D-bar problems and the solution of the sine-Gordon equation in time-dependent convex domains
Beatrice Pelloni · 2002
: We solve a class of initial boundary value problems posed in a timedependent convex domain for the sine-Gordon equation and for its linearized version. We give an explicit integral representation of the solution by using the Fokas transform method; this representation, which has an explicit exponential x and t dependence, is obtained by solving a d-bar problem in the complex plane. This d-bar problem is scalar for the linear equation, and matrix-valued for the nonlinear equation. In the linear case, the method also identifies all well-posed boundary value problems in the given domain and yields a rigorous proof of the existence of the solution. AMS (MOS) Subject Classification. 35K60, 35K57 1. INTRODUCTION The inverse scattering (or inverse spectral) method is a nonlinear Fourier transform method for solving initial value problems for integrable nonlinear PDE's. The Fokas transform method is an extension of the inverse scattering method for studying initial boundary value problem...