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 { t∈[0, T]}∪{ x∈[0,∞]} are prescribed in a class of functions that vanish (mod2 π) as x→∞ at initial time. The method is based on the inverse spectral transform (IST) for the Schrödinger spectral problem on the semi-line x>0 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).