Higher dimensional integrable deformations of the modified KdV equation

TL;DR

This paper introduces a method to construct higher-dimensional integrable equations analogous to the modified KdV equation by using conservation forms and deformation mappings, expanding the understanding of integrability in multiple dimensions.

Contribution

It presents a novel approach to derive higher-dimensional integrable equations from the modified KdV equation using conservation forms and deformation mappings, including explicit Lax pairs.

Findings

Constructed higher-dimensional integrable equations from modified KdV.

Provided Lax pairs for the new higher-dimensional equations.

Extended the modified KdV hierarchy to higher dimensions.

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.