Symmetry Analysis, One-dimensional Optimal System, and Group Invariant Solutions for the Shallow Water Waves of Finite Amplitude in (1+1)-dimensions
Pratik Majhi *
Department of Mathematics, Visva-Bharati University, Santiniketan 731235, West Bengal, India.
Sharda Kumari
Department of Mathematics, Visva-Bharati University, Santiniketan 731235, West Bengal, India.
Madan Mohan Panja
Department of Mathematics, Visva-Bharati University, Santiniketan 731235, West Bengal, India.
*Author to whom correspondence should be addressed.
Abstract
The Lie group theoretic approach is employed here to obtain inequivalent group invariant solutions of a system of nonlinear partial differential equations (SNLPDEs) in (1+1)-dimensions appearing in the mathematical analysis of shallow water waves of finite amplitude, tsunami waves in particular. It is found that the system of equations admits a five-parameter Lie group of symmetry transformations with a Lie algebra \(\mathcal{G}\) of infinitesimal generators. An optimal set \(\mathcal{OG}_1\) of eighteen one-dimensional subalgebras of \(\mathcal{G}\) has been obtained. Similarity variables corresponding to each member of \(\mathcal{OG}_1\) have been determined and used to reduce the SNLPDEs to the system of ordinary differential equations. Some constants of the motion and inequivalent group invariant solutions have been provided whenever the system of reduced equations is solvable. The solutions to some of the reduced equations and the constants of motion provided here seem novel. Results obtained here may be used to check the accuracy of approximate solutions (grid formation) in the approximation (numerical) scheme.
Keywords: Shallow water waves of finite amplitude, optimal one-dimensional subalgebras, group-invariant solutions, Casimir- and Killing polynomials, differential invariants