New Exact Solutions of Ion-Acoustic Wave Equations by (𝐺^′/𝐺)-Expansion Method

Abstract

The (𝐺^′/𝐺)-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.

1. Introduction

The ion-acoustic solitary wave is one of the fundamental nonlinear waves phenomena appearing in fluid dynamics [1] and plasma physics [2]. To allowing for the trapping of some of the electrons on ion-acoustic waves, Schamel proposed a modified equation for ion-acoustic waves [3] given by 𝑢_𝑡+𝑢^1/2𝑢_𝑥+𝛿𝑢_𝑥𝑥𝑥=0, (1) where 𝑢 is the wave potential and 𝛿 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]: 𝑢_𝑡+𝛼𝑢^1/2+𝛽𝑢𝑢_𝑥+𝛿𝑢_𝑥𝑥𝑥=0,𝛿𝛽≠0, (2) where 𝛽, 𝛼, and 𝛿 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 𝛼=0 and the Schamel equation when 𝛽=0 for 𝛿=1. 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 [6] 𝑢_𝑡+𝛼𝑢^1/2𝑢_𝑥+𝛽𝑢_{𝑥𝑥𝑥𝑥}+𝛿𝑢_𝑦𝑦=0. (3)

Equation (3) was firstly derived by Chakraborty and Das [7]; 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 [8].

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 [13], exp-function method [14], 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 [21], 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 sech² or may be kinks in terms of tanh. We believe that the (𝐺^′/𝐺)-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 (𝐺^′/𝐺)-expansion method, which was first proposed by Wang et al. [22], 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 (𝐺^′/𝐺)-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 (𝐺^′/𝐺)-expansion method is without approximation. (iv)The solution procedure can be easily implemented in Mathematica or Maple.

In fact, the (𝐺^′/𝐺)-expansion method has been successfully applied to obtain exact solution for a variety of NLPDE [23–34].

In this paper, the (𝐺^′/𝐺)-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 (𝐺^′/𝐺)-expansion method, and Section 3 describes the applications of the (𝐺^′/𝐺)-expansion method for Schamel equation, S-KdV equation, and modified KP equation, and lastly, conclusions are given in Section 4.

2. Summary of the (𝐺^′/𝐺)-Expansion Method

In this section, we describe the (𝐺^′/𝐺)-expansion method for finding traveling wave solutions of NLPDE. Suppose that a nonlinear partial differential equation in two independent variables, 𝑥 and 𝑡, is given by 𝑝𝑢,𝑢_𝑡,𝑢_𝑥,𝑢_𝑥𝑡,𝑢_𝑡𝑡,𝑢_𝑥𝑥,…=0, (4) where 𝑢=𝑢(𝑥,𝑡) is an unknown function, 𝑃 is a polynomial in 𝑢=𝑢(𝑥,𝑡) and its various partial derivatives, in which highest order derivatives and nonlinear terms are involved.

The summary of the (𝐺^′/𝐺)-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: 𝑢(𝑥,𝑡)=𝑢(𝜁),𝜁=(𝑥−𝑐𝑡), (5) where the constant 𝑐 is generally termed the wave velocity. Substituting (5) into (4), we obtain the following ordinary differential equations (ODE) in 𝜁 (which illustrates a principal advantage of a traveling wave solution; i.e., a PDE is reduced to an ODE): 𝑝𝑢,𝑐𝑢^′,𝑢^′,𝑐𝑢^′′,𝑐²𝑢^′′,𝑢^′′,…=0. (6)

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 (𝐺^′/𝐺) as 𝑢(𝜁)=^𝑚_𝑖=0𝑎_𝑖𝐺^′𝐺^𝑖, (7) where 𝐺=𝐺(𝜁) satisfies the second-order linear ordinary differential equation 𝐺^′′(𝜁)+𝜆𝐺^′(𝜁)+𝜇𝐺(𝜁)=0, (8) where 𝐺^′=𝑑𝐺/𝑑𝜁, 𝐺^′′=𝑑²𝐺/𝑑𝜁², and 𝑎_𝑖, 𝜆, and 𝜇 are real constants with 𝑎_𝑚≠0. Here, the prime denotes the derivative with respect to 𝜁. Using the general solutions of (8), we have

𝐺^′𝐺=−𝜆2+𝜆²−4𝜇2×𝑐₁sinh𝜆²−4𝜇/2𝜁+𝑐₂cosh𝜆²−4𝜇/2𝜁𝑐₁cosh𝜆²−4𝜇/2𝜁+𝑐₂sinh𝜆²𝜁,𝜆−4𝜇/2₂−4𝜇>0,−𝜆2+4𝜇−𝜆²2×−𝑐₁sin4𝜇−𝜆²/2𝜁+𝑐₂cos4𝜇−𝜆²/2𝜁𝑐₁cos4𝜇−𝜆²/2𝜁+𝑐₂sin4𝜇−𝜆²𝜁,𝜆/2₂𝑐−4𝜇<0,₂𝑐₁+𝑐₂𝜁−𝜆2,𝜆²−4𝜇=0. (9)

