The deterministic and stochastic solutions of the Schrodinger equation with time conformable derivative in birefrigent fibers

1.   Introduction

In the recently, there have been many studies and developments in the fractional calculus field so far ^{[1,2,3,4,5,6,7,8,9]}. Random models that depend upon uncertainty in differential equations or partial differential equations have been analysed in many years ago in many fields of applied science such as sociology, physics, chemistry, sociology, economics, medicine and biology. Randomness effect in the input may arise because of several faults in the noted or measured data and empirical states ^{[10,11,12,13,14,15]}. Due to the truth that the stochastic types are more naturalistic than the deterministic types, we concentrate our work in this article on the wick-type stochastic the time nonlinear Schrödinger equation with conformable fractional derivatives. A lot of research on stochastic fractional differential equations has been done recently ^[16,17,18]. Ghany and Zakarya ^[16] studied exact traveling wave solutions for wick-type stochastic Schamel KdV equation, in ^[17] is obtained exact solutions for the stochastic time fractional Gardner equation, in ^[18] is studied white noise functional process for the fractional coupled KdV equations and is found some soliton solutions.

Birefringence in optical fibers is a natural phenomenal that occurs because of some components such as twists, bends and anisotropic stress of fibers. These cause to differential set delay and thus the splitting of pulses occur. An addition to detrimental positions of the random birefringence in optical fibers, a ruled and artificial birefringence presented in the fiber is sometimes studied to realise in-line fiber optical factors and tools. Artificially presented birefringence in optical fibers may be used to advance of high-birefringence (Hi-Bi) fibers. Such Hi-Bi fibers are realised by introducing a permanent high birefringence into the fiber during fabrication. The nonlinear Schrödinger's equation plays a vital role in various areas of biological, physical and engineering sciences. It appears in many applied fields, including fluid dynamics, nonlinear optics, plasma physics and proteinchemistry. In this article, we will study the nonlinear Schrödinger equation with the aid of time conformable derivative.

The presented equations for perturbation solitons in birefringent fibers given with Kerr law nonlinearity are expressed with the vector coupled nonlinear Schrödinger's equations (NLSE). The coupled NLSEs given with the spatio-temporal dispersion and Kerr law nonlinearity are as below ^[19,20].

Here $q = q(x, \tau)$ and $r = r(x, \tau)$ define wave profiles of two split pulses that are the function of spatial variable $x$ and the temporal variable $\tau$ For $l = 1, 2, \,$ ${a_l}$ defines the coefficients of group-velocity dispersion, while ${b_l}$ gives the coefficients of spatio-temporal dispersion, ${c_l}$ defines the coefficients of cross-phase modulation, ${d_l}$ defines the coefficients of self-phase modulations. Where the perturbation conditions ${\alpha _l}$ defines the inter-modal dispersion, ${\lambda _l}$ is the self-steepening expression, ${v_l}$ and ${\theta _l}$ are nonlinear dispersions and finally ${\gamma _l}$ is the third order dispersion that must be taken into account in case the group-velocity dispersion is small.

We assume the following types without Spatio-temporal dispersion and Kerr Law nonlinearity without perturbation conditions, Hence ${\alpha _l} = 0, {\lambda _l}\; = 0, {v_l} = 0, {\theta _l} = 0, {\gamma _l} = 0,$${b_l} = 0,$ Then the presented equation is modeled by the following dimensionless vector coupled nonlinear Schrödinger equations,

The space-time coupled nonlinear Schrödinger equations with the aid of conformable derivative are given by ^{[21,22,23,24]}:

where $\vartheta \in (0, 1)$, ${c_1}, {c_2}, {d_1}$ and ${d_2}$ are nonzero constants, $q$ and $r$ are complex functions of $x$ and $\tau$ that represent the amplitudes of circularly-polarized waves in a nonlinear optical fiber ^[21]. Eq (1.3) are developed by ^[21] and are named to playing great role in the pulse propagation through a two-mode optical fiber and the soliton wavelength division multiplexing. When $\vartheta = 1$, we have the original (1+1)-dimensional coupled nonlinear Schrödinger equations ^[21].

The nonlinear Schrödinger equation is an example of a universal nonlinear version that defines many physical nonlinear systems. The equation can be studied to nonlinear optics, hydrodynamics, quantum condensates, nonlinear acoustics, heat pulses in solids and various other nonlinear instability phenomenal. Such equations have been expressed with govern pulse propagation along orthogonal polarization axes in nonlinear optical fibers and in wavelength-division-multiplexed systems. These equations also type beam propagation inside crystals or photorefractive as well as water wave interactions. Solitary waves in these equations are often named vector solitons in the literature as they generally comprise two elements. In all the above physical cases, collision of vector solitons is an significant effect ^{[25,26,27,28,29,30,31]}.

