Direct construction method for conservation laws of partial differential equations Part I: Examples of conservation law classifications

TLDR

The paper is part of a two‑part series; Part I demonstrates the method on nonlinear wave equations, while Part II will provide a general treatment. The authors present an effective algorithmic method for finding local conservation laws of partial differential equations with any number of independent and dependent variables. The method reduces the calculation of conservation laws to solving a system of linear determining equations, without requiring a variational principle. An explicit construction formula is derived that yields a conservation law for each solution of the determining system, and classification results for the conservation laws of the example nonlinear wave equations are obtained.

Abstract

An effective algorithmic method is presented for finding the local conservation laws for partial differential equations with any number of independent and dependent variables. The method does not require the use or existence of a variational principle and reduces the calculation of conservation laws to solving a system of linear determining equations similar to that for finding symmetries. An explicit construction formula is derived which yields a conservation law for each solution of the determining system. In the first of two papers (Part I), examples of nonlinear wave equations are used to exhibit the method. Classification results for conservation laws of these equations are obtained. In a second paper (Part II), a general treatment of the method is given.