Quasi self-adjoint nonlinear wave equations

Abstract

In this paper we generalize the classification of self-adjoint second order linear partial differential equation to a family of nonlinear wave equations with two independent variables. We find a class of quasi self-adjoint nonlinear equations which includes the self-adjoint linear equations as a particular case. The property of a differential equation to be quasi self-adjoint is important, e.g. for constructing conservation laws associated with symmetries of the differential equation.

New conservation laws for inviscid Burgers equation

“…Conservation laws for the Camassa-Holm equation was obtained by Ibragimov, Khamitova and Valenti in [20]. Further examples can be found in [4,5,15].…”

“…In addition, from Ibragimov's theorem on conservation laws [13], conservation laws for projectable Lie point symmetries (see [21]) of (1) were established. Recently Maria Luz Gandarias [8] and Nail Ibragimov [17,18,19] have generalized the previous concepts of self-adjoint equations [12,13,14].…”

On the Incompleteness of Ibragimov’s Conservation Law Theorem and Its Equivalence to a Standard Formula Using Symmetries and Adjoint-Symmetries

“…This generator (20) defines the characteristic form for the infinitesimal symmetry. The symmetry invariance (16) of the DE system can then be expressed by…”

“…In recent years, a similar conservation law formula has been popularized by Ibragimov [9][10][11][12][13] and subsequently extended by others [14][15][16][17], where a "nonlinear self-adjointness" condition is required to hold for the given DE system. However, in several papers [17][18][19], this formula sometimes is seen to produce only trivial conservation laws, and sometimes, the formula does not produce all admitted conservation laws.…”

“…However, in several papers [17][18][19], this formula sometimes is seen to produce only trivial conservation laws, and sometimes, the formula does not produce all admitted conservation laws. Furthermore, in a number of papers [14][15][16][17][20][21][22], the use of translation symmetries is mysteriously avoided, and other more complicated symmetries are used instead, without explanation.…”