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Advances in Nonlinear Analysis

Editor-in-Chief: Radulescu, Vicentiu / Squassina, Marco


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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.

Keywords: Boundary value problems; elliptic PDEs; sinh-Gordon equation; integrable equations

MSC 2010: 47K15; 35Q55

1 Introduction

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

qxx+qyy=sinhq,(x,y)Ω,(1.1)

where Ω={(x,y)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 {aj(k),bj(k)}, j=1,2,3. These spectral functions are defined by boundary values {q(x,0),qy(x,0)}, {q(0,y),qx(0,y)} and {q(x,L),qy(x,L)}, respectively. It should be remarked that the spectral functions satisfy the global algebraic relation that involves all boundary values:

a3(k)=a1(k)a2(-k)+b1(k)b2(-k),k-,(1.2a)b3(k)e-2ω2(k)L=a2(k)b1(k)-a1(k)b2(k),k-.(1.2b)

(ii) Given boundary values {g0(x),g1(x)}, {f0(y),f1(y)} and {h0(x),h1(x)}, where

q(x,0)=g0(x),qy(x,0)=g1(x),(1.3a)q(0,y)=f0(y),qx(0,y)=f1(y),(1.3b)q(x,L)=h0(x),qy(x,L)=h1(x),(1.3c)

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]:

μx+ω1(k)[σ3,μ]=Q(x,y,k)μ,(2.1a)μy+ω2(k)[σ3,μ]=iQ~(x,y,k)μ,(2.1b)

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

ω1(k)=-12i(k-14k),ω2(k)=-12(k+14k),Q(x,y,k)=14(i2k(coshq-1)-(r+sinhq2k)r-sinhq2k-i2k(coshq-1))

with

Q~(x,y,k)=Q(x,y,-k),r(x,y)=iqx(x,y)+qy(x,y),σ3=(100-1),

and the matrix commutator defined by [σ3,A]=σ3A-Aσ3. We write σ^3A=[σ3,A] for the matrix commutator. It is convenient to use the notation

eσ^3ξA=eσ3ξAe-σ3ξ=(a11e2ξa12e-2ξa21a22).

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

d[e(ω1(k)x+ω2(k)y)σ^3μ(x,y,k)]=e(ω1(k)x+ω2(k)y)σ^3W(x,y,k),

where the differential 1-form W is given by

W(x,y,k)=Q(x,y,k)μ(x,y,k)dx+iQ~(x,y,k)μ(x,y,k)dy.(2.2)

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

μj(x,y,k)=I+(xj,yj)(x,y)e-(ω1(k)(x-ξ)+ω2(k)(y-η))σ^3Wj(ξ,η,k),(2.3)

where Wj is the differential form given in (2.2) with μj. 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 (xj,yj), j=1,2,3 (see Figure 1),

(x1,y1)=(0,L),(x2,y2)=(0,0),(x3,y3)=(,y).

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

μ1(x,y,k)=I+0xe-ω1(x-ξ)σ^3(Qμ1)(ξ,y,k)𝑑ξ-iyLe-(ω1(k)x+ω2(k)(y-η))σ^3(Q~μ1)(0,η,k)𝑑η,μ2(x,y,k)=I+0xe-ω1(k)(x-ξ)σ^3(Qμ2)(ξ,y,k)𝑑ξ+i0ye-(ω1(k)x+ω2(k)(y-η))σ^3(Q~μ2)(0,η,k)𝑑η,μ3(x,y,k)=I-xe-ω1(k)(x-ξ)σ^3(Qμ3)(ξ,y,k)𝑑ξ.

The three distinct points for the eigenfunctions μj{\mu_{j}}, j=1,2,3{j=1,2,3}.
Figure 1

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

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

Reω1(k)<0for Imk>0,Reω2(k)<0for Rek>0.

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

D1={k:Reω1(k)<0}{k:Reω2(k)<0},D2={k:Reω1(k)<0}{k:Reω2(k)>0},D3={k:Reω1(k)>0}{k:Reω2(k)>0},D4={k:Reω1(k)>0}{k:Reω2(k)<0}.

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

  • μ1(x,y,k) is analytic and bounded for k(D2,D4),

  • μ2(x,y,k) is analytic and bounded for k(D1,D3),

  • μ3(x,y,k) is analytic and bounded for k(D3D4,D1D2).

