Soliton Solutions of the Coupled Schrödinger-Boussinesq Equations for Kerr Law Nonlinearity Anwar Ja ’ afar

and Applied Analysis 3 Substitute (12) in (10), then ξ = (x − 2kt + χ) . (14) Integrating (13) twice with zero constant, (13) can be written as [12k2 + α2] V − V + 3V2 − u2 = 0. (15) 5.1. Bright Soliton. Seeking the solution by sech function method as in (6) u (ξ) = A1 sech1 (μξ) , V (ξ) = A2 sech2 (μξ) , (16) the system of equations in (11) and (15) becomes, respectively, β1μ2 [(β1 + 1) sech1 (μξ) − β1 sech1 (μξ)] + [ω + k2 − α1] sech1 (μξ) − A2 sech12 (μξ) = 0, (17) [12k2 + α2] A2 sech2 (μξ) + A2β2μ2 [(β2 + 1) sech2 (μξ) − β2 sech2 (μξ)] + 3A22 sech2 (μξ) − A12 sech1 (μξ) = 0. (18) Equating the exponents and the coefficients of each pair of the sech functions, we find 2β1 = β2 + 2, β1 + β2 = β1 + 2, then β1 = β2 = 2. (19) Thus setting coefficients of (17)-(18) to zero yields set system of equations: 4μ2 − [ω + k2 − α1] = 0, [12k2 + α2] − 4μ2 = 0, 6μ2 − A2 = 0, 6A2μ2 + 3A22 − A12 = 0. (20) Solving the system of equations in (20), we get A1 = ∓3 [12k2 + α2] , A2 = 3 2 [12k2 + α2] , (21) E1(x, t)


Introduction
All optical communications are being used for transcontinental and transoceanic data transfer, through long-haul optical fibers, at the present time.There are various aspects of soliton communication that still need to be addressed.One of the features is the dispersive optical solitons.In presence of higher order dispersion terms, soliton communications are sometimes a hindrance as these dispersion terms produce soliton radiation.Nonlinear evolution equations have a major role in various scientific and engineering fields, such as optical fibers.Nonlinear wave phenomena of dispersion, dissipation, diffusion, reaction, and convection are very important in nonlinear wave equations.In recent years, quite a few methods for obtaining explicit traveling and solitary wave solutions of nonlinear evolution equations have been proposed.In recent years, exact homoclinic and heteroclinic solutions were proposed for some NEEs like nonlinear Schrödinger equation, Sine-Gordon equation, Davey-Stewartson equation, Zakharov equation, and Boussinesq equation [1][2][3][4][5][6][7].
In particular, the study of the coupled Schrödinger-Boussinesq equations has attracted much attention of mathematicians and physicists [8][9][10].The existence of the global solution of the initial boundary problem for the equations was investigated in [8].The existence of a periodic solution for the equations was considered in [9].Kilicman and Abazari [10] used the (  /)-expansion method to construct periodic and soliton solutions for the Schrödinger-Boussinesq.The investigation of nonlinear partial differential equations plays an important role in the study of nonlinear physical phenomena [9][10][11][12].
The nonlinear coupled Schrödinger-Boussinesq equation (SBE) governs the propagation of optical solitons in a dispersive optical fiber and is a very important equation in the area of theoretical and mathematical physics.This paper is going to take a look at the bright, dark, and singular soliton solutions for Kerr law nonlinearity media.

Governing Equations
Consider the coupled Schrödinger-Boussinesq equations (SBE).They appeared in [13] as a special case of general systems governing the stationary propagation of coupled nonlinear upper hybrid and magneto sonic waves in magnetized plasma.These equations were in the form [14]

The Traveling 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.

Hyperbolic Function Methods
The solutions of many nonlinear equations can be expressed in the following form.

Dark Soliton.
Seeking the solution by tanh function method as in ( 7) the system of equations in ( 11) and ( 15) becomes, respectively, Equating the exponents and the coefficients of each pair of the sech functions, we find Thus setting coefficients of ( 26)-(27) to zero yields set system of equations: Solving the system of equations in (29), we get

Singular Soliton.
Seeking the solution by sech function method as in ( 8) the system of equations in (11) and ( 15) becomes, respectively, Equating the exponents and the coefficients of each pair of the sech functions, we find Thus setting coefficients of ( 35)-(36) to zero yields set system of equations: Solving the system of equations in (38), we get

Modified Simple Equation
Method.This section will analyze (11) and ( 15) by the modified simple equation method; assume that solutions are of the form [23]

𝑢 (𝜉) =
where the parameters ,  can be found by balancing the highest-order linear term with the nonlinear terms in (11) and (15), respectively.In (11), we balance   with V, to obtain +2 = +, and then  = 2.While in (15), We balance V  with  2 , to obtain  + 2 = 2, and then  = 2. Then where  0 ,  1 ,  2 ,  0 ,  1 , and  2 are constants to be calculated.Substitute (43) in ( 11) and (15), respectively, to get Abstract and Applied Analysis 7 In (44) equating expressions at (   = 0, −1, −2, −3, −4) to zero, we get the following system of equations: Obviously when solving the system of (45), we conclude that equations can be satisfied simultaneously for the following constraints.Hence, the modified simple equation method does not produce the soliton solution in general case: Then we will solve the following ordinary differential equation: and therefore where  0 ,  1 are arbitrary constants.(50)

Conclusion
In this paper the dispersive bright, dark, and singular soliton solutions to SBE with Kerr law of nonlinearity were studied.The sech, tanh, csch, and the modified simplest equation method have been successfully applied to find solitons solutions for the coupled Schrödinger-Boussinesq equations.Several constraint conditions were assuring the existence of such solitons with Kerr law nonlinearity.The modified simple equation method does not produce the soliton solution in general case.Solutions by three methods are plotted in figures for the real and imaginary parts for (, ) and (, ).Compatibility in figures shape between the solutions of (, ) and (, ) by the same method sometimes appeared.Solutions may be important for the conservation laws for dispersive optical solitons.Those research outcomes will be soon disseminated.