Direct Methods and Symbolic Software for Conservation Laws of Nonlinear Equations

TL;DR

This paper introduces symbolic methods and software for computing conservation laws of nonlinear PDEs and DDEs, including polynomial and transcendental cases, with implementations in Mathematica and Maple.

Contribution

It presents a new direct method using leading order analysis for conservation laws, applicable to both PDEs and DDEs, overcoming limitations of previous undetermined coefficients methods.

Findings

Successfully applied to classical nonlinear PDEs like KdV, Boussinesq, sine-Gordon.

Implemented in Mathematica and Maple for practical computation.

Extended to nonlinear DDEs including lattices like Toda and Ablowitz-Ladik.

Abstract

We present direct methods and symbolic software for the computation of conservation laws of nonlinear partial differential equations (PDEs) and differential-difference equations (DDEs).The methods are applied to nonlinear PDEs in (1+1) dimensions with polynomial nonlinearities which include the Korteweg-de Vries (KdV), Boussinesq, and Drinfel'd-Sokolov-Wilson equations. An adaptation of the methods is applied to PDEs with transcendental nonlinearities. Examples include the sine-Gordon, sinh-Gordon, and Liouville equations. With respect to nonlinear DDEs, our methods are applied to Kac-van Moerbeke, Toda, and Ablowitz-Ladik lattices. To overcome the shortcomings of the undetermined coefficients method, we designed a new direct method which uses leading order analysis. That method is applied to discretizations of the KdV and modified KdV equations, and a combination thereof. Additional examples include lattices due to Bogoyavlenskii, Belov-Chaltikian, and Blaszak-Marciniak. The undetermined coefficient methods for PDEs and DDEs have been implemented in Mathematica. The code "TransPDEDensityFlux.m" computes densities and fluxes of systems of PDEs with or without transcendental nonlinearities. The code "DDEDensityFlux.m" does the same for polynomial nonlinear DDEs. Starting from the leading order terms, the new Maple library "discrete" computes densities and fluxes of nonlinear DDEs.