Abstract
In this paper, we have described two dreadfully important methods to solve nonlinear partial differential equations which are known as exp-function and the exp(−ϕ(ξ)) -expansion method. Recently, there are several methods to use for finding analytical solutions of the nonlinear partial differential equations. The methods are diverse and useful for solving the nonlinear evolution equations. With the help of these methods, we are investigated the exact travelling wave solutions of the Vakhnenko- Parkes equation. The obtaining soliton solutions of this equation are described many physical phenomena for weakly nonlinear surface and internal waves in a rotating ocean. Further, three-dimensional plots of the solutions such as solitons, singular solitons, bell type solitary wave i.e. non-topological solitons solutions and periodic solutions are also given to visualize the dynamics of the equation.
Highlights
The effort in finding exact solutions to nonlinear equations is witnessed significant curiosity and progress in finding solutions to nonlinear partial differential equations (NPDEs) that resemble physical phenomena
The nonlinear wave phenomena observed in fluid dynamics, plasma and optical fibers are often modeled by the bell shaped sech solutions and the kink shaped tanh solutions
We have achieved a family of solutions via exp-function method
Summary
The effort in finding exact solutions to nonlinear equations is witnessed significant curiosity and progress in finding solutions to nonlinear partial differential equations (NPDEs) that resemble physical phenomena. −3a21b−1b0 þ a30 þ 3a−1a21b0−18a0a1b−1 þ 3a20a1b0 −3a20b0 þ 3a0a1b20; C5 1⁄4 a30b0 þ 3a20a1b−1 þ 6a0a1a−1b0 þ 12a−1a1b20 þ 3a20a−1−12a20b−1 þ 3a21a−1b‐1 þ 3a1a2−1 C6 1⁄4 −18a0a−1b−1 þ 3a0a1b2−1 þ 3a1a2−1b0 þ 18a−1a1b0b−1 þ 3a0a2−1 þ 6a0a−1a1b−1 þ a30b−1−3a2−1b0−3a20b−1b0 þ 3a20a−1b0 þ 3a0a1b20; C7 1⁄4 12a−1a1b2−1 þ 3a20a−1b−1 þ a3−1 þ 3a0a2−1b0 þ 3a2−1a1b‐1−12a2−1b−1; C8 1⁄4 3a0a−1b2−1 þ 3a0a2−1b−1 þ a3−1b0−3a2−1b−1b0 and C9 1⁄4 a3−1b−1 Setting these equations to zero and solving the system of algebraic equations with the aid of commercial software Maple-13, we achieve the following solution. Solutions of Vakhnenko- Parkes equation via the exp (−φ(ξ)) -expansion method Balance the highest order derivate term uu′′ with the highest nonlinear terms u3 in Eq (16), we obtain m = 2, so assume the equation Eq (1) has the solution uðξÞ 1⁄4 l0 þ l1ð expð−φðξÞÞÞ þ l2ð expð−φðξÞÞÞ2 ð31Þ. Substituting the values of l0, l1, l2 in the general solutions of Eq (9) achieve more traveling wave solutions of nonlinear evolution equation Eq (1) as follows: When λ2 − 4μ > 0, μ ≠ 0,
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