The ‘breather-train’ solution of the Sine-Gordon equation with the ‘periodic wave–zero’ boundary conditions

Article type: Research Article

Authors: Novokshenov, V.Yu.

Affiliations: Institute of Mathematics, Ufa Science Center, Academy of Sciences of Russia, 450000, Chernyshevsky str. 112, Ufa, Russia, E-mail: [email protected]

Abstract: The large-time asymptotic solution of the Sine-Gordon equation is considered describing the decay of a step-like initial condition with nonidentical finite-gap limits as x→±∞. The leading term of asymptotics is found as a finite-gap quasi periodic solution with its phase vectors modulated by a slow space-like variable. The Whitham equations describing the modulation are studied in detail for the case of complex-valued group velocities, which arise for the problem above. The existence and uniqueness theorems for these equations are proved and a complete picture of the. Whitham deformation is given for the case of a periodic one-gap boundary condition at the left infinity and zero at the right infinity.

DOI: 10.3233/ASY-1996-12201

Journal: Asymptotic Analysis, vol. 12, no. 2, pp. 77-89, 1996