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Volume 8, Issue 1

The elliptic sinh-Gordon equation in a semi-strip

Guenbo Hwang
Published Online: 2017-06-01 | DOI: https://doi.org/10.1515/anona-2016-0206

Abstract

We study the elliptic sinh-Gordon equation posed in a semi-strip by applying the so-called Fokas method, a generalization of the inverse scattering transform for boundary value problems. Based on the spectral analysis for the Lax pair formulation, we show that the spectral functions can be characterized from the boundary values. We express the solution of the equation in terms of the unique solution of the matrix Riemann–Hilbert problem whose jump matrices are defined by the spectral functions. Moreover, we derive the global algebraic relation that involves the boundary values. In addition, it can be verified that the solution of the elliptic sinh-Gordon equation posed in the semi-strip exists if the spectral functions defined by the boundary values satisfy this global relation.

MSC 2010: 47K15; 35Q55

1 Introduction

We study the boundary problem for the elliptic sinh-Gordon equation posed in a semi-strip,

${q}_{xx}+{q}_{yy}=\mathrm{sinh}q,\left(x,y\right)\in \mathrm{\Omega },$(1.1)

where $\mathrm{\Omega }=\left\{\left(x,y\right)\in {ℝ}^{2}:0. It is well known that the elliptic sinh-Gordon equation is completely integrable [1, 3] and hence equation (1.1) can be analyzed by the inverse scattering transform. Indeed, the classical inverse scattering transform was applied to solve the elliptic sinh-Gordon equation posed in the entire plane $\left\{-\mathrm{\infty } (see [3, 15]). For more complicated or general domains, the so-called Fokas method can be applied to solve boundary value problems [2, 4, 5, 6, 7] (see also the monograph [9] and references therein). It should be noted that the method can be considered as a significant extension of the inverse scattering transform for boundary value problems. Regarding the boundary value problem for the elliptic sinh-Gordon equation, the Fokas method has been applied to solve problem (1.1) posed in the half plane $\left\{-\mathrm{\infty } (see [13]) and the quarter plane $\left\{0 (see [14]). It has been shown in [13, 14] that the solution of the equation can be expressed in terms of the unique solution of the matrix Riemann–Hilbert problem defined by the spectral functions. It also has been shown that the solution of the equation exists if the spectral functions determined by the boundary values satisfy the so-called global relation.

In this paper, we extend the results presented in [13, 14] to the semi-strip (see also [11, 10, 16] for analogous results). The rigorous analysis of the Fokas method involves the following steps:

(i) Assuming that a smooth solution $q\left(x,y\right)$ exists, we express the solution $q\left(x,y\right)$ in terms of the unique solution of the matrix Riemann–Hilbert problem defined by the spectral functions $\left\{{a}_{j}\left(k\right),{b}_{j}\left(k\right)\right\}$, $j=1,2,3$. These spectral functions are defined by boundary values $\left\{q\left(x,0\right),{q}_{y}\left(x,0\right)\right\}$, $\left\{q\left(0,y\right),{q}_{x}\left(0,y\right)\right\}$ and $\left\{q\left(x,L\right),{q}_{y}\left(x,L\right)\right\}$, respectively. It should be remarked that the spectral functions satisfy the global algebraic relation that involves all boundary values:

${a}_{3}\left(k\right)={a}_{1}\left(k\right){a}_{2}\left(-k\right)+{b}_{1}\left(k\right){b}_{2}\left(-k\right),$$\mathrm{ }k\in {ℂ}^{-},$(1.2a)${b}_{3}\left(k\right){e}^{-2{\omega }_{2}\left(k\right)L}={a}_{2}\left(k\right){b}_{1}\left(k\right)-{a}_{1}\left(k\right){b}_{2}\left(k\right),$$\mathrm{ }k\in {ℂ}^{-}.$(1.2b)

(ii) Given boundary values $\left\{{g}_{0}\left(x\right),{g}_{1}\left(x\right)\right\}$, $\left\{{f}_{0}\left(y\right),{f}_{1}\left(y\right)\right\}$ and $\left\{{h}_{0}\left(x\right),{h}_{1}\left(x\right)\right\}$, where

$q\left(x,0\right)={g}_{0}\left(x\right),$$\mathrm{ }{q}_{y}\left(x,0\right)={g}_{1}\left(x\right),$(1.3a)$q\left(0,y\right)={f}_{0}\left(y\right),$$\mathrm{ }{q}_{x}\left(0,y\right)={f}_{1}\left(y\right),$(1.3b)$q\left(x,L\right)={h}_{0}\left(x\right),$$\mathrm{ }{q}_{y}\left(x,L\right)={h}_{1}\left(x\right),$(1.3c)

we define the spectral functions and $q\left(x,y\right)$ in terms of the solution for the Riemann–Hilbert problem formulated in step (i). Assuming that the spectral functions satisfy the global relation (1.2), we prove that the function $q\left(x,y\right)$ defined by the solution of the Riemann–Hilbert problem solves equation (1.1) and satisfy the boundary values (1.3).

The outline of this work is the following. In Section 2, we introduce the Lax pair for the elliptic sinh-Gordon equation and eigenfunctions that satisfy both parts of the Lax pair. In Section 3, we discuss the spectral functions defined by the boundary values and the global algebraic relation that involves all boundary values. In Section 4, we formulate the matrix Riemann–Hilbert problem whose jump matrices are uniquely defined by the spectral functions. Analyzing the Riemann–Hilbert problem, we show that the solution of the equation exists if the boundary values satisfy the global relation. We end with concluding remarks in Section 5.

2 Preliminaries

It is well known that the elliptic sinh-Gordon equation (1.1) admits the following compatibility condition of the Lax pair [3, 13, 15]:

${\mu }_{x}+{\omega }_{1}\left(k\right)\left[{\sigma }_{3},\mu \right]=Q\left(x,y,k\right)\mu ,$(2.1a)${\mu }_{y}+{\omega }_{2}\left(k\right)\left[{\sigma }_{3},\mu \right]=i\stackrel{~}{Q}\left(x,y,k\right)\mu ,$(2.1b)

where $k\in ℂ$ is a spectral parameter, μ is a $2×2$ matrix-valued eigenfunction and

${\omega }_{1}\left(k\right)=-\frac{1}{2i}\left(k-\frac{1}{4k}\right),{\omega }_{2}\left(k\right)=-\frac{1}{2}\left(k+\frac{1}{4k}\right),Q\left(x,y,k\right)=\frac{1}{4}\left(\begin{array}{cc}\hfill \frac{i}{2k}\left(\mathrm{cosh}q-1\right)\hfill & \hfill -\left(r+\frac{\mathrm{sinh}q}{2k}\right)\hfill \\ \hfill r-\frac{\mathrm{sinh}q}{2k}\hfill & \hfill -\frac{i}{2k}\left(\mathrm{cosh}q-1\right)\hfill \end{array}\right)$

with

$\stackrel{~}{Q}\left(x,y,k\right)=Q\left(x,y,-k\right),r\left(x,y\right)=i{q}_{x}\left(x,y\right)+{q}_{y}\left(x,y\right),{\sigma }_{3}=\left(\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -1\hfill \end{array}\right),$

and the matrix commutator defined by $\left[{\sigma }_{3},A\right]={\sigma }_{3}A-A{\sigma }_{3}$. We write ${\stackrel{^}{\sigma }}_{3}A=\left[{\sigma }_{3},A\right]$ for the matrix commutator. It is convenient to use the notation

${e}^{{\stackrel{^}{\sigma }}_{3}\xi }A={e}^{{\sigma }_{3}\xi }A{e}^{-{\sigma }_{3}\xi }=\left(\begin{array}{cc}\hfill {a}_{11}\hfill & \hfill {e}^{2\xi }{a}_{12}\hfill \\ \hfill {e}^{-2\xi }{a}_{21}\hfill & \hfill {a}_{22}\hfill \end{array}\right).$

