Abstract
In this study, the Benney-Luke equation is considered. In order to derive new type of solutions, the sn-ns method is applied to this equation. Then, we introduce trigonometric and elliptic functions solutions in addition to the hyperbolic ones which are gained by tanh-coth. Three types of solutions are derived at the same time with the help of this method. Therefore, it can be said that this method is convenient to obtain more solutions to many kinds of nonlinear partial differential equations.
Highlights
The equation (1.2) is called Benney-Luke equation
Comparison we have compared the solutions gained by the sn-ns method in this work with those obtained by the tanh-coth method in [21]
We have found exact solutions of Benney-Luke equation given in (1.2) by the sn-ns method
Summary
This method has been introduced by H. Alvaro [11] in detail In this method, we have searched for the traveling wave solutions to nonlinear partial differential equation of the form. Θ = θ(ξ) is a suitable function that makes the transformation simple. For this purpose, we take θ(ξ) as the identity function, i.e. θ(ξ) = ξ. Using the transformation (2.2) and putting ordinary derivative of v(ξ) instead of the partial derivatives of u(x, t), (3.1) converts to an ordinary differential equation (ODE) with respect to the function v(ξ). With Q being a polynomial with respect to variables v, v′, v′′,. In this method, we seek the traveling wave solutions to (2.3) in the form v(ξ) = a0 +.
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