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

μ1=(μ1(2),μ1(4)),μ2=(μ2(1),μ2(3)),μ3=(μ3(34),μ3(12)),

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

μj(x,y,k)=I+O(1k)as k.(2.4)

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

μ11(x,y,k)=μ22(x,y,-k),μ21(x,y,k)=-μ12(x,y,-k),(2.5)

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

Regarding the boundary values, we assume that gj, hjH1(+) and fjH1([0,L]) for j=0,1.

The domain Dj{D_{j}} and the oriented contour Lj{L_{j}}, j=1,…,4{j=1,\ldots,4}.
Figure 2

The domain Dj and the oriented contour Lj, j=1,,4.

3 Spectral analysis

3.1 Spectral functions

Note that the matrix eigenfunctions μ1, μ2 and μ3 are fundamental solutions of the Lax pair (2.1) and μ1(0,L,k)=I and μ2(0,0,k)=I. Hence, the eigenfunctions are related by the so-called spectral functions, also known as the scattering matrices, S1(k), S2(k) and S3(k):

μ3(x,y,k)=μ2(x,y,k)e-(ω1(k)x+ω2(k)y)σ^3S1(k),k(+,-),(3.1a)μ1(x,y,k)=μ2(x,y,k)e-(ω1(k)x+ω2(k)y)σ^3S2(k),k(i+,i-),(3.1b)μ3(x,y,k)=μ1(x,y,k)e-(ω1(k)x+ω2(k)(y-L))σ^3S3(k),k(-,+).(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

S1(k)=μ3(0,0,k),S2(k)=μ1(0,0,k),S3(k)=μ3(0,L,k).

From the symmetry (2.5) of the eigenfunctions, we write the spectral functions S1(k), S2(k) and S3(k) as

Sj(k)=(aj(k)-bj(-k)bj(k)aj(-k)),j=1,2,3.

Since detSj(k)=1, j=1,2,3, we find the following identities:

aj(k)aj(-k)+bj(k)bj(-k)=1,j=1,2,3.

Motivated by the above arguments, we define

Φ1(x,k)=μ3(x,0,k),Φ2(y,k)=μ1(0,y,k),Φ3(x,k)=μ3(x,L,k).

More specifically, the functions Φj, j=1,2,3, satisfy the integral equations

Φ1(x,k)=I-xe-ω1(k)(x-ξ)σ^3(Q0Φ1)(ξ,k)𝑑ξ,k(-,+), 0x<,Φ2(y,k)=I-iyLe-ω2(k)(y-η)σ^3(Q~0Φ2)(η,k)𝑑η,k, 0yL,Φ3(x,k)=I-xe-ω1(k)(x-ξ)σ^3(QLΦ3)(ξ,k)𝑑ξ,k(-,+), 0x<,

where Q0(x,k)=Q(x,0,k), Q~0(y,k)=Q(0,y,-k) and QL(x,k)=Q(x,L,k). Note that

Sj(k)=Φj(0,k),j=1,2,3,

which immediately imply that the spectral functions aj(k) and bj(k), j=1,3, are analytic and bounded for Imk<0, while the spectral functions a2(k) and b2(k) are analytic and bounded for k except for the essential singularities k=0 and . Moreover, due to the symmetry (2.5), the functions Φj, j=1,2,3, can also be written as

Φj(x,k)=(Aj(x,k)-Bj(x,-k)Bj(x,k)Aj(x,-k)),j=1,3,Φ2(y,k)=(A2(y,k)-B2(y,-k)B2(y,k)A2(y,-k)).

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

Definition 3.1.

Given q(x,0)=g0(x) and qy(x,0)=g1(x), the map

{g0(x),g1(x)}{a1(k),b1(k)}

is defined by

a1(k)=1-140{i2k(coshg0(ξ)-1)A1(ξ,k)-(ig˙0(ξ)+g1(ξ)+12ksinhg0(ξ))B1(ξ,k)}dξ,Imk<0,(3.2a)b1(k)=-140e-2ω1(k)ξ{(ig˙0(ξ)+g1(ξ)-12ksinhg0(ξ))A1(ξ,k)-i2k(coshg0(ξ)-1)B1(ξ,k)}dξ,Imk<0,(3.2b)

where the functions A1 and B1 are the solutions of the x-part of the Lax pair (2.1a) with y=0, i.e., A1 and B1 solve the following system of ordinary differential equations:

A1x=14[i2k(coshg0(x)-1)A1-(ig˙0(x)+g1(x)+12ksinhg0(x))B1],B1x-2ω1(k)B1=14[(ig˙0(x)+g1(x)-12ksinhg0(x))A1-i2k(coshg0(x)-1)B1]

with limx(A1,B1)=(1,0).

