New Wave Solutions for Nonlinear Differential Equations using an Extended Bernoulli Equation as a New Expansion Method

In this paper, we presented a new expansion method constructed by taking inspiration for the Kudryashov method. Bernoulli equation is chosen in the form of n F BF AF    and some expansions are made on the auxiliary Bernoulli equation which is used in this method. In this auxiliary Bernoulli equation some wave solutions are obtained from the shallow water wave equation system in the general form of “n-order”. The obtained new results are simulated by graphically in 3D and 2D. To sum up, it is considered that this method can be applied to the several of nonlinear evolution equations in mathematics physics.


Introduction
Mathematical studies from nonlinear phenomena to particular case works play a significant role in applied mathematics, physics, engineering, education and so on [1].Where at the laws of the natural and physical world are generally modeled by partial differential equations.Searching the solutions of partial differential equations is become one of the main areas of scientific works [2].Many analytic methods have been introduced in literature.Some of them are: the Jacobi elliptic-function method [3], the sine-Gordon expansion method [4], the improved Bernoulli sub-equation function method [5] and so on.
The aim of this work, to investigate solutions of the system of the Shallow Water Wave equation with the help of the new expansion method which is a useful and popular tool for constructing exact solutions.

1 The Kudryashov Method
This section is cover the Kudryashov method for finding wave solutions of nonlinear partial differential equations [6][7].Accordingly, a general form of nonlinear PDE is defined as follows   1 , , , , , , 0 where k and w are respectively the number of waves and the wave velocity.After (2.2) transformation is applied to the Eq.(2.1), we obtain a nonlinear ordinary differential It is assumed that the solution   y  of the nonlinear equation given in Eq. (2.3) can be written in the following , where , 1 are constants and M is positive integers.M will be determine with homogeneous balance principle.  F  is the following function This function satisfies to the first order ordinary differential equation Eq. (2.6) is necessary to calculate the derivatives of function  .
y  Differentiating the expression (2.6) with respect to  and taking into account (2.6) we have We substitute expressions (2.7) in Eq. (2.1).After it we take   y  from (2.4) into account.
Thus Eq. (2.1) takes the form where is a polynomial of function  .
F  Then we collect all terms with the same powers of function   F  and equate this expression equal to zero.As a result we obtain system of algebraic equations.Solving this system we get the values of unknown parameters.

2 New Expansion Method
Instead of the auxiliary equation given in the Kudryashov method, we have specifically defined a new equation as follows. (where where A  ; c is an arbitrary constant from Eq. (2.8).

Exact solitary wave solutions of the system of the shallow water wave equation
In this example we consider the system of the shallow water wave equation [8] 0, 0.
Using Eq. (3.2) on Eq. (3.1) and integrating with respect to  yields, we yield following equation, The solutions of the systems, we are looking for in stated in the form , .
Balancing the highest linear terms with the highest nonlinear terms Eq. ( 3.3) we can found Case II:

Solution 1:
We obtain the solution of Eq. (3.1) for case 1 as following

Solution 2:
We obtain the solution of Eq. (3.1) for case 2 as following Specifically, in the solution (3.8) we obtained solution below for (3.10)  .
) Substituting (3.5) into Eq.(3.3) yields a set of algebraic equations for the aid of Mathematica from the solutions of the system, we can found Case I:

Fig. 1 .
Fig. 1.   , u x t and   , v x t three dimensional wave view, by substituting the values 1, 1, 1 w c     respectively for solution (3.10) of the Eq.(3.1).

4 ITMand 1 t
Fig. 2.   , u x t and   , v x t two dimensional wave view, by substituting the values