J. d'Analyse Math. 116, 163-218 (2012) [DOI: 10.1007/s11854-012-0005-7]

Long-Time Asymptotics of Perturbed Finite-Gap Korteweg-de Vries Solutions

Alice Mikikits-Leitner and Gerald Teschl

We apply the method of nonlinear steepest descent to compute the long-time asymptotics of solutions of the Korteweg-de Vries equation which are decaying perturbations of a quasi-periodic finite-gap background solution. We compute a nonlinear dispersion relation and show that the x/t plane splits into g+1 soliton regions which are interlaced by g+1 oscillatory regions, where g+1 is the number of spectral gaps.

In the soliton regions the solution is asymptotically given by a number of solitons travelling on top of finite-gap solutions which are in the same isospectral class as the background solution. In the oscillatory region the solution can be described by a modulated finite-gap solution plus a decaying dispersive tail. The modulation is given by a phase transition on the isospectral torus and is, together with the dispersive tail, explicitly characterized in terms of Abelian integrals on the underlying hyperelliptic curve.

MSC2000: Primary 35Q53, 37K40; Secondary 35Q15, 37K20
Keywords: Riemann-Hilbert problem, Korteweg-de Vries equation, Solitons

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