Lax pairs for higher-dimensional evolution PDEs and a 3+1 dimensional integrable generalization of the Burgers equation

Abstract

We construct Lax pairs for general d + 1 dimensional evolution equations in the form ut = F[u], where F[u] depends on the field u and its space derivatives. As an example we study a 3 + 1 dimensional integrable generalization of the Burgers equation. We develop a procedure to generate some exact solutions of this equation, based on a class of discrete symmetries of the Darboux transformation type. In the one-dimensional limit, these symmetries reduce to the Cole-Hopf substitution for the Burgers equation. It is discussed how the technique can be used to construct exact solutions for higher-dimensional evolution PDEs in a broader context. © 2006 American Mathematical Society.