%0 Journal Article %T The elliptic sinh-Gordon equation in the half plane %A Hwang, Guenbo %J Journal of Nonlinear Sciences and Applications %D 2015 %V 8 %N 2 %@ ISSN 2008-1901 %F Hwang2015 %X 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 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 Riemann-Hilbert problem uniquely defined 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 modified Helmhotz equation. %9 journal article %R 10.22436/jnsa.008.02.08 %U http://dx.doi.org/10.22436/jnsa.008.02.08 %P 163--173 %0 Book %T Solitons, Nonlinear Evolution Equations and Inverse Scattering %A M. J. Ablowitz %A P. A. Clarkson %D 1991 %I Cambridge University Press %C Cambridge %F Ablowitz1991 %0 Journal Article %T Initial-boundary value problems for discrete evolution equations: discrete linear Schrödinger and integrable discrete nonlinear Schrödinger equations %A G. Biondini %A G. Hwang %J Inverse. Problems %D 2004 %V 24 %F Biondini2004 %0 Journal Article %T Initial-boundary-value problems for discrete linear evolution equations %A G. Biondini %A D.Wang %J IMA J. Appl. Math. %D 2010 %V 75 %F Biondini2010 %0 Journal Article %T Integrable two-dimensional generalisation of the sine- and sinh-Gordon equations %A M. Boiti %A J. J-P. Leon %A F. Pempinelli %J Inverse Problems %D 1987 %V 3 %F Boiti1987 %0 Journal Article %T A steepest descent method for oscillatory Riemann-Hilbert problems %A P. Deift %A X. Zhou %J Bull. Amer. Math. Soc. %D 1992 %V 26 %F Deift1992 %0 Journal Article %T New results in small dispersion KdV by an extension of the steepest descent method for Riemann-Hilbert problems %A P. Deift %A S. Venakides %A X. Zhou %J Int. Math. Res. Notices %D 1997 %V 6 %F Deift1997 %0 Journal Article %T A unified transform method for solving linear and certain nonlinear PDEs %A A. S. Fokas %J Proc. Roy. Soc. London A %D 1997 %V 453 %F Fokas1997 %0 Journal Article %T On the integrability of certain linear and nonlinear partial differential equations %A A. S. Fokas %J J. Math. Phys. %D 2000 %V 41 %F Fokas2000 %0 Journal Article %T Two dimensional linear PDEs in a convex polygon %A A. S. Fokas %J Proc. Roy. Soc. London A %D 2001 %V 457 %F Fokas2001 %0 Journal Article %T Integrable nonlinear evolution equations on the half-line %A A. S. Fokas %J Comm. Math. Phys. %D 2002 %V 230 %F Fokas2002 %0 Journal Article %T The generalized Dirichlet-to-Neumann map for certain nonlinear evolution PDEs %A A. S. Fokas %J Comm. Pure Appl. Math. %D 2005 %V LVIII %F Fokas2005 %0 Book %T A Unified Approach to Boundary Value Problems %A A. S. Fokas %D 2008 %I CBMS-NSF regional conference series in applied mathematics %C SIAM %F Fokas2008 %0 Journal Article %T The unified method: I. Non-linearizable problems on the half-line %A A. S. Fokas %A J. Lenells %J J. Phys. A: Math. Theor. %D 2012 %V 45 %F Fokas2012 %0 Journal Article %T The Dirichlet-to-Nemann map for the elliptic sine-Gordon equation %A A. S. Fokas %A B. Pelloni %J Nonlinearity %D 2012 %V 25 %F Fokas2012 %0 Journal Article %T The Fokas method: The Dirichlet to Neumann map for the sine-Gordon equation %A G. Hwang %J Stud. Appl. Math. %D 2014 %V 132 %F Hwang 2014 %0 Journal Article %T A perturbative approach for the asymptotic evaluation of the Neumann value corresponding to the Dirichlet datum of a single periodic exponential for the NLS %A G. Hwang %J J. Nonlinear Math. Phys. %D 2014 %V 21 %F Hwang2014 %0 Journal Article %T The modified Korteweg-de Vries equation on the half-line with a sine-wave as Dirichlet datum %A G. Hwang %A A. S. Fokas %J J. Nonlinear Math. Phys. %D 2013 %V 20 %F Hwang2013 %0 Journal Article %T Direct and inverse scattering problem associated with the elliptic sinh-Gordon equation %A M. Jaworski %A D. Kaup %J Inverse Problems %D 1990 %V 6 %F Jaworski1990 %0 Journal Article %T The elliptic sine-Gordon equation in a half plane %A B. Pelloni %A D. A. Pinotsis %J Nonlinearity %D 2010 %V 23 %F Pelloni2010