Note that the Lax pair (2.1) can be written as the simple formulation

$d\left[{e}^{\left({\omega }_{1}\left(k\right)x+{\omega }_{2}\left(k\right)y\right){\stackrel{^}{\sigma }}_{3}}\mu \left(x,y,k\right)\right]={e}^{\left({\omega }_{1}\left(k\right)x+{\omega }_{2}\left(k\right)y\right){\stackrel{^}{\sigma }}_{3}}W\left(x,y,k\right),$

where the differential 1-form W is given by

$W\left(x,y,k\right)=Q\left(x,y,k\right)\mu \left(x,y,k\right)dx+i\stackrel{~}{Q}\left(x,y,k\right)\mu \left(x,y,k\right)dy.$(2.2)

Hence, we define eigenfunctions that satisfy both parts of the Lax pair (2.1) as

${\mu }_{j}\left(x,y,k\right)=I+{\int }_{\left({x}_{j},{y}_{j}\right)}^{\left(x,y\right)}{e}^{-\left({\omega }_{1}\left(k\right)\left(x-\xi \right)+{\omega }_{2}\left(k\right)\left(y-\eta \right)\right){\stackrel{^}{\sigma }}_{3}}{W}_{j}\left(\xi ,\eta ,k\right),$(2.3)

where ${W}_{j}$ is the differential form given in (2.2) with ${\mu }_{j}$. Since the differential 1-form $W\left(x,y,k\right)$ is closed, the integration in (2.3) does not depend on paths [6, 14]. We choose three distinct points $\left({x}_{j},{y}_{j}\right)$, $j=1,2,3$ (see Figure 1),

$\left({x}_{1},{y}_{1}\right)=\left(0,L\right),\left({x}_{2},{y}_{2}\right)=\left(0,0\right),\left({x}_{3},{y}_{3}\right)=\left(\mathrm{\infty },y\right).$

More specifically, the eigenfunctions ${\mu }_{j}\left(x,y,k\right)$, $j=1,2,3$, satisfy the following integral equations:

${\mu }_{1}\left(x,y,k\right)=I+{\int }_{0}^{x}{e}^{-{\omega }_{1}\left(x-\xi \right){\stackrel{^}{\sigma }}_{3}}\left(Q{\mu }_{1}\right)\left(\xi ,y,k\right)𝑑\xi -i{\int }_{y}^{L}{e}^{-\left({\omega }_{1}\left(k\right)x+{\omega }_{2}\left(k\right)\left(y-\eta \right)\right){\stackrel{^}{\sigma }}_{3}}\left(\stackrel{~}{Q}{\mu }_{1}\right)\left(0,\eta ,k\right)𝑑\eta ,$${\mu }_{2}\left(x,y,k\right)=I+{\int }_{0}^{x}{e}^{-{\omega }_{1}\left(k\right)\left(x-\xi \right){\stackrel{^}{\sigma }}_{3}}\left(Q{\mu }_{2}\right)\left(\xi ,y,k\right)𝑑\xi +i{\int }_{0}^{y}{e}^{-\left({\omega }_{1}\left(k\right)x+{\omega }_{2}\left(k\right)\left(y-\eta \right)\right){\stackrel{^}{\sigma }}_{3}}\left(\stackrel{~}{Q}{\mu }_{2}\right)\left(0,\eta ,k\right)𝑑\eta ,$${\mu }_{3}\left(x,y,k\right)=I-{\int }_{x}^{\mathrm{\infty }}{e}^{-{\omega }_{1}\left(k\right)\left(x-\xi \right){\stackrel{^}{\sigma }}_{3}}\left(Q{\mu }_{3}\right)\left(\xi ,y,k\right)𝑑\xi .$

Figure 1

The three distinct points for the eigenfunctions ${\mu }_{j}$, $j=1,2,3$.

Note that the off-diagonal components of the matrix-valued eigenfunctions ${\mu }_{j}$ involve the explicit exponential terms and

Thus, let the domains ${D}_{j}$ in the complex k-plane, $j=1,\mathrm{\dots },4$ (see Figure 2), be defined by

${D}_{1}=\left\{k\in ℂ:\mathrm{Re}{\omega }_{1}\left(k\right)<0\right\}\cap \left\{k\in ℂ:\mathrm{Re}{\omega }_{2}\left(k\right)<0\right\},$${D}_{2}=\left\{k\in ℂ:\mathrm{Re}{\omega }_{1}\left(k\right)<0\right\}\cap \left\{k\in ℂ:\mathrm{Re}{\omega }_{2}\left(k\right)>0\right\},$${D}_{3}=\left\{k\in ℂ:\mathrm{Re}{\omega }_{1}\left(k\right)>0\right\}\cap \left\{k\in ℂ:\mathrm{Re}{\omega }_{2}\left(k\right)>0\right\},$${D}_{4}=\left\{k\in ℂ:\mathrm{Re}{\omega }_{1}\left(k\right)>0\right\}\cap \left\{k\in ℂ:\mathrm{Re}{\omega }_{2}\left(k\right)<0\right\}.$

As a result, the domains of analyticity and boundedness for the eigenfunctions can be determined:

• ${\mu }_{1}\left(x,y,k\right)$ is analytic and bounded for $k\in \left({D}_{2},{D}_{4}\right)$,

• ${\mu }_{2}\left(x,y,k\right)$ is analytic and bounded for $k\in \left({D}_{1},{D}_{3}\right)$,

• ${\mu }_{3}\left(x,y,k\right)$ is analytic and bounded for $k\in \left({D}_{3}\cup {D}_{4},{D}_{1}\cup {D}_{2}\right)$.

For convenience, we write each column of ${\mu }_{j}\left(x,y,k\right)$ as follows:

${\mu }_{1}=\left({\mu }_{1}^{\left(2\right)},{\mu }_{1}^{\left(4\right)}\right),{\mu }_{2}=\left({\mu }_{2}^{\left(1\right)},{\mu }_{2}^{\left(3\right)}\right),{\mu }_{3}=\left({\mu }_{3}^{\left(34\right)},{\mu }_{3}^{\left(12\right)}\right),$

where the superscripts indicate the analytic and bounded domains ${D}_{j}$, $j=1,\mathrm{\dots },4$, for the columns of the matrix-valued eigenfunctions. Using integration by parts, in the appropriate domain, we know

(2.4)

Since $\mathrm{trace}\left(Q\right)=\mathrm{trace}\left(\stackrel{~}{Q}\right)=0$, equation (2.4) implies that $det{\mu }_{j}=1$, $j=1,2,3$. The eigenfunctions enjoy the same symmetry as Q and $\stackrel{~}{Q}$:

${\mu }_{11}\left(x,y,k\right)={\mu }_{22}\left(x,y,-k\right),{\mu }_{21}\left(x,y,k\right)=-{\mu }_{12}\left(x,y,-k\right),$(2.5)

where the subscripts denote the $\left(i,j\right)$-component of the matrix.

Regarding the boundary values, we assume that ${g}_{j}$, ${h}_{j}\in {H}^{1}\left({ℝ}^{+}\right)$ and ${f}_{j}\in {H}^{1}\left(\left[0,L\right]\right)$ for $j=0,1$.

Figure 2

The domain ${D}_{j}$ and the oriented contour ${L}_{j}$, $j=1,\mathrm{\dots },4$.

