D-bar Problems and the Solution of the Sine-gordon Equation in Time-dependent Convex Domains
نویسنده
چکیده
We solve a class of initial boundary value problems posed in a time-dependent 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 identiies all well-posed boundary value problems in the given domain and yields a rigorous proof of the existence of the solution.
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تاریخ انتشار 2007