An investigation of abundant traveling wave solutions of complex nonlinear evolution equations: The perturbed nonlinear Schrodinger equation and the cubic-quintic Ginzburg-Landau equation

In this article, the two variables (G∕G, 1∕G)-expansion method is suggested to obtain abundant closed form wave solutions to the perturbed nonlinear Schrodinger equation and the cubic-quintic Ginzburg-Landau equation arising in the analysis of various problems in mathematical physics. The wave solutions are expressed in terms of hyperbolic function, the trigonometric function, and the rational functions. The method can be considered as the generalization of the familiar (G′/G)-expansion method established by Wang et al. The approach of this method is simple, standard, and computerized. It is also powerful, reliable, and effective. Subjects: Science; Mathematics & Statistics; Applied Mathematics


Introduction
Investigations of exact wave solutions to nonlinear evolution equations (NLEEs) play the central role in the study the complicated tangible phenomena. The exact solutions provide much information

PUBLIC INTEREST STATEMENT
The modeling of most of the real-world phenomena lead to nonlinear evolution equations (NLEEs). In order for better understanding the complex phenomena, exact solutions play a vital role. Therefore, diverse group of researchers developed and extended different methods for investigating closed form solutions to NLEEs. In the present article, we use the two variables-expansion method to investigate closed form wave solutions of the perturbed nonlinear Schrodinger equation and the cubic-quintic Ginzburg-Landau equation. Consequently, we obtain abundant closed form wave solutions of these two equations among them some are new solutions. We expect that the new closed form solutions will be helpful to explain the associated phenomena. and help to understand the inner composition that governs physical phenomena, such as plasma physics, optical fibers, biology, solid state physics, fluid mechanics, chemical reaction, and so on. Therefore, during the last several decades mathematicians, physicists, and engineers tried their best to find closed form solutions, but due to the rapidly growing complexity in changing the real parameters including time, it is not easy to control all problems by a unique method. Consequently, several direct methods for obtaining exact solutions to NLEEs have been developed, such as the tanh-function method (Fan, 2000;Parkes & Duffy, 1996;Yan, 2001), the Jacobi elliptic function expansion method (Fu, Liu, & Shi-Da, 2003;Liu, Fu, Liu, & Zhao, 2001;Yan, 2003), the homogeneous balance method (Wang, 1995(Wang, , 1996Wang, Zhou, & Li, 1996), the F-expansion method and its extension (Wang & Li, 2005a, 2005bWang, Li, & Zhang, 2007a;Wang & Zhou, 2003;Zhang, Wang, & Li, 2006;Zhou, Wang, & Miao, 2004), the Sub-ODE method (Li & Wang, 2007;Wang, Li, & Zhang, 2007b), the auxiliary differential equation method (Guo & Lai, 2010;Guo & Wang, 2011), the exp-function method (Akbar & Ali, 2011;He & Wu, 2006), the trial function method (Kudryashov, 1990;Zhang, 2008), the modified simple equation method (Jawad, Petkovic, & Biswas, 2010;Khan & Akbar, 2013), the (G′/G)-expansion method (Akbar, Ali, & Zayed, 2012a;Wang, Li, & Zhang, 2008), etc.
Owing to show the effectiveness and to enhance the range of applicability of the (G′/G)-expansion method several extensions have been introduced by the researchers, for instance, (Zhang, Jiang, and Zhao (2010) proposed an improved (G′/G)-expansion method to examine general traveling wave solutions, Li, Li, and Wang (2010) established a two variables (G � ∕G, 1∕G)-expansion method, Zayed (Zayed, 2011) developed a further alternative approach of this method, in which G(ξ) satisfies the Riccati equation. Akbar, Ali, and Zayed (2012b) presented a generalized and improved (G′/G)expansion method which provides further general wave solutions to NLEEs (Akbar & Ali, 2016). In this article, we introduce and implement the two variables (G � ∕G, 1∕G)-expansion method to the perturbed nonlinear Schrodinger equation in the form (Zhang, 2008).
as well as the nonlinear cubic-quintic Ginzburg-Landau equation (Shi, Dai, & Li, 2009). where in Equation (1.1) γ 1 , γ 2 , γ 3 , and α are constants and in Equation (1.2), β < 0 is a real constant, z and x are the propagation and transverse coordinates, respectively, r 1 , r 2 are constants. Abundant exact wave solutions including solitary wave, shock wave, and periodic wave solutions with arbitrary parameter of Equations (1.1) and (1.2) are successfully obtained. In previous literature exact solutions of Equations (1.1) and (1.2) have been sought by using the extended modified trigonometric function series method (Zang, Li, Liu, & Miao, 2010), the modified (G′/G)-expansion method (Shehata, 2010), the extended tanh-function method (Dai & Zhang, 2006), etc. But no one studied solutions of the above-mentioned equations through the two variables (G � ∕G, 1∕G)-expansion method. As a pioneer work, Li et al. (2010) applied the two variables (G � ∕G, 1∕G)-expansion method and found the exact solutions of Zakharov equations. Then , Zayed, Hoda Ibrahim, and Abdelaziz (2012), Demiray, Ünsal, and Bekir (2015) determined exact solution of nonlinear evolution equations by using this method.
The rest of this article is organized as follows: In Section 2, we describe the two variables (G � ∕G, 1∕G)-expansion method. In Section 3, the perturbation nonlinear Schrodinger Equation (1.1) is investigated by the proposed method. In Section 4, we utilize the proposed method to examine Equation (1.2). In Section 5, conclusions are given.

