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

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.