On the Cauchy Problem for the Korteweg-de Vries Equation with Steplike Finite-Gap Initial Data I. Schwartz-Type Perturbations

We solve the Cauchy problem for the Korteweg-de Vries equation with initial conditions which are steplike Schwartz-type perturbations of finite-gap potentials under the assumption that the respective spectral bands either coincide or are disjoint.Comment: 29 page

Riemann-Hilbert problem to the modified Korteveg-de Vries equation: Long-time dynamics of the steplike initial data

We consider the modified Korteveg de Vriez equation on the whole line. Initial data is real and step-like, i.e. $q(x,0)=0$ for $x\geq0$ and $q(x,0)=c$ for $x<0$, where c is arbitrary real number. The goal of this paper is to study the asymptotic behavior of the initial-value problem's solution by means of the asymptotic behavior of the some Riemann\textendash Hilbert problem. In this paper we show that the solution of this problem has different asymptotic behavior in different regions. In the region $x4c^2t$ the solution is tend to $c$ and 0 correspondingly. In the region $-6c^2t<x<4c^2t$ the solution takes the form of a modulated elliptic wave.Comment: 31 pages, 8 figure