New Exact Solutions of Ion-Acoustic Wave Equations by ()-Expansion Method
The ()-expansion method is used to study ion-acoustic waves equations in plasma physic for the first time. Many new exact traveling wave solutions of the Schamel equation, Schamel-KdV (S-KdV), and the two-dimensional modified KP (Kadomtsev-Petviashvili) equation with square root nonlinearity are constructed. The traveling wave solutions obtained via this method are expressed by hyperbolic functions, the trigonometric functions, and the rational functions. In addition to solitary waves solutions, a variety of special solutions like kink shaped, antikink shaped, and bell type solitary solutions are obtained when the choice of parameters is taken at special values. Two- and three-dimensional plots are drawn to illustrate the nature of solutions. Moreover, the solution obtained via this method is in good agreement with previously obtained solutions of other researchers.
The ion-acoustic solitary wave is one of the fundamental nonlinear waves phenomena appearing in fluid dynamics  and plasma physics . To allowing for the trapping of some of the electrons on ion-acoustic waves, Schamel proposed a modified equation for ion-acoustic waves  given by where is the wave potential and is a constant, this equation describing the ion-acoustic wave, where the electrons do not behave isothermally during their passage of the wave in a cold-ion plasma. Then, combining the equations of Schamel and the KdV equation, one obtains the so-called one-dimensional form of the Schamel-KdV (S-KdV) equation equation [4, 5]: where , , and are constants. This equation is established in plasma physics in the study of ion acoustic solitons when electron trapping is present, and also it governs the electrostatic potential for a certain electron distribution in velocity space. Note that we obtain the KdV equation when and the Schamel equation when for . Due to the wide range of applications of (2), it is important to find new exact wave solutions of the Schamel-KdV (S-KdV) equation. Another equation arising in the study of ion-acoustic waves is the so-called modified Kadomtsev-Petviashvili (KP) equation given by 
Equation (3) was firstly derived by Chakraborty and Das ; the modified KP equation containing a square root nonlinearity is a very attractive model for the study of ion-acoustic waves in a multispecies plasma made up of non-isothermal electrons in plasma physics.
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 [9, 10], sine-cosine method [11, 12], tanh-coth method , exp-function method , homogeneous-balance method [15, 16], Jacobi-elliptic function method [17, 18], and first-integral method [19, 20] to solve analytically nonlinear equations such as the above ion-acoustic wave equations.
Moreover, in the standard tanh method developed by Malfliet in 1992 , the tanh is used as a new variable. Since all derivatives of a tanh are represented by tanh itself, the solution obtained by this method may be solitons in terms of or may be kinks in terms of tanh. We believe that the -expansion method is more efficient than the tanh method. Moreover, the tanh method may yield more than one soliton solution, a capability which the tanh method does not have. The sine-cosine method yields a solution in trigonometric form. The Exp-function method leads to both generalized solitary solution and periodic solutions. The homogeneous-balance method is a generalized tanh function method for many nonlinear PDEs. The first integral method, which is based on the ring theory of commutative algebra, was first proposed by Feng. There is no general theory telling us how to find its first integrals in a systematic way; so, a key idea of this approach to find the first integral is to utilize the division theorem. The traveling wave solutions expressed by the -expansion method, which was first proposed by Wang et al. , transform the given difficult problem into a set of simple problems which can be solved easily to get solutions in the forms of hyperbolic, trigonometric, and rational functions. The main merits of the -expansion method over the other methods are as follows. (i)Higher-order nonlinear equations can be reduced to ODEs of order greater than 3. (ii)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. (iii)The general solution obtained by the -expansion method is without approximation. (iv)The solution procedure can be easily implemented in Mathematica or Maple.
In this paper, the -expansion method is used to study ion-acoustic waves equations in plasma physic for the first time. We obtain many new exact traveling wave solutions for the Schamel equation, S-KdV, and the two-dimensional modified KP equation. The traveling wave solutions obtained via this method are expressed by hyperbolic functions, the trigonometric functions, and the rational functions. In addition to solitary waves solutions, a variety of special solutions like kink shaped, antikink shaped, and bell type solitary solutions are obtained when the choice of parameters is taken at special values. Two- and three-dimensional plots are drawn to illustrate the nature of solutions. Moreover, the solution obtained via this method is in good agreement with previously obtained solutions of other researchers.
Our paper is organized as follows: in Section 2, we present the summary of the -expansion method, and Section 3 describes the applications of the -expansion method for Schamel equation, S-KdV equation, and modified KP equation, and lastly, conclusions are given in Section 4.
2. Summary of the -Expansion Method
In this section, we describe the -expansion method for finding traveling wave solutions of NLPDE. Suppose that a nonlinear partial differential equation in two independent variables, and , is given by where is an unknown function, is a polynomial in and its various partial derivatives, in which highest order derivatives and nonlinear terms are involved.
The summary of the -expansion method can be presented in the following six steps.
Step 1. To find the traveling wave solutions of (4), we introduce the wave variable: where the constant is generally termed the wave velocity. Substituting (5) into (4), we obtain the following ordinary differential equations (ODE) in (which illustrates a principal advantage of a traveling wave solution; i.e., a PDE is reduced to an ODE):
Step 2. If necessary, we integrate (6) as many times as possible and set the constants of integration to be zero for simplicity.
