Abstract
We construct the traveling wave solutions of the (1+1)-dimensional modified Benjamin-Bona-Mahony equation, the (2+1)-dimensional typical breaking soliton equation, the (1+1)-dimensional classical Boussinesq equations, and the (2+1)-dimensional Broer-Kaup-Kuperschmidt equations by using an extended -expansion method, whereGsatisfies the second-order linear ordinary differential equation. By using this method, new exact solutions involving parameters, expressed by three types of functions which are hyperbolic, trigonometric and rational function solutions, are obtained. When the parameters are taken as special values, some solitary wave solutions are derived from the hyperbolic function solutions.
Highlights
The investigation of the traveling wave solutions of nonlinear partial differential equations NPDEs plays an important role in the study of nonlinear physical phenomena
The exact solutions have been investigated by many authors see, e.g., 1–27 who are interested in nonlinear physical phenomena
Many powerful methods have been presented such as the homogeneous balance method 13, the tanh method 4, 15, 24, the inverse scattering transform 1, the expfunction expansion method 2, 6, 20, the Jacobi elliptic function expansion, the Backlund transform 8, 9, the generalized Riccati equation, the modified extended Fan subequation method, the truncated Painleveexpansion 27, and the auxiliary equation method 10, 11
Summary
The investigation of the traveling wave solutions of nonlinear partial differential equations NPDEs plays an important role in the study of nonlinear physical phenomena. Mathematical Problems in Engineering mathematicians Wang et al 14 for which the traveling wave solutions of the nonlinear evolution equations are obtained. This method has been extended to solve differencedifferential equations 28, 29. Gou and Zhou 5 have obtained the exact traveling wave solutions of some nonlinear PDEs using an extended G /G -expansion method. We use the extended G /G -expansion method which is proposed in 5 to derive traveling wave solutions for some nonlinear PDEs in mathematical physics namely; the 1 1 -dimensional modified Benjamin-Bona-Mahony equation, the 2 1 -dimensional typical breaking soliton equation, the 1 1 -dimensional classical Boussinesq equations, and the 2 1 -dimensional Broer-Kaup-Kuperschmidt equations. Suppose the solution of 2.3 can be expressed in G /G as follows:
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