The above results can be written in simplified forms as 𝐺^′𝐺=−𝜆2+𝜆²−4𝜇2tanh𝜆²−4𝜇2𝜁,𝜆²−4𝜇>0,−𝜆2+4𝜇−𝜆²2tan4𝜇−𝜆²2𝜁,𝜆²𝑐−4𝜇<0,₂𝑐₁+𝑐₂𝜁−𝜆2,𝜆²−4𝜇=0. (10)

Step 4. The positive integer 𝑚 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 𝑢(𝜁) as 𝐷[𝑢(𝜁)]=𝑚, then the degree of other expressions is defined by 𝐷𝑑^𝑞𝑢𝑑𝜁^𝑞𝐷𝑢=𝑚+𝑞,_𝑟𝑑^𝑞𝑢𝑑𝜁^𝑞𝑠=𝑚𝑟+𝑠𝑞+𝑚. (11)
Therefore, we can get the value of 𝑚 in (7).

Step 5. Substituting (7) into (6), useing general solutions of (8), and collecting all terms with the same order of (𝐺^′/𝐺) together, then setting each coefficient of this polynomial to zero yields a set of algebraic equations for 𝑎_𝑖, 𝑐, 𝜆, and 𝜇.

Step 6. Substitute 𝑎_𝑖, 𝑐, 𝜆, and 𝜇 obtained in Step 5 and the general solutions of (8) into (7). Next, depending on the sign of the discriminant 𝐴=𝜆²−4𝜇, we can obtain the explicit solutions of (4) immediately.

3. Applications of the (𝐺^′/𝐺)-Expansion Method

3.1. Schamel Equation

In order to find the solitary wave solution of (1), we use the transformations 𝑢(𝑥,𝑡)=𝑣²(𝑥,𝑡),𝑣(𝑥,𝑡)=𝑣(𝜁),𝜁=𝑘𝑥−𝑐𝑡. (12) Then, (1) becomes −𝑐𝑣𝑣^′+𝑘𝑣²𝑣^′+𝛿𝑘³𝑣𝑣^′′′+3𝑣^′𝑣^′′=0. (13) Integrating (13) with respect to 𝜁 and setting the integration constant equal to zero, we have −𝑐2𝑣²+𝑘3𝑣³+𝑘³𝛿𝑣^′2+𝑘³𝛿𝑣𝑣^′′=0. (14) According to the previous steps, using the balancing procedure between 𝑣³ with 𝑣𝑣^′′ in (14), we get 3𝑚=2𝑚+2 so that 𝑚=2. Now, assume that (14) has the following solution: 𝑣(𝜁)=𝑎₀+𝑎₁𝐺^′𝐺+𝑎₂𝐺^′𝐺²,𝑎₂≠0, (15) where 𝑎₀, 𝑎₁, and 𝑎₂ are unknown constants to be determined later. Substituting (15) along with (8) into (14) and collecting all terms with the same order of (𝐺^′/𝐺), the left hand side of (14) is converted into a polynomial in (𝐺^′/𝐺). Equating each coefficient of the resulting polynomials to zero yields a set of algebraic equations for 𝑎₀, 𝑎₁, 𝑎₂, 𝛿, 𝜆, 𝑐, 𝑘, and 𝜇 as follows: 𝐺^′𝐺⁶∶13𝑘𝑎³₂+10𝑘³𝛿𝑎²₂𝐺=0,_′𝐺⁵∶18𝑘³𝛿𝑎²₂𝜆+12𝑘³𝛿𝑎₁𝑎₂+𝑘𝑎₁𝑎²₂𝐺=0,_′𝐺⁴∶6𝑘³𝛿𝑎₀𝑎₂+16𝑘³𝛿𝑎²₂𝜇+8𝑘³𝛿𝑎²₂𝜆²−12𝑐𝑎²₂+21𝑘³𝛿𝑎²₁+𝑘𝑎₀𝑎²₂+𝑘𝑎²₁𝑎₂𝐺=0,_′𝐺³∶14𝑘³𝛿𝑎²₂𝜆𝜇+5𝑘³𝛿𝑎²₁𝜆+18𝑘³𝛿𝑎₁𝑎₂𝜇+9𝑘³𝛿𝑎₁𝜆²𝑎₂+2𝑘𝑎₀𝑎₁𝑎₂−𝑐𝑎₁𝑎₂+13𝑘𝑎³₁+2𝑘³𝛿𝑎₀𝑎₁+10𝑘³𝛿𝑎₀𝑎₂𝐺𝜆=0,_′𝐺²∶2𝑘³𝛿𝑎²₁𝜆²+4𝑘³𝛿𝑎²₁𝜇+8𝑘³𝛿𝑎₀𝑎₂1𝜇−2𝑐𝑎²₁−𝑐𝑎₀𝑎₂+15𝑘³𝛿𝑎₁𝑎₂𝜆𝜇+𝑘𝑎₀𝑎²₁+4𝑘³𝛿𝑎₀𝑎₂𝜆²+6𝑘³𝛿𝑎²₂𝜇²+3𝑘𝛿𝑎₀𝑎₁𝜆+𝑘𝑎²₀𝐺=0,_′𝐺¹∶2𝑘³𝑎₀𝑎₁𝜇+𝑘³𝛿𝑎₀𝑎₁𝜆²+6𝑘³𝛿𝑎₀𝑎₂𝜆𝜇+𝑘𝑎²₀−𝑐𝑎₀𝑎₁+6𝑘³𝑎₁𝑎₂𝛿𝜇²+3𝑘³𝛿𝜆𝜇𝑎²₁𝐺=0,_′𝐺⁰∶𝑘³𝛿𝑎²₁𝜇²−12𝑐𝑎²₀+13𝑘𝑎³₀+𝑘³𝛿𝜆𝜇𝑎₀𝑎₁+2𝑘³𝛿𝜇²𝑎₀𝑎₂=0. (16) On solving the above set of algebraic equations by Maple, we have 𝑎₀=−30𝜇𝛿𝑘²,𝑎₁=−30𝑘²𝑎𝛿𝜆,₂=−30𝑘²𝛿,𝑐=𝛿𝑘³4𝜆²−16𝜇. (17) Now, (15) becomes 𝑣(𝜁)=−30𝜇𝛿𝑘²−30𝑘²𝐺𝛿𝜆^′𝐺−30𝑘²𝛿𝐺^′𝐺². (18) Substituting the general solution of (8) into (18), we obtain the three types of traveling wave solutions depending on the sign of 𝐴=𝜆²−4𝜇.

