi )=A_{1,2} e^{-\sqrt {\alpha _2}\,\chi }+B_{1,2}e^{\sqrt {\alpha _2}\,\chi }, \end{equation}

where the constants $A_{1,2}$ and $B_{1,2}$ are determined to ensure that $u_{p,0}(\chi )$ is continuously differentiable at $\chi =\pm L$ . Thus, the composite solution can be defined by

(3.18) \begin{equation} u_{p,0}(\chi ) = \begin{cases} u_{h,-}(\chi ) \,,\,\,\,\,\,\,\,\,\,\,\, \quad &\chi \, \in \,(\!-\infty ,-L), \\[5pt] u_{i,-}(\chi )\,, &\chi \, \in (\!-L,0),\quad \quad \\[5pt] u_{i,+}(\chi )\,, \quad &\chi \in \,(\,0,\,L\,),\quad \,\quad \,\\[5pt] u_{h,+}(\chi )\,,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, &\chi \,\in \,(L,+\infty ). \end{cases} \end{equation}

Hence, the smooth matching condition at $\chi =\pm L$ is given by

(3.19) \begin{equation} \begin{pmatrix} e^{\sqrt {\alpha _2}\,L} & e^{-\sqrt {\alpha _2}\,L} & 0 & 0\\[5pt] -\sqrt {\alpha _2} e^{\sqrt {\alpha _2}\,L} &\sqrt {\alpha _2} e^{-\sqrt {\alpha _2}\,L} &0 &0\\[5pt] 0 & 0 & e^{-\sqrt {\alpha _2}\,L} & e^{\sqrt {\alpha _2}\,L}\\[5pt] 0 &0 &-\sqrt {\alpha _2} e^{-\sqrt {\alpha _2}\,L} &\sqrt {\alpha _2} e^{\sqrt {\alpha _2}\,L}\\[5pt] \end{pmatrix} \begin{pmatrix} A_{1}\\[5pt] B_{1}\\[5pt] A_{2}\\[5pt] B_{2} \end{pmatrix} =\begin{pmatrix} u_{in}\\[5pt] p_{in}\\[5pt] u_{out}\\[5pt] p_{out} \end{pmatrix}. \end{equation}

Solving (3.19) directly yields

(3.20) \begin{equation} \begin{aligned} A_{1}&=B_{2}=\frac {(\sqrt {\alpha _2} \,u_{in}-p_{in})e^{-\sqrt {\alpha _2}L}}{2\sqrt {\alpha _2}} =\frac {\alpha _2-\sqrt {\alpha _1\alpha _2}}{2\alpha _2}\sqrt {\frac {2h }{\alpha _1-\alpha _2}}e^{-\sqrt {\alpha _2}L}, \\[5pt] B_{1}&=A_{2}=\frac {(\sqrt {\alpha _2} u_{in}+p_{in})e^{\sqrt {\alpha _2}L}}{2\sqrt {\alpha _2}} =\frac {\alpha _2+\sqrt {\alpha _1\alpha _2}}{2\alpha _2}\sqrt {\frac {2h }{\alpha _1-\alpha _2}}\,e^{\sqrt {\alpha _2}L}. \end{aligned} \end{equation}

Theorem 3. Let $\varepsilon \gt 0$ be sufficiently small, and $\alpha _1$ , $\alpha _2,\,\sigma ,\,L\gt 0$ be given. Assume that (3.13) admits a non-degenerate root $h$ satisfying $\textrm {sign}(h)=\textrm { sign}(\alpha _1-\alpha _2)$ and that (3.14) holds, then system (3.1) with $g(\chi )\equiv 0$ admits a pinned pulse solution $(u_{h}(\xi ), v_{h}(\xi ))$ with

(3.21) \begin{equation} |v_{h}(\xi )-v_{0}(\xi;\;u_0(h_0))|,\,\,|u_{h}(\xi )-u_0(h_0)|=\mathcal{O}(\varepsilon ), \end{equation}

for $\xi \in I_f$ , where $v_{0}(\xi;\;u_0(h))$ and $u_0(h)$ are given by (3.4) and (3.12), respectively. For $\xi \in I_s^\pm$ , we have

(3.22) \begin{equation} |v_{h}(\xi )|,\,\,|u_{h}(\chi )-u_{p,0}(\chi )|=\mathcal{O}(\varepsilon ), \end{equation}

where $u_{p,0}(\chi )$ is defined in (3.18).

3.2. Case 2: $g(\chi )\neq 0$ .

In the previous subsection, we study the $g(\chi )\equiv 0$ case; that is, the slow limiting system is piecewise linear. When $g(\chi ) \neq 0$ , the system becomes piecewise nonlinear, leading to significantly increased complexity.

Outside the heterogeneous region ( $|\chi | \gt L$ ), equation (3.6) turns out to be

(3.23) \begin{equation} u_{\chi \chi }=\alpha _1 u -\gamma _1 u^d. \end{equation}

When $\alpha _1, \gamma _1 \gt 0$ , this equation admits a homoclinic orbit given by

(3.24) \begin{equation} u_{h,0}(\chi )=\left [\frac {\alpha _1(d+1)}{2\gamma _1}\textrm {sech}^2\left ( \frac {d-1}{2}\sqrt {\alpha _1} \chi \right )\right ]^{\frac {1}{d-1}}. \end{equation}

Therefore, the unstable and stable manifolds of the saddle $(0, 0)$ outside the heterogeneous region can be, respectively, parameterised by

(3.25) \begin{equation} \begin{aligned} u_{h,-}(\chi )&=\left [\frac {\alpha _1(d+1)}{2\gamma _1}\textrm {sech}^2\left ( \frac {d-1}{2}\sqrt {\alpha _1} (\chi -\textrm {sign}(p_{in})x_{*} ) \right )\right ]^{\frac {1}{d-1}},\quad \quad \chi \lt -L\\[5pt] u_{h,+}(\chi )&=\left [\frac {\alpha _1(d+1)}{2\gamma _1}\textrm {sech}^2\left ( \frac {d-1}{2}\sqrt {\alpha _1} (\chi +\textrm {sign}(p_{in})x_{*} )\right )\right ]^{\frac {1}{d-1}},\quad \quad \,\,\,\chi \gt \,L, \end{aligned} \end{equation}

where $x_{*} \gt 0$ is chosen such that $u_{h,\pm }(\chi =\pm L)=u_{in}$ .

