Solutions to the modified Korteweg-de Vries equation

TL;DR

This paper reviews and derives explicit Wronskian solutions for the modified Korteweg-de Vries equation, including solitons, breathers, rational solutions, and their dynamics, using an auxiliary matrix approach.

Contribution

It introduces a novel auxiliary matrix method to solve the matrix differential equations for Wronskian solutions of the mKdV equation, expanding solution types and analytical techniques.

Findings

Complete solution expressions for solitons and breathers.

Rational solutions derived via Galilean transformation.

Illustration of solution dynamics and behaviors.

Abstract

This is a continuation of Ref.[1](arXiv:nlin.SI/0603008). In the present paper we review solutions to the modified Korteweg-de Vries equation in terms of Wronskians. The Wronskian entry vector needs to satisfy a matrix differential equation set which contains complex operation. This is different from the case of the Korteweg-de Vries equation. We introduce an auxiliary matrix to deal with the complex operation and then we are able to give complete solution expressions for the matrix differential equation set. The obtained solutions to the modified Korteweg-de Vries equation can simply be categorized by two types: solitons and breathers, together with their limit cases. Besides, we give rational solutions to the modified Korteweg-de Vries equation in Wromskian form. This is derived with the help of the Galilean transformed modified Korteweg-de Vries equation. Finally, typical dynamics of the obtained solutions is analyzed and illustrated. We list out the obtained solutions and their corresponding basic Wronskian vectors in the conclusion part.