The elliptic sinh-Gordon equation in a semi-strip

1 Introduction

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

where Ω = { ( x , y ) ∈ ℝ 2 : 0 < x < ∞ , 0 < y < L } . 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 { - ∞ < x , y < ∞ } (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 { - ∞ < x < ∞ ,  0 < y < ∞ } (see [13]) and the quarter plane { 0 < x , y < ∞ } (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 ⁢ ( x , y ) exists, we express the solution q ⁢ ( x , y ) in terms of the unique solution of the matrix Riemann–Hilbert problem defined by the spectral functions { a j ⁢ ( k ) , b j ⁢ ( k ) } , j = 1 , 2 , 3 . These spectral functions are defined by boundary values { q ⁢ ( x , 0 ) , q y ⁢ ( x , 0 ) } , { q ⁢ ( 0 , y ) , q x ⁢ ( 0 , y ) } and { q ⁢ ( x , L ) , q y ⁢ ( x , L ) } , respectively. It should be remarked that the spectral functions satisfy the global algebraic relation that involves all boundary values:

(1.2)

(ii) Given boundary values { g 0 ⁢ ( x ) , g 1 ⁢ ( x ) } , { f 0 ⁢ ( y ) , f 1 ⁢ ( y ) } and { h 0 ⁢ ( x ) , h 1 ⁢ ( x ) } , where

(1.3)

we define the spectral functions and q ⁢ ( x , y ) 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 ⁢ ( x , y ) 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]:

(2.1)

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

with

and the matrix commutator defined by [ σ 3 , A ] = σ 3 ⁢ A - A ⁢ σ 3 . We write σ ^ 3 ⁢ A = [ σ 3 , A ] for the matrix commutator. It is convenient to use the notation

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

where the differential 1-form W is given by

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

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

More specifically, the eigenfunctions μ j ⁢ ( x , y , k ) , j = 1 , 2 , 3 , satisfy the following integral equations:

Figure 1

The three distinct points for the eigenfunctions μ j , j = 1 , 2 , 3 .

Note that the off-diagonal components of the matrix-valued eigenfunctions μ j involve the explicit exponential terms and

Thus, let the domains D j in the complex k-plane, j = 1 , … , 4 (see Figure 2), be defined by

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

1. μ 1 ⁢ ( x , y , k ) is analytic and bounded for k ∈ ( D 2 , D 4 ) ,

2. μ 2 ⁢ ( x , y , k ) is analytic and bounded for k ∈ ( D 1 , D 3 ) ,

3. μ 3 ⁢ ( x , y , k ) is analytic and bounded for k ∈ ( D 3 ∪ D 4 , D 1 ∪ D 2 ) .

For convenience, we write each column of μ j ⁢ ( x , y , k ) as follows:

where the superscripts indicate the analytic and bounded domains D j , j = 1 , … , 4 , for the columns of the matrix-valued eigenfunctions. Using integration by parts, in the appropriate domain, we know

Since trace ⁡ ( Q ) = trace ⁡ ( Q ~ ) = 0 , equation (2.4) implies that det ⁡ μ j = 1 , j = 1 , 2 , 3 . The eigenfunctions enjoy the same symmetry as Q and Q ~ :

where the subscripts denote the ( i , j ) -component of the matrix.

Regarding the boundary values, we assume that g j , h j ∈ H 1 ⁢ ( ℝ + ) and f j ∈ H 1 ⁢ ( [ 0 , L ] ) for j = 0 , 1 .

Figure 2

The domain D j and the oriented contour L j , j = 1 , … , 4 .

3 Spectral analysis