We denote the segments of the pinned pulse within the heterogeneous region ( $|\chi |\lt L$ ) by $u_{i,\pm }(\chi ,h)$ , which are the solutions to

(3.26) \begin{equation} \left \{ {\begin{array}{c}\begin{aligned} u_{\chi }&=p,\\[5pt] p_{\chi }&=\alpha _2 u -\gamma _2 u^d,\\[5pt] (u,p)\,(\,0\,)&=(u_0(h),p_0(h)), \\[5pt] (u,p)(\pm L)&=(u_{in}(h),\mp p_{in}(h)). \end{aligned} \end{array}} \right . \end{equation}

If $u_{i,\pm }(\chi ,h)$ are the homoclinic orbits of the 2nd system, then their expressions are given by

(3.27) \begin{equation} \begin{aligned} u_{i,-}(\chi ,0)&=\left [\frac {\alpha _2(d+1)}{2\gamma _2} \textrm {sech}^2\left ( \frac {d-1}{2}\sqrt {\alpha _2}( \chi -\textrm {sign}(\sigma )y_{*} ) \right )\right ]^{\frac {1}{d-1}},\quad \quad -L\lt \chi \lt 0,\\[5pt] u_{i,+}(\chi ,0)&=\left [\frac {\alpha _2(d+1)}{2\gamma _2} \textrm {sech}^2\left ( \frac {d-1}{2}\sqrt {\alpha _2} ( \chi +\textrm {sign}(\sigma )y_{*} ) \right )\right ]^{\frac {1}{d-1}},\quad \quad \,\,\,\,0\lt \chi \lt \,L, \end{aligned} \end{equation}

where the constant $y_{*}\gt 0$ is determined by the matching condition

(3.28) \begin{equation} u_{h,-}(0)=\left [\frac {\alpha _2(d+1)}{2\gamma _2} \textrm {sech}^2\left ( \frac {d-1}{2}\sqrt {\alpha _2}\cdot \textrm {sign}(\sigma )y_{*} \right )\right ]^{\frac {1}{d-1}}=u_{0}. \end{equation}

As a result, equation (3.8) reduces to the algebraic relation

(3.29) \begin{equation} \begin{aligned} 1 - \frac {2(\gamma _1 - \gamma _2)}{(d+1)(\alpha _1 - \alpha _2)} u_{in}^{d-1} = 0, \end{aligned} \end{equation}

which admits a unique positive solution

(3.30) \begin{equation} u_{in} (h=0)= \left [\frac {(d+1)(\alpha _1 - \alpha _2)}{2(\gamma _1 - \gamma _2)}\right ]^{\frac {1}{d-1}} \end{equation}

provided $\textrm {sign}(\gamma _1 - \gamma _2) = \textrm {sign}(\alpha _1 - \alpha _2)$ .

Theorem 4. Let $\varepsilon \gt 0$ be sufficiently small, and let the parameters $ \alpha _1, \alpha _2 \gt 0$ , $ \gamma _1, \gamma _2 \gt 0$ , $ d \gt 1$ , $ \sigma \gt 0$ and $ L \gt 0$ be given. Suppose that equations (3.8) and (3.9) admit non-degenerate solutions $ (u_{in}(h), p_{in}(h))$ and $ (u_0(h), p_0(h))$ , respectively, with $ u_0 \gt u_{in}$ , then the following statements hold:

• • When $ p_{in} \gt 0$ .

  1. – If

     (3.31) \begin{equation} \int _{u_{in}}^{u_0} \frac {\,\textrm {d}u}{\sqrt {2h + \alpha _2 u^2 - \frac {2\gamma _2}{d+1}u^{d+1}}} = L \end{equation}
     admits a non-degenerate root $ h \gt 0$ , then system (3.1) admits a pinned pulse solution shown in Figure 3a , b .

  2. – If

     (3.32) \begin{equation} \int _{u_{in}}^{u_0} \frac {\,\textrm {d}u}{\sqrt {2h + \alpha _2 u^2 - \frac {2\gamma _2}{d+1}u^{d+1}}}= L\pmod { T} \end{equation}
     admits a non-degenerate root $ h \lt 0$ , where
     \begin{equation*} T \;:\!=\; \int _{u_{l}}^{u_{r}} \frac {1}{\sqrt {\alpha _2 u^2 - \frac {2\gamma _2}{d+1}u^{d+1} + 2h}} \,\textrm {d}u, \end{equation*}
     in which $ u_{l}$ and $ u_{r}$ are the intersections of the periodic orbit of the 2nd system with the $ u$ -axis, then system (3.1) admits a pinned $\left ( 2\left \lfloor \frac {L}{T} \right \rfloor + 1 \right )$ -hump pulse solution shown in Figure 3c .

• • When $ p_{in} \lt 0$ .

  1. – If

     (3.33) \begin{equation} \int _{u_{l}}^{u_{in}} \frac {\,\textrm {d}u}{\sqrt {2h + \alpha _2 u^2 - \frac {2\gamma _2}{d+1}u^{d+1}}} + \int _{u_{l}}^{u_0} \frac {\,\textrm {d}u}{\sqrt {2h + \alpha _2 u^2 - \frac {2\gamma _2}{d+1}u^{d+1}}}= L\pmod { T}, \end{equation}
     admits a non-degenerate root $ h \lt 0$ , then system (3.1) admits a pinned $\left ( \left \lfloor \frac {L}{2T} \right \rfloor + 3 \right )$ -hump pulse solution shown in Figure 3d .

Furthermore, to leading order, the pinned pulse $ \big (u_{h}(\xi ), v_{h}(\xi )\big )$ is given by

(3.34) \begin{equation} u_{p,0}(\xi )= \left \{ \begin{aligned} &u_{0,-}(\varepsilon ^2\xi ),\quad \quad \quad &\xi \in I_{s}^{-},\\[5pt] & u_0 , &\xi \in I_{f} ,\\[5pt] &u_{0,+}(\varepsilon ^2\xi ),\quad \quad \quad &\xi \in I_{s}^{+}, \end{aligned}\right . \quad \quad \quad v_{h,0}( \xi )= \left \{ \begin{aligned} &0,\quad \quad \quad &\xi \in I_{s}^{-},\\[5pt] & v_0( \xi ) , &\xi \in I_{f},\\[5pt] &0,\quad \quad \quad &\xi \in I_{s}^{+}, \end{aligned}\right . \end{equation}

where $v_0( \xi )$ is defined by (3.4), $u_0$ is solved from (3.9) and

(3.35) \begin{equation} u_{0,-}(\chi )= \left \{ \begin{aligned} &u_{h,-}(\chi ),\quad \quad \quad &\chi \lt -L,\\[5pt] &u_{i,-}(\chi ),\quad \quad \quad &-L\lt \chi \lt 0, \end{aligned}\right . \quad \quad \quad u_{0,+}(\chi )= \left \{ \begin{aligned} &u_{h,+}(\chi ),\quad \quad \quad &\chi \gt L,\\[5pt] &u_{i,+}(\chi ),\quad \quad \quad &0\lt \chi \lt L, \end{aligned}\right . \end{equation}

where $u_{h,\pm }(\chi )$ and $u_{i,\pm }(\chi ,h)$ are defined in (3.25) and (3.26), respectively.

