The (

The ion-acoustic solitary wave is one of the fundamental nonlinear waves phenomena appearing in fluid dynamics [

Equation (

In the literature, the KP equation is also known as the two-dimensional KdV equation [

It has lately become more interesting to obtain exact analytical solutions to nonlinear partial differential equations such as the one arising from the ion-acoustic wave phenomena, by using appropriate techniques. Several important techniques have been developed such as the tanh-method [

Moreover, in the standard tanh method developed by Malfliet in 1992 [

Higher-order nonlinear equations can be reduced to ODEs of order greater than 3.

There is no need to apply the initial and boundary conditions at the outset. The method yields a general solution with free parameters which can be identified by the above conditions.

The general solution obtained by the

The solution procedure can be easily implemented in Mathematica or Maple.

In fact, the

In this paper, the

Our paper is organized as follows: in Section

In this section, we describe the

The summary of the

To find the traveling wave solutions of (

If necessary, we integrate (

Suppose that the solution of nonlinear partial differential equation can be expressed by a polynomial in

The above results can be written in simplified forms as

The positive integer

Therefore, we can get the value of

Substituting (

Substitute

In order to find the solitary wave solution of (

If

If

If

Writing

Note that Khater and Hassan [

To find the general exact solutions of (

If we set

(a) 2D profile of (

But if

As mentioned before, the

The modified KP equation (

Solving this system by Maple gives two sets of solutions.

We have

If

We have

We remark that our results in (

Bell type solitary (a) 2D profile and (b) corresponding 3D plot of (

The

Moreover, all the methods have some limitations in their applications. In fact, there is no unified method that can be used to handle all types of nonlinear partial differential equations (NLPDE). Certainly, each investigator in the field of differential equations has his own experience to choose the method depending on form of the nonlinear differential equation and the pole of its solution. So, the limitations of the

In our future works, we can extend our method by introducing a more generalized ansätz

This work is financially supported by Universiti Kebangsaan Malaysia Grant: UKM-DIP-2012-31.