Backlund Transformations and Darboux Integrability for Nonlinear Wave Equations

Details Backlund Transformations and Darboux Integrability for Nonlinear Wave Equations Backlund Transformations and Darboux Integrability for Nonlinear Wave Equations

2007-07-30

arxiv.org/abs/0707.4408v2

Mathematics

Differential Geometry

48 pages; submitted, in revised form, to Asian J. Math

Scientific paper

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.

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