Three Different Methods for New Soliton Solutions of the Generalized NLS Equation Anwar Ja ’ afar

and Applied Analysis 3 Then u󸀠2 + c2u2 + c4u4 + c6u6 = 0. (15) 4. Methodology In this section we will apply three different methods to solve (15). These methods are Csch method, Extended Tanh-Coth method, and the modified simple equationmethod (MSEM). 4.1. Csch Function Method. The solution of many nonlinear equations can be expressed in the form [11] u (ξ) = A cschτ (μξ) (16) and their derivative u󸀠 (ξ) = −Aτμ cschτ (μξ) ⋅ coth (μξ) , u󸀠󸀠 (ξ) = Aτμ2 [(τ + 1) cschτ+2 (μξ) + τ cschτ (μξ)] , (17) whereA, μ, and τ are parameters to be determined and μ and λ are the wave number and the wave speed, respectively. We substitute (16)-(17) into the reduced equation (15); we get A2τ2μ2csch2τ (μξ) + A2τ2μ2csch2τ+2 (μξ) + c2A2csch2τ (μξ) + c4A4csch4τ (μξ) + c6A6csch6τ (μξ) = 0. (18) Balance the terms of the Csch functions to find τ 2τ + 2 = 6τ, Then τ = 12 . (19) We next collect all terms in (18) with the same power in cschk(μξ) and set their coefficients to zero to get a system of algebraic equations among the unknowns A, μ, and τ and solve the subsequent system A2 14μ2 + c2A2 = 0, A2 14μ2 + c6A6 = 0. (20) Solving the system of equations in (20), we get μ = 2i√c2 = 2i k √βα [αk13 − k12 − k2], A = ∓ 4 √c2 c6 = ∓ 4 √3β γ [αk13 − k12 − k2]; (21) then u (ξ) = ∓ 4 √3β γ [αk13 − k12 − k2]√csch (μξ) (22)


Introduction
Partial differential equations describe various nonlinear phenomena in natural and applied sciences such as fluid dynamics, plasma physics, solid state physics, optical fibers, acoustics, biology, and mathematical finance.Partial differential equations which arise in real-world physical problems are often too complicated to be solved exactly.It is of significant importance to solve nonlinear partial differential equations (NLPDEs) from both theoretical and practical points of view.The analysis of some physical phenomena is investigated by the exact solutions of nonlinear evolution equations (NLEEs) [1][2][3][4][5][6][7][8][9].
In this paper, the third-order generalized NLS equation is studied, which is proposed by Radhakrishnan, Kundu, and Lakshmanan (RKL) [10].The normalized RKL model can be written as Equation ( 1) describes the propagation of femtosecond optical pulses, (, ) represents normalized complex slowly varying amplitude of the pulse envelope, and , , , and  are real constants.Some solitary wave solutions and combined Jacobian elliptic function solution were constructed by different methods [3,4].The Csch method is used to carry out the solutions.Then, the Extended Tanh-Coth method and the modified simple equation method are used to obtain the soliton solutions of this equation.

Traveling Wave Solution
Consider the nonlinear partial differential equation in the form where (, ) is a traveling wave solution of nonlinear partial differential equation (2).We use the transformations, where  =  − .This enables us to use the following changes: Using (4) to transfer the nonlinear partial differential equation (2) to nonlinear ordinary differential equation,  (,   ,   ,   , . ..) = 0. ( The ordinary differential equation ( 5) is then integrated as long as all terms contain derivatives, where we neglect the integration constants.

The Generalized NLS Equation (RKL)
In this section, the generalized third-order NLS equation (RKL) ( 1) is chosen to illustrate the effectiveness of three methods.
The solution of (1) may be supposed as where Substituting ( 6) into (1) and by defining the derivatives, then decomposing (1) into real and imaginary parts yields a pair of relations which represented nonlinear ordinary differential equations.The real part is while the imaginary part is Integrating ( 9) once and setting the integration constant to zero, we obtain Equations ( 8) and (10) will be equivalent, provided that from which we get the parametric constraints multiplying both sides of (10) by   and integrating with respect to  with zero constant, we get Then (15)

Methodology
In this section we will apply three different methods to solve (15).These methods are Csch method, Extended Tanh-Coth method, and the modified simple equation method (MSEM).

Csch Function Method.
The solution of many nonlinear equations can be expressed in the form [11] and their derivative where , , and  are parameters to be determined and  and  are the wave number and the wave speed, respectively.We substitute ( 16)-( 17) into the reduced equation (15); we get Balance the terms of the Csch functions to find We next collect all terms in (18) with the same power in csch  () and set their coefficients to zero to get a system of algebraic equations among the unknowns , , and  and solve the subsequent system (20) Solving the system of equations in (20), we get then therefore

Tanh-Coth Method.
The key step is to introduce the ansatz, the new independent variable [12,13] that leads to the change of variables: Assume Equation ( 15) can be written as The next step is that the solution of ( 29) is expressed in the form where the parameter  can be found by balancing the highest-order linear term with the nonlinear terms in (29).

The Modified Simple Equation Method.
We look for solutions of (29) in the form [14] Then (29) can be written as Then (41) can be written as Equating expressions in (42) at  −1 ,  −2 ,  −3 , and  −4 to zero, we have the following system of equations: Solving the system of equations in (43), Family 1 Family 8 Family 9 Family 10 Family 11

Conclusion
In this paper, series of new traveling wave solutions have been obtained.The Csch method and the Extended Tanh-Coth method and modified simple equation method are used to carry out the integration of the generalized NLS equation, which is RKL.These methods can be also applied to solve other types of the generalized nonlinear evolution equations with complex coefficients.The solitary waves in Figures 1 and  2 obtained by the Csch and Tanh-Coth methods, respectively, are identical in form and behavior.The obtained solutions are very useful and may be important to explain some physical phenomena and find applications in the nonlinear pulse propagation through optical fibers.