On stationary solutions of KdV and mKdV equations

TL;DR

This paper constructs periodic stationary solutions for the KdV and mKdV equations on bounded intervals, establishing exact conditions relating interval length and equation coefficients for their existence.

Contribution

It introduces a method based on conservative systems to explicitly construct solutions and derive necessary and sufficient conditions for their existence.

Findings

Solutions are periodic and explicitly constructed.

Exact relations between interval length and coefficients are established.

Conditions for existence of nontrivial solutions are provided.

Abstract

Stationary solutions on a bounded interval for an initial-boundary value problem to Korteweg--de~Vries and modified Korteweg--de~Vries equation (for the last one both in focusing and defocusing cases) are constructed. The method of the study is based on the theory of conservative systems with one degree of freedom. The obtained solutions turn out to be periodic. Exact relations between the length of the interval and coefficients of the equations which are necessary and sufficient for existence of nontrivial solutions are established.