Solution of the Dirichlet boundary value problem for the Sine-Gordon equation

Abstract

The sine-Gordon equation in light cone coordinates is solved when Dirichlet conditions on the L-shape boundaries of the strip [0,T]X[0,infinity) are prescribed in a class of functions that vanish (mod 2 pi) for large x at initial time. The method is based on the inverse spectral transform (IST) for the Schroedinger spectral problem on the semi-line solved as a Hilbert boundary value problem. Contrarily to what occurs when using the Zakharov-Shabat eigenvalue problem, the spectral transform is regular and in particular the discrete spectrum contains a finite number of eigenvalues (and no accumulation point).