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 JHEP03(2023)018, applied to Lax integrable (1 + 1)-dimensional systems, produces Lax integrable higher dimensional systems. The same property is enjoyed by the generalized deformation algorithm introduced in [Chinese Phys. Lett 40(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.