The Analysis of Ansatzes for Optical Fibers Models

Optical fibers have a major role in the industry and daily life. To simulate the pulse propagation in optical fibers, numerous models are seen in the literature. In this work, the Kundu-Mikherjee-Naskar model and the Schrödinger equation with anti -cubic nonlinearity are considered. To solve these models, ansatz-based methods are considered and also, the problem about the determination of ansatz is replied so that it is not clear in the literature. The results of the proposed models will play a key role due to the applications in nonlinear optics, fluid dynamics, quantum mechanics and various other branches of science.


Introduction
Solitons in optics are the basic structure of soliton transmission technology, data transmission, transoceanic and intercontinental distances, telecommunication, as well as molecules or pulses that form the base fabric as optical fibers in the world within seconds. [1]. Optical fibers have played an important role in the industry and in daily life, particularly in the telecommunications industry, for the last few decades, providing the latest technology in modern fiber optic communication technology. Soliton molecules have various aspects studied in various optoelectronic devices. One of the avenues of attention is the nonlinearity effects in a fiber such as ultrashort pulses, optical solitons, second harmonic generation, four-wave mixing, self-phase modulation, stimulated Raman scattering, etc. [2]. In the literature, many models such as Kaup-Newell equation [11], Lakshmanan-Porsezian-Daniel model [10], complex Ginzburg-Landau equation [14], Radhakrishnan-Kundu-Lakshmanan equation [15], Fokas-Lenells equation [12,13] are seen to simulate the pulse propagation in optical fibers [1]. In 2014, Kundu et al. [3] proposed a new model for oceanic rogue waves as well as hole waves in deep sea. Additionally, the model can be applied to study optical wave propagation by means of consistent excited resonance waveguides contributed by soliton pulses, Erbium atoms [4,5] and bending phenomena of light beams. With another view, the model is known as a new extension of the non-linear Schrödinger equation by incorporating non-linearities in different forms relative to the Kerr and non-Kerr law nonlinearites to investigate soliton pulses in (2 + 1) -dimensions [1,6]. Kundu-Mukherjee-Naskar (KMN) model [5] is given The two-dimensional soliton propagation dynamics and the temporal variable t along an optical fiber with the spatial variables are x and y , while the dependent variable ( , , ) u x y t represents the nonlinear wave envelope. The first term in (1) refers to temporal evolution of the wave followed by the dispersion term given by the coefficient of a . The coefficient of nonlinear term b is different from the conventional Kerr law nonlinearity or from any known non-Kerr law media. In (1), this nonlinear term accounts for "current-like" nonlinearity resulting from chirality.
It is known that Schrödinger equation is the main equation generally used to model phenomenon especially in quantum mechanics, energy and energy quantization. The linear Schrödinger equation defines the time evolution of a quantum state. The nonlinear Schrödinger equation is known as one of the universal equations that describe the evolution of slowly changing quasimonochromatic wave packets in weakly nonlinear media that have dispersion [7]. Anti-Cubic nonlinearity differs from nonlinearity with respect to Kerr and non-Kerr law nonlinearities and is common to the dynamics of propagation of fibers through optical fibers for polarization preserving fibers.
The nonlinear Schrödinger's equation (NLSE) with anti-cubic nonlinearity is proposed ( ) where the coefficient of group velocity dispersion is a . The nonlinearities stem out from the coefficients of ( 1,2,3) i i  = . In particular, 1  gives the effect of anti-cubic nonlinearity. In case 1 0  = , it is parabolic law nonlinearity that kicks in [8,19].
The unclear problem about the ansatz is tried to reply so in the literature various determinations of ansatz are seen: Kudryashov [9] considers the ansatz as In this work, the most important models of optical fibers are solved via Bernoulli approximation method [16,17,18,19] after reduction with the mentioned ansatz and the obtained solutions are compared.

Reductions and Results of the models:
In this section, for the considered two models of the optical fibers, analytical solutions are obtained via different ansatz proposed in the literature.

The nonlinear Schrödinger's equation (NLSE) with anti-cubic nonlinearity
In the first equation of the system, assuming ( ) With the balancing principle, the solution is assumed as a finite series where 0 g and 1 g are parameters, ( ) z  is the solution of the variable coefficient Bernoulli type differential equation Substituting the finite series (Eq. (6)) and Eq. (7) into Eq. (5), equaling each coefficients of the power of ( ) z  , the algebraic system is obtained to determine the parameters.
In case are hold from the algebraic system. Therefore, the solution of Eq (7) with the obtained parameters is  ( ) In the first equation of the system, assuming The parameters are obtained as a result of the considered procedure for 2 k = in Eq. (7); are hold from the algebraic system. Therefore, the solution of Eq. (7) with the obtained parameters is In the first equation of the system, assuming  As it is seen from Figure 2, the ansatz is reduced to each other and have similar behavior. In the following, it is proved that when the ansatz is chosen as , the behavior depends on the parameter m that is the wave speed.
In the first equation of the system, assuming Substituting Eq. (15) into the second equation of Eq. (14), the nonlinear ordinary differential equation is obtained for () y  2 0 Applying the procedure 2 k = for Eq. (7), are hold from the algebraic system. Therefore, the solution of Eq. (7) with the obtained parameters is In the first equation of the system, assuming In the first equation of the system, assuming