Malaysian Journal of Mathematical Sciences, January 2019, Vol. 13, No. 1
Travelling Wave Group-Invariant Solutions and Conservation Laws for \(\theta\)-Equation
Johnpillai, A. G., Khalique, C. M., and Mahomed, F. M.
Corresponding Email: Masood.Khalique@nwu.ac.za
Received date: 17 June 2017
Accepted date: 18 January 2019
Abstract:
We study a class of nonlinear dispersive models called the \(\theta\)-equations from the Lie group-theoretic point of view. The Lie point symmetry generators of the class of equations are derived. Using the optimal system of one-dimensional subalgebras constructed from these symmetry generators, we obtain symmetry reductions and travelling wave group invariant solutions for the underlying equation. Moreover, we construct conservation laws for the class of equations by making use of the partial Lagrangian approach and the multiplier method. The underlying equation is of odd order and thus not variational. To apply the partial variational method a nonlocal transformation \(u=v_x\) is used to raise the order of the given class of equations. We show that the existence of nonlocal conservation laws for underlying equation is possible only if \(\theta=1/3\). In the multiplier approach, we obtain conservation laws for the given class of equations and a special case of the equation when \(\theta=1/3\) in which the first-order multipliers are considered.
Keywords: \(\theta\)-equation, Lie point symmetries, optimal system, groupinvariant solutions, partial Lagrangian, multiplier method, conservation laws
Indexing