3.1 Spectral functions

Note that the matrix eigenfunctions ${\mu }_{1}$, ${\mu }_{2}$ and ${\mu }_{3}$ are fundamental solutions of the Lax pair (2.1) and ${\mu }_{1}\left(0,L,k\right)=I$ and ${\mu }_{2}\left(0,0,k\right)=I$. Hence, the eigenfunctions are related by the so-called spectral functions, also known as the scattering matrices, ${S}_{1}\left(k\right)$, ${S}_{2}\left(k\right)$ and ${S}_{3}\left(k\right)$:

${\mu }_{3}\left(x,y,k\right)={\mu }_{2}\left(x,y,k\right){e}^{-\left({\omega }_{1}\left(k\right)x+{\omega }_{2}\left(k\right)y\right){\stackrel{^}{\sigma }}_{3}}{S}_{1}\left(k\right),$$\mathrm{ }k\in \left({ℝ}^{+},{ℝ}^{-}\right),$(3.1a)${\mu }_{1}\left(x,y,k\right)={\mu }_{2}\left(x,y,k\right){e}^{-\left({\omega }_{1}\left(k\right)x+{\omega }_{2}\left(k\right)y\right){\stackrel{^}{\sigma }}_{3}}{S}_{2}\left(k\right),$$\mathrm{ }k\in \left(i{ℝ}^{+},i{ℝ}^{-}\right),$(3.1b)${\mu }_{3}\left(x,y,k\right)={\mu }_{1}\left(x,y,k\right){e}^{-\left({\omega }_{1}\left(k\right)x+{\omega }_{2}\left(k\right)\left(y-L\right)\right){\stackrel{^}{\sigma }}_{3}}{S}_{3}\left(k\right),$$\mathrm{ }k\in \left({ℝ}^{-},{ℝ}^{+}\right).$(3.1c)

Let $x=0$, $y=0$ in (3.1a) and (3.1b) and $x=0$, $y=L$ in (3.1c). Then the spectral functions are given by

${S}_{1}\left(k\right)={\mu }_{3}\left(0,0,k\right),{S}_{2}\left(k\right)={\mu }_{1}\left(0,0,k\right),{S}_{3}\left(k\right)={\mu }_{3}\left(0,L,k\right).$

From the symmetry (2.5) of the eigenfunctions, we write the spectral functions ${S}_{1}\left(k\right)$, ${S}_{2}\left(k\right)$ and ${S}_{3}\left(k\right)$ as

${S}_{j}\left(k\right)=\left(\begin{array}{cc}\hfill {a}_{j}\left(k\right)\hfill & \hfill -{b}_{j}\left(-k\right)\hfill \\ \hfill {b}_{j}\left(k\right)\hfill & \hfill {a}_{j}\left(-k\right)\hfill \end{array}\right),j=1,2,3.$

Since $det{S}_{j}\left(k\right)=1$, $j=1,2,3$, we find the following identities:

${a}_{j}\left(k\right){a}_{j}\left(-k\right)+{b}_{j}\left(k\right){b}_{j}\left(-k\right)=1,j=1,2,3.$

Motivated by the above arguments, we define

${\mathrm{\Phi }}_{1}\left(x,k\right)={\mu }_{3}\left(x,0,k\right),{\mathrm{\Phi }}_{2}\left(y,k\right)={\mu }_{1}\left(0,y,k\right),{\mathrm{\Phi }}_{3}\left(x,k\right)={\mu }_{3}\left(x,L,k\right).$

More specifically, the functions ${\mathrm{\Phi }}_{j}$, $j=1,2,3$, satisfy the integral equations

${\mathrm{\Phi }}_{1}\left(x,k\right)=I-{\int }_{x}^{\mathrm{\infty }}{e}^{-{\omega }_{1}\left(k\right)\left(x-\xi \right){\stackrel{^}{\sigma }}_{3}}\left({Q}_{0}{\mathrm{\Phi }}_{1}\right)\left(\xi ,k\right)𝑑\xi ,\mathrm{ }k\in \left({ℂ}^{-},{ℂ}^{+}\right), 0\le x<\mathrm{\infty },$${\mathrm{\Phi }}_{2}\left(y,k\right)=I-i{\int }_{y}^{L}{e}^{-{\omega }_{2}\left(k\right)\left(y-\eta \right){\stackrel{^}{\sigma }}_{3}}\left({\stackrel{~}{Q}}_{0}{\mathrm{\Phi }}_{2}\right)\left(\eta ,k\right)𝑑\eta ,\mathrm{ }k\in ℂ, 0\le y\le L,$${\mathrm{\Phi }}_{3}\left(x,k\right)=I-{\int }_{x}^{\mathrm{\infty }}{e}^{-{\omega }_{1}\left(k\right)\left(x-\xi \right){\stackrel{^}{\sigma }}_{3}}\left({Q}_{L}{\mathrm{\Phi }}_{3}\right)\left(\xi ,k\right)𝑑\xi ,\mathrm{ }k\in \left({ℂ}^{-},{ℂ}^{+}\right), 0\le x<\mathrm{\infty },$

where ${Q}_{0}\left(x,k\right)=Q\left(x,0,k\right)$, ${\stackrel{~}{Q}}_{0}\left(y,k\right)=Q\left(0,y,-k\right)$ and ${Q}_{L}\left(x,k\right)=Q\left(x,L,k\right)$. Note that

${S}_{j}\left(k\right)={\mathrm{\Phi }}_{j}\left(0,k\right),j=1,2,3,$

which immediately imply that the spectral functions ${a}_{j}\left(k\right)$ and ${b}_{j}\left(k\right)$, $j=1,3$, are analytic and bounded for $\mathrm{Im}k<0$, while the spectral functions ${a}_{2}\left(k\right)$ and ${b}_{2}\left(k\right)$ are analytic and bounded for $k\in ℂ$ except for the essential singularities $k=0$ and $\mathrm{\infty }$. Moreover, due to the symmetry (2.5), the functions ${\mathrm{\Phi }}_{j}$, $j=1,2,3$, can also be written as

${\mathrm{\Phi }}_{j}\left(x,k\right)=\left(\begin{array}{cc}\hfill {A}_{j}\left(x,k\right)\hfill & \hfill -{B}_{j}\left(x,-k\right)\hfill \\ \hfill {B}_{j}\left(x,k\right)\hfill & \hfill {A}_{j}\left(x,-k\right)\hfill \end{array}\right),j=1,3,{\mathrm{\Phi }}_{2}\left(y,k\right)=\left(\begin{array}{cc}\hfill {A}_{2}\left(y,k\right)\hfill & \hfill -{B}_{2}\left(y,-k\right)\hfill \\ \hfill {B}_{2}\left(y,k\right)\hfill & \hfill {A}_{2}\left(y,-k\right)\hfill \end{array}\right).$

In summary, we will define the integral representations for the spectral functions below.

Definition 3.1.

