Nonlinear Schrödingers equations with cubic nonlinearity : M-derivative soliton solutions by exp ( − Φ ( ξ ) )-Expansion method

This paper uses the exp (−Φ(ξ ))-Expansion method to investigate solitons to the M-fractional nonlinear Schrödingers equation with cubic nonlinearity. The results obtained are dark solitons, trigonometric function solutions, hype rbolic solutions and rational solutions. Thus, the constraint relations between the model coe ffi ients and the traveling wave frequency coefficient for the existence of solitons solutio ns are also derived.


Introduction
During the past two decades, fractional calculus have advanced in analytical solution of nonlinear partial differential equation.On this way, a lot off attention has been place for investigation an exact traveling wave solutions of fractional models which yields to fractional differential equations.In addition, fractional calculus can provide mathematical formulas to transform the nonlinear partial differential equation to the nonlinear ordinary equation to handle them by some tractable integration tools.Also, it is very important to use the fractional derivatives which can provides an excellent implementation for the description of memory and hereditary properties [1].Moreover, conformable fractional versions of some nonlinear system were investigate [2][3][4].Thus, investigation of optical soliton with fractional time evolution, become very important due to its application in secure communication system of analog and digital signals, and to carry out hight speed data transmission over distance of several thousands of kilometers [5][6][7][8][9][10][11][12].Recently, some effective integration methods have been used to construct exact solutions for PDEs, such as semi-inverse variational principe [21], the simplest equation approach [22], the first integral method [23], ansatz scheme [24] and the generalized tanh method [26] and so on.On this way, exact optical solitons in metamaterials with different nonlinearities have been reported [13][14][15].The present paper will consider the M-fractional nonlinear Schrödingers equation with cubic nonlinearity.To construct soliton solutions, the exp(−Φ(ξ ))-Expansion method is used to derived the ordinary differential equation obtained.

M-fractional preliminaries
During the last decade, several definitions of fractional derivatives have been used in literature such as Atangana-Baleanu derivative in Caputo direction, Atangana-Baleanu fractional derivative in Riemann-Liouville sense, the new truncated M-fractional derivative of , just to name a few.This section will highlight some basic definitions and theorem of M-derivative of order α ∈ (0, 1).

M-fractional nonlinear Schrödingers equation with cubic nonlinearity
The proposed equation has been studied and many exact solutions were obtained [19,20] iD Where ψ is a complex valued function of the spatial coordinate x and time t, while Ω is the coefficient of the nonlinearity To establish solutions of (2), we surmise that ψ = ψ(x,t) can be expressed as follows ψ(x,t) = v(x,t) + iu(x,t), Substitute (3) in (2), the following system is obtained To transform the system of equation obtained, we used the following variable where κ and ω are real constants.Considering v(x,t) = V (ξ ) and u(x,t) = U(ξ ), it is obtained the following ODE

exp(−Φ(ξ ))-Expansion method
This section will be used the exp(−Φ(ξ ))-Expansion method to construct solutions to (6).Thereby, solution of ( 6) can be expressed as follows and Φ(ξ ) satisfies the following ODE By using the homogeneous balance principle, it is recovered from (6) N=M=1.Hence, (7) gives Substitute ( 9) and ( 8) into ( 6), it is obtained a system of algebraic equations.After solving the set of algebraic equations by aid of MAPLE, we get the following results.

Conclusion and remarks
In this paper, we investigate soliton solutions to the M-fractional nonlinear Schrödingers equation with cubic nonlinearity.The exp(−Φ(ξ ))-Expansion method is used derived the couple of nonlinear ordinary differential equation obtained.As a result, Dark solitons, trigonometric function solutions, hyperbolic function solutions and rational solutions have been obtained.These results obtained may be helpful in explaining communication system and nonlinear complex system.Figures (1) and (2) illustrated the (3D) plot of dark (10) solitary waves and trigonometric function solutions (12).

1 .
Department of Physics, Faculty of Science, the University of Maroua, P.O Box 814, Cameroon 2.Faculty of Sciences and Techniques Errachidia, University Moulay Ismail, Morocco., 3.Department of Physics, Faculty of Science,the University of Ngaoundere, P.O Box 454, Cameroon.