Temporal asymptotics for soliton equations in problems with step initial conditions |
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Authors: | V Yu Novokshenov |
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Institution: | (1) Mat. Inst., VTs, Ufa |
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Abstract: | In this survey, we present modern approaches to the construction and justification of large time asymptotics for solutions of main soliton equations with step-like initial condition whose boundary conditions as x ± are finite-gap, quasi-periodic solutions. The principal term of the asymptotic is also a finite-gap, quasi-periodic solution whose phase vectors are modulated with respect to the slow space-like variable. The Whitham equations describing this modulation are studied in detail. For the KdV equations, we construct and justify the principal term of the asymptotic for arbitrary finite-gap boundary conditions. By examining the sine-Gordon equation, we study the case of boundary conditions with complex-valued, self-conjugated quasi-periods. We prove the existence and uniqueness theorems in the case of the complex-valued group velocities that appear in the Whitham equations in the case considered. We present a complete picture (uniform in x) of the Whitham deformation for the case of 1-gap boundary conditions in the sine-Gordon equation.Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 5, Asymptotic Methods, 2003. |
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