Notes on solutions in Wronskian form to soliton equations: KdV-type

TL;DR

This paper reviews Wronskian and Casoratian solutions to KdV-type soliton equations, providing explicit solutions, limit relations, and a four-step Wronskian technique, with examples including KdV, Toda lattice, and KP equations.

Contribution

It introduces explicit general solutions to condition equations for Wronskian/Casoratian entries and formulates a four-step Wronskian solution method for KdV-type equations.

Findings

Derived explicit solutions for matrices commuting with Jordan blocks.

Established limit relations between different solution forms.

Presented a four-step Wronskian solution technique.

Abstract

This paper can be an overview on solutions in Wronskian/Casoratian form to soliton equations with KdV-type bilinear forms. We first investigate properties of matrices commuting with a Jordan block, by which we derive explicit general solutions to equations satisfied by Wronskian/Casoratian entry vectors, which we call condition equations. These solutions are given according to the coefficient matrix in the condition equations taking diagonal or Jordan block form. Limit relations between these different solutions are described. We take the KdV equation and the Toda lattice to serve as two examples for solutions in Wronskian form and Casoratian form, respectively. We also discuss Wronskian solutions for the KP equation. Finally, we formulate the Wronskian technique as four steps.