Remark 10. By comparing with the autonomous case, that is, $f(\chi )\equiv \alpha _1$ and $g(\chi )\equiv \beta _1$ in equation (3.1), which had been studied in Veerman and Doelman [Reference Veerman and Doelman37], where

(3.36) \begin{equation} \begin{aligned} u_{h,-}(\chi )&=\left [\frac {\alpha _1(d+1)}{2\gamma _1}\textrm {sech}^2\left ( \frac {d-1}{2}\sqrt {\alpha _1} (\chi -\textrm {sign}(\sigma )x_{*} ) \right )\right ]^{\frac {1}{d-1}},\quad \quad \chi \lt \,0,\\[5pt] u_{h,+}(\chi )&=\left [\frac {\alpha _1(d+1)}{2\gamma _1}\textrm {sech}^2\left ( \frac {d-1}{2}\sqrt {\alpha _1} (\chi +\textrm {sign}(\sigma )x_{*} )\right )\right ]^{\frac {1}{d-1}},\quad \quad \chi \gt \,0, \end{aligned} \end{equation}

satisfy $u_{h,\pm }(0)=u_0$ , the existence results in [Reference Veerman and Doelman37, Theorem 2.1] can be reproduced if we set $L=0$ and $h=0$ in Theorem 4 and Remark 9. This comparison demonstrates the validity of the results in this paper.

Figure 3.

(a)–(d) Sketched plots on the different types of pinned pulses $u_{p,0}(\chi )$ in the $(u, p)$ plane, where $\sigma \gt 0$ and $u_0 \gt u_{in}$ . (a) The homoclinic orbit of the 1st system connects to the open orbit of the 2nd system defined by the energy level $H=h$ ( $h\gt 0$ ). (b) The homoclinic orbits of 1st system connects to the homoclinic orbit of the 2nd system defined by $H=h$ ( $h=0$ ). (c) The homoclinic orbits of 1st system connects to the periodic orbit of the 2nd system defined by $H=h$ ( $h\lt 0$ ) with $p_{in}\gt 0$ . (d) The homoclinic orbits of 1st system connects to the periodic orbit the 2nd system defined by $H=h$ ( $h\lt 0$ ) with $p_{in}\lt 0$ . (e) $-$ (g) Singular pulse orbits $u_{p,0}(\chi )$ in the $(\chi , u)$ plane, in which (f) $\lfloor \frac {L}{T} \rfloor =1$ , (g) $\lfloor \frac {L}{2T} \rfloor =1$ .

Figure 3 Long description

Panel A: A phase plane diagram shows a homoclinic orbit of the first system connecting to an open orbit of the second system defined by the energy level H=h with h greater than 0. The axes are labeled u and p. Panel B: A phase plane diagram shows homoclinic orbits of the first system connecting to a homoclinic orbit of the second system defined by H=h with h equal to 0. The axes are labeled u and p. Panel C: A phase plane diagram shows homoclinic orbits of the first system connecting to a periodic orbit of the second system defined by H=h with h less than 0 and pin greater than 0. The axes are labeled u and p. Panel D: A phase plane diagram shows homoclinic orbits of the first system connecting to a periodic orbit of the second system defined by H=h with h less than 0 and pin less than 0. The axes are labeled u and p. Panel E: A graph shows a singular pulse orbit in the (x, u) plane for h greater than or equal to 0. The axes are labeled x and u. Panel F: A graph shows a singular pulse orbit in the (x, u) plane for h less than 0 and pin greater than 0. The axes are labeled x and u. Panel G: A graph shows a singular pulse orbit in the (x, u) plane for h less than 0 and pin less than 0. The axes are labeled x and u.

Figure 4.

(a)–(d) Sketched plots on the pinned singular homoclinic orbits $u_{p,0}(\chi )$ in the $(u,p)$ plane with $\sigma \lt 0$ and $u_0 \gt u_{in}$ . (e)–(g) Sketched plots on the pinned singular homoclinic orbits in the $(\chi ,u)$ plane.

4.1. The Evans function

Define the complement of the essential spectrum in the complex plane $\mathbb{C}$ by

\begin{equation*} \mathcal{C}_e\;:\!=\;\mathbb{C}\backslash \Sigma _{\textrm {ess}}(\mathcal{L} ). \end{equation*}

Since $|\Lambda _{2,3}(\lambda )|\ll |\Lambda _{1,4}(\lambda )|$ , the solutions to the eigenvalue problem (4.3) also exhibit a slow-fast nature.

Lemma 1. For any $\lambda \in \mathcal{C}_e$ , system (4.3) has four solutions $ \{\varphi _{\pm ,v}(\xi ,\lambda ),\, \varphi _{\pm ,u}(\xi ,\lambda )\}$ satisfying the asymptotic conditions as follows:

(4.10) \begin{equation} \begin{aligned} \lim _{\xi \rightarrow -\infty }\varphi _{+,v}(\xi ,\lambda )e^{- \Lambda _{1}(\lambda )\,\xi }=&Y_{1}(\lambda ),\\[5pt] \lim _{\xi \rightarrow -\infty }\varphi _{+,u}(\xi ,\lambda )e^{-\Lambda _{2}(\lambda )\,\xi } =& Y_{2}(\lambda ),\\[5pt] \lim _{\xi \rightarrow +\infty }\,\varphi _{-,v}(\xi ,\lambda )e^{-\Lambda _{ 4}(\lambda )\,\xi }=&Y_{4}(\lambda ),\\[5pt] \lim _{\xi \rightarrow +\infty }\,\,\varphi _{-,u}(\xi ,\lambda )e^{-\Lambda _{3}(\lambda )\,\xi } =& Y_{3}(\lambda ). \end{aligned} \end{equation}

Proof. Since $A (\xi ,\lambda ,\varepsilon )\rightarrow A_{\infty }(\lambda ,\varepsilon )$ as $\xi \rightarrow \pm \infty$ and

\begin{equation*} \int ^{\pm \infty }_0\,\|A (\xi ,\lambda ,\varepsilon )- A_{\infty }(\lambda ,\varepsilon )\|\,\textrm {d}\xi \lt \infty , \end{equation*}

the conclusion follows directly from [Reference Coppel9, Theorem 4.1].