Given $q\left(x,0\right)={g}_{0}\left(x\right)$ and ${q}_{y}\left(x,0\right)={g}_{1}\left(x\right)$, the map

$\left\{{g}_{0}\left(x\right),{g}_{1}\left(x\right)\right\}\to \left\{{a}_{1}\left(k\right),{b}_{1}\left(k\right)\right\}$

is defined by

${a}_{1}\left(k\right)=1-\frac{1}{4}{\int }_{0}^{\mathrm{\infty }}\left\{\frac{i}{2k}\left(\mathrm{cosh}{g}_{0}\left(\xi \right)-1\right){A}_{1}\left(\xi ,k\right)$$-\left(i{\stackrel{˙}{g}}_{0}\left(\xi \right)+{g}_{1}\left(\xi \right)+\frac{1}{2k}\mathrm{sinh}{g}_{0}\left(\xi \right)\right){B}_{1}\left(\xi ,k\right)\right\}d\xi ,$$\mathrm{ }\mathrm{Im}k<0,$(3.2a)${b}_{1}\left(k\right)=-\frac{1}{4}{\int }_{0}^{\mathrm{\infty }}{e}^{-2{\omega }_{1}\left(k\right)\xi }\left\{\left(i{\stackrel{˙}{g}}_{0}\left(\xi \right)+{g}_{1}\left(\xi \right)-\frac{1}{2k}\mathrm{sinh}{g}_{0}\left(\xi \right)\right){A}_{1}\left(\xi ,k\right)$$-\frac{i}{2k}\left(\mathrm{cosh}{g}_{0}\left(\xi \right)-1\right){B}_{1}\left(\xi ,k\right)\right\}d\xi ,$$\mathrm{ }\mathrm{Im}k<0,$(3.2b)

where the functions ${A}_{1}$ and ${B}_{1}$ are the solutions of the x-part of the Lax pair (2.1a) with $y=0$, i.e., ${A}_{1}$ and ${B}_{1}$ solve the following system of ordinary differential equations:

${A}_{1x}=\frac{1}{4}\left[\frac{i}{2k}\left(\mathrm{cosh}{g}_{0}\left(x\right)-1\right){A}_{1}-\left(i{\stackrel{˙}{g}}_{0}\left(x\right)+{g}_{1}\left(x\right)+\frac{1}{2k}\mathrm{sinh}{g}_{0}\left(x\right)\right){B}_{1}\right],$${B}_{1x}-2{\omega }_{1}\left(k\right){B}_{1}=\frac{1}{4}\left[\left(i{\stackrel{˙}{g}}_{0}\left(x\right)+{g}_{1}\left(x\right)-\frac{1}{2k}\mathrm{sinh}{g}_{0}\left(x\right)\right){A}_{1}-\frac{i}{2k}\left(\mathrm{cosh}{g}_{0}\left(x\right)-1\right){B}_{1}\right]$

with ${lim}_{x\to \mathrm{\infty }}\left({A}_{1},{B}_{1}\right)=\left(1,0\right)$.

Definition 3.2.

Given $q\left(0,y\right)={f}_{0}\left(y\right)$ and ${q}_{x}\left(0,y\right)={f}_{1}\left(y\right)$, the map

$\left\{{f}_{0}\left(y\right),{f}_{1}\left(y\right)\right\}\to \left\{{a}_{2}\left(k\right),{b}_{2}\left(k\right)\right\}$

is defined by

${a}_{2}\left(k\right)=1+\frac{i}{4}{\int }_{0}^{L}\left\{\frac{i}{2k}\left(\mathrm{cosh}{f}_{0}\left(\eta \right)-1\right){A}_{2}\left(\eta ,k\right)$$+\left(i{f}_{1}\left(\eta \right)+{\stackrel{˙}{f}}_{0}\left(\eta \right)-\frac{1}{2k}\mathrm{sinh}{f}_{0}\left(\eta \right)\right){B}_{2}\left(\eta ,k\right)\right\}d\eta ,$$\mathrm{ }k\in ℂ,$(3.3a)${b}_{2}\left(k\right)=-\frac{i}{4}{\int }_{0}^{L}{e}^{-2{\omega }_{2}\left(k\right)\eta }\left\{\left(i{f}_{1}\left(\eta \right)+{\stackrel{˙}{f}}_{0}\left(\eta \right)+\frac{1}{2k}\mathrm{sinh}{f}_{0}\left(\eta \right)\right){A}_{2}\left(\eta ,k\right)$$+\frac{i}{2k}\left(\mathrm{cosh}{f}_{0}\left(\eta \right)-1\right){B}_{2}\left(\eta ,k\right)\right\}d\eta ,$$\mathrm{ }k\in ℂ,$(3.3b)

where the functions ${A}_{2}$ and ${B}_{2}$ are the solutions of the y-part of the Lax pair (2.1b) with $x=0$, i.e., ${A}_{2}$ and ${B}_{2}$ solve the following system of ordinary differential equations:

${A}_{2y}=-\frac{i}{4}\left[\frac{i}{2k}\left(\mathrm{cosh}{f}_{0}\left(y\right)-1\right){A}_{2}+\left(i{f}_{0}\left(y\right)+{\stackrel{˙}{f}}_{1}\left(y\right)-\frac{1}{2k}\mathrm{sinh}{f}_{0}\left(y\right)\right){B}_{2}\right],$${B}_{2y}-2{\omega }_{2}\left(k\right){B}_{2}=\frac{i}{4}\left[\left(i{f}_{0}\left(y\right)+{\stackrel{˙}{f}}_{1}\left(y\right)+\frac{1}{2k}\mathrm{sinh}{f}_{0}\left(y\right)\right){A}_{2}+\frac{i}{2k}\left(\mathrm{cosh}{f}_{0}\left(y\right)-1\right){B}_{2}\right]$

with ${lim}_{y\to \mathrm{\infty }}\left({A}_{2},{B}_{2}\right)=\left(1,0\right)$.

Definition 3.3.

Given $q\left(x,L\right)={h}_{0}\left(x\right)$ and ${q}_{y}\left(x,L\right)={h}_{1}\left(x\right)$, the map

$\left\{{h}_{0}\left(x\right),{h}_{1}\left(x\right)\right\}\to \left\{{a}_{3}\left(k\right),{b}_{3}\left(k\right)\right\}$

is defined by

${a}_{3}\left(k\right)=1-\frac{1}{4}{\int }_{0}^{\mathrm{\infty }}\left\{\frac{i}{2k}\left(\mathrm{cosh}{h}_{0}\left(\xi \right)-1\right){A}_{3}\left(\xi ,k\right)$$-\left(i{\stackrel{˙}{h}}_{0}\left(\xi \right)+{h}_{1}\left(\xi \right)+\frac{1}{2k}\mathrm{sinh}{h}_{0}\left(\xi \right)\right){B}_{3}\left(\xi ,k\right)\right\}d\xi ,$$\mathrm{ }\mathrm{Im}k<0,$(3.4a)${b}_{3}\left(k\right)=-\frac{1}{4}{\int }_{0}^{\mathrm{\infty }}{e}^{-2{\omega }_{1}\left(k\right)\xi }\left\{\left(i{\stackrel{˙}{h}}_{0}\left(\xi \right)+{h}_{1}\left(\xi \right)-\frac{1}{2k}\mathrm{sinh}{h}_{0}\left(\xi \right)\right){A}_{3}\left(\xi ,k\right)$$-\frac{i}{2k}\left(\mathrm{cosh}{h}_{0}\left(\xi \right)-1\right){B}_{3}\left(\xi ,k\right)\right\}d\xi ,$$\mathrm{ }\mathrm{Im}k<0,$(3.4b)

where the functions ${A}_{3}$ and ${B}_{3}$ are the solutions of the x-part of the Lax pair (2.1a) with $y=L$, i.e., ${A}_{3}$ and ${B}_{3}$ solve the following system of ordinary differential equations:

${A}_{3x}=\frac{1}{4}\left[\frac{i}{2k}\left(\mathrm{cosh}{h}_{0}\left(x\right)-1\right){A}_{3}-\left(i{\stackrel{˙}{h}}_{0}\left(x\right)+{h}_{1}\left(x\right)+\frac{1}{2k}\mathrm{sinh}{h}_{0}\left(x\right)\right){B}_{3}\right],$${B}_{3x}-2{\omega }_{1}\left(k\right){B}_{3}=\frac{1}{4}\left[\left(i{\stackrel{˙}{h}}_{0}\left(x\right)+{h}_{1}\left(x\right)-\frac{1}{2k}\mathrm{sinh}{h}_{0}\left(x\right)\right){A}_{3}-\frac{i}{2k}\left(\mathrm{cosh}{h}_{0}\left(x\right)-1\right){B}_{3}\right]$

with ${lim}_{x\to \mathrm{\infty }}\left({A}_{3},{B}_{3}\right)=\left(1,0\right)$.