The stochastic coupled nonlinear Schrödinger equations with the aid of time conformable derivative are given by:

where ${a_1}(\tau),$ ${a_2}(\tau),$ ${c_1}(\tau)$ and ${c_2}(\tau)$ are limited mensurable or integrable functions on ${{\rm{R}}_ + }$. $D_\tau ^\vartheta p(x, \tau)$ and $D_\tau ^\vartheta r(x, \tau)$ are the time conformable derivative operator and ${d_1}, {d_2}$ are real valued constants.

The conformable derivative operator was exposed in ^[32]. This derivative operator can reform the failures of the other definitions. This important operator is the easiest, most ordinary and effectual description of the fractional derivative for order $\vartheta \in (0, 1]$. We should note that the description can be generalised to need any $\vartheta$. All the same, the order $\vartheta \in (0, 1]$ is the most influential order.

The conformable derivative of order $\vartheta \in (0, 1)$ described as the following statement ^[32]

The description represents a ordinary formation of standard derivatives. Furthermore, the expression of the description represents that it is the most natural, and the most effectual definition. The description for $0 \leqslant \vartheta < 1$ gives with the standard expressions on polynomials (up to a constant).

Some characteristics of the conformable derivative are given by ^[32,33].

This derivative is more advantageous than others because it's easy to apply. In the recently, there has been some researces on the conformable form of fractional computations ^{[33,34,35,36,37]}.

The wick sense stochastic model of Eq (1.4) given with conformable derivatives expressed as below,

where "$\blacklozenge$" give the Wick product on the Kondratiev distribution space ${(S)_{ - 1}}$, ${a_1}(\tau),$ ${a_2}(\tau),$ ${c_1}(\tau)$ and ${c_2}(\tau)$ are ${(S)_{ - 1}}$-valued functions.

We only consider it in a white noise environment, that is, we will discuss the Wick-type stochastic coupled nonlinear Schrödinger equations for time conformable derivative (1.5) to obtain the exact solutions of the stochastic coupled nonlinear Schrödinger equations with time conformable derivative.

Our focus in this study are to analyse new soliton and periodic wave solutions of coupled nonlinear Schrödinger equations by using time conformable derivative and to analyse new stochastic soliton and periodic wave solutions of the Wick-type stochastic coupled nonlinear Schrödinger equations by using time conformable derivatives. We use Hermite transform, white noise theory and the modified fractional sub-equation method ^[38,39]. In addition to, we use the inverse Hermite transform to find stochastic soliton and periodic wave solutions of the Wick-type stochastic coupled nonlinear Schrödinger equations by using time conformable derivative. Eventually, we give an application example to show how the stochastic solutions can be expression as Brownian motion functional solutions.

2.   Deterministic case applications

In this section, we will obtain exact solutions of nonlinear Schrödinger equations. By using the Hermite transform of Eq (1.3), we define the deterministic equation as below

where $z = ({z_1}, {z_2}, ...) \in {\left({{{\rm{C}}^N}} \right)_c}$ is a argument. To find travelling wave solutions to Eq (2.1), we give the transformations below

with,

where $\theta \ne 0,$ $\varpi, k$ are arbitrary constants. So, Eq (2.1) can be reborn to NODEs. These equations are resolved it into real and imaginary sections as follows,

The real sections;

and by integrating with respect to $\varphi$ once of imaginary sections and by integration constant neglect gives:

From Eq (2.3) we obtain $\theta = \frac{{2{k^2}{a_1}(\tau, z)}}{\varpi } = \frac{{2{k^2}{a_2}(\tau, z)}}{\varpi }$ and ${a_1}(\tau, z) = {a_2}(\tau, z).$ We consider ${a_1}(\tau, z) = {a_2}(\tau, z) = a(\tau, z),$ we can write $\theta = \frac{{2{k^2}a(\tau, z)}}{\varpi }.$

$\bullet$Assume the solutions of Eq (2.2) can express as a series expansion solution as below,

where ${\alpha _i}$ $(i = 0, 1, ..., n), {\beta _i}$ $(i = 1, 2, ..., n)$ are functions to be determined later and $G(\varsigma)$ satisfies the fractional Riccati equation as