Definition 3.2.

Given q(0,y)=f0(y) and qx(0,y)=f1(y), the map

{f0(y),f1(y)}{a2(k),b2(k)}

is defined by

a2(k)=1+i40L{i2k(coshf0(η)-1)A2(η,k)+(if1(η)+f˙0(η)-12ksinhf0(η))B2(η,k)}dη,k,(3.3a)b2(k)=-i40Le-2ω2(k)η{(if1(η)+f˙0(η)+12ksinhf0(η))A2(η,k)+i2k(coshf0(η)-1)B2(η,k)}dη,k,(3.3b)

where the functions A2 and B2 are the solutions of the y-part of the Lax pair (2.1b) with x=0, i.e., A2 and B2 solve the following system of ordinary differential equations:

A2y=-i4[i2k(coshf0(y)-1)A2+(if0(y)+f˙1(y)-12ksinhf0(y))B2],B2y-2ω2(k)B2=i4[(if0(y)+f˙1(y)+12ksinhf0(y))A2+i2k(coshf0(y)-1)B2]

with limy(A2,B2)=(1,0).

Definition 3.3.

Given q(x,L)=h0(x) and qy(x,L)=h1(x), the map

{h0(x),h1(x)}{a3(k),b3(k)}

is defined by

a3(k)=1-140{i2k(coshh0(ξ)-1)A3(ξ,k)-(ih˙0(ξ)+h1(ξ)+12ksinhh0(ξ))B3(ξ,k)}dξ,Imk<0,(3.4a)b3(k)=-140e-2ω1(k)ξ{(ih˙0(ξ)+h1(ξ)-12ksinhh0(ξ))A3(ξ,k)-i2k(coshh0(ξ)-1)B3(ξ,k)}dξ,Imk<0,(3.4b)

where the functions A3 and B3 are the solutions of the x-part of the Lax pair (2.1a) with y=L, i.e., A3 and B3 solve the following system of ordinary differential equations:

A3x=14[i2k(coshh0(x)-1)A3-(ih˙0(x)+h1(x)+12ksinhh0(x))B3],B3x-2ω1(k)B3=14[(ih˙0(x)+h1(x)-12ksinhh0(x))A3-i2k(coshh0(x)-1)B3]

with limx(A3,B3)=(1,0).

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

S3(k)=μ2(0,L,k)e-ω2(k)Lσ^3S1(k),I=μ2(0,L,k)e-ω2(k)Lσ^3S2(k).

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

eω2(k)Lσ^3S3(k)=S2-1(k)S1(k).(3.5)

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

a3(k)=a1(k)a2(-k)+b1(k)b2(-k),k-,(3.6a)b3(k)e-2ω2(k)L=a2(k)b1(k)-a1(k)b2(k),k-.(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

{a1(k),b1(k)}{q(x,0),qy(x,0)}

to the map defined in Definition 3.1 is given by

coshq(x,0)=1-8ilimkkM11x(x)-8limk(kM21(x))2,iqx(x,0)+qy(x,0)=-4ilimkkM21(x),

where M(x) is the solution of the matrix Riemann–Hilbert problem

M-(x)(x,k)=M+(x)(x,k)J(x)(x,k),k(3.7)

with the jump matrix J(x) given by

J(x)(x,k)=(1b1(-k)a1(k)e-2ω1(k)xb1(k)a1(-k)e2ω1(k)x1a1(k)a1(-k)),k.(3.8)

Proof.

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

μ3(x,0,k)=μ2(x,0,k)e-ω1(k)xσ^3S1(k),k(+,-), 0x<.

Note that the eigenfunction μ2(1)(x,0,k) is analytic and bounded for kD1D2 and μ2(3)(x,0,k) is analytic and bounded for kD3D4. Thus, we formulate the matrix Riemann–Hilbert problem (3.7) with the jump matrix J(x)(x,k) given by (3.8), where the sectionally meromorphic functions M±(x) are defined by

M+(x)(x,k)=(μ2(1)(x,0,k)a1(-k),μ3(12)(x,0,k)),Imk>0,M-(x)(x,k)=(μ3(34)(x,0,k),μ2(3)(x,0,k)a1(k)),Imk<0

with detM±(x)=1 and M±(x)=I+O(1k) as k. The remaining proof follows arguments similar to arguments in [14]. ∎

Proposition 3.5.