In what follows, we derive the global relation that involves all boundary values. It should be noted that the global relation plays a crucial role in the implementation of the Fokas method for boundary value problems. Evaluating equations (3.1a) and (3.1b) at $x=0$ and $y=L$, we find

${S}_{3}\left(k\right)={\mu }_{2}\left(0,L,k\right){e}^{-{\omega }_{2}\left(k\right)L{\stackrel{^}{\sigma }}_{3}}{S}_{1}\left(k\right),$$I={\mu }_{2}\left(0,L,k\right){e}^{-{\omega }_{2}\left(k\right)L{\stackrel{^}{\sigma }}_{3}}{S}_{2}\left(k\right).$

We then obtain the following global relation in terms of the scattering matrices:

${e}^{{\omega }_{2}\left(k\right)L{\stackrel{^}{\sigma }}_{3}}{S}_{3}\left(k\right)={S}_{2}^{-1}\left(k\right){S}_{1}\left(k\right).$(3.5)

In particular, the global relation (3.5) can be written as

${a}_{3}\left(k\right)={a}_{1}\left(k\right){a}_{2}\left(-k\right)+{b}_{1}\left(k\right){b}_{2}\left(-k\right),$$\mathrm{ }k\in {ℂ}^{-},$(3.6a)${b}_{3}\left(k\right){e}^{-2{\omega }_{2}\left(k\right)L}={a}_{2}\left(k\right){b}_{1}\left(k\right)-{a}_{1}\left(k\right){b}_{2}\left(k\right),$$\mathrm{ }k\in {ℂ}^{-}.$(3.6b)

3.2 Spectral analysis at boundary values

In this section we characterize the boundary values in terms of the spectral functions. This can be done by the spectral analysis for the Lax pair at $x=0$, $y=0$ and $y=L$, respectively.

Proposition 3.4.

The inverse map

$\left\{{a}_{1}\left(k\right),{b}_{1}\left(k\right)\right\}\to \left\{q\left(x,0\right),{q}_{y}\left(x,0\right)\right\}$

to the map defined in Definition 3.1 is given by

$\mathrm{cosh}q\left(x,0\right)=1-8i\underset{k\to \mathrm{\infty }}{lim}k{M}_{11x}^{\left(x\right)}-8\underset{k\to \mathrm{\infty }}{lim}{\left(k{M}_{21}^{\left(x\right)}\right)}^{2},$$i{q}_{x}\left(x,0\right)+{q}_{y}\left(x,0\right)=-4i\underset{k\to \mathrm{\infty }}{lim}k{M}_{21}^{\left(x\right)},$

where ${M}^{\mathrm{\left(}x\mathrm{\right)}}$ is the solution of the matrix Riemann–Hilbert problem

${M}_{-}^{\left(x\right)}\left(x,k\right)={M}_{+}^{\left(x\right)}\left(x,k\right){J}^{\left(x\right)}\left(x,k\right),k\in ℝ$(3.7)

with the jump matrix ${J}^{\mathrm{\left(}x\mathrm{\right)}}$ given by

${J}^{\left(x\right)}\left(x,k\right)=\left(\begin{array}{cc}\hfill 1\hfill & \hfill \frac{{b}_{1}\left(-k\right)}{{a}_{1}\left(k\right)}{e}^{-2{\omega }_{1}\left(k\right)x}\hfill \\ \hfill \frac{{b}_{1}\left(k\right)}{{a}_{1}\left(-k\right)}{e}^{2{\omega }_{1}\left(k\right)x}\hfill & \hfill \frac{1}{{a}_{1}\left(k\right){a}_{1}\left(-k\right)}\hfill \end{array}\right),k\in ℝ.$(3.8)

Proof.

Evaluating equation (3.1a) at $y=0$, we find

${\mu }_{3}\left(x,0,k\right)={\mu }_{2}\left(x,0,k\right){e}^{-{\omega }_{1}\left(k\right)x{\stackrel{^}{\sigma }}_{3}}{S}_{1}\left(k\right),k\in \left({ℝ}^{+},{ℝ}^{-}\right), 0\le x<\mathrm{\infty }.$

Note that the eigenfunction ${\mu }_{2}^{\left(1\right)}\left(x,0,k\right)$ is analytic and bounded for $k\in {D}_{1}\cup {D}_{2}$ and ${\mu }_{2}^{\left(3\right)}\left(x,0,k\right)$ is analytic and bounded for $k\in {D}_{3}\cup {D}_{4}$. Thus, we formulate the matrix Riemann–Hilbert problem (3.7) with the jump matrix ${J}^{\left(x\right)}\left(x,k\right)$ given by (3.8), where the sectionally meromorphic functions ${M}_{±}^{\left(x\right)}$ are defined by

${M}_{+}^{\left(x\right)}\left(x,k\right)=\left(\frac{{\mu }_{2}^{\left(1\right)}\left(x,0,k\right)}{{a}_{1}\left(-k\right)},{\mu }_{3}^{\left(12\right)}\left(x,0,k\right)\right),\mathrm{Im}k>0,$${M}_{-}^{\left(x\right)}\left(x,k\right)=\left({\mu }_{3}^{\left(34\right)}\left(x,0,k\right),\frac{{\mu }_{2}^{\left(3\right)}\left(x,0,k\right)}{{a}_{1}\left(k\right)}\right),\mathrm{Im}k<0$

with $det{M}_{±}^{\left(x\right)}=1$ and ${M}_{±}^{\left(x\right)}=I+O\left(\frac{1}{k}\right)$ as $k\to \mathrm{\infty }$. The remaining proof follows arguments similar to arguments in [14]. ∎

Proposition 3.5.

The inverse map

$\left\{{a}_{2}\left(k\right),{b}_{2}\left(k\right)\right\}\to \left\{q\left(0,y\right),{q}_{x}\left(0,y\right)\right\}$

to the map defined in Definition 3.2 is given by

$\mathrm{cosh}q\left(0,y\right)=1+8\underset{k\to \mathrm{\infty }}{lim}k{M}_{11y}^{\left(y\right)}+8\underset{k\to \mathrm{\infty }}{lim}{\left(k{M}_{21}^{\left(y\right)}\right)}^{2},$$i{q}_{x}\left(0,y\right)+{q}_{y}\left(0,y\right)=-4i\underset{k\to \mathrm{\infty }}{lim}k{M}_{21}^{\left(y\right)},$

where ${M}^{\mathrm{\left(}y\mathrm{\right)}}$ is the solution of the matrix Riemann–Hilbert problem

${M}_{-}^{\left(y\right)}\left(y,k\right)={M}_{+}^{\left(y\right)}\left(y,k\right){J}^{\left(y\right)}\left(y,k\right),k\in iℝ$

with the jump matrix ${J}^{\mathrm{\left(}y\mathrm{\right)}}$ given by

${J}^{\left(y\right)}\left(y,k\right)=\left(\begin{array}{cc}\hfill 1\hfill & \hfill \frac{{b}_{2}\left(-k\right)}{{a}_{2}\left(k\right)}{e}^{-2{\omega }_{2}\left(k\right)y}\hfill \\ \hfill \frac{{b}_{2}\left(k\right)}{{a}_{2}\left(-k\right)}{e}^{2{\omega }_{2}\left(k\right)y}\hfill & \hfill \frac{1}{{a}_{2}\left(k\right){a}_{2}\left(-k\right)}\hfill \end{array}\right),k\in iℝ.$(3.10)

Proof.

