Backlund Transformations and Darboux Integrability for Nonlinear Wave Equations

TL;DR

This paper establishes a precise link between Backlund transformations and Darboux integrability for second-order hyperbolic Monge-Ampere equations, providing explicit transformations for several integrable cases.

Contribution

It proves that such equations are connected to the wave equation via Backlund transformations if and only if they are Darboux integrable, and constructs explicit transformations for key examples.

Findings

Backlund transformations correspond to Darboux integrability for these equations.

Explicit transformations are provided for several classical Darboux-integrable equations.

The paper develops a method to construct Backlund transformations from Darboux integrability.

Abstract

We prove that second-order hyperbolic Monge-Ampere equations for one function of two variables are connected to the wave equation by a Backlund transformation if and only if they are integrable by the method of Darboux at second order. One direction of proof, proving Darboux integrability, follows the implications of the wave equation for the invariants of the G-structure associated to the Backlund transformation. The other direction constructs Backlund transformations for Darboux integrable equations as solutions of an involutive exterior differential system. Explicit transformations are given for several equations on the Goursat-Vessiot list of Darboux-integrable equations.