If 𝐴>0, we have the following general hyperbolic traveling wave solutions of (1):𝑣(𝑥,𝑡)=−30𝜇𝛿𝑘²−30𝑘²×𝛿𝜆−𝜆2+√𝐴2×𝑐₁sinh√𝐴/2𝜁+𝑐₂cosh√𝐴/2𝜁𝑐₁cosh√𝐴/2𝜁+𝑐₂sinh√𝐴/2𝜁−30𝑘²𝛿−𝜆2+√𝐴2×𝑐₁sinh√𝐴/2𝜁+𝑐₂cosh√𝐴/2𝜁𝑐₁cosh√𝐴/2𝜁+𝑐₂sinh√𝐴/2𝜁², (19)where 𝑐₁ and 𝑐₂ are arbitrary constants.

If 𝐴<0, we have the following general trigonometric function solutions of (1):𝑣(𝑥,𝑡)=−30𝜇𝛿𝑘²−30𝑘²×𝛿𝜆−𝜆2+√𝐴2×−𝑐₁sin√𝐴/2𝜁+𝑐₂cos√𝐴/2𝑐₁cos√𝐴/2𝜁+𝑐₂sin√𝐴/2𝜁−30𝑘²𝛿−𝜆2+√𝐴2×−𝑐₁sin√𝐴/2𝜁+𝑐₂cos√𝐴/2𝜁𝑐₁cos√𝐴/2𝜁+𝑐₂sin√𝐴/2𝜁². (20)

If 𝐴=0, we have the following general rational function solutions of (1): 𝑣(𝑥,𝑡)=−30𝜇𝛿𝑘²−30𝑘²−𝜆𝛿𝜆2+𝑐₂𝑐₁+𝑐₂𝜁−30𝑘²𝛿−𝜆2+𝑐₂𝑐₁+𝑐₂𝜁², (21) where 𝜁=𝑘𝑥−𝛿𝑘³(4𝜆²−16𝜇)𝑡.

Writing 𝑢(𝑥,𝑡)=𝑣²(𝑥,𝑡) and setting 𝑐₂=𝜇=0 and 𝜆=2 in (19), we reproduce the result of Khater and Hassan [35] (see their Equation (4.7)), 𝑢(𝑥,𝑡)=4900𝑘⁴𝛿²sech⁴𝜁, (22) where 𝜁=𝑘𝑥−16𝛿𝑘³𝑡.

