Solutions to the modified Korteweg-de Vries equation

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.