Investigation of Soliton Solutions for Some Important Nonlinear Evolution Equations Via Exp(-Φ(ζ)) Expansion Method

One effective technique for finding some exact traveling wave solutions of nonlinear partial differential equations (NPDEs) is the exp expansion method. In this paper, the exact traveling wave solutions for the nonlinear coupled Whitham-Broer-Kaup equation and the new coupled Korteweg-de Vries (KdV) equation are obtained by applying the exp expansion method. The numerical results of these solutions by using Maple have been presented graphically and discussed. The obtained traveling wave solutions include exponential functions, hyperbolic functions, trigonometric functions, and rational functions. Moreover, 3D graphics of solutions like the bell-shaped soliton solution, kink-type, periodic traveling waves, singular kink-type, singular cuspon type, as well as plane-wave solutions are presented to illustrate the dynamics of the equations. Comparing the results of the proposed method with the results of the homotopy analysis method shows that the proposed method is a strong and attractive method for solving systems of nonlinear partial differential equations. The results demonstrated the efficiency and simplicity of this method in extracting these exact solutions. The effectiveness of this method in solving nonlinear coupled partial differential equations that arise in mathematical physics and engineering has been shown.