Note that Khater and Hassan [35] 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 𝑢(𝑥,𝑡)=𝑣²(𝑥,𝑡) to transform (2) into 𝑣𝑣_𝑡+𝛼𝑣+𝛽𝑣²𝑣_𝑥+𝛿𝑣𝑣_𝑥𝑥𝑥=0. (23) Assume the traveling wave solution of (23) in the form 𝑣(𝑥,𝑡)=𝑉(𝜁),𝜁=𝑘(𝑥−𝑐𝑡). (24) Hence, (23) becomes −𝑐𝑉𝑉^′+𝛼𝑉²+𝛽𝑉³𝑉^′+𝑘²𝛿𝑉𝑉^′′′+3𝑉^′𝑉^′′=0. (25) Suppose that the solution of (25) can be expressed by a polynomial in (𝐺^′/𝐺) as 𝑉(𝜁)=^𝑚_𝑖=0𝑎_𝑖𝐺^′𝐺^𝑖,𝑎_𝑖≠0 (26) and 𝐺(𝜁) satisfies (8). The homogeneous balance between the highest order derivative 𝑉𝑉^′′′ and the nonlinear term 𝑉³𝑉^′ appearing in (25) yields 𝑚=1, and hence, we take the following formal solution: 𝑉(𝜁)=𝑎₀+𝑎₁𝐺^′𝐺, (27) where the positive integers 𝑎₀ and 𝑎₁ are to be determined later. Substituting (27) along with (8) into (25), collecting all the terms with the same power of (𝐺^′/𝐺), and equating each coefficient to zero yield a set of simultaneous algebraic equations for 𝑎₀, 𝑎₁, 𝑐, 𝑘, 𝛼, 𝛽, and 𝛿 as follows: 𝐺^′𝐺⁵∶−𝛽𝑎⁴₁−12𝛿𝑘²𝑎²₁𝐺=0,_′𝐺⁴∶−3𝛽𝑎³₁𝑎₀−𝛽𝑎⁴₁𝜆−27𝛿𝑘²𝑎²₁𝜆−𝛼𝑎³₁−6𝛿𝑘²𝑎₀𝑎₁𝐺=0,_′𝐺³∶−3𝛽𝑎³₁𝜆𝑎₀−19𝛿𝑘²𝑎²₁𝜆²−12𝛿𝑘²𝑎₀𝑎₁𝜆−20𝛿𝑘²𝑎²₁𝜇−𝛼𝑎³₁𝜆−𝛽𝑎⁴₁𝜇+𝑐𝑎²₁−2𝛼𝑎²₁𝑎₀−3𝛽𝑎²₁𝑎²₀𝐺=0,_′𝐺²∶𝑐𝑎₀𝑎₁−3𝛽𝑎²₁𝜆𝑎²₀−2𝛼𝑎²₁𝜆𝑎₀+𝑐𝑎²₁𝜆−26𝛿𝑘²𝑎²₁𝜆𝜇−8𝛿𝑘²𝑎₀𝑎₁𝜇−7𝛿𝑘²𝑎₀𝑎₁𝜆²−3𝛽𝑎³₁𝜇𝑎₀−𝛽𝑎₁𝑎³₀−𝛼𝑎₁𝑎²₀−𝛼𝑎³₁𝐺𝜇=0,_′𝐺¹∶−𝛼𝜆𝑎₁𝑎²₀+𝑐𝑎₀𝑎₁𝜆−8𝛿𝑘²𝑎²₁𝜇²−7𝛿𝜇𝑘²𝑎²₁𝜆²−3𝛽𝜇𝑎²₁𝑎²₀+𝑐𝑎²₁𝜇−𝛽𝜆𝑎₁𝑎³₀−8𝛿𝜆𝜇𝑘²𝑎₀𝑎₁−2𝛼𝜇𝑎²₁𝑎₀𝐺=0,_′𝐺⁰∶𝑐𝜇𝑎₀𝑎₁−𝛼𝜇𝑎₁𝑎²₀−𝛽𝜇𝑎₁𝑎³₀−3𝛿𝜆𝑘²𝑎²₁𝜇²−𝛿𝜇𝜆²𝑘²𝑎₀𝑎₁−2𝛿𝜇²𝑘²𝑎₀𝑎₁=0. (28) The above system admits the following sets of solutions: 𝑎₀=0,𝑎₁=4𝛼5𝛽𝜆,𝜇=0,𝑐=−16𝛼²275𝛽,𝑘=±1/−75𝛿𝛽𝛼𝜆,𝑎₀=−4𝛼5𝛽,𝑎₁=−4𝛼5𝛽𝜆,𝜇=0,𝑐=−16𝛼²275𝛽,𝑘=±1/−75𝛿𝛽𝛼𝜆,𝑎₀=5𝛽𝑎₁𝜆−4𝛼10𝛽,𝜇=25𝛽²𝑎²₁𝜆²−16𝛼²100𝛽²𝑎²₁,𝑐=−16𝛼²75𝛽,𝑘=±𝛽𝑎−12𝛿¹. (29)(30)(31) Now, substituting (29)-(30) into (27) gives, respectively, 𝑉₁(𝜁)=4𝛼𝐺5𝛽𝜆𝐺^′,𝑉₂(𝜁)=−4𝛼−5𝛽4𝛼𝐺5𝛽𝜆𝐺^′,𝑉₃(𝜁)=5𝛽𝑎₁𝜆−4𝛼10𝛽+𝑎₁𝐺^′𝐺. (32) When substituting the general solutions (9) into (32), we obtain the following three types of traveling wave solutions:

Case 𝐴>0: (hyperbolic type) 𝑣₁(𝑥,𝑡)=−2𝛼+5𝛽2𝛼𝑐5𝛽¹sinh(𝜆/2)𝜁+𝑐₂cosh(𝜆/2)𝜁𝑐₁cosh(𝜆/2)𝜁+𝑐₂)𝜁,2sinh(𝜆/2𝜁=±1/−75𝛿𝛽𝛼𝜆𝑥+16𝛼²𝑣75𝛽𝑡,₂(𝑥,𝑡)=−2𝛼−5𝛽2𝛼𝑐5𝛽¹sinh(𝜆/2)𝜁+𝑐₂cosh(𝜆/2)𝜁𝑐₁cosh(𝜆/2)𝜁+𝑐₂)𝜁,2sinh(𝜆/2𝜁=±1/−75𝛿𝛽𝛼𝜆𝑥+16𝛼²𝑣75𝛽𝑡,₃(𝑥,𝑡)=−2𝛼+5𝛽2𝛼×𝑐5𝛽₁sinh2𝛼/5𝛽𝑎₁𝜁+𝑐₂cosh2𝛼/5𝛽𝑎₁𝜁𝑐₁cosh2𝛼/5𝛽𝑎₁𝜁+𝑐₂sinh2𝛼/5𝛽𝑎₁𝜁,𝜁=±𝛽𝑎−12𝛿¹𝑥+16𝛼²75𝛽𝑡. (33)

If we set 𝑐₂=0 and write 𝑢(𝑥,𝑡)=𝑣²(𝑥,𝑡), then the above solutions can be written as 𝑢₁(𝑥,𝑡)=4𝛼²25𝛽²𝛼−1±tanh5−3𝛽𝛿𝑥+16𝛼²𝑡75𝛽²,𝑢₂(𝑥,𝑡)=4𝛼²25𝛽²𝛼1±tanh5−3𝛽𝛿𝑥+16𝛼²𝑡75𝛽²,𝑢₃(𝑥,𝑡)=4𝛼²25𝛽²𝛼−1±2tanh5−3𝛽𝛿𝑥+16𝛼²𝑡75𝛽². (34)(35)(36) Note that (35) is exactly the same solution of Khater and Hassan [35] as given in their first equation of (3.9) with 𝜁₀=0. Similarly we can obtain the second solution of (36) in Hassan [5] if we set 𝑐₁=0 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 𝑐₁=1:

(a)

(b)

Figure 1 

(a) 2D profile of (35): kink shaped solitary (𝑢+, blue line), anti-kink shaped solitary (𝑢−, black line). (b) Corresponding 3D plots when + sign is taken and when −ve sign is taken, with 𝛼=1, 𝛽=−1, and 𝛿=1.

Case 𝐴<0: (trigonometric type) 𝑉₁(𝑥,𝑡)=−2𝛼+5𝛽2𝛼𝑖5𝛽−𝑐₁sin(𝑖𝜆/2)𝜁+𝑐₂cos(𝑖𝜆/2)𝜁𝑐₁cos(𝑖𝜆/2)𝜁+𝑐₂)𝜁,2sin(𝑖𝜆/2𝜁=±1/−75𝛿𝛽𝛼𝜆𝑥+16𝛼²𝑉75𝛽𝑡,₂(𝑥,𝑡)=−2𝛼−5𝛽2𝑖𝛼×5𝛽−𝑐₁sin(𝑖𝜆/2)𝜁+𝑐₂cos(𝑖𝜆/2)𝜁𝑐₁cos(𝑖𝜆/2)𝜁+𝑐₂)𝜁,2sin(𝑖𝜆/2𝜁=±1/−75𝛿𝛽𝛼𝜆𝑥+16𝛼²𝑉75𝛽𝑡,₃(𝑥,𝑡)=−2𝛼+5𝛽2𝑖𝛼×5𝛽−𝑐₁sin2𝑖𝛼/5𝛽𝑎₁𝜁+𝑐₂cos2𝑖𝛼/5𝛽𝑎₁𝜁𝑐₁cos2𝑖𝛼/5𝛽𝑎₁𝜁+𝑐₂sin2𝑖𝛼/5𝛽𝑎₁𝜁,𝜁=±𝛽𝑎−12𝛿¹𝑥+16𝛼²75𝛽𝑡. (37)

But if 𝑐₂=0 and 𝑢(𝑥,𝑡)=𝑣²(𝑥,𝑡), then trigonometric type solution becomes 𝑢₁(𝑥,𝑡)=4𝛼²25𝛽²𝛼−1±𝑖tan53𝛽𝛿𝑥+16𝛼²𝑡75𝛽²,𝑢₂(𝑥,𝑡)=4𝛼²25𝛽²𝛼1±𝑖tan53𝛽𝛿𝑥+16𝛼²𝑡75𝛽²,𝑢₃(𝑥,𝑡)=4𝛼²25𝛽²𝛼−1±2𝑖tan53𝛽𝛿𝑥+16𝛼²𝑡75𝛽². (38)

Case 𝐴=0: (rational type) 𝑢₁(𝑥,𝑡)=4𝛼²25𝛽²𝑐−1+2²𝑐₁+𝑐₂𝜁²,𝑢₂(𝑥,𝑡)=4𝛼²25𝛽²𝑐1+2²𝑐₁+𝑐₂𝜁²,𝑢₃(𝑥,𝑡)=2𝛼5𝛽+𝑎₁𝑐₂𝑐₁+𝑐₂𝜁². (39)

As mentioned before, the (𝐺^′/𝐺)-expansion method gives more general types of solutions than that found by Khater and Hassan [35] and Hassan [5].