The inverse map

{a2(k),b2(k)}{q(0,y),qx(0,y)}

to the map defined in Definition 3.2 is given by

coshq(0,y)=1+8limkkM11y(y)+8limk(kM21(y))2,iqx(0,y)+qy(0,y)=-4ilimkkM21(y),

where M(y) is the solution of the matrix Riemann–Hilbert problem

M-(y)(y,k)=M+(y)(y,k)J(y)(y,k),ki

with the jump matrix J(y) given by

J(y)(y,k)=(1b2(-k)a2(k)e-2ω2(k)yb2(k)a2(-k)e2ω2(k)y1a2(k)a2(-k)),ki.(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(y)(y,k) given in (3.10) and the sectionally meromorphic functions M±(y) given by

M+(y)(y,k)=(μ2(1)(0,y,k)a2(-k),μ1(4)(0,y,k)),Rek>0,M-(y)(y,k)=(μ1(2)(0,y,k),μ2(3)(0,y,k)a2(k)),Rek<0

with detM±(y)=1 and M±(y)=I+O(1k) as k. Note that the eigenfunction μ2(1)(0,y,k) is analytic and bounded for kD1D4 and the function μ2(3)(0,y,k) is analytic and bounded for kD2D3. ∎

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

Proposition 3.6.

The inverse map

{a3(k),b3(k)}{q(x,L),qy(x,L)}

to the map defined in Definition 3.3 is given by

coshq(x,L)=1-8ilimkkM11x(L)-8limk(kM21(L))2,iqx(x,L)+qy(x,L)=-4ilimkkM21(L),

where M(L) is the solution of the matrix Riemann–Hilbert problem

M-(L)(x,k)=M+(L)(x,k)J(L)(x,k),k(3.12)

with the jump matrix J(L) given by

J(L)(x,k)=(1b3(-k)a3(k)e-2ω1(k)xb3(k)a3(-k)e2ω1(k)x1a3(k)a3(-k)),k.(3.13)

4 Riemann–Hilbert problem

In this section, we first formulate the matrix Riemann–Hilbert problem under the assumption that the solution q(x,y) of (1.1) exists and then we discuss the existence of the solution q(x,y) 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-(x,y,k)=M+(x,y,k)J(x,y,k),k,(4.1)

where the oriented contours =L1L2L3L4 are given by (cf. Figure 2)

L1=D1D2,L2=D2D3,(4.2a)L3=D3D4,L4=D4D1,(4.2b)

and the jump matrices are defined by

J1(x,y,k)=(10b2(k)a1(-k)a3(-k)e2θ(x,y,k)1),kL1,(4.3a)J2(x,y,k)=(a2(k)a1(k)a3(-k)-b1(-k)a1(k)e-2θ(x,y,k)-b3(k)a3(-k)e2(θ(x,y,k)-ω2(k)L)1),kL2,(4.3b)J3(x,y,k)=(1-b2(-k)a1(k)a3(k)e-2θ(x,y,k)01),kL3,(4.3c)J4(x,y,k)=J1J2-1J3=(1b3(-k)a3(k)e-2(θ(x,y,k)-ω2(k)L)b1(k)a1(-k)e2θ(x,y,k)a2(-k)a1(-k)a3(k)),kL4(4.3d)

with θ(x,y,k)=ω1(k)x+ω2(k)y. The matrix-valued functions M± are sectionally meromorphic and defined as follows:

M+(x,y,k)={(μ2(1)a1(-k),μ3(12))for kD1,(μ3(34),μ2(3)a1(k))for kD3,M-(x,y,k)={(μ1(2)a3(-k),μ3(12))for kD2,(μ3(34),μ1(4)a3(k))for kD4.

Note that detM±=1 and M±=I+O(1k) as k. Indeed, using the global relation (3.6), we find

a2(k)-b1(-k)b3(k)e-2ω2(k)L=a1(k)a3(-k),

which implies that det(J2)=det(J4)=1.

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

M+(x,y,k)-M-(x,y,k)=M+(x,y,k)J~(x,y,k).

