Alice Mikikits-Leitner and Gerald Teschl
Long-Time Asymptotics of Perturbed Finite-Gap Korteweg-de Vries Solutions
(132K, LaTeX2e)
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 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.