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 [8]. We will obtain more general exact solutions of the modified KP equation. In order to find the traveling wave solution of (3), we let 𝑣𝑥,𝑦,𝑡=𝑣(𝜁),𝜁=𝑥+𝑘𝑦−𝑐𝑡. (40) Now, taking 𝑢(𝑥,𝑦,𝑡)=𝑣²(𝑥,𝑦,𝑡), (3) becomes −𝑐+𝛿𝑘²𝑣𝑣^′′+−𝑐+𝛿𝑘²𝑣^′2+𝛼𝑣²𝑣^′′+2𝛼𝑣𝑣^′2+𝛽𝑣𝑣^′′′′+4𝛽𝑣^′𝑣^′′′+3𝛽𝑣^′′2=0, (41) where 𝑘, 𝑐, 𝛽, 𝛿, and 𝛼 are constants and the prime denotes differentiation with respect to 𝜁. Integrating (41) with respect to 𝜁 and setting the integration constant equal to zero, we obtain −𝑐+𝛿𝑘²𝑣𝑣^′+𝛼𝑣²𝑣^′+3𝛽𝑣^′𝑣^′′+𝛽𝑣𝑣^′′′=0. (42) Balancing 𝑣²𝑣^′ with 𝑣𝑣^′′′ gives 𝑚=2. Therefore, we can write the solution of (42) in the form 𝑣(𝜁)=𝑎₀+𝑎₁𝐺^′𝐺+𝑎₂𝐺^′𝐺², (43) where 𝑎₀, 𝑎₁, and 𝑎₂ are constants to be determined later. Substituting (43) along with (8) into (42) and collecting all terms with the same order of (𝐺^′/𝐺), the left hand sides of (42) are converted into a polynomial in (𝐺^′/𝐺). Setting each coefficient of each polynomial to zero, we derive a set of algebraic equations for 𝑎₀, 𝑎₁, 𝑎₂, 𝛿, 𝜆, 𝛼, 𝛽, 𝑐, 𝑘, and 𝜇 as follows: 𝐺^′𝐺⁷∶−60𝛽𝑎²₂−2𝛼𝑎³₂𝐺=0,_′𝐺⁶∶−2𝛼𝑎³₂𝜆−5𝛼𝑎₁𝑎²₂−60𝛽𝑎₁𝑎₂−150𝑎²₂𝐺𝜆=0,_′𝐺⁵∶−12𝛽𝑎²₁−5𝛼𝑎₁𝑎²₂𝜆−124𝛽𝑎²₂𝜆²−2𝑘²𝛿𝑎²₂−24𝛽𝑎₀𝑎₂−4𝛼𝑎₀𝑎₂−144𝛽𝑎₁𝑎₂𝜆+2𝑐𝑎²₂−4𝛼𝑎²₁𝑎₂−2𝛼𝑎³₂𝐺𝜇=0,_′𝐺⁴∶−196𝛽𝑎²₂𝜆𝜇−6𝛼𝑎₀𝑎₁𝑎₂−111𝛽𝑎₁𝑎₂𝜆²−4𝛼𝑎₀𝑎²₂𝜆−32𝛽𝑎²₂𝜆³−𝛼𝑎³₁−2𝑘²𝛿𝑎²₂𝜆+2𝑐𝑎²₂𝜆−54𝛽𝑎₀𝑎₂𝜆−5𝛼𝑎₁𝑎²₂𝜇−4𝛼𝑎²₁𝑎₂𝜆+3𝑐𝑎₁𝑎₂−6𝛽𝑎₀𝑎₁−3𝑘²𝛿𝑎₁𝑎₂𝐺=0,_′𝐺³∶2𝑐𝑎₀𝑎₂−76𝛽𝑎²₂𝜇²−4𝛼𝑎₀𝑎²₂𝜇−𝑘²𝛿𝑎²₁−27𝛽𝑎₁𝜆³𝑎₂−74𝛽𝑎²₂𝜆²𝜇+𝑐𝑎²₁−6𝛼𝑎₀𝑎₁𝑎₂𝜆+3𝑐𝑎₁𝑎₂𝜆−40𝛽𝑎₀𝑎₂𝜇−38𝛽𝑎₀𝑎₂𝜆²−19𝛽𝑎²₁𝜆²−2𝛼𝑎₀𝑎²₁−4𝛼𝑎²₁𝑎₂𝜇−168𝛽𝑎₁𝑎₂𝜆𝜇−3𝑘²𝛿𝑎₁𝑎₂𝜆+2𝑐𝑎²₂𝜇−2𝛼𝑎²₀𝑎₂−20𝛽𝑎²₁𝜇−𝛼𝑎³₁𝜆−12𝛽𝑎₀𝑎₁𝜆−2𝑘²𝛿𝑎₀𝑎₂𝐺=0,_′𝐺²∶2𝑐𝑎₀𝑎₂𝜆−4𝛽𝑎²₁𝜆³−8𝛽𝑎₀𝑎₁𝜇−3𝑘²𝛿𝑎₁𝑎₂𝜇−𝛼𝑎³₁𝜇−52𝛽𝑎₀𝑎₂𝜆𝜇+𝑐𝑎₀𝑎₁+3𝑐𝑎₁𝑎₂𝜇−6𝛼𝑎₀𝑎₁𝑎₂𝜇−𝑘²𝛿𝑎²₁𝜆−60𝛽𝑎₁𝑎₂𝜇²−𝛼𝑎²₀𝑎₁−2𝛼𝑎₀𝑎²₁𝜆−2𝑘³𝛿𝑎₀𝑎₂𝜆−2𝛼𝑎²₀𝑎₂𝜆−7𝛽𝑎₀𝑎₁𝜆²−26𝛽𝑎²₁𝜆𝜇+𝑐𝑎²₁𝜆−57𝛽𝑎₁𝜆²𝑎₂𝜇−8𝛽𝑎₀𝑎₂𝜆³−𝑘²𝛿𝑎₀𝑎₁−54𝛽𝑎²₂𝜆𝜇²𝐺=0,_′𝐺¹∶−2𝛼𝑎²₀𝜇+2𝑐𝑎₀𝑎₂𝜇−𝛼𝑎²₀𝑎₁𝜆−2𝑘²𝛿𝑎₀𝑎₂𝜇−8𝛽𝑎²₁𝜇²+𝑐𝑎₀𝑎₁𝜆−16𝛽𝑎₀𝑎₂𝜇²−8𝛽𝑎₀𝑎₁𝜆𝜇−𝛽𝑎₀𝑎₁𝜆³−14𝛽𝑎₀𝑎₂𝜆²𝜇−36𝛽𝜆𝜇²𝑎₁𝑎₂−12𝛽𝑎²₂𝜇³−𝑘²𝛿𝑎₀𝑎₁𝜆−𝑘²𝛿𝑎²₁𝜇+𝑐𝑎²₁𝜇−2𝛼𝑎₀𝑎²₁𝜇−7𝛽𝜇𝜆²𝑎²₁𝐺=0,_′𝐺⁰∶−6𝛽𝜇³𝑎₁𝑎₂+𝑐𝑎₀𝑎₁𝜇−𝛼𝜇𝑎²₀𝑎₁−3𝛽𝜆𝜇²𝑎²₁−𝛽𝜇𝜆²𝑎₀𝑎₁−6𝛽𝜆𝜇²𝑎₀𝑎₂−2𝛽𝑎₀𝑎₁𝜇²−𝑘²𝛿𝜇𝑎₀𝑎₁=0. (44)

