Analytical Solutions of Nonlinear Coupled Schrodinger–KdV Equation via Advance Exponential Expansion

This research work is to represent an advance exp(-Φ(ξ))-expansion method with nonlinear ordinary differential equation for constructing interacting analytical solutions of nonlinear coupled physical models arising in science and engineering. It is capable of determining all branches of interacting analytical solutions simultaneously and this difficult to discriminate with numerical technique. To verify its computational potentiality, the coupled Schrodinger-KdV equation is considered. The obtained solutions in this work reveal that the method is a very effective and easily applicable of formulating the scattered exact traveling wave solutions of many nonlinear coupled wave equations. It is investigated the scattered wave solutions may be useful in understanding the behavior of physical structures in any varied instances, where the coupled Schrodinger-KdV equation is occurred.


Introduction
In this article, the advance exp(-( )) ξ Φ -expansion method are mainly highlighted for finding more valuable explicit solutions of NLEEs. The valuable explicit form solution provides a means to determine the salient features in various science, technology and engineering applications. It can be serve as a basis for perfecting and testing computer algebraic software, such as Maple, Mathematica, MatLab etc for solving NLEEs. It is noted that several types of nonlinear partial differential equations (NPDEs) of physics, chemistry and biology hold unknown parameters and unknown functions. Analytical solutions allow researchers to design and perform experiments, by creating suitable natural situations, to determine these functions and parameters. There are several types of well-established methods that have been devoted to evaluate analytical solutions of NPDEs, such as the modified simple equation method [1,2], the ( / ) G G ′ expansion method [3,4], the tanh method [5,6], the Homotopy perturbation technique [7], the homogeneous balance method [8,9], the Hirota method [10], the Expfunction method [11,12], the exp ( ( )) ϕ ξ − -expansion method [13][14][15][16][17][18], the modified Kudryashov method [19], the generalized exp ( ( )) ϕ ξ − -expansion method [20,21], and so on. Due to the effectiveness of mathematical approaches, the advance exp(-( )) ξ Φ -expansion method may be easily applicable with the aid of symbolic computational software to find more general solitary and periodic wave solutions of NPDEs in mathematical physics and engineering. The main idea of this technique is to express the exact traveling wave solutions of NLEEs that satisfy the nonlinear ordinary differential equation where λ and µ real parameters. The advantage of this method over the other existing methods is that it provides some simple form exact traveling wave solutions to the nonlinear PDEs. Algebraic manipulations of this method is also much easier rather than the others existing methods. However, many types of coupled NLEEs that appeared as model equations for describing the interacting wave phenomena in mathematical physics, chemistry and biology. For instance, the coupled Schrodinger-KdV equation are appeared as model equation to describe the interacting wave dynamics in Langmuir wave, dust-acoustic wave and electromagnetic wave in plasma physics, also appeared as a model equation to describe various types of wave phenomena in mathematical physics and so on. The existence and appearance of solitary waves in intricate physical issues apart from the model equations of mathematical physics must be analyzed with sufficient accuracy. Therefore, the work is to explore a study linking to the advance exp(-( )) ξ Φ expansion method for solving the coupled Schrodinger-KdV equation to demonstrate the effectiveness and truthfulness of this method.

-Expansion Method
Let us consider the following coupled Schrodinger -KdV equation as ( ) Here ( ) , u x t present the complex function while ( ) , v x t present the real-valued function. The coupled Schrodinger -KdV equation [22][23][24] appeared as model equation for describing various types of wave propagation such as Langmuir wave, dust-acoustic wave and electromagnetic waves in plasma physics. From the article [22][23][24], it is observed that some of the traveling wave solutions have been analyzed using different methods. In this article, the advance exp(-( )) ξ Φ -expansion method is employed for finding more contented explicit from solutions to the coupled Schrodinger -KdV equation.
Besides, it is well known that the exp (-Φ (ξ))-expansion method [13][14][15][16][17] has been employed to look into exact solutions of the nonlinear evolution equations, wherein the nonlinear ODE ( ) exp( ( )) exp( ( )) λ µ ∈ℜ provides only a few traveling wave solutions to the nonlinear NLEEs. Therefore, in order to get more traveling wave solutions and to understand the inner structure apparently of the nonlinear physical phenomena, in this article we choose the following ODE as auxiliary equation: It is notable that eq. (4) has the following six kinds of general solutions as follows [18]: where 0 ξ is the integrating constant and 0 λµ > or 0 λµ < depends on sign of λ . Now, one can consider following travelling wave transform: where α , β and V are constants. By substituting eq. (4) into eq. (1), one obtains that 2 V α = and , U v satisfy the following coupled nonlinear ordinary differential equations: where primes denotes the differentiation with regards to ξ .
The pole of the coupled Eq. (5) are 2 N = , 2 M = . Therefore, the advance exp( ( )) ϕ ξ − -expansion method allows us to use the solution in the following form: where 0 1 2 , , , a a a b are unknowns constants to be determined. By substituting eq. (6) in the eq. (5) and collecting all terms with same power of the coefficient of ( ) e ϕ ξ − together, we obtain a system of algebraic equations. The system of algebraic equations is ignored for convenience. Solving the obtaining system of algebraic equations, we obtain the following set of solutions: From the solutions (9) to (17), it is observed that the method according to subsidiary equation (2) 11 , v x t with 5 , 5 x t − ≤ ≤ .  12 , v x t with 3 , 3 x t − ≤ ≤ . , | u x t and (b) non-topological bell type for ( ) 13 , v x t with 5 , 5 x t − ≤ ≤ . ( ) 23 , v x t with 5 , 5 x t − ≤ ≤ .

Result and Discussions
In this article, the ODE as in Eq. (2) has been considered as auxiliary equation and their solutions have been used. The main advantage of the introduced method is that it offers more simple form general exact traveling wave solutions with some free parameters. The exact solutions have its extensive importance to interpret the inner structures of the natural phenomena in mathematical physics, chemistry and biology. The explicit solutions represented various types of solitary wave solutions according to the variation of the physical parameters. In this article, some of types of solitary and periodic wave solutions are displayed graphically in Figures 1 to 4. The advantage of this method is that sometimes gives solutions in disguised versions of known solutions that may be found by other methods. It is worth noted that the (G'/G)-expansion method is special case of the extended G'/G)-expansion method. N. A. Kudryashov [25] have been shown that the ( / ) G G ′ -expansion method is equivalent to the well known tanh-method. Besides, the solutions are achieved via the advance exp (-Φ (ξ))expansion method with the auxiliary ODE ( ) exp( ( )) exp( ( )) 0 expansion method and the extended (G'/G)-expansion method performed with others. It is concluded that some of our solutions may be coincided with already published results, if the parameters taken particular values which authenticate our solutions. Therefore, it can be decided that the method is powerful mathematical tool for easily solving nonlinear evolutions equations and all kinds of NLEEs may be solved through this method. It is also predicted form this investigations that the obtained results may be useful for better understanding interacting wave phenomena in any varied instance, where the considered coupled equations are applicable.

Conclusions
The advance exp(-( )) ξ Φ -expansion method has been successfully employed to obtain generalized traveling wave solutions for describing the interacting wave phenomena in the vicinity of the coupled Schrodinger-KdV equation. The obtained solutions in this article are defined in the simple forms involving of hyperbolic functions, trigonometric functions and rational functions. It is found that the advance exp(-( )) ξ Φ -expansion method changes the given difficult problems into simple problems which may be solved easily. Hence, this method may be more easily used to many others NLEEs arising in mathematical physics and engineering.