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.
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