Solving this system by Maple gives two sets of solutions.

Case 1. We have 𝑎₀=−30𝛽𝜇𝛼,𝑎₁=−30𝛽𝜆𝛼,𝑎₂=−30𝛽𝛼,𝑐=−16𝛽𝜇+4𝛽𝜆²+𝑘²𝛿. (45) Substituting the above case and the general solution (8) into (43) and according to (42), we obtain three types of traveling wave solutions of (3) as follows.

If 𝐴>0, we have the hyperbolic type𝑣=𝑥,𝑦,𝑡−30𝛽𝜇𝛼−30𝛽𝜆𝛼×−𝜆2+√𝐴2×𝑐₁sinh√𝐴/2𝜁+𝑐₂cosh√𝐴/2𝜁𝑐₁cosh√𝐴/2𝜁+𝑐₂sinh√𝜁−𝐴/230𝛽𝛼−𝜆2+√𝐴2×𝑐₁sinh√𝐴/2𝜁+𝑐₂cosh√𝐴/2𝜁𝑐₁cosh√𝐴/2𝜁+𝑐₂sinh√𝐴/2𝜁². (46)In particular, if 𝑐₁≠0, 𝑐₂=0, 𝜆>0, and 𝜇=0, then 𝑢(𝑥,𝑦,𝑡) becomes 𝑢𝑥,𝑦,𝑡=225𝛽²𝜆⁴4𝛼²sech⁴𝜆2𝜁,𝜁=𝑥+𝑘𝑦−4𝛽𝜆²+𝑘²𝛿𝑡. (47) If 𝐴<0, we have the trigonometric type 𝑣=𝑥,𝑦,𝑡−30𝛽𝜇𝛼−30𝛽𝜆𝛼×−𝜆2+√𝐴2×−𝑐₁sin√𝐴/2𝜁+𝑐₂cos√𝐴/2𝜁𝑐₁cos√𝐴/2𝜁+𝑐₂sin√𝜁−𝐴/230𝛽𝛼−𝜆2+√𝐴2×−𝑐₁sin√𝐴/2𝜁+𝑐₂cos√𝐴/2𝜁𝑐₁cos√𝐴/2𝜁+𝑐₂sin√𝐴/2𝜁². (48) So, the traveling wave solutions of (3) in this case are 𝑢𝑥,𝑦,𝑡=225𝛽²𝜆⁴4𝛼²sec⁴−𝜆²2𝜁,𝜁=𝑥+𝑘𝑦−4𝛽𝜆²+𝑘²𝛿𝑡. (49)

