Exact solutions and conservation laws for a

generalized double sinh-Gordon equation

GABRIEL MAGALAKWEIISAMM

North West University, Mafikeng Campus

Supervisor: Prof C.M.KHALIQUE

Energy Postgraduate Conference 2013

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

Introduction

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).

Exact solutions of a GD SH-G equation

Application of simplest equation method

Application of Exp-function method

Conservation laws

Conservation laws

Conservation laws

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.

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.

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.

Thank you