The elliptic sinhGordon equation in the half plane
Authors
Guenbo Hwang
 Department of Mathematics, Daegu University, Gyeongsan Gyeongbuk 712714, Korea.
Abstract
Boundary value problems for the elliptic sinhGordon equation formulated in the half plane are studied
by applying the socalled Fokas method. The method is a significant extension of the inverse scattering
transform, based on the analysis of the Lax pair formulation and the global relation that involves all known
and unknown boundary values. In this paper, we derive the formal representation of the solution in terms
of the solution of the matrix RiemannHilbert problem uniquely defined by the spectral functions. We also
present the global relation associated with the elliptic sinhGordon equation in the half plane. We in turn
show that given appropriate initial and boundary conditions, the unique solution exists provided that the
boundary values satisfy the global relation. Furthermore, we verify that the linear limit of the solution
coincides with that of the linearized equation known as the modified Helmhotz equation.
Keywords
 Boundary value problems
 elliptic PDEs
 sinhGordon equation
 integrable equation.
MSC
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