Case 2. We have 𝑎₀=−5𝛽𝜆²+2𝜇𝛼,𝑎₁=−30𝛽𝜆𝛼,𝑎₂=−30𝛽𝛼,𝑐=16𝛽𝜇−4𝛽𝜆²+𝑘²𝛿. (50) If 𝐴>0, we have the hyperbolic type𝑣𝜆𝑥,𝑦,𝑡=−5𝛽²+2𝜇𝛼−30𝛽𝜆𝛼×−𝜆2+√𝐴2×𝑐₁sinh√𝐴/2𝜁+𝑐₂cosh√𝐴/2𝜁𝑐₁cosh√𝐴/2𝜁+𝑐₂sinh√𝜁−𝐴/230𝛽𝛼−𝜆2+√𝐴2×𝑐₁sinh√𝐴/2𝜁+𝑐₂cosh√𝐴/2𝜁𝑐₁cosh√𝐴/2𝜁+𝑐₂sinh√𝐴/2𝜁². (51)However, if 𝑐₁≠0, 𝑐₂=0, 𝜆>0, and 𝜇=0, then 𝑢(𝑥,𝑦,𝑡) becomes 𝑢𝑥,𝑦,𝑡=25𝛽²𝜆⁴4𝛼²2−3sech²𝜆2𝜁²,𝜁=𝑥+𝑘𝑦−−4𝛽𝜆²+𝑘²𝛿𝑡. (52) If 𝐴<0, we have the trigonometric type 𝑣𝜆𝑥,𝑦,𝑡=−5𝛽²+2𝜇𝛼−30𝛽𝜆𝛼×−𝜆2+√𝐴2×−𝑐₁sin√𝐴/2𝜁+𝑐₂cos√𝐴/2𝜁𝑐₁cos√𝐴/2𝜁+𝑐₂sin√𝜁−𝐴/230𝛽𝛼−𝜆2+√𝐴2×−𝑐₁sin√𝐴/2𝜁+𝑐₂cos√𝐴/2𝜁𝑐₁cos√𝐴/2𝜁+𝑐₂sin√𝐴/2𝜁². (53) In the particular case when 𝑐₁≠0, 𝑐₂=0, 𝜆>0, and 𝜇=0, 𝑢(𝑥,𝑦,𝑡) becomes 𝑢𝑥,𝑦,𝑡=25𝛽²𝜆⁴4𝛼²2−3sec²−𝜆²2𝜁²,𝜁=𝑥+𝑘𝑦−−4𝛽𝜆²+𝑘²𝛿𝑡. (54) If 𝐴=0, we have the rational type 𝑣𝜆𝑥,𝑦,𝑡=−5𝛽²+2𝜇𝛼−30𝑘²−𝜆𝛿𝜆2+𝑐₂𝑐₁+𝑐₂𝜁−30𝑘²𝛿−𝜆2+𝑐₂𝑐₁+𝑐₂𝜁², (55) where 𝜁=𝑥+𝑘𝑦−(16𝛽𝜇−4𝛽𝜆²+𝑘²𝛿)𝑡.

We remark that our results in (47) and (52), when 𝑐₁≠0, 𝑐₂=0, 𝜆>0, and 𝜇=0, match those of Khater et al. [6] (2.19) when 𝑎=1. In Figure 2, we plot the bell type solitary for 2D profile and the corresponding 3D plot of (47) for parameters 𝛼=2, 𝛽=0.4 and 𝛿=0.1, 𝜆=1, 𝑘=1, and 𝑡=0.5.

(a)

(b)

Figure 2 

Bell type solitary (a) 2D profile and (b) corresponding 3D plot of (47) for parameters 𝛼=2, 𝛽=0.4 and 𝛿=0.1, 𝜆=1, 𝑘=1, and 𝑡=0.5.

4. Conclusion

The (𝐺^′/𝐺)-expansion was applied to solve the model of ion-acoustic waves in plasma physics where these equations each contain a square root nonlinearity. The (𝐺^′/𝐺)-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 𝑐₁, 𝑐₂, 𝜆, and 𝜇. We note that the special case 𝑐₁≠0, 𝑐₂=0, 𝜆>0, and 𝜇=0 reproduced the results of Khater and Hassan [35], Hassan [5] and Khater et al. [6]. 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 (𝐺^′/𝐺)-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 𝐺^′2=𝑑₂𝐺²+𝑑₃𝐺³+𝑑₄𝐺⁴, where 𝐺=𝐺(𝜁), to solve Schamel equation, Schamel-KdV (S-KdV) equation, and modified Kadomtsev-Petviashvili (KP) equation.

Acknowledgment

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