According to [Reference Kapitula and Promislow20], $\lambda$ is an eigenvalue of $\mathcal{L}$ if and only if there exists a non-trivial function $\varphi \in \textrm{C}^1(\mathbb{R}, \mathbb{R}^4)$ solving system (4.4). By Lemma subsection 1, the solution $\varphi _{+,v/u}(\xi , \lambda )$ decays exponentially to $(0, 0, 0, 0)^T$ as $\xi \to -\infty$ , and $\varphi _{-,v/u}(\xi , \lambda )$ decays exponentially to $(0, 0, 0, 0)^T$ as $\xi \to +\infty$ . Thus, $\{\varphi _{+,u}, \, \varphi _{+,v}\}$ and $\{\varphi _{-,u}, \, \varphi _{-,v}\}$ span the unstable subspace $\Phi _+(\xi , \lambda )$ and the stable subspace $\Phi _-(\xi , \lambda )$ , respectively. The eigenfunctions of (4.4) correspond to the intersections of these subspaces. This geometric viewpoint provides the motivation for the construction of the Evans function $E(\lambda , \varepsilon )$ , which is analytic in $\lambda$ , and whose zeros correspond to the eigenvalues of $\mathcal{L}$ (counted with multiplicity). These observations lead to the following definition of the Evans function:

(4.11) \begin{align} E(\lambda ,\varepsilon )=\det [\varphi _{+,v}(\xi ,\lambda ),\varphi _{+,u}(\xi ,\lambda ),\varphi _{-,u}(\xi ,\lambda ), \varphi _{-,v}(\xi ,\lambda )]. \end{align}

Consequently, the problem of identifying the eigenvalues is reduced to the location the zeros of the analytic function $E(\lambda , \varepsilon )$ (see [Reference Kapitula and Promislow20, Lemma 9.3.4]). Since $\Lambda _{1,4}(\lambda ) = \mathcal{O}(1)$ and $\Lambda _{2,3}(\lambda ) = \mathcal{O}(\varepsilon ^2) \ll 1$ , the functions $\varphi _{+,v}(\xi , \lambda )$ and $\varphi _{-,v}(\xi , \lambda )$ are uniquely determined by their asymptotic behaviour as $\xi \to -\infty$ and $\xi \to \infty$ .

Since $v_{h}(\xi )$ vanishes at leading order in the slow fields, it follows that $A (\xi ,\lambda ,\varepsilon )$ approaches a slowly varying intermediate matrix, namely,

(4.12) \begin{equation} A_{s}(\xi ,\lambda ,\varepsilon )= \begin{pmatrix} 0\;\;\;\; & \varepsilon\;\;\;\; & 0\;\;\;\; & 0\\[9pt] \varepsilon ^3\left (\dfrac {\partial ^2 }{\partial u^2}W(u_{p,0}(\xi ),\varepsilon ^2\xi )+\lambda \right )\;\;\;\; & 0\;\;\;\; &0\;\;\;\; &0\\[9pt] 0\;\;\;\; & 0\;\;\;\; & 0\;\;\;\; & 1\\[5pt] 0\;\;\;\; &0\;\;\;\; &G_v(u_{p,0}(\xi ),0,0)+\lambda\;\;\;\; &0 \end{pmatrix}. \end{equation}

We observe in (4.12) that the dynamics of the slow and fast variables have been separated. Furthermore, the distance between $A_{s}(\xi , \lambda ,\varepsilon )$ and $A_{\infty }(\lambda ,\varepsilon )$ is exponentially small.

Lemma 2. Consider

(4.13) \begin{equation} \frac{\text{d}}{\text{d} \xi }\psi =A_{s}(\xi ,\lambda ,\varepsilon )\psi ,\quad \end{equation}

where $\psi =(u,p,v,q)^T$ and $A_{s}(\xi ,\lambda ,\varepsilon )$ is given in (4.12). Let $\lambda \in \Omega$ , and then for $\xi \lt -\frac {1}{\sqrt {\varepsilon }}$ , the stable and unstable subspaces of (4.13) are, respectively, spanned by $\psi ^-_{+,u,v}(\xi ,\lambda )$ and $\psi ^-_{-,u,v}(\xi ,\lambda )$ , which are given by

(4.14) \begin{equation} \psi ^-_{\pm ,u}(\xi ,\lambda )=\left (u_{\pm }(\xi ,\lambda ),\frac {1}{\varepsilon }\frac {du_{\pm }}{d \xi }(\xi ,\lambda ),0,0\right )^T,\quad \psi ^-_{\pm ,v}(\xi ,\lambda )=\left (0,0,v_{\pm }(\xi ,\lambda ),\frac {dv_{\pm }}{d \xi }(\xi ,\lambda )\right )^T, \end{equation}

enjoying the following properties:

(4.15) \begin{equation} \begin{aligned} \lim _{\xi \rightarrow -\infty } \psi ^-_{+,u}(\xi ,\lambda )e^{-\Lambda _{2}(\lambda )\,\xi } =& Y_{2}(\lambda ),\\[5pt] \lim _{\xi \rightarrow +\infty }\psi ^-_{-,u}(\xi ,\lambda )e^{-\Lambda _{3}(\lambda )\,\xi } =& Y_{3}(\lambda ),\\[5pt] \lim _{\xi \rightarrow -\infty } \psi ^-_{+,v}(\xi ,\lambda )e^{-\Lambda _{1}(\lambda )\,\xi } =& Y_{1}(\lambda ),\\[5pt] \lim _{\xi \rightarrow +\infty }\psi ^-_{-,v}(\xi ,\lambda )e^{-\Lambda _{4}(\lambda )\,\xi } =& Y_{4}(\lambda ). \end{aligned} \end{equation}

For $\xi \gt \frac {1}{\sqrt {\varepsilon }}$ , by the reversibility symmetry, the solution spaces of (4.13) now are spanned by $\psi ^+_{\pm ,u,v}(\xi ,\lambda )$ , where

(4.16) \begin{equation} \begin{aligned} \psi ^+_{\pm ,u}(\chi;\;\lambda )&=\left (u_{\mp }(\!-\xi;\;\lambda ),-\frac {1}{\varepsilon }\frac {du_{\mp }}{d \xi }(\!-\xi;\;\lambda ),0,0\right )^T,\\[5pt] \psi ^+_{\pm ,v}(\chi;\;\lambda )&=\left (0,0,v_{\mp }(\!-\xi;\;\lambda ),-\frac {dv_{\mp }}{d \xi }(\!-\xi;\;\lambda )\right )^T. \end{aligned} \end{equation}

Theorem 5. The eigenvalues are given by the zeros of the Evans function

(4.17) \begin{align} E(\lambda ,\varepsilon )&=4\,\varepsilon t_{s,+}(\lambda ,\varepsilon )t_{f,+}(\lambda ,\varepsilon )\sqrt {G_v(0,0,\varepsilon )+\lambda }\sqrt {W^{\prime \prime }_1(0)+\lambda }, \end{align}

where $t_{f,+}(\lambda ,\varepsilon )$ is an analytic transmission function and $t_{s,+}(\lambda ,\varepsilon )$ is meromorphic function in $\lambda$ . These functions are defined via the asymptotic behaviour

(4.18) \begin{equation} \begin{aligned} \lim _{\xi \rightarrow +\infty }\varphi _{+,v}(\xi ,\lambda )\,e^{- \Lambda _{1}(\lambda )\,\xi }\,=t_{f,+}(\lambda ,\varepsilon )Y_{1}(\lambda ),\\[5pt] \lim _{\xi \rightarrow +\infty }\varphi _{+,u}(\xi ,\lambda )\,e^{- \Lambda _{2}(\lambda )\,\xi }\,=t_{s,+}(\lambda ,\varepsilon )Y_{2}(\lambda ), \end{aligned} \end{equation}

where $\varphi _{+,v}(\xi ,\lambda )$ has been stated in Lemma subsection 1 and $\varphi _{+,u}(\xi ,\lambda )$ is the unique solution to (4.4), provided $t_{f,+}(\lambda ,\varepsilon )\neq 0$ , satisfying the boundary conditions

