Optical singular and dark solitons to the nonlinear Schrödinger equation in magneto-optic waveguides with anti-cubic nonlinearity

The present paper aims to investigate the coupled nonlinear Schrödinger equation in magneto-optic waveguides having anti-cubic (AC) law nonlinearity. The solitons secured to magneto-optic waveguides with AC law nonlinearity are extremely useful to fiber-optic transmission technology. Three constructive techniques, namely, the (G′/G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(G^{\prime }/G)$$\end{document}-expansion method, the modified simple equation method, and the extended tanh method are utilized to find the exact soliton solutions of this model. Consequently, dark, singular, combined dark-singular and periodic soliton solutions are obtained. The behaviours of soliton solutions are presented by 3D and 2D plots.


Introduction
Many nonlinear physical phenomena arise in various fields of science and engineering such as quantum mechanics, fluid dynamics, molecular biology, nuclear dynamics plasma physics, solid state physics and optical fibres. To express these complex physical phenomena methods are devoted to Sect. 3. The Physical explanation of the obtained solutions are provided in Sect. 4. Finally, the last Section is reserved for the concluding remarks.

Solitons to the NLSE in magneto-optic waveguides with anti-cubic nonlinearity
In this section, we apply the proposed three methods to find the soliton solutions of NLSE in magneto-optic waveguides having AC nonlinearity.
If we choose c 1 = 0 and c 2 ≠ 0 in Eq. (27), then the periodic solitary wave solutions falls out: = 0, we obtain the rational function solutions as follows: and v(x, t) = Ku(x, t), where c 1 and c 2 are constants. where A 0 and A 1 ≠ 0 are constants to be determined later. Inserting Eq. (31) into Eq. (18), and then setting the coefficients Ψ( ) −j , (j = 0, 1, 2, 3, 4) equal to zero, we find a set of algebraic equations as follows:

Applying the modified simple equation method
Solving above algebraic equations in Eq. (32), we obtain:  (4) with (17), we find the exact solution to coupled pair of Eqs. (1) and (2) as follows: and v(x, t) = Ku(x, t) . As a particular selection, if we take c 1 = 0 and Page 9 of 16 722

Applying the extended tanh method
According to the proposed method (Wazwaz 2007;Abdou 2007), we assume that which gives to the change of variables  (18), and making all the coefficients of powers Y equal to zero, we get a set of equations as follows: Solving the above system, we find the following three sets of solutions: 1. The first set: (1) and (2) read as follows: and v(x, t) = Ku(x, t).
Remark 1 Note that the first two solutions are equivalent to the previous solutions Eqs. (25) and (26) respectively.

Physical explanation of the solutions
In this section, we present the physical interpretation of the obtained exact soliton solutions of the coupled NLSE in magneto-optic waveguides having AC law nonlinearity. The graphical illustrations of 3D and 2D plots of some solutions are given in Figs  in this paper gives us a different physical interpretation for the coupled NLSE in magnetooptic waveguides having AC law nonlinearity..

Conclusions
This study investigates the optical solitons solution of the nonlinear Schrödinger equation in magneto-optic waveguides with AC nonlinearity. The solitons secured to magneto-optic waveguides with AC law nonlinearity will be extremely advantageous in fiber-optic transmission technology. Therefore, based on three effective methods, namely the (G � ∕G)-expansion method, the modified simple equation method and the extended tanh method, we have successfully obtained the dark, singular and combined dark-singular soliton solutions of the above model. To our best knowledge, the application of proposed methods to the model, and the received combined soliton solutions are new, which have not been reported earlier. The obtained solutions are illustrated by 3D and 2D graphs to express the dynamical behaviour of solutions.