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

M(x,y,k)=I+12iπM+(x,y,k)J~(x,y,k)dkk-k.

Expanding the integrand for k, we find

M(x,y,k)=I-12iπkM+(x,y,k)J~(x,y,k)𝑑k+O(1k2),k.

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

M(x,y,k)=I+M(1)(x,y)k+M(2)(x,y)k2+O(1k3),k.(4.4)

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

iqx(x,y)+qy(x,y)=-4iM21(1)(x,y)(4.5)

and the (1,1)-component at O(1k) implies

M11x(1)(x,y)=-18i(coshq(x,y)-1)-14(iqx(x,y)+qy(x,y))M21(1)(x,y).

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

coshq(x,y)=1-8iM11x(1)(x,y)-8(M21(1))2.

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

coshq(x,y)=1+8M11y(1)(x,y)+8(M21(1))2.

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

Theorem 4.1.

Let the functions gj(x), hj(x)H1(R+) and fj(y)H1([0,L]) , j=0,1, be given with the sufficiently small H1 norms. Let the functions aj(k), bj(k), j=1,2,3, 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 aj and bj, j=1,2,3, satisfy the global relation (3.6). Let M(x,y,k) be the solution of the following matrix Riemann–Hilbert (RH) problem

M-(x,y,k)=M+(x,y,k)J(x,y,k),k,(4.6)

where det(M±)=1, M±=I+O(1k) as k, the oriented contours 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(x,y) defined by

iqx+qy=-4ilimkkM21,coshq(x,y)=1-8ilimkkM11x-8limk(kM21)2(4.7)

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

q(x,0)=g0(x),qy(x,0)=g1(x),(4.8a)q(0,y)=f0(y),qx(0,y)=f1(y),(4.8b)q(x,L)=h0(x),qy(x,L)=h1(x).(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(x,y) defined in (4.7) solves the elliptic sinh-Gordon equation (1.1).

The proof that q(x,y) 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

M(L)(x,k)={M(x,L,k)J1(x,L,k)for kD1,M(x,L,k)for kD2,M(x,L,k)F(x,k)for kD3,M(x,L,k)J3-1(x,L,k)F(x,k)for kD4,(4.9)

where

F(x,k)=(1-b2(-k)a1(k)a3(k)e-2(ω1(k)x+ω2(k)L)01).

It is convenient to denote M(x,L,k) and M(L)(x,k) for kDj, j=1,,4, by Mj(x,L,k) and Mj(L)(x,k), respectively. Then, equations (4.6) and (4.9) can be written as

M2(x,L,k)=M1J1(x,L,k),M2(x,L,k)=M3J2(x,L,k),(4.10a)M4(x,L,k)=M3J3(x,L,k),M4(x,L,k)=M1J4(x,L,k),(4.10b)

and

M1(L)(x,k)=M1(x,L,k)J1(x,L,k),M2(L)(x,k)=M2(x,L,k),(4.11a)M3(L)(x,k)=M3(x,L,k)F(x,k),M4(L)(x,k)=M4(x,L,k)J3-1(x,L,k)F(x,k).(4.11b)

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

M2(L)(x,k)=M1(L)(x,k),M3(L)(x,k)=M2(L)(x,k)J2-1(x,L,k)F(x,k),M4(L)(x,k)=M3(L)(x,k),M4(L)(x,k)=M1(L)(x,k)J2-1(x,L,k)F(x,k).

Note that no jumps occur along the contours L1 and L3 and that J2-1(x,L,k)F(x,k)=J(L)(x,k), where J(L)(x,k) is given in (3.13). Thus, we define

M(L)(x,k)=M+(L)(x,k),kD1D2,M(L)(x,k)=M-(L)(x,k),kD3D4

and then equations (4.10) are equivalent to the Riemann–Hilbert problem (3.12) with the jump matrix J(L)(x,k). Thus, evaluating (4.7) at y=L, we prove that the function q(x,y) 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 {aj(k),bj(k)}, j=1,2,3, can be determined by the boundary values {q(x,0),qy(x,0)}, {q(0,y),qx(0,y)} and {q(x,L),qy(x,L)}. 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.

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About the article

Received: 2016-09-19

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, DOI: https://doi.org/10.1515/anona-2016-0206.

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© 2019 Walter de Gruyter GmbH, Berlin/Boston. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0

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