(4.19) \begin{equation} \begin{aligned} \lim _{x\rightarrow -\infty }\varphi _{+,u}(\xi ,\lambda )e^{-\Lambda _{2}(\lambda )\,\,\xi }&= Y_{2}(\lambda ),\\[5pt] \lim _{x\rightarrow +\infty }\varphi _{+,u}(\xi ,\lambda )e^{- \Lambda _{1}(\lambda )\,\xi }&=(0,0,0,0)^T. \end{aligned} \end{equation}

Moreover, there exists a meromorphic transmission function $t_{s,-}(\lambda ,\varepsilon )$ such that, to leading order,

(4.20) \begin{equation} \varphi _{+,u}(\xi ,\lambda )= \begin{cases} \psi ^-_{+,u}(\xi ,\lambda ), \quad \quad \quad &\xi \lt -\dfrac {1}{\sqrt {\varepsilon }},\\[10pt] t_{s,+}(\lambda ,\varepsilon )\psi ^-_{-,u}(\!-\xi ,\lambda ) +t_{s,-}(\lambda ,\varepsilon )\psi ^-_{+,u}(\!-\xi ,\lambda ), \quad &\xi \,\,\gt \,\,\dfrac {1}{\sqrt {\varepsilon }}. \end{cases} \end{equation}

Proof. It follows from (3.34) that there are two positive, $\mathcal{O}(1)$ constants $C_1$ and $C_2$ such that

\begin{equation*}\|A (\xi ,\lambda ,\varepsilon )- A_{s}(\xi ,\lambda ,\varepsilon )\|\leq C_1\textrm {e}^{-C_2|\xi |}\end{equation*}

for $\xi \in I_{s}^\pm$ . Hence, to leading order, $\psi _{\pm ,v}(\xi , \lambda )$ and $\psi _{\pm ,u}(\xi , \lambda )$ can be selected as basis solutions to (4.4). According to the asymptotics of $\varphi _{+,v}(\xi ,\lambda )$ as $\xi \rightarrow -\infty$ , we know that it remains exponentially close to $\psi _{+,v}(\xi ,\lambda )$ for $\xi \lt -\frac {1}{\sqrt {\varepsilon }}$ . Although the exact form of $\varphi _{+,v}(\xi ,\lambda )$ is unknown, to leading order, it can be written as

\begin{equation*} \varphi _{+,v}(\xi ,\lambda )=t_{f,+}(\lambda )\psi ^+_{+,v}(\xi )+t_{f,-}(\lambda )\psi ^+_{-,v}(\xi )+t_{+}(\lambda )\psi ^+_{+,u}(\xi )+t_{-}(\lambda )\psi ^+_{-,u}(\xi ) \end{equation*}

for $\xi \gt \frac {1}{\sqrt {\varepsilon }}$ . This proves the first result in (4.18). In order to determine $\varphi _{+,u}(\xi )$ uniquely, it can be found that $\varphi _{+,u}(\xi ,\lambda )$ grows obeying the speed of $\mathcal{O}(\textrm {e}^{\Lambda _{2}(\lambda )\,\,\xi })$ . That is, $\varphi _{+,u}(\xi ,\lambda )$ must satisfy (4.19) if $t_{f,+}(\lambda )\neq 0$ . Hence, similar to $\varphi _{+,v}(\xi ,\lambda )$ , we obtain (4.20) accordingly.

By Liouville’s formula and $\textrm {Tr}\textrm {A}(x)=\Sigma _{i=1}^4 \Lambda _{i}(\lambda )$ , the Evans function turns out to be

\begin{align*} E(\lambda ,\varepsilon )&=\lim _{\xi \rightarrow +\infty }\det [\varphi _{+,v}(\xi ,\lambda ),\varphi _{+,u}(\xi ,\lambda ),\varphi _{-,u}(\xi ,\lambda ), \varphi _{-,v}(\xi ,\lambda )]e^{-\int ^{x}_{0}Tr \textrm {A}(x)dx}\\[5pt] &=\lim _{\xi \rightarrow +\infty }\det \big[\varphi _{+,v}(\xi ,\lambda )e^{-\Lambda _{1}(\lambda )\xi },\varphi _{+,u}(\xi ,\lambda )e^{-\Lambda _{2}(\lambda )\xi }, \varphi _{-,u}(\xi ,\lambda )e^{-\Lambda _{3}(\lambda )\xi },\varphi _{-,v}(\xi ,\lambda )e^{-\Lambda _{4}(\lambda )\xi }\big]\\[5pt] &=\det [t_{f,+}(\lambda ,\varepsilon )Y_{1}(\lambda ),t_{s,+}(\lambda ,\varepsilon )Y_{2}(\lambda ),Y_{3}(\lambda ),Y_{4}(\lambda )]\\[5pt] &=4\varepsilon t_{s,+}(\lambda ,\varepsilon )t_{f,+}(\lambda ,\varepsilon ) \sqrt {G_v(0,0,\varepsilon )+\lambda }\sqrt {W^{\prime \prime }_1(0)+\lambda }. \end{align*}

As a consequence, the eigenvalues of (4.3) are exactly identical with the roots of $t_{s,+}(\lambda ,\varepsilon )$ or $t_{f,+}(\lambda ,\varepsilon )$ . Nevertheless, we will also see that some roots of $t_{f,+}(\lambda ,\varepsilon )$ may not correspond to the true eigenvalues, as $t_{s,+}(\lambda ,\varepsilon )$ may possess poles at these values, which cancel the zeros.

4.2. The fast transmission function $t_{f,+}(\lambda ,\varepsilon )$

In this section, we determine the zeros of the fast transmission function $t_{f,+}(\lambda ,\varepsilon ).$

Let $\varepsilon \rightarrow 0$ in system (4.3), we get

(4.21) \begin{equation} (\mathcal{L}_f-\lambda ) v=G_u(u_{0},v_{0}(\xi ),0)\,\bar {u}(0),\quad \quad \mathcal{L}_fv:v_{\xi \xi }-G_v(u_{0},v_{0}(\xi ),0)\,v, \end{equation}

which is a second-order inhomogeneous differential equation. Here, we have used the fact that $u_h(\xi )=u_0$ and $v_h(\xi )=v_{0}(\xi )$ for $\xi \in I_f$ as $\varepsilon \rightarrow 0$ . Equation (4.21) corresponds to the fast limiting eigenvalue problem in the singular limit of system (4.3), which is a classical Sturm–Liouville eigenvalue problem. Thus, by Sturm–Liouville theory, the eigenvalues of $\mathcal{L}_f$ can be enumerated in a strictly descending order,

\begin{equation*}\lambda ^f_{0}\gt \lambda ^f_{1}=0\gt \cdots \gt \lambda ^f_{N}\gt -G_v(u_{0},0,0).\end{equation*}

Moreover, the eigenfunction $\bar {v}_i(\xi )$ corresponding to $\lambda ^f_{i}$ has exactly $i$ distinct zeroes and is even or odd depending on whether $i$ is even or odd, respectively.

