 10128 Alice MikikitsLeitner and Gerald Teschl
 LongTime Asymptotics of Perturbed FiniteGap Kortewegde Vries Solutions
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Aug 22, 10

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Abstract. We apply the method of nonlinear steepest descent to compute the longtime asymptotics of
solutions of the Kortewegde Vries equation which are decaying perturbations of a quasiperiodic
finitegap 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 finitegap solutions which are in the same isospectral class as the background solution. In the
oscillatory region the solution can be described by a modulated finitegap solution plus a decaying
dispersive tail. The modulation is given by 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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