The proof is similar to the arguments discussed in Proposition 3.4. From the spectral relation (3.1b) with $x=0$, we find the jump matrix ${J}^{\left(y\right)}\left(y,k\right)$ given in (3.10) and the sectionally meromorphic functions ${M}_{±}^{\left(y\right)}$ given by

${M}_{+}^{\left(y\right)}\left(y,k\right)=\left(\frac{{\mu }_{2}^{\left(1\right)}\left(0,y,k\right)}{{a}_{2}\left(-k\right)},{\mu }_{1}^{\left(4\right)}\left(0,y,k\right)\right),\mathrm{Re}k>0,$${M}_{-}^{\left(y\right)}\left(y,k\right)=\left({\mu }_{1}^{\left(2\right)}\left(0,y,k\right),\frac{{\mu }_{2}^{\left(3\right)}\left(0,y,k\right)}{{a}_{2}\left(k\right)}\right),\mathrm{Re}k<0$

with $det{M}_{±}^{\left(y\right)}=1$ and ${M}_{±}^{\left(y\right)}=I+O\left(\frac{1}{k}\right)$ as $k\to \mathrm{\infty }$. Note that the eigenfunction ${\mu }_{2}^{\left(1\right)}\left(0,y,k\right)$ is analytic and bounded for $k\in {D}_{1}\cup {D}_{4}$ and the function ${\mu }_{2}^{\left(3\right)}\left(0,y,k\right)$ is analytic and bounded for $k\in {D}_{2}\cup {D}_{3}$. ∎

The spectral analysis at the boundary $y=L$ is similar to Proposition 3.4.

Proposition 3.6.

The inverse map

$\left\{{a}_{3}\left(k\right),{b}_{3}\left(k\right)\right\}\to \left\{q\left(x,L\right),{q}_{y}\left(x,L\right)\right\}$

to the map defined in Definition 3.3 is given by

$\mathrm{cosh}q\left(x,L\right)=1-8i\underset{k\to \mathrm{\infty }}{lim}k{M}_{11x}^{\left(L\right)}-8\underset{k\to \mathrm{\infty }}{lim}{\left(k{M}_{21}^{\left(L\right)}\right)}^{2},$$i{q}_{x}\left(x,L\right)+{q}_{y}\left(x,L\right)=-4i\underset{k\to \mathrm{\infty }}{lim}k{M}_{21}^{\left(L\right)},$

where ${M}^{\mathrm{\left(}L\mathrm{\right)}}$ is the solution of the matrix Riemann–Hilbert problem

${M}_{-}^{\left(L\right)}\left(x,k\right)={M}_{+}^{\left(L\right)}\left(x,k\right){J}^{\left(L\right)}\left(x,k\right),k\in ℝ$(3.12)

with the jump matrix ${J}^{\mathrm{\left(}L\mathrm{\right)}}$ given by

${J}^{\left(L\right)}\left(x,k\right)=\left(\begin{array}{cc}\hfill 1\hfill & \hfill \frac{{b}_{3}\left(-k\right)}{{a}_{3}\left(k\right)}{e}^{-2{\omega }_{1}\left(k\right)x}\hfill \\ \hfill \frac{{b}_{3}\left(k\right)}{{a}_{3}\left(-k\right)}{e}^{2{\omega }_{1}\left(k\right)x}\hfill & \hfill \frac{1}{{a}_{3}\left(k\right){a}_{3}\left(-k\right)}\hfill \end{array}\right),k\in ℝ.$(3.13)

4 Riemann–Hilbert problem

In this section, we first formulate the matrix Riemann–Hilbert problem under the assumption that the solution $q\left(x,y\right)$ of (1.1) exists and then we discuss the existence of the solution $q\left(x,y\right)$ satisfying the boundary conditions. Using equations (3.1) and the global relation (3.6), after tedious but straightforward calculations, we formulate the following matrix Riemann–Hilbert problem:

${M}_{-}\left(x,y,k\right)={M}_{+}\left(x,y,k\right)J\left(x,y,k\right),k\in \mathcal{ℒ},$(4.1)

where the oriented contours $\mathcal{ℒ}={L}_{1}\cup {L}_{2}\cup {L}_{3}\cup {L}_{4}$ are given by (cf. Figure 2)

${L}_{1}={D}_{1}\cap {D}_{2},{L}_{2}={D}_{2}\cap {D}_{3},$(4.2a)${L}_{3}={D}_{3}\cap {D}_{4},{L}_{4}={D}_{4}\cap {D}_{1},$(4.2b)

and the jump matrices are defined by

${J}_{1}\left(x,y,k\right)=\left(\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill \frac{{b}_{2}\left(k\right)}{{a}_{1}\left(-k\right){a}_{3}\left(-k\right)}{e}^{2\theta \left(x,y,k\right)}\hfill & \hfill 1\hfill \end{array}\right),\mathrm{ }k\in {L}_{1},$(4.3a)${J}_{2}\left(x,y,k\right)=\left(\begin{array}{cc}\hfill \frac{{a}_{2}\left(k\right)}{{a}_{1}\left(k\right){a}_{3}\left(-k\right)}\hfill & \hfill -\frac{{b}_{1}\left(-k\right)}{{a}_{1}\left(k\right)}{e}^{-2\theta \left(x,y,k\right)}\hfill \\ \hfill -\frac{{b}_{3}\left(k\right)}{{a}_{3}\left(-k\right)}{e}^{2\left(\theta \left(x,y,k\right)-{\omega }_{2}\left(k\right)L\right)}\hfill & \hfill 1\hfill \end{array}\right),\mathrm{ }k\in {L}_{2},$(4.3b)${J}_{3}\left(x,y,k\right)=\left(\begin{array}{cc}\hfill 1\hfill & \hfill -\frac{{b}_{2}\left(-k\right)}{{a}_{1}\left(k\right){a}_{3}\left(k\right)}{e}^{-2\theta \left(x,y,k\right)}\hfill \\ \hfill 0\hfill & \hfill 1\hfill \end{array}\right),\mathrm{ }k\in {L}_{3},$(4.3c)${J}_{4}\left(x,y,k\right)={J}_{1}{J}_{2}^{-1}{J}_{3}=\left(\begin{array}{cc}\hfill 1\hfill & \hfill \frac{{b}_{3}\left(-k\right)}{{a}_{3}\left(k\right)}{e}^{-2\left(\theta \left(x,y,k\right)-{\omega }_{2}\left(k\right)L\right)}\hfill \\ \hfill \frac{{b}_{1}\left(k\right)}{{a}_{1}\left(-k\right)}{e}^{2\theta \left(x,y,k\right)}\hfill & \hfill \frac{{a}_{2}\left(-k\right)}{{a}_{1}\left(-k\right){a}_{3}\left(k\right)}\hfill \end{array}\right),\mathrm{ }k\in {L}_{4}$(4.3d)

with $\theta \left(x,y,k\right)={\omega }_{1}\left(k\right)x+{\omega }_{2}\left(k\right)y$. The matrix-valued functions ${M}_{±}$ are sectionally meromorphic and defined as follows:

Note that $det{M}_{±}=1$ and ${M}_{±}=I+O\left(\frac{1}{k}\right)$ as $k\to \mathrm{\infty }$. Indeed, using the global relation (3.6), we find

${a}_{2}\left(k\right)-{b}_{1}\left(-k\right){b}_{3}\left(k\right){e}^{-2{\omega }_{2}\left(k\right)L}={a}_{1}\left(k\right){a}_{3}\left(-k\right),$

which implies that $det\left({J}_{2}\right)=det\left({J}_{4}\right)=1$.

