Symbolic computation of conservation laws for nonlinear partial differential equations in multiple space dimensions

TL;DR

This paper introduces an algorithmic symbolic method, implemented in Mathematica, for computing conservation laws of nonlinear PDEs in multiple space dimensions, demonstrated on several complex equations.

Contribution

It presents a novel, algorithmic approach using variational calculus and linear algebra, with a software package for symbolic conservation law computation in multi-dimensional PDEs.

Findings

Successfully applied to (2+1)-dimensional equations

Automates conservation law computation for polynomial PDEs

Implemented in a publicly available Mathematica package

Abstract

A method for symbolically computing conservation laws of nonlinear partial differential equations (PDEs) in multiple space dimensions is presented in the language of variational calculus and linear algebra. The steps of the method are illustrated using the Zakharov-Kuznetsov and Kadomtsev-Petviashvili equations as examples. The method is algorithmic and has been implemented in Mathematica. The software package, ConservationLawsMD.m, can be used to symbolically compute and test conservation laws for polynomial PDEs that can be written as nonlinear evolution equations. The code ConservationLawsMD.m has been applied to (2+1)-dimensional versions of the Sawada-Kotera, Camassa-Holm, and Gardner equations, and the multi-dimensional Khokhlov-Zabolotskaya equation.