Soliton Solutions of Non-linear Evolution Equations

Abstract

Soliton solutions of three typical non-linear evolutions—the Kadomtsev–Petviashvili equation, the non-linear Schrödinger equation and the Davey–Stewartson equation—are shown to be representable in terms of Wronskian determinants. Using this more compact notation, it is demonstrated how the evolution equations and the auto-Bäcklund transformations are satisfied by the soliton solutions. Difficulties arise in the Davey-Stewartson equation where the phase variables depend explicitly on the number of solitons and Bäcklund rather than auto-Bäcklund transformations must be invoked.