Step 3. Suppose that the solution of nonlinear partial differential equation can be expressed by a polynomial in as where satisfies the second-order linear ordinary differential equation where , , and , , and are real constants with . Here, the prime denotes the derivative with respect to . Using the general solutions of (8), we have
The above results can be written in simplified forms as
Step 4. The positive integer can be accomplished by considering the homogeneous balance between the highest order derivatives and nonlinear terms appearing in (6) as follows: if we define the degree of as , then the degree of other expressions is defined by
Therefore, we can get the value of in (7).
Step 5. Substituting (7) into (6), useing general solutions of (8), and collecting all terms with the same order of together, then setting each coefficient of this polynomial to zero yields a set of algebraic equations for , , , and .
3. Applications of the -Expansion Method
3.1. Schamel Equation
In order to find the solitary wave solution of (1), we use the transformations Then, (1) becomes Integrating (13) with respect to and setting the integration constant equal to zero, we have According to the previous steps, using the balancing procedure between with in (14), we get so that . Now, assume that (14) has the following solution: where , , and are unknown constants to be determined later. Substituting (15) along with (8) into (14) and collecting all terms with the same order of , the left hand side of (14) is converted into a polynomial in . Equating each coefficient of the resulting polynomials to zero yields a set of algebraic equations for , , , , , , , and as follows: On solving the above set of algebraic equations by Maple, we have Now, (15) becomes Substituting the general solution of (8) into (18), we obtain the three types of traveling wave solutions depending on the sign of .
If , we have the following general hyperbolic traveling wave solutions of (1):where and are arbitrary constants.
If , we have the following general trigonometric function solutions of (1):
If , we have the following general rational function solutions of (1): where .
Note that Khater and Hassan  obtained only hyperbolic solutions, but in this work, we found two additional types of solutions, that is, trigonometric and rational solutions.
3.2. S-KdV Equation
To find the general exact solutions of (2), we first write to transform (2) into Assume the traveling wave solution of (23) in the form Hence, (23) becomes Suppose that the solution of (25) can be expressed by a polynomial in as and satisfies (8). The homogeneous balance between the highest order derivative and the nonlinear term appearing in (25) yields , and hence, we take the following formal solution: where the positive integers and are to be determined later. Substituting (27) along with (8) into (25), collecting all the terms with the same power of , and equating each coefficient to zero yield a set of simultaneous algebraic equations for , , , , , , and as follows: The above system admits the following sets of solutions: Now, substituting (29)-(30) into (27) gives, respectively, When substituting the general solutions (9) into (32), we obtain the following three types of traveling wave solutions:
Case : (hyperbolic type)
If we set and write , then the above solutions can be written as Note that (35) is exactly the same solution of Khater and Hassan  as given in their first equation of (3.9) with . Similarly we can obtain the second solution of (36) in Hassan  if we set in our solution (35). The solution (35) represents kink shaped solitary and antikink shaped solitary solutions (depending upon the choice of sign) which are shown graphically in Figure 1 for the case :
Case : (trigonometric type)
But if and , then trigonometric type solution becomes
Case : (rational type)
3.3. The Modified Two-Dimensional KP (Kadomtsev-Petviashvili) Equation
The modified KP equation (3) containing a square root nonlinearity is a very attractive model for the study of ion-acoustic waves in plasma physic . We will obtain more general exact solutions of the modified KP equation. In order to find the traveling wave solution of (3), we let Now, taking , (3) becomes where , , , , and are constants and the prime denotes differentiation with respect to . Integrating (41) with respect to and setting the integration constant equal to zero, we obtain Balancing with gives . Therefore, we can write the solution of (42) in the form where , , and are constants to be determined later. Substituting (43) along with (8) into (42) and collecting all terms with the same order of , the left hand sides of (42) are converted into a polynomial in . Setting each coefficient of each polynomial to zero, we derive a set of algebraic equations for , , , , , , , , , and as follows:
Solving this system by Maple gives two sets of solutions.
If , we have the hyperbolic typeIn particular, if , , , and , then becomes If , we have the trigonometric type So, the traveling wave solutions of (3) in this case are
Case 2. We have If , we have the hyperbolic typeHowever, if , , , and , then becomes If , we have the trigonometric type In the particular case when , , , and , becomes If , we have the rational type where .
We remark that our results in (47) and (52), when , , , and , match those of Khater et al.  (2.19) when . In Figure 2, we plot the bell type solitary for 2D profile and the corresponding 3D plot of (47) for parameters , and , , , and .
The -expansion was applied to solve the model of ion-acoustic waves in plasma physics where these equations each contain a square root nonlinearity. The -expansion has been successfully used to obtain some exact traveling wave solutions of the Schamel equation, Schamel-KdV (S-KdV) equation, and modified KP (Kadomtsev-Petviashvili) equation. Moreover, the reliability of the method and the reduction in the size of computational domain give this method a wider applicability. This fact shows that our algorithm is effective and more powerful for NLPDE. In all the general solutions (22), (35), (47), and (52), we have the additional arbitrary constants , , , and . We note that the special case , , , and reproduced the results of Khater and Hassan , Hassan  and Khater et al. . Many different new forms of traveling wave solutions such as the kink shaped, antikink shaped, and bell type solitary solutions were obtained. Finally, numerical simulations are given to complete the study.
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 -expansion method used a rise only when the equation has the traveling wave and becomes powerful in finding traveling wave solutions of NLPDE only.
In our future works, we can extend our method by introducing a more generalized ansätz , where , to solve Schamel equation, Schamel-KdV (S-KdV) equation, and modified Kadomtsev-Petviashvili (KP) equation.
This work is financially supported by Universiti Kebangsaan Malaysia Grant: UKM-DIP-2012-31.
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