Symmetry analysis of a system of modified shallow-water equations

TL;DR

This paper revises the symmetry analysis of a modified shallow-water system, identifying its infinite-dimensional Lie algebra, linearizing it via hodograph transformation, and deriving new solutions through Lie reductions and linearization.

Contribution

It provides a complete symmetry analysis of the modified shallow-water equations, including the construction of an optimal subalgebra list and the derivation of non-Lie solutions.

Findings

MSWE has an infinite-dimensional Lie symmetry algebra.

The MSWE can be linearized with a hodograph transformation.

New solutions are obtained from the linearized equations.

Abstract

We revise the symmetry analysis of a modified system of one-dimensional shallow-water equations (MSWE) recently considered by Raja Sekhar and Sharma [Commun. Nonlinear Sci. Numer. Simulat. 20 (2012) 630-636]. Only a finite dimensional subalgebra of the maximal Lie invariance algebra of the MSWE, which in fact is infinite dimensional, was found in the aforementioned paper. The MSWE can be linearized using a hodograph transformation. An optimal list of inequivalent one-dimensional subalgebras of the maximal Lie invariance algebra is constructed and used for Lie reductions. Non-Lie solutions are found from solutions of the linearized MSWE.