Solutions to the modified Korteweg–de Vries equation

Abstract

This is a continuation of [Notes on solutions in Wronskian form to soliton equations: Korteweg–de Vries-type, 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 fact makes the analysis of the modified Korteweg–de Vries to be different from the case of the Korteweg–de Vries equation. To derive complete solution expressions for the matrix differential equation set, we introduce an auxiliary matrix to deal with the complex operation. As a result, the obtained solutions to the modified Korteweg–de Vries equation are categorized into two types: solitons and breathers, together with their limit cases. Besides, we give rational solutions to the modified Korteweg–de Vries equation in Wronskian form. This is derived with the help of a Galilean transformed version of the modified Korteweg–de Vries equation. Finally, typical dynamics of the obtained solutions are analyzed and illustrated. We also list out the obtained solutions and their corresponding basic Wronskian vectors in the conclusion part.