Description of the (G � ∕G, 1∕G)-expansion method
In this section, we depict the main steps of the (G � ∕G, 1∕G)-expansion method for finding traveling wave solutions to NLEEs. Let us consider the second-order linear ordinary differential equation (LODE): and for minimalism here and later on, we let By means of (2.1) and (2.2), we obtain The general solution of LODE (2.1) depends on the sign of and thus we obtain the following three types of solutions: Case 1: When < 0, the general solution of LODE (2.1) is given as, where A 1 and A 2 are two arbitrary constants and Case 2: When > 0, the general solution of LODE (2.1) is as follows: where A 1 and A 2 are two arbitrary constants and Case 3: When = 0, the general solution of LODE (2.1) is as follows: where A 1 and A 2 are two arbitrary constants and Let us consider a general nonlinear evolution equation (NLEE), in three independent variables say, x, y, and t, Usually, the left-hand side of Equation (2.7) is a polynomial in u(x, y, t) and its different partial derivatives. In order to investigate exact traveling wave solutions of NLEEs by means of the two variables (G � ∕G, 1∕G)-expansion method, the following steps need to be performed: Step 1: By means of the wave variable = x + y − v t, u(x, y, t) = u( ), Equation (2.7) can be reduced to an ODE as follows: (2.1) (2.7) F(u, u t , u x , u y , u tt , u xx , u xt , u yy , …) = 0.

Application of the method to the perturbed nonlinear Schrodinger equation
In this section, we implement the (G � ∕G, 1∕G)-expansion method to extract traveling wave solutions to the perturbed nonlinear Schrodinger Equation (1.1). Since u(x, t) in Equation (1.1) is a complex function, so we assume that are real function and 1 , , k, c are arbitrary constants to be calculated. Substituting (3.1) into Equation (1.1), we have two ODEs for δ(ξ): and Integrating (3.3) with respect to ξ and setting the constant of integration to be zero yields Now the necessary and sufficient condition for a nontrivial solution of the function δ = δ(ξ) satisfying both (3.2) and (3.4) is that, the coefficients of (3.2) and (3.4) should be proportional.
Therefore, we get (Shehata, 2010): (3.5) By balancing the highest order derivatives term δ″ with the nonlinear term of the highest order δ 3 appearing in (3.5), we obtain the balance number N = 1. So, the solution of Equation (3.5) has the form: where a 0 , a 1 and b 1 are constants to be estimated afterward. There are three cases, we have discussed earlier and we give the related theorems.
Case 1: When < 0 (Hyperbolic function solutions), substituting (3.6) into (3.5) and with the help of (2.3) and (2.4), the left-hand side of (3.5) be converted into a polynomial in φ and ψ. Setting the coefficients of the similar power to zero yield a system of algebraic equations in a 0 , a 1 , b 1 , ( < 0), and σ: Therefore, we obtain Hence, for this case we have the exact solution of (1.1) in the following form In particular, if we take A 1 = 0 and A 2 > 0 in (3.13), then we have solitary wave solution Again, if we take A 2 = 0 and A 1 > 0 in (3.13), then we have solitary wave solution Case 2: When > 0 (Trigonometric function solutions), similar to case 1, after solving a system of algebraic equations by using Mathematica, we have the three solutions: (a) Using Equations (3.1) and (3.6), we get the solution of the perturbed nonlinear Schrodinger Equation (1.1) for A < 0 and B > 0: Using Equations (3.1) and (3.6), we get the solution of the perturbed nonlinear Schrodinger Equation (1.1) for A = 0 and B < 0: Therefore, we get Hence, we have for this case the exact solution of (1.1) in the form For all cases, from Equations (3.1)-(3.25), A 1 , A 2 are arbitrary constants, = A 2 1 − A 2 2 , = A 2 1 + A 2 2

