Conservation laws for self-adjoint first order evolution equations

TL;DR

This paper classifies first order evolution equations based on their self-adjoint properties and derives conservation laws using Ibragimov's Theorem, exemplified by the inviscid Burgers' equation, revealing new symmetries and conservation laws.

Contribution

It introduces a classification of first order evolution equations into quasi-self-adjoint and self-adjoint subclasses and applies Ibragimov's Theorem to derive new conservation laws.

Findings

Identification of self-adjoint subclasses of equations

Derivation of conservation laws for these subclasses

Discovery of infinite symmetries in inviscid Burgers' equation

Abstract

In this work we consider the problem on group classification and conservation laws of the general first order evolution equations. We obtain the subclasses of these general equations which are quasi-self-adjoint and self-adjoint. By using the recent Ibragimov's Theorem on conservation laws, we establish the conservation laws of the equations admiting self-adjoint equations. We illustrate our results applying them to the inviscid Burgers' equation. In particular an infinite number of new symmetries of these equations are found and their corresponding conservation laws are established.