Higher dimensional integrable deformations of the modified KdV equation

Abstract

The derivation of nonlinear integrable evolution partial differential equations in higher dimensions has always been the holy grail in the field of integrability. The well-known modified KdV equation is a prototypical example of integrable evolution equations in one spatial dimension. Do there exist integrable analogs of modified KdV equation in higher spatial dimensions? In what follows, we present a positive answer to this question. In particular, rewriting the (1+1)-dimensional integrable modified KdV equation in conservation forms and adding deformation mappings during the process allow one to construct higher dimensional integrable equations. Further, we illustrate this idea with examples from the modified KdV hierarchy, also present the Lax pairs of these higher dimensional integrable evolution equations.