The homogeneous problem associated with system (4.21) can be written as

(4.22) \begin{equation} \frac{\text{d}}{\text{d} \xi }\phi =A_{f}(\xi ,\lambda )\phi ,\quad \end{equation}

where $\phi =(v,q)^T$ and $A_{f}(\xi , \lambda )$ is a $2\times 2$ matrix, which is exactly the lower $2\times 2$ block of $A(\xi , \lambda ,\varepsilon )$ in the limit $\varepsilon \rightarrow 0.$ Obviously, system (4.22) has two solutions $\phi _{+}(\xi , \lambda )$ , $\phi _{-}(\xi , \lambda )$ satisfying

(4.23) \begin{equation} \begin{aligned} \lim _{\xi \rightarrow -\infty }\phi _{+}(\xi , \lambda )e^{- \Lambda _{1}(\lambda )\,\xi }=(1,\Lambda _{1}(\lambda ))^T, \quad \quad \quad \lim _{\xi \rightarrow +\infty }\,\phi _{-}(\xi , \lambda )e^{-\Lambda _{ 4}(\lambda )\,\xi }=(1,\Lambda _{4}(\lambda ))^T, \end{aligned} \end{equation}

and there exists an analytic function $t_{f}(\lambda )$ such that

(4.24) \begin{equation} \lim _{\xi \rightarrow +\infty }\,\phi _{+}(\xi ,\lambda )e^{-\Lambda _{ 1}(\lambda )\,\xi }=t_{f}(\lambda )(1,\Lambda _{1}(\lambda ))^T. \end{equation}

The Evans function associated with this problem is given by

(4.25) \begin{align} \begin{aligned} E_f(\lambda )=\lim _{\xi \rightarrow +\infty }\det [\phi _{+}(\xi ,\lambda ),\phi _{-}(\xi ,\lambda )] =\det [t_{f}(\lambda )(1,\Lambda _{1}(\lambda ))^T,(1,\Lambda _{4}(\lambda ))^T]=-2t_{f}(\lambda )\Lambda _{1}(\lambda ). \end{aligned} \end{align}

By construction, the leading order behaviour of $t_{f,+}(\lambda ,\varepsilon )$ is determined by $t_{f}(\lambda )$ . It thus follows from (4.25) that $t_{f}(\lambda )=0$ if and only if $E_{f}(\lambda )=0$ . Hence, the eigenvalues of (4.22) correspond to the zeros of $t_{f}(\lambda )=0$ . That is, the roots of the function $t_{f}(\lambda )$ are given by $\lambda ^f_{i}$ , $i=0,1,\ldots ,N$ .

Lemma 3. Let $0\lt \varepsilon \ll 1$ , then the fast transmission function is given by

(4.26) \begin{equation} t_{f,+}(\lambda ,\varepsilon )=\tilde {t}_f(\lambda ,\varepsilon ) \prod _{i=1}^{n}\left (\lambda -\lambda ^f_{i}(\varepsilon )\right ) \end{equation}

with $\tilde {t}_f(\lambda ,\varepsilon )\neq 0$ , and $\lambda ^f_{i}(\varepsilon )$ has the following regular expansion:

(4.27) \begin{equation} \lambda ^f_{i}(\varepsilon )=\lambda ^f_{i}+\varepsilon ^2 \lambda ^f_{i,1}+\mathcal{O}(\varepsilon ^4), \end{equation}

for $i=0,1,\ldots ,N.$

The proof of this lemma is analogous to those in [Reference Doelman, Gardner and Kaper12–Reference Doelman, Iron and Nishiura14] by using the ‘elephant trunk’ procedure [Reference Alexander, Gardner and Jones1, Reference Gardner and Jones18]. It is important to note that $\lambda ^f_{1}(\varepsilon )\equiv 0$ is no longer valid since system (1.2) now is non-autonomous, leading to the loss of translational invariance. As a result, under the perturbation, the eigenvalue $\lambda ^f_{1}(\varepsilon )$ may be shifted to the one having the positive real part. Under certain situations, standard perturbation methods such as the Lin–Sandstede method [Reference Bastiaansen, Chirilus-Bruckner and Doelman3, Reference Kapitula and Promislow20, Reference Li and Yu23] can be applied to estimate the location of these eigenvalues. However, due to the piecewise-smooth nature of system (1.2), no effective methods currently exist for precisely determining the location of these eigenvalues.

Although $\lambda ^f_0(\varepsilon )$ is a simple zero of the function $t_{f,+}(\lambda , \varepsilon )$ , this does not imply that the function $E(\lambda , \varepsilon )$ also vanishes at this point. This is due to the fact that $\lambda = \lambda ^f_0(\varepsilon )$ is a first-order pole of the transmission function $t_{s,+}(\lambda )$ and therefore cannot be an eigenvalue of the full system. This phenomenon is known as the ‘NLEP Paradox’ in literature.

According to (4.23)–(4.24), we know that $\phi _{+}(\xi ,\lambda )$ and $\phi _{-}(\xi ,\lambda )$ form a fundamental set of linearly independent solutions to the homogeneous system (4.22) for $\lambda \neq \lambda ^f_{i}$ , $i=0,1,\ldots N$ . Consequently, by employing the method of constant variation we can derive a bounded solution to the heterogeneous problem (4.21).

Lemma 4. For $\lambda \neq \lambda ^f_{i}$ , $i=0,1,\ldots N$ , the unique bounded solution $v_{in}(\xi ,\lambda )$ to the inhomogeneous problem (4.21) is given by

(4.28) \begin{align} \begin{aligned} v_{in}(\xi ,\lambda )&= \frac {\bar {u}(0)}{E_f(\lambda )} \left (v_{-}(\xi , \lambda )\int ^{\xi }_{-\infty } G_u(u_{0},v_{0}(s),0)v_{+}(s, \lambda )\, \textrm {d}s\right .\\[5pt] & \quad + \left .v_{+}(\xi , \lambda )\int _{\xi }^{+\infty } G_u(u_{0},v_{0}(s),0)v_{-}(s, \lambda )\,\textrm {d}s\right ), \end{aligned} \end{align}

where $v_{\pm }(\xi , \lambda )$ is the $v$ -component of $\phi _{\pm }(\xi , \lambda )=(v_{\pm }(\xi , \lambda ),q_{\pm }(\xi , \lambda ))$ , and $E_f(\lambda )$ is defined as in (4.25).

