Z. Naturforsch. **68a,** 510 – 514
(2013)

doi:10.5560/ZNA.2013-0027

Exact Group Invariant Solutions and Conservation Laws of the Complex Modified Korteweg–de Vries Equation

Andrew
G. Johnpillai^{1},
Abdul
H. Kara^{2}, and
Anjan
Biswas^{3,4}

^{1 }Department of Mathematics, Eastern University, Sri Lanka

^{2 }School of Mathematics and Centre for Differential Equations, Continuum Mechanics and Applications, University of the Witwatersrand, Wits 2050, Johannesburg, South Africa

^{3 }Department of Mathematical Sciences, Delaware State University, Dover, DE 19901-2277, USA

^{4 }Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia

Received November 27, 2012 / revised March 10, 2013 / published online May 1, 2013

We study the scalar complex modified Korteweg–de Vries (cmKdV) equation by analyzing a system of partial differential equations (PDEs) from the Lie symmetry point of view. These systems of PDEs are obtained by decomposing the underlying cmKdV equation into real and imaginary components. We derive the Lie point symmetry generators of the system of PDEs and classify them to get the optimal system of one-dimensional subalgebras of the Lie symmetry algebra of the system of PDEs. These subalgebras are then used to construct a number of symmetry reductions and exact group invariant solutions to the system of PDEs. Finally, using the Lie symmetry approach, a couple of new conservation laws are constructed. Subsequently, respective conserved quantities from their respective conserved densities are computed.

*Key words:* Complex Modified KdV Equation; Solitons; Lie Symmetries; Optimal System; Symmetry Reduction; Group Invariant Solutions; Conservation Laws.

*Mathematics Subject Classification 2000:* 35Q55