The Riemann–Hilbert problem (4.1) can be solved by a Cauchy-type integral equation. Let $\stackrel{~}{J}=I-J$. Then equation (4.1) becomes

${M}_{+}\left(x,y,k\right)-{M}_{-}\left(x,y,k\right)={M}_{+}\left(x,y,k\right)\stackrel{~}{J}\left(x,y,k\right).$

Applying the Plemelj formula [9], the solution M of the Riemann–Hilbert problem (4.1) can be expressed as

$M\left(x,y,k\right)=I+\frac{1}{2i\pi }{\int }_{\mathcal{ℒ}}{M}_{+}\left(x,y,{k}^{\prime }\right)\stackrel{~}{J}\left(x,y,{k}^{\prime }\right)\frac{d{k}^{\prime }}{{k}^{\prime }-k}.$

Expanding the integrand for k, we find

$M\left(x,y,k\right)=I-\frac{1}{2i\pi k}{\int }_{\mathcal{ℒ}}{M}_{+}\left(x,y,{k}^{\prime }\right)\stackrel{~}{J}\left(x,y,{k}^{\prime }\right)𝑑{k}^{\prime }+O\left(\frac{1}{{k}^{2}}\right),k\to \mathrm{\infty }.$

Thus, we expand the solution M of the Riemann–Hilbert problem (4.1) as

$M\left(x,y,k\right)=I+\frac{{M}^{\left(1\right)}\left(x,y\right)}{k}+\frac{{M}^{\left(2\right)}\left(x,y\right)}{{k}^{2}}+O\left(\frac{1}{{k}^{3}}\right),k\to \mathrm{\infty }.$(4.4)

If we substitute the expansion (4.4) into the x-part of the Lax pair (2.1a), the $\left(2,1\right)$-component at $O\left(1\right)$ yields

$i{q}_{x}\left(x,y\right)+{q}_{y}\left(x,y\right)=-4i{M}_{21}^{\left(1\right)}\left(x,y\right)$(4.5)

and the $\left(1,1\right)$-component at $O\left(\frac{1}{k}\right)$ implies

${M}_{11x}^{\left(1\right)}\left(x,y\right)=-\frac{1}{8i}\left(\mathrm{cosh}q\left(x,y\right)-1\right)-\frac{1}{4}\left(i{q}_{x}\left(x,y\right)+{q}_{y}\left(x,y\right)\right){M}_{21}^{\left(1\right)}\left(x,y\right).$

Simplifying the above equation with (4.5), we obtain the reconstruction formula for $q\left(x,y\right)$ in terms of the solution of the Riemann–Hilbert problem

$\mathrm{cosh}q\left(x,y\right)=1-8i{M}_{11x}^{\left(1\right)}\left(x,y\right)-8{\left({M}_{21}^{\left(1\right)}\right)}^{2}.$

Similarly, if we substitute the expansion (4.4) into the y-part of the Lax pair, the solution $q\left(x,y\right)$ can be written equivalently as

$\mathrm{cosh}q\left(x,y\right)=1+8{M}_{11y}^{\left(1\right)}\left(x,y\right)+8{\left({M}_{21}^{\left(1\right)}\right)}^{2}.$

Let us now state the existence theorem for the elliptic sinh-Gordon equation in the semi-strip.

Theorem 4.1.

Let the functions ${g}_{j}\mathit{}\mathrm{\left(}x\mathrm{\right)}$, ${h}_{j}\mathit{}\mathrm{\left(}x\mathrm{\right)}\mathrm{\in }{H}^{\mathrm{1}}\mathit{}\mathrm{\left(}{\mathrm{R}}^{\mathrm{+}}\mathrm{\right)}$ and ${f}_{j}\mathit{}\mathrm{\left(}y\mathrm{\right)}\mathrm{\in }{H}^{\mathrm{1}}\mathit{}\mathrm{\left(}\mathrm{\left[}\mathrm{0}\mathrm{,}L\mathrm{\right]}\mathrm{\right)}$ , $j\mathrm{=}\mathrm{0}\mathrm{,}\mathrm{1}$, be given with the sufficiently small ${H}^{\mathrm{1}}$ norms. Let the functions ${a}_{j}\mathit{}\mathrm{\left(}k\mathrm{\right)}$, ${b}_{j}\mathit{}\mathrm{\left(}k\mathrm{\right)}$, $j\mathrm{=}\mathrm{1}\mathrm{,}\mathrm{2}\mathrm{,}\mathrm{3}$, be given by (3.2), (3.3) and (3.4) in Definitions 3.1, 3.2 and 3.3, respectively. Suppose that the spectral functions ${a}_{j}$ and ${b}_{j}$, $j\mathrm{=}\mathrm{1}\mathrm{,}\mathrm{2}\mathrm{,}\mathrm{3}$, satisfy the global relation (3.6). Let $M\mathit{}\mathrm{\left(}x\mathrm{,}y\mathrm{,}k\mathrm{\right)}$ be the solution of the following matrix Riemann–Hilbert (RH) problem

${M}_{-}\left(x,y,k\right)={M}_{+}\left(x,y,k\right)J\left(x,y,k\right),k\in \mathcal{ℒ},$(4.6)

where $\mathrm{det}\mathit{}\mathrm{\left(}{M}_{\mathrm{±}}\mathrm{\right)}\mathrm{=}\mathrm{1}$, ${M}_{\mathrm{±}}\mathrm{=}I\mathrm{+}O\mathit{}\mathrm{\left(}\frac{\mathrm{1}}{k}\mathrm{\right)}$ as $k\mathrm{\to }\mathrm{\infty }$, the oriented contours $\mathcal{L}$ are defined in (4.2) and the jump matrices J are given in (4.3).

Then the Riemann–Hilbert problem is uniquely solvable and the function $q\mathit{}\mathrm{\left(}x\mathrm{,}y\mathrm{\right)}$ defined by

$i{q}_{x}+{q}_{y}=-4i\underset{k\to \mathrm{\infty }}{lim}k{M}_{21},\mathrm{cosh}q\left(x,y\right)=1-8i\underset{k\to \mathrm{\infty }}{lim}k{M}_{11x}-8\underset{k\to \mathrm{\infty }}{lim}{\left(k{M}_{21}\right)}^{2}$(4.7)

solves the elliptic sinh-Gordon equation (1.1) satisfying the boundary conditions

$q\left(x,0\right)={g}_{0}\left(x\right),$$\mathrm{ }{q}_{y}\left(x,0\right)={g}_{1}\left(x\right),$(4.8a)$q\left(0,y\right)={f}_{0}\left(y\right),$$\mathrm{ }{q}_{x}\left(0,y\right)={f}_{1}\left(y\right),$(4.8b)$q\left(x,L\right)={h}_{0}\left(x\right),$$\mathrm{ }{q}_{y}\left(x,L\right)={h}_{1}\left(x\right).$(4.8c)

Proof.

Using the dressing method and the vanishing lemma presented in [7, 9], we can prove the unique solvability of the Riemann–Hilbert problem (4.6) and then we can also verify that the $q\left(x,y\right)$ defined in (4.7) solves the elliptic sinh-Gordon equation (1.1).