Proof. Since $\mathcal{L}_f - \lambda$ is a Fredholm operator of index zero for $\lambda \in \mathcal{C}_e$ , and $\mathcal{L}_f - \lambda$ is invertible if and only if $\lambda \in \mathcal{C}_e\backslash \{\lambda ^f_{i}\}_{i=0}^N$ , it follows that $(\mathcal{L}_f - \lambda )^{-1}$ is analytic on $\mathcal{C}_e\backslash \{\lambda ^f_{i}\}_{i=0}^N$ and meromorphic on the entire region $\mathcal{C}_e$ . Since $\mathcal{L}_f - \lambda$ is a self-adjoint operator, it follows from the Fredholm alternative [Reference Kapitula and Promislow20, Theorem 2.2.1] that a bounded solution to the inhomogeneous equation exists at $\lambda \in \{\lambda ^f_{i}\}_{i=0}^N$ if and only if the following solvability condition is satisfied:

\begin{equation*} \int _{-\infty }^{+\infty } G_u(u_{0},v_{0}(\xi ),0)\bar {v}_{i}(\xi , \lambda ) \,\textrm {d}\xi =0, \end{equation*}

where $\bar {v}_{i}(\xi , \lambda )\in \ker (\mathcal{L}_f-\lambda )$ . According to Sturm–Liouville theory, if $i$ is even, the corresponding eigenfunction $\bar {v}_i(\xi )$ is an even function, and if $i$ is odd, it is an odd function. Since the homoclinic solution $v_0(\xi )$ is even, the solvability condition does not hold when $i$ is even. Therefore, $\lambda = \lambda ^f_{i}$ is a pole of $v_{in}(\xi , \lambda )$ . When $i$ is odd, the solvability condition holds, and $\lambda =\lambda ^f_{i}$ is a removable singularity.

4.3. The slow transmission function $t_{s,+}(\lambda ,\varepsilon )$

We will see that the slow transmission function $t_{s,+}(\lambda ,\varepsilon )$ can be determined by matching the slow and fast orbits, respectively, of the following slow limit eigenvalue problem:

(4.29) \begin{equation} \left \{ {\begin{array}{l} u_{\chi }=p,\\[5pt] p_{\chi }=\left (W(u_{p,0}(\chi ),\chi )+\lambda \right )u, \end{array}} \right . \end{equation}

and the fast limit eigenvalue problem (4.21).

Recall that, on the slow intervals $I_s^{\pm }$ ,

\begin{equation*} \psi ^+_{\pm ,u}(\xi ,\lambda ) = \left ( u_{\pm }(\xi ,\lambda ), \frac {1}{\varepsilon } \frac {du_{\pm }}{d \xi }(\xi ,\lambda ), 0, 0 \right ).\end{equation*}

To simplify the notation and to enhance the clarity of formulation, we utilise the variable $\chi =\varepsilon ^2 \xi$ and define $ (u_{\pm }(\chi , \lambda ), p_{\pm }(\chi , \lambda ))$ .

Denote

\begin{equation*} \psi ^+_{\pm ,u}(\chi ,\lambda ) = \left ( u_{\pm }(\chi , \lambda ), \varepsilon p_{\pm }(\chi , \lambda ), 0, 0 \right ) \end{equation*}

with $p_{\pm }(\chi , \lambda )=\frac{\text{d}}{\text{d}\chi }u_{\pm }(\chi , \lambda )$ ; thus, $\left (u_{\pm }(\chi , \lambda ), p_{\pm }(\chi , \lambda )\right )$ is the unique solution to the slow limit eigenvalue problem (4.29) satisfying

\begin{equation*} \lim _{\chi \rightarrow \mp \infty }( u_{\pm }(\chi , \lambda ), p_{\pm }(\chi , \lambda ))\textrm {e}^{\mp \Lambda _s(\lambda )\chi } =(1,\,\pm \Lambda _s(\lambda )). \end{equation*}

Lemma 6. Define $u_{\pm }(0, \lambda )=u_{\pm }(\chi =0)$ and $p_{\pm }(0, \lambda )=p_{\pm }(\chi =0)$ , then the leading order of the slow transmission function can be given by

(4.30) \begin{equation} \begin{aligned} t_{s,+}(\lambda ) =-\frac {\Big (u_+(0, \lambda ) \mathcal{G}(\lambda )+2p_+(0, \lambda )\Big )u_+(0, \lambda )}{u_+(0, \lambda ) p_-(0, \lambda )-u_-(0, \lambda )p_+(0, \lambda )}, \end{aligned} \end{equation}

where

(4.31) \begin{equation} \mathcal{G}(\lambda )=\int ^{+\infty }_{-\infty }\int ^{\xi }_{-\infty }\Big (F_u(u_{0},v_{0}(\xi )+\frac {F_v(u_{0},v_{0}(\xi ))}{E_f(\lambda )} G_u(u_{0},v_{0}(s),0)v_{+}(s, \lambda )v_{-}(\xi , \lambda )\Big )\,\textrm {d}s\,\textrm {d}\xi . \end{equation}

Proof. It follows from Theorem 5 that there exists a positive constant $K$ independent of $\varepsilon$ such that

\begin{equation*} \varphi _{+,u}(\xi , \lambda )=\psi ^-_{+,u}(\xi ,\lambda )+\mathcal{O}(e^{-K\xi }), \end{equation*}

for $\xi \in I_s^-$ , and

\begin{equation*} \varphi _{+,u}(\xi , \lambda )=t_{s,+}(\lambda , \varepsilon )\psi ^+_{+,u}(\xi , \lambda ) +t_{s,-}(\lambda , \varepsilon )\psi ^+_{-,u}(\xi , \lambda )+\mathcal{O}(e^{-K\xi }) \end{equation*}

for $\xi \in I_s^+$ . Accordingly, we can determine the explicit representation of $t_{s,+}(\lambda ,\varepsilon )$ by matching the fast and slow orbits. More precisely, to compute $t_{s,+}(\lambda , \varepsilon )$ , we need to track the changes of $\bar {u}_{+}(\xi , \lambda )$ and $\bar {p}_{+}(\xi , \lambda )$ during the fast transition, that is, $\xi \in I_f$ .

