The elliptic sinh-Gordon equation in the half plane

Boundary value problems for the elliptic sinh-Gordon equation formulated in the half plane are studied by applying the so-called Fokas method. The method is a signi cant 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 Riemann-Hilbert problem uniquely de ned by the spectral functions. We also present the global relation associated with the elliptic sinh-Gordon 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 modi ed Helmhotz equation.