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Long-time asymptotics of perturbed finite-gap Korteweg-de Vries solutions

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Abstract

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.

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Authors and Affiliations

  1. Faculty of Mathematics, University of Vienna, Nordbergstrasse 15, 1090, Wien, Austria

    Alice Mikikits-Leitner & Gerald Teschl

  2. International Erwin Schrödinger Institute for Mathematical Physics, Boltzmanngasse 9, 1090, Wien, Austria

    Gerald Teschl

Authors
  1. Alice Mikikits-Leitner
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  2. Gerald Teschl
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Correspondence to Alice Mikikits-Leitner.

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Research supported by the Austrian Science Fund (FWF) under Grant No. Y330.

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Mikikits-Leitner, A., Teschl, G. Long-time asymptotics of perturbed finite-gap Korteweg-de Vries solutions. JAMA 116, 163–218 (2012). https://doi.org/10.1007/s11854-012-0005-7

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  • DOI: https://doi.org/10.1007/s11854-012-0005-7

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