The solitary solutions for the stochastic fractional Chen Lee Liu model perturbed by multiplicative noise in optical fibers and plasma physics

In this paper, we consider the stochastic fractional Chen Lee Liu model (SFCLLM). We apply the mapping method in order to get hyperbolic, elliptic, rational and trigonometric stochastic fractional solutions. These solutions are important for understanding some fundamentally complicated phenomena. The acquired solutions will be very helpful for applications such as fiber optics and plasma physics. Finally, we show how the conformable derivative order and stochastic term affect the exact solution of the SFCLLM.

where G is the the normalized electric-field envelope, D α x is a conformable fractional derivative (CFD), a, b and ρ are positive constants, and B t = ∂B ∂t is the derivative of the Brownain motion B(t).Due to the importance of the Chen Lee Liu model in fiber optics and plasma physics, many authors have used several methods in order to acquire the analytical solutions for this model such as Laplace Adomian decomposition method 26 , chirped W shaped optical solitons 27 , Darboux transformation 28 , modified extended tanh-expansion method 29 , Sardar sub-equation method 30 , Riccati-Bernoulli and generalized tanh methods 31 , ( G ′ /G, 1/G)-expansion approach 32 , extended direct algebraic method 33 , and modified Khater method 34 . (1) The purpose of this paper is to create the exact solutions of the SFCLLM (1).We apply the mapping method to produce a variety of solutions for instance hyperbolic, trigonometric, rational, and elliptic functions.Furthermore, we use Matlab program to create 2D and 3D graphs for some of the analytical solutions established in this paper to address the impact of the conformable fractional derivative and time-dependent coefficient on the acquired solutions of the SFCLLM (1).
The paper is organized as described below.In "Conformable derivative", we define the CFD and describe some of its features.To attain the wave equation of the SFCLLM (1), we utilize an appropriate wave transformation in "Wave equation for SFCLLM".In "The solutions of the SFCLLM", we find the exact solutions of the SFCLLM (1) using the mapping method .In "Impacts of CD and noise", we address the impact of the CFD and stochastic term on the attained solutions.Finally, the conclusion of the paper is introduced.

Conformable derivative
Fractional calculus operators are an effective tool for modeling and evaluating complicated processes that cannot be effectively explained using regular integer-order calculus.Several types of fractional derivative operators have been suggested in the literature, including the Katugampola derivative, the Jumarie derivative, the Hadamard derivative, the Riemann-Liouville derivative, the Caputo derivative, and the Grünwald-Letnikov derivative [35][36][37][38] .In recent years, Khalil et al. 40 proposed the conformable derivative (CD), which has features similar to Newton derivative.From here, let us define the CD for the function P : (0, ∞) → R of order α ∈ (0, 1] as follows: The CD fulfills the next properties for any constant a and b:

The solutions of the SFCLLM
To find the solutions of Eq. ( 7), we use the mapping method, which is stated in 40 .Assuming the solutions of Eq. ( 7) take the form where a i (t) are undefined functions in t for i = 0, 1, ...., M , and X is the solution of where b 1 , b 2 andb 3 are real constants.

Discussion and physical interpretation
This work aimed to get exact solutions of the SFCLLM (1).We used the mapping approach, which yielded a variety of solutions, including periodic solutions, kink solutions, brilliant solutions, dark optical solutions, solitary solutions, and so on.Physically, dark optical soliton denotes waves with lower intensities than the backdrop.Singular solitons are solitary waves with discontinuous derivatives, including compactions with limited (compact) support or peakons with discontinuous first derivatives.These kinds of solitary waves are very important owing to their efficiency and of course flexibility in long distance optical communication.We investigated the influence of conformable derivatives on the obtained solutions and concluded that as the order of fractional derivatives increases, the surface shrinks, as depicted in Figs. 1 and 2. Furthermore, we examined the impact of  15) with α = 1, 0.8, 0.6 (iv) depict 2D-profile of Eq. ( 15) with various value of α. noise on the solutions and observed that when the noise strength increases, the surface stabilizes around zero as shown in Figs. 3 and 4.

Conclusions
In this paper, we introduced a large variety of exact solutions to the stochastic fractional Chen Lee Liu model (SFCLLM) (1) forced by multiplicative noise in the Itô sense.By using the mapping approach, rational, elliptic, hyperbolic, and trigonometric stochastic fractional solutions were obtained.These solutions are important for understanding some fundamentally complicated phenomena.The attained solutions are very helpful for applications such as optics, plasma physics and nonlinear quantum mechanics.Finally, we show how the conformable derivative order and the stochastic term affect the exact solution of the SFCLLM (1).  14with α = 1,and different ρ (iv) depict 2D-profile of Eq. ( 14).  15with α = 1,and different ρ (iv) depict 2D-profile of Eq. ( 15).