Implementation of the method to the nonlinear cubic-quintic Ginzburg-Landau equation
For the nonlinear cubic-quintic Ginzburg-Landau equation, we assume that is called the reduced time, t is the physical time, v 0 is the group velocity of the carrier wave, k is the real parameter, a (x, ) and (x, ) are real functions.
By means of (4.1), Equation (1.2) transformations into a complex function and splitting real and imaginary parts, we get and where l 0 , l 1 , h 0 , h 1 are all constants to be determined. , so we substitute a ( ) = 1 2 ( ) which yields Balancing the highest order derivative terms and the highest order nonlinear terms, we obtain N = 1. Therefore, solution shape of Equation (4.7) has the form,
Consequently, the traveling wave solution of the nonlinear cubic-quintic Ginzburg-Landau equation (1.2) becomes, To fix, if we choose A 1 = 0 and A 2 > 0 in (4.15), we obtain the following solitary wave solution Again, if we choose A 2 = 0 and A 1 > 0 in (4.15), we obtain the following solitary wave solution Case 2: When > 0, similar to case 1, after solving a system of algebraic equations by using Mathematica, we obtain three kind of solutions: (a) By using the Equations (4.1) and (4.8), we get the solution of Equation (1.2) by choosing the following set of solution for r 2 < 0 and l 1 < 0, And so, the solution of Equation (4.7) turns into, As a result, we obtain the subsequent wave solution of the nonlinear cubic-quintic Ginzburg-Landau Equation (1.2), In particular, if we take A 1 = 0 and A 2 > 0 in (4.18), we obtain the subsequent wave solution, Alternatively, if we take A 2 = 0 and A 1 > 0 in (4.18) we obtain the subsequent wave solution, (b) By using the Equations (4.1) and (4.8), we get the solution of Equation (1.2) by choosing the following set of solution for r 2 > 0 and ρ > 0, Hence, the solution of Equation (4.7) becomes, Therefore, for this case, we obtain the solution as follows: In particular, if we consider A 1 = 0 and A 2 > 0 in (4.21), the following wave solution can be found, Once again, if we consider A 2 = 0 and A 1 > 0 in (4.21), the following wave solution can be found, (c) By using the Equations (4.1) and (4.8), we get the solution of Equation (1.2) by considering the following set of solution for r 2 < 0 and l 1 < 0, . (4.20)

Results and discussions
The G � ∕G, 1∕G-expansion method has been used to obtain exact traveling wave solutions involving arbitrary parameters of the perturbed nonlinear Schrodinger equation and the cubic-quintic Ginzburg-Landau equation arising in the analysis of wave in complex media. We get many familiar solitary waves as the two parameters A 1 and A 2 receive special values. The key point of this method is that, using the wave variable we transform the NLEE into an ODE. When we take μ = 0 in Equation (2.1) and b i = 0 in (2.9), then the two variables G � ∕G, 1∕G expansion method transforms to the modified G � ∕G-expansion method. By this method, we get a solution of the polynomial form in two variables G � ∕G and 1∕G in that case G = G( ) is the general solution of (2.1).
In this article, nineteen traveling wave solutions of the perturbed nonlinear Schrodinger equation and twenty new traveling wave solutions of the cubic-quintic Ginzburg-Landau equation have been successfully obtained by using the G � ∕G, 1∕G expansion method. The solutions of the Schrodinger Equation (1.1) and the cubic-quintic Ginzburg-Landau Equation (1.2) depend on the chosen constants A, B, D and ρ, σ, as less than zero (> 0 or = 0), respectively. The six solutions (3.7)-(3.9) and (3.16)-(3.18) of the perturbed nonlinear Schrodinger equation are identical to the solutions obtained in (Shehata, 2010), if we set 1 0 and 2 0 is equal to zero. Other thirteen solutions of the Schrodinger equation are new which might be important in the wave analysis. We see that by using the two variables G � ∕G, 1∕G expansion method we get abundant closed form wave solutions.

Conclusion
In this article, the two variables G � ∕G, 1∕G-expansion method has been suggested and used to obtain the exact traveling wave solution to the perturbed nonlinear Schrodinger equation and the cubic-quintic Ginzburg-Landau equation. It is seen that three types of traveling wave solution in terms of hyperbolic, trigonometric, and rational functions of these equations have successfully been found ( ) = 3r 1 4r 2 .  . by using this method. This expansion method changes the difficult problems into simple problems which can be examined easily. In physical science, the solutions of these nonlinear equations have many applications. Usually, it is very difficult to study the perturbed nonlinear Schrodinger equation and the cubic-quintic Ginzburg-Landau equation by the traditional methods. On comparing to other methods this expansion method is powerful, effective, and convenient to investigate complex nonlinear evolution equations. Additionally, this method is reliable, simple, and gives many new exact solutions. It is also standard and computerized method which allows one to solve more complicated nonlinear evolution equations in mathematical physics.