Optical solitons perturbation and traveling wave solutions in magneto-optic waveguides with the generalized stochastic Schrödinger-Hirota equation

The main purpose of this work is to study the optical soliton solutions and single traveling wave solutions of the generalized stochastic Schr¨odinger-Hirota equation in magneto-optic waveguides. With the help of the complete discriminant system technique and symbolic computation, a range of new single traveling wave solutions and optcial solitons are derived, which include Jacobian elliptic function solutions, dark solitons, trigonometric function solutions, singular solitons, rational function solutions, hyperbolic function solutions, periodic wave solutions and solitary wave solutions. Lastly, in order to understand mechanisms of complex physical phenomena and dynamical processes for the generalized stochastic Schr¨odinger-Hirota equation in magneto-optic waveguides, two-dimensional and three-dimensional diagrams are also drawn.


Introduction
The propagation of optical solitons through waveguides or other types of optical fibers are mainly controlled by nonlinear systems such as complex Ginzburg-Landau equation (CGLE) [1,2], Lakshmanan-Porsezian-Daniel equation (LPDE) [3][4][5], Radhakrishnan-Kundu-Lakshmanan equation (RKLE) [6], Schrödinger-Hirota equation (SHE) [7][8][9] and so on.As we all know, due to the delicate balance between self-phase modulation (SPM) and Group velocity dispersion (GVD) , solitons are stable waves or pulses that can easily carry and transmit information over a long distance through optical fibers in the femto-second scale [10][11][12][13][14][15][16].When the chromatic dispersion (CD) of the NLSE systems decreases, the balance is fails.Then the modified model for handling NLSE with low CD is proposed in aid of substituting three-order dispersion (3OD) and fourth-order dispersion (4OD) to GVD, which have gained great interest and been implemented to retrieve so-called analytical cubic-quartic (CQ) solitons.
In the current work, the generalized stochastic Schrödinger-Hirota equation (SSHE) in magnetooptic waveguides is considered as follows [17,18] here φ = φ(x, t) and ϕ = ϕ(x, t) are complex functions, which represent the wave profiles.Nonzero constants a l (l = 1, 2) stand for the coefficients of chromatic dispersion (CD) terms, the coefficients b l (l = 1, 2) and c l (l = 1, 2) denote the self-phase modulation (SPM) and cross-phase modulation (XPM) terms, respectively, moreover, nonzero constants d l (l = 1, 2) represent the third order dispersion 3OD.The coefficients of e l , f l (l = 1, 2) represent the nonlinear terms.In addition, σ stands for the coefficient of noise strength and W (t) stands for the standard Wiener process, dW dt such that denotes the white noise.R l (l = 1, 2) represent the magneto-optic wave-guides terms.Finally, the coefficients α l , γ l , β l , (l = 1, 2) stand for the inter-modal dispersion (IMD), the nonlinear dispersion terms and self-steepening (SS) terms, respectively.
The outline of this work is arranged as follows.In Sect.2, we simplify system into an equivalent second-order ordinary differential system by using the traveling wave transformations and some other suitable transformations.Next, we derive a range of new optical soliton solutions and traveling wave solutions for system (1.1) through the complete discriminant system technique as well as symbolic computation.In Sect.3, numerical simulations are presented to visualize the mechanism of system (1.1) by selecting some suitable parameters.The last section summarizes the results of the current work.

Mathematical analysis and optical soliton solutions for the SSHE
In order to construct the traveling wave solutions and optical solitons of Eq. (1.1), we firstly take the following wave transformations into consideration and where nonzero real-valued constants µ, ω 0 and λ represent the frequency of the soliton, the velocity of soliton and the wave number, respectively.In addition, Φ 1 (ξ) and Φ 2 (ξ) denote the real-functions which stand for the amplitude portions of the solitons.Inserting (2.1) and (2.2) into Eq.(1.1), the real part and the imaginary part of Eq. (1.1) can be derived respectively, and and the imaginary parts and Assume that there exists a linear relationship between Φ 1 and Φ Integrating (2.9) and (2.10) once, one has According to Eqs. (2.11) and (2.12), we get It follows from (2.13), we deduce the velocity of the soliton (2.14) For Eq. (2.7), let that n = 1, one has Next, we will devote ourselves to seeking some new traveling wave solutions and optical soliton solutions for the generalized stochastic Schrödinger-Hirota equation through the complete discrim-

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inant system method and symbolic computation.As is known to all that professor Yang and his team [19] has introduced an algorithm in 1996, which could calculate a complete discrimination of a high-order polynomial via the computer algebra.Many of optical soliton solutions of different types have been derived in recent years [20][21][22][23][24][25].
By using the transformation in the following form, we deduce (2.17) Thus, the (2.16) can be transformed as follows By integrating (2.18) once, one has where ξ 0 represents the integration constant.For convenience, letting H(Ψ 1 ) = Ψ 2 1 + b 1 Ψ 1 + b 0 , thus the complete discrimination system can be deduced as follows After that, according to the root-classifications of (2.20), there will be four cases to be discussed.
(2.41)  of the SSHE in magneto-optic waveguides.Finally, three-dimensional and two-dimensional diagrams are also presented. 65