1. Exact solutions and conservation laws for a generalized double sinh- Gordon equation GABRIEL MAGALAKWE IISAMM North West University, Mafikeng Campus Supervisor: Prof C.M.KHALIQUE Energy Postgraduate Conference 2013

2. Preamble In this talk, we study the generalized double Sinh-Gordon equation, which has applications in various fields, such as, fluid dynamics, integrable quantum field theory, kink dynamics. Lie symmetry analysis together with the simplest equation method and the Exp-function method are used to obtain exact solutions for this equation. We derive conservation laws for this equation by using the direct method. Lastly concluding remarks are given

3. Introduction

4. Introduction In physics a wave is an oscillation that travels through space and matter, accompanied by a transfer of energy. Conservation laws play a vital role in the solution process of differential equations (DEs). It is well known that, the existence of a large number of conservation laws of a system of partial differential equations (PDEs) is an indication of its integrability [6] (Bluman and Kumei). Recently, conserved vector was used to determine the unknown exponent in the similarity solution which cannot be obtained from the homogeneous boundary conditions [7] (Naz, Mahomed and Mason).

5. Exact solutions of a GD SH-G equation

6. Application of simplest equation method

7. Application of Exp-function method

8. Conservation laws

9. Conservation laws

10. Conservation laws

11. Concluding remarks We have studied the generalized double sinh-Gordon equation using the Lie symmetry analysis. Simplest equation method and Exp-function method were used to obtain exact solutions of (1). Finally conservation laws for (1) were derive by employing the direct method.

12. References [1] A. M.Wazwaz, Exact solutions to the double sinh-Gordon equation by the tanh method and variable separated ODE method, Comp. and Mathematics with applications. 50 (2005) 1685-1696. [2] S. Tang, W. Huang, Bifurcation of travelling wave solutions for the generalized double sinh-Gordon equation, Applied Mathematics and Computation. 189 (2007) 1774-1781. [3] A. M.Wazwaz, Exact solutions to the double sinh-Gordon equation by the tanh method and variable separated ODE method, Comp. and Mathematics with applications. 50 (2005) 1685-1696.

13. References [4] S. Tang and W. Huang, Bifurcation of travelling wave solutions for the generalized double sinh-Gordon equation, Applied Mathematics and Computation.189 (2007) 1774-1781. [5] H. Kheiri and A. Jabbari, Exact solutions for the double sinh- Gordon and the generalized form of the double sinh-Gordon equations by (G′/G)-expansion method, Turkish Journal of Phys. 34 (2010) 73-82. [6] G. W. Bluman and S. Kumei, Symmetries and Differential Equations, Applied Mathematical Sciences, 81, Springer- Verlag, New York, 1989. [7] R. Naz, F. M. Mahomed and D. P. Mason, Comparison of different approaches to conservation laws for some partial differential equations in fluid mechanics, Applied Mathematics and Computation. 205 (2008) 212-230.

14. Thank you