Séminaire EDP-Analyse ICJ

Asymptotic soliton-like solutions to the Korteweg-de Vries equation with variable coefficients and singular perturbation

by Yuliia Samoilenko (Taras Shevchenko National University of Kyiv)

Europe/Paris
Fokko (ICJ)

Fokko

ICJ

Description

The talk deals with the Korteweg-de Vries equation with variable coefficients and a small parameter at the highest derivative. These problems arise while studying wave processes in media with variable characteristics and a small dispersion.

An approach to the construction of asymptotic soliton-like solution of the equation is presented. The approach is based on the non-linear WKB technique. The algorithm for constructing the asymptotic soliton-like solution is described in detail, as well as its justification.

The constructed asymptotic solution is similar to the soliton solution of the Korteweg-de Vries equation $ u_t+uu_x+u_{xxx}=0 $. The solution contains regular and singular parts of asymptotic. The regular part of the solution forms the background of the wave process, and its singular part reflects the features associated with the specific properties of the soliton. Differential equations for the terms of the asymptotics are given and their solving is discussed.

The problem of the existence of a global asymptotic soliton-like solution of the equation is also considered.

The obtained results are illustrated by examples.