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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 204746, 5 pages
http://dx.doi.org/10.1155/2013/204746
Research Article

Symmetry Reductions, Exact Solutions, and Conservation Laws of a Modified Hunter-Saxton Equation

1Department of Mathematics, Eastern University, Sri Lanka
2International Institute for Symmetry Analysis and Mathematical Modelling, Department of Mathematical Sciences, North-West University, Mafikeng Campus, Private Bag X 2046, Mmabatho 2735, South Africa

Received 29 June 2013; Accepted 14 August 2013

Academic Editor: Maria Gandarias

Copyright © 2013 Andrew Gratien Johnpillai and Chaudry Masood Khalique. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study a modified Hunter-Saxton equation from the Lie group-theoretic point of view. The Lie point symmetry generators of the underlying equation are derived. We utilize the Lie algebra admitted by the equation to obtain the optimal system of one-dimensional subalgebras of the Lie algebra of the equation. These subalgebras are then used to reduce the underlying equation to nonlinear third-order ordinary differential equations. Exact traveling wave group-invariant solutions for the equation are constructed by integrating the reduced ordinary differential equations. Moreover, using the variational method, we construct infinite number of nonlocal conservation laws by the transformation of the dependent variable of the underlying equation.