Particular solutions to multidimensional PDEs with KdV-type nonlinearity

TL;DR

This paper introduces a method for finding particular solutions to multidimensional PDEs with KdV-type nonlinearity, reducing them to ODEs, and demonstrates solutions including elementary functions and cnoidal waves.

Contribution

It presents a novel approach based on characteristic deformation to solve higher-dimensional PDEs with KdV-type nonlinearity, including explicit solutions for specific cases.

Findings

Solutions include elementary functions for n=1

Cnoidal wave solutions for n=2

Method applicable to higher-dimensional PDEs with linear parts

Abstract

We consider a class of particular solutions to the (2+1)-dimensional nonlinear partial differential equation (PDE) $u_t +\partial_{x_2}^n u_{x_1} - u_{x_1} u =0$ (here $n$ is any integer) reducing it to the ordinary differential equation (ODE). In a simplest case, $n=1$, the ODE is solvable in terms of elementary functions. Next choice, $n=2$, yields the cnoidal waves for the special case of Zakharov-Kuznetsov equation. The proposed method is based on the deformation of the characteristic of the equation $u_t-uu_{x_1}=0$ and might also be useful in study the higher dimensional PDEs with arbitrary linear part and KdV-type nonlinearity (i.e. the nonlinear term is $u_{x_1} u$).