We define the slow difference function $\Delta _s$ by

(4.32) \begin{equation} \Delta _s\left (\begin{array}{l} \bar {u }\\[5pt] \bar {p} \\[5pt] \bar {v }\\[5pt] \bar {q} \end{array}\right )=\lim _{\xi \downarrow \frac {1}{\sqrt {\varepsilon }}}\Big (t_{s,+}(\lambda ,\varepsilon )\psi ^-_{-,u}(\!-\xi;\;\lambda ) +t_{s,-}(\lambda ,\varepsilon )\psi ^-_{+,u}(\!-\xi;\;\lambda )\Big ) -\lim _{\xi \uparrow -\frac {1}{\sqrt {\varepsilon }}} \psi ^-_{+,u}(\xi ,\lambda ). \end{equation}

First, it follows from (4.4) that $\bar {u}_+(\xi , \lambda )$ remains constant to leading order during the fast field $I_f$ . As a consequence,

(4.33) \begin{equation} \bar {u}_+\left (\!-\frac {1}{\sqrt {\varepsilon }}\right )= \bar {u}_+\left (\frac {1}{\sqrt {\varepsilon }}\right )+\mathcal{O}(\varepsilon ^{3/2}). \end{equation}

This together with

(4.34) \begin{equation} \bar {u}\left (\!-\frac {1}{\sqrt {\varepsilon }}\right )=u_+(0,\lambda )+\mathcal{O}(\varepsilon ^{3/2}) \end{equation}

gives

(4.35) \begin{equation} \bar {u}\left (\frac {1}{\sqrt {\varepsilon }}\right ) =t_{s,+}(\lambda )u_-(0,\lambda )+t_{s,-}(\lambda )u_+(0,\lambda )+\mathcal{O}(\varepsilon ^{3/2}). \end{equation}

Thus, the first matching condition (4.33) can be written as

(4.36) \begin{equation} u_+(0,\lambda ) =t_{s,+}(\lambda )u_-(0,\lambda )+ t_{s,-}(\lambda )u_+(0,\lambda ). \end{equation}

So the first relation between $t_{s,+}(\lambda )$ and $t_{s,-}(\lambda )$ can be given by

(4.37) \begin{equation} t_{s,-}(\lambda )=1-\frac {u_-(0,\lambda )}{u_+(0,\lambda )}t_{s,+}(\lambda ). \end{equation}

On the other hand, the derivative $\bar {p}_{+}(\xi , \lambda )$ changes rapidly in the fast field $I_f$ . It can be determined through the following two ways. First, we use the information on $\varphi _{+,u}(\xi , \lambda )$ in the slow field to compute it. The change $\Delta _{s}\bar {p}$ , to leading order, can be approximated by the difference of $\bar {p}_{+}(\xi , \lambda )$ between two ends of the slow field, that is,

(4.38) \begin{align} \begin{aligned} \Delta _{s}\bar {p}&= \bar {p}_{+}\left (\frac {1}{\sqrt {\varepsilon }}\right )- \bar {p}_{+}\left (\!-\frac {1}{\sqrt {\varepsilon }}\right )\\[5pt] &=\frac {1}{\varepsilon }\left (t_{s,+}(\lambda )\frac{\text{d}}{\text{d}\xi }u_-(\!-\xi )+ t_{s,-}(\lambda )\left .\frac{\text{d}}{\text{d}\xi }u_+(\!-\xi )\right )\right |_{\xi =\frac {1}{\sqrt {\varepsilon }}} -\frac {1}{\varepsilon } \left .\frac{\text{d}}{\text{d}\xi }u_+(\xi ) \right |_{\xi =-\frac {1}{\sqrt {\varepsilon }}}\\[5pt] &=-\frac {1}{\varepsilon }\left (t_{s,+}(\lambda )\frac{\text{d}}{\text{d}\xi }u_-(\xi )+ t_{s,-}(\lambda )\left .\frac{\text{d}}{\text{d}\xi }u_+(\xi ) +\frac{\text{d}}{\text{d}\xi }u_+(\xi )\right )\right |_{\xi =-\frac {1}{\sqrt {\varepsilon }}}\\[5pt] &=-\frac {1}{\varepsilon }\left (t_{s,+}(\lambda )\frac{\text{d}}{\text{d}\xi }u_-(\xi )+ \left (1-\frac {u_-(0)}{u_+(0)}t_{s,+}(\lambda )\right )\left .\frac{\text{d}}{\text{d}\xi }u_+(\xi ) +\frac{\text{d}}{\text{d}\xi }u_+(\xi )\right )\right |_{\xi =-\frac {1}{\sqrt {\varepsilon }}}\\[5pt] &=-\varepsilon \left [\left (\frac{\text{d}}{\text{d}\chi }u_-(\chi ) -\frac {u_-(0)\frac{\text{d}}{\text{d}\chi }u_+(\chi )}{u_+(0)}t_{s,+}(\lambda )\right )\left . +2\frac{\text{d}}{\text{d}\chi }u_+(\chi )\right ]\right |_{\chi =-\varepsilon ^{3/2}}\\[5pt] &=-\varepsilon \left ( \frac {u_+(0)p_-(0)-u_-(0)p_+(0)}{u_+(0)}t_{s,+}(\lambda ) +2p_+(0)\right ). \end{aligned} \end{align}

Second, the accumulated jump concerning the first-order derivative can be obtained by integrating $\overline {u}_{\xi \xi }$ over the fast field, that is, to leading order,

(4.39) \begin{equation} \begin{aligned} \Delta _{f}\,\bar {p}&=\frac {1}{\varepsilon }\int _{I_f}\bar {u}_{\xi \xi }\,\textrm {d}\xi \\[5pt] &=\varepsilon \int ^{+\infty }_{-\infty }\Big ( F_u(u_{0},v_{0}(\xi )\bar {u}(0)+F_v(u_{0},v_{0}(\xi ))v_{in}(\xi , \lambda )\Big )\,\textrm {d}\xi ,\\[5pt] &= \varepsilon u_+(0)\int ^{+\infty }_{-\infty }\int ^{\xi }_{-\infty }\Big (F_u(u_{0},v_{0}(\xi )+\frac {F_v(u_{0},v_{0}(\xi ))}{E_f(\lambda )} G_u(u_{0},v_{0}(s),0)v_{+}(s;\;\lambda )v_{-}(\xi , \lambda )\Big )\,\textrm {d}s\,\textrm {d}\xi . \end{aligned} \end{equation}

By combining (4.38) and (4.39), one can get (4.30) up to the leading order.

Remark 14. If $F_v(u,v)\equiv 0$ for all $u,\,v\gt 0$ , then

\begin{equation*} \mathcal{G}(\lambda )=\int ^{+\infty }_{-\infty }\int ^{\xi }_{-\infty } F_u(u_{0},v_{0}(\xi )) \,\textrm {d}s \,\textrm {d}\xi , \end{equation*}

which implies that the transmission function $t_{s,+}(\lambda ,\varepsilon )$ is only determined by the slow reduced eigenvalues problem (4.29). As a consequence, $t_{s,+}(\lambda ,\varepsilon )$ is analytic on $\mathcal{C}_e$ , and the zeros of $t_{f,+}(\lambda ,\varepsilon )$ cannot be cancelled by the poles of $t_{s,+}(\lambda ,\varepsilon )$ . In this case, we conclude that the linearised eigenvalue problem can be completely separated into the slow and fast reduced eigenvalue problems without the interaction between them. Thus, the zeros of $t_{f,+}(\lambda ,\varepsilon )$ with positive real parts yield the spectral instability directly.