Extended group analysis and conservation laws of a class of variable coefficient generalized Kawahara equations

Abstract

We review and extend the results on the group analysis of a class of generalized Kawahara equations with time-dependent coefficients. First, we provide an overview of the existing literature on Lie symmetries and Lie-invariant solutions of such equations. We then present a complete description of the transformation properties of these equations, including admissible, equivalence, and Lie symmetry transformations. For practical applications, the results are further extended by presenting a complete Lie symmetry classification without simplifying the coefficients by equivalence transformations. The Lie reductions are then thoroughly performed and some Lie exact solutions are constructed. The local conservation laws of low order are classified: every equation from the class admits the mass and the $L^2$-norm conservation laws, whereas energy-type conservation laws exist only for distinguished coefficient branches that agree with the cases singled out by the symmetry classification. The classification results are enhanced by a study of contractions, which link cases of Lie symmetry extensions together with the associated reductions and conservation laws.