A nonlinear second-order evolution equation – classical and nonclassical symmetry analyses and their physical implications

Abstract

The classical Lie group formalism as well as the nonclassical procedure is applied to study a nonlinear partial differential equation of the second order. It is important to stress that until now no symmetry calculation is available. Therefore it seems indispensable to apply this method yielding a deeper insight into the behaviour of the solution manifold.

Firstly we determine the classical Lie point symmetries including algebraic properties.

Similarity solutions in a most general form and nonlinear transformations are obtained.

Also a statement relating to potential symmetries is performed. Then we show how the equation leads to approximate symmetries and we apply the method for the first time.

Secondly some important hints relating to different alge braic solution techniques are given in order to construct further closed-form solutions. We finally show how this less-studied equation admits solitary and peakon classes of solutions of practical relevance.