New Exact Solutions of a Perturbed Nonlinear Schrodingers in Nonlinear Optics

Here, new exact travelling wave solutions of a perturbed nonlinear Schrodingers equation arising in nonlinear optics are successfully obtained via the generalized Kudryashov method. The proposed method has been successfully implemented to seek exact solutions for the modeles coming for describing nonlinear optics. The results obtained by the generalized Kudryashov method is straightforward and concise mathematical tool to establish the exact analytical solutions of nonlinear equations. The solutions obtained here are new and have not been reported in former literature.


Introduction
The investigation of the travelling wave solutions for nonlinear evolution equations arising in mathematical physics plays an important role in the study of nonlinear physical phenomena. The nonlinear evolution equations are major subjects in physical science,appears in various scientific and engineering fields, such as fluid mechanics, plasma physics, optical fibers, biology, solid state physics, chemical kinematics, chemical physics and ochemistry. Nonlinear wave phenomena of dissipation, diffusion, reaction and convection are very important in nonlinear wave equations. In the past several decades,new exact solutions may help to find new phenomena. A variety of powerful methods for obtaining the exact solutions of nonlinear evolution equations have been presented .
The nonlinear Schrodinger equation has a central importance in many natural sciences as well as engineering with numerous interpretations and applications concerning e.g. nonlinear optics, protein chemistry, plasma physics and fluid dynamics. This paper will consider the perturbed NLSE which governs the dynamics of solitons in negativeindex material with non-Kerr nonlinearity and third-order dispersion, and the dimensionless form of the equation is given by [24][25][26][27][28][29] where u(x, t) is the complex field amplitude. a, b, and c are the coefficients of group velocity dispersion, spatial temporal dispersion and non-Kerr nonlinearity, and d, s, μ, θ and γ account for the inter-modal dispersion, selfsteepening, Raman effect, nonlinear dispersion and third order dispersion, respectively. The last three terms appear in the context of negative-index material.
The main aim of this study is to extract exact solitons to Eq. (1) using the proposed method. Four different kinds of nonlinearity are considered for Eq. (1). They are Kerr law, power law, parabolic law and dual-power law.

Analysis of the Method
In what follows the properties of the the generalized Kudryashov method [30,31] as: For a given the general nonlinear partial differential equation of the type where u(x, t) is an unknown function, x is the spatial variable and t is the time variable, ψ is a polynomial in u and its derivatives, in which the highest order derivatives and nonlinear terms are involved.
Step 1 The travelling wave variable ξ = x − νt transform Eq. (2) into ODE as where the prime denotes to the differenation with respect to ξ .
Step 2 Considering trial equation of solution in Eq. (3), it can be written as where A and B are polynomial of Q(ξ ). Therefore, we can find the value of N and M, where Q = Q(ξ ) satisfies the following ODE: The Riccati Eq. (5) admits the following exact solution as follows: where is an constant.
Step 3 The positive integer N and M appearing in Eq. (4) can be determined by considering the balancing between the highest order derivative and the nonlinear term comes from Eq. (2) via the relations where p, q, s are integer numbers. Therefore, we can find the value of N and M in Eq. (4).

The New Exact Solutions of Eq. (1)
To solve Eq. (1), we use the wave transformations as where P(ξ ) represents the shape of the pulse and In Eq. (9), φ(x, t) gives the phase component of the soliton. Then, in Eq. (10), k, w and ξ 0 respectively represent the frequency, wave number and phase constant and in Eq. (10), v shows the velocity of the soliton.

For Kerr Law
For Kerr law nonlinearity as F(q) = q, then Eq. (14) becomes In this section the propsed methed will be used for construction the new exact solution of Eq. (15). Now balancing the highest order derivative P and nonlinear term P 3 , we get 3N − 3M = N − M + 2 or equivalent N = M + 1. Setting M = 1, we obtain N = 2. Then Eq. (14) reads Making use of Eq. (16) into (15) with Eq. (5), and collecting all power of Q(ξ ), we get a system of algebraic equations. By solving this algebraic system of equations, we have case (1) In view of Eq. (17), inserting Eq. (17) into (16), admits to the new exact solution of Eq. (15) as where w, ν and k are given in Eq. (17),

For Power Law
In this case, we assume as F(q) = q n , then Eq. (14) becomes In the same manner, to solve for obtaining the exact solution of Eq. (24), we use Then Eq. (24) reads Balancing the highest order derivative UU and nonlinear term U 3 , we get 3N − 3M = N − M + 2 or equivalent N = M + 2. For M = 1, we obtain N = 3. Then Eq. (26) reads Inserting Eq. (27) into (26) with Eq. (5), and collecting all power of Q(ξ ), we get a system of algebraic equations. By solving this algebraic system of equations, we have case (1) Using Eq. (28) into (27), one can directly obtined the new exact traveling wave solution as where w, ν and k are given in Eq. (28),

Remark 2
The new exact traveling wave solution obtained here via the proposed method are new and have not been reported in former literature. For simplicity case (2) should be omitted here.

For Parabolic Law
Here, we set F(q) = q + ηq 2 , then Eq. (14) becomes To solve the reduced Eq. (36) for constructing the new exact solutions of Eq. (36), we use Then Eq. (36) can be rewritten as Balancing the highest order derivative UU and nonlinear term U 4 , we get N = M + 1 and Inserting Eq. (39) into (38) with Eq. (5), and collecting all power of Q(ξ ), we get a system of algebraic equations. By solving this algebraic system of equations, we have Making use Eq. (40) into (39), admits to where w, ν and k are given in Eq. (40),

Conclusion and Summary
In this work, we investigated analytically solutions the nonlinear physical model Eq. (1) arising in nonlinear optics vis the generalized Kudryashov method. Different kinds of nonlinearities including Kerr law, power law, parabolic law and dual-power law are taken into accountd. The generalized Kudryashov method has been successfully implemented to seek exact solutions for nonlinear differential equations coming for describing nonlinear optics. An extended methods will be used such as [32][33][34][35][36][37][38][39][40][41] for constructing the travelling wave solutions of a fractional order space-time nonlinear evolution equations arising in nonlinear optics. This our task in future.
We conclude that the studied method can be more effectively applied to investigate other nonlinear partial differential equations which frequently arise in mathematical physics and other scientific application fields.
To our knowledge, these new solutions have not been reported in former literature, they may be of significant importance for the explanation of some special physical phenomena.