The proof that $q\left(x,y\right)$ given in (4.7) satisfies the boundary values is similar to the argument presented in [14]. Here, we only state the proof of (4.8c). We first define

(4.9)

where

$F\left(x,k\right)=\left(\begin{array}{ccc}\hfill 1\hfill & \hfill \hfill & \hfill -\frac{{b}_{2}\left(-k\right)}{{a}_{1}\left(k\right){a}_{3}\left(k\right)}{e}^{-2\left({\omega }_{1}\left(k\right)x+{\omega }_{2}\left(k\right)L\right)}\hfill \\ \hfill 0\hfill & \hfill \hfill & \hfill 1\hfill \end{array}\right).$

It is convenient to denote $M\left(x,L,k\right)$ and ${M}^{\left(L\right)}\left(x,k\right)$ for $k\in {D}_{j}$, $j=1,\mathrm{\dots },4$, by ${M}_{j}\left(x,L,k\right)$ and ${M}_{j}^{\left(L\right)}\left(x,k\right)$, respectively. Then, equations (4.6) and (4.9) can be written as

${M}_{2}\left(x,L,k\right)={M}_{1}{J}_{1}\left(x,L,k\right),{M}_{2}\left(x,L,k\right)={M}_{3}{J}_{2}\left(x,L,k\right),$(4.10a)${M}_{4}\left(x,L,k\right)={M}_{3}{J}_{3}\left(x,L,k\right),{M}_{4}\left(x,L,k\right)={M}_{1}{J}_{4}\left(x,L,k\right),$(4.10b)

and

${M}_{1}^{\left(L\right)}\left(x,k\right)={M}_{1}\left(x,L,k\right){J}_{1}\left(x,L,k\right),$${M}_{2}^{\left(L\right)}\left(x,k\right)={M}_{2}\left(x,L,k\right),$(4.11a)${M}_{3}^{\left(L\right)}\left(x,k\right)={M}_{3}\left(x,L,k\right)F\left(x,k\right),$${M}_{4}^{\left(L\right)}\left(x,k\right)={M}_{4}\left(x,L,k\right){J}_{3}^{-1}\left(x,L,k\right)F\left(x,k\right).$(4.11b)

Combining (4.11) and (4.10), we find the jump conditions as

${M}_{2}^{\left(L\right)}\left(x,k\right)={M}_{1}^{\left(L\right)}\left(x,k\right),{M}_{3}^{\left(L\right)}\left(x,k\right)={M}_{2}^{\left(L\right)}\left(x,k\right){J}_{2}^{-1}\left(x,L,k\right)F\left(x,k\right),$${M}_{4}^{\left(L\right)}\left(x,k\right)={M}_{3}^{\left(L\right)}\left(x,k\right),{M}_{4}^{\left(L\right)}\left(x,k\right)={M}_{1}^{\left(L\right)}\left(x,k\right){J}_{2}^{-1}\left(x,L,k\right)F\left(x,k\right).$

Note that no jumps occur along the contours ${L}_{1}$ and ${L}_{3}$ and that ${J}_{2}^{-1}\left(x,L,k\right)F\left(x,k\right)={J}^{\left(L\right)}\left(x,k\right)$, where ${J}^{\left(L\right)}\left(x,k\right)$ is given in (3.13). Thus, we define

${M}^{\left(L\right)}\left(x,k\right)={M}_{+}^{\left(L\right)}\left(x,k\right),\mathrm{ }k\in {D}_{1}\cup {D}_{2},$${M}^{\left(L\right)}\left(x,k\right)={M}_{-}^{\left(L\right)}\left(x,k\right),\mathrm{ }k\in {D}_{3}\cup {D}_{4}$

and then equations (4.10) are equivalent to the Riemann–Hilbert problem (3.12) with the jump matrix ${J}^{\left(L\right)}\left(x,k\right)$. Thus, evaluating (4.7) at $y=L$, we prove that the function $q\left(x,y\right)$ defined in (4.7) satisfies the boundary values (4.8c).

In a similar way, equations (4.8a) and (4.8b) can be proved. ∎

5 Concluding remarks

In conclusion, we have studied the boundary value problem for the elliptic sinh-Gordon equation formulated in the semi-strip by the Fokas method, a generalization of the inverse scattering transform for boundary value problems. In particular, we have characterized the spectral functions in terms of the boundary values and we have derived the global algebraic relation that involves the boundary values. Furthermore, we have shown that the solution of the elliptic sinh-Gordon equation posed in the semi-strip exists provided that the spectral function defined by the boundary values satisfy the global relation. This solution can be expressed in terms of the unique solution of the matrix Riemann–Hilbert problem whose jump matrices are uniquely defined by the spectral functions. These spectral functions denoted by $\left\{{a}_{j}\left(k\right),{b}_{j}\left(k\right)\right\}$, $j=1,2,3$, can be determined by the boundary values $\left\{q\left(x,0\right),{q}_{y}\left(x,0\right)\right\}$, $\left\{q\left(0,y\right),{q}_{x}\left(0,y\right)\right\}$ and $\left\{q\left(x,L\right),{q}_{y}\left(x,L\right)\right\}$. However, for a well-posed boundary value problem, it is required that a subset of the boundary values should be prescribed. In this respect, it is necessary to characterize unknown boundary values, called the Dirichlet-to-Neumann map [8]. This characterization can be done by analyzing the global relation as discussed in [11, 10, 12] and we will address this issue in the near future.

References

• [1]

M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge, Cambridge University Press, 1991.  Google Scholar

• [2]

G. Biondini and G. Hwang, Initial-boundary value problems for discrete evolution equations: Discrete linear Schrödinger and integrable discrete nonlinear Schrödinger equations, Inverse Problems 24 (2008), Article ID 065011.  Google Scholar

• [3]

M. Boiti, J.-P. Leon and F. Pempinelli, Integrable two-dimensional generalisation of the sine- and sinh-Gordon equations, Inverse Problems 3 (1987), 37–49.

• [4]

A. S. Fokas, A unified transform method for solving linear and certain nonlinear PDEs, Proc. Roy. Soc. Lond. Ser. A 453 (1997), 1411–1443.

• [5]

A. S. Fokas, On the integrability of certain linear and nonlinear partial differential equations, J. Math. Phys. 41 (2000), 4188–4237.

• [6]

A. S. Fokas, Two-dimensional linear PDEs in a convex polygon, Proc. Roy. Soc. Lond. Ser. A 457 (2001), 371–393.

• [7]

A. S. Fokas, Integrable nonlinear evolution equations on the half-line, Comm. Math. Phys. 230 (2002), 1–39.

• [8]

A. S. Fokas, The generalized Dirichlet-to-Neumann map for certain nonlinear evolution PDEs, Comm. Pure Appl. Math. 58 (2005), 639–670.

• [9]

A. S. Fokas, A Unified Approach to Boundary Value Problems, CBMS-NSF Regional Conf. Ser. in Appl. Math. 78, SIAM, Philadelphia, 2008.  Google Scholar

• [10]

A. S. Fokas, J. Lenells and B. Pelloni, Boundary value problems for the elliptic sine-Gordon equation in a semi-strip, J. Nonlinear Sci. 23 (2013), 241–282.

• [11]

A. S. Fokas and B. Pelloni, The Dirichlet-to-Neumann map for the elliptic sine-Gordon equation, Nonlinearity 25 (2012), 1011–1031.

• [12]

G. Hwang, The Fokas method: The Dirichlet to Neumann map for the sine-Gordon equation, Stud. Appl. Math. 132 (2014), 381–406.

• [13]

G. Hwang, The elliptic sinh-Gordon equation in the half plane, J. Nonlinear Sci. Appl. 8 (2015), 163–173.

• [14]

G. Hwang, The elliptic sinh-Gordon equation in the quarter plane, J. Nonlinear Math. Phys. 23 (2016), 127–140.

• [15]

M. Jaworski and D. Kaup, Direct and inverse scattering problem associated with the elliptic sinh-Gordon equation, Inverse Problems 6 (1990), 543–556.

• [16]

B. Pelloni and D. A. Pinotsis, The elliptic sine-Gordon equation in a half plane, Nonlinearity 23 (2010), 77–88.

Accepted: 2017-04-09

Published Online: 2017-06-01

Funding Source: Daegu University

Award identifier / Grant number: Research Grant

Award identifier / Grant number: 2014

The work is supported by the Daegu University Research Grant, 2014.

Citation Information: Advances in Nonlinear Analysis, Volume 8, Issue 1, Pages 533–544, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496,

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