Differential reductions of the Kadomtsev-Petviashvili equation and associated higher dimensional nonlinear PDEs

TL;DR

This paper introduces an algorithm to generate new classes of partially integrable multidimensional nonlinear PDEs, including variants of KP and KdV equations, by combining Frobenius and differential reductions.

Contribution

The paper presents a novel algorithm for constructing higher-dimensional nonlinear PDEs with partial integrability, expanding the class of known integrable and partially integrable equations.

Findings

Derived four-dimensional nonlinear matrix PDEs with arbitrary functions in solutions

Examples include variants of KP and KdV equations

The PDEs admit solutions with arbitrary functions of two variables

Abstract

We represent an algorithm allowing one to construct new classes of partially integrable multidimensional nonlinear partial differential equations (PDEs) starting with the special type of solutions to the (1+1)-dimensional hierarchy of nonlinear PDEs linearizable by the matrix Hopf-Cole substitution (the B\"urgers hierarchy). We derive examples of four-dimensional nonlinear matrix PDEs together with they scalar and three-dimensional reductions. Variants of the Kadomtsev-Petviashvili type and Korteweg-de Vries type equations are represented among them. Our algorithm is based on the combination of two Frobenius type reductions and special differential reduction imposed on the matrix fields of integrable PDEs. It is shown that the derived four-dimensional nonlinear PDEs admit arbitrary functions of two variables in their solution spaces which clarifies the integrability degree of these PDEs.