Wronskian solutions of integrable systems

TL;DR

This paper reviews the Wronskian technique for constructing solutions to integrable systems, highlighting its historical development, mathematical foundations, and applications to equations like KdV and mKdV.

Contribution

It provides a comprehensive review of the Wronskian method for integrable systems, including new examples and detailed explanations of the underlying linear differential equations.

Findings

Wronskian solutions effectively solve various integrable equations.

The technique simplifies verifying bilinear equations via Plücker relations.

Examples include solutions for KdV, mKdV, and related hierarchies.

Abstract

Wronski determinant (Wronskian) provides a compact form for $\tau$-functions that play roles in a large range of mathematical physics. In 1979 Matveev and Satsuma, independently, obtained solutions in Wronskian form for the Kadomtsev-Petviashvili equation. Later, in 1981 these solutions were constructed from Sato's approach. Then in 1983, Freeman and Nimmo invented the so-called Wronskian technique, which allows directly verifying bilinear equations when their solutions are given in terms of Wronskians. In this technique the considered bilinear equation is usually reduced to the Pl\"ucker relation on Grassmannians, and finding solutions of the bilinear equation is transferred to find a Wronskian vector that is defined by a linear differential equation system. General solutions of such differential equation systems can be constructed by means of triangular Toeplitz matrices. In this monograph we review the Wronskian technique and solutions in Wronskian form, with supporting instructive examples, including the Korteweg-de Vries (KdV) equation, the modified KdV equation, the Ablowitz-Kaup-Newell-Segur hierarchy and reductions, and the lattice potential KdV equation. (Dedicated to Jonathan J C Nimmo).