Multidimensional Integrable Deformations of Integrable PDEs

Abstract

In a recent series of papers by Lou et al., it was conjectured that higher dimensional integrable equations may be constructed by utilizing some conservation laws of (1 + 1)-dimensional systems. We prove that the deformation algorithm introduced in arXiv:2211.06844, applied to Lax integrable $(1+1)$-dimensional systems, produces Lax integrable higher imensional systems. The same property is enjoyed by the generalized deformation algorithm introduced in [Lou,Jia,Hao. Chinese Phys. Lett. 2023]; we present a novel example of a $(2+1)$-dimensional deformation of KdV equation obtained by generalized deformation. The deformed systems obtained by such procedure, however, pose a serious challenge because most of the mathematical structures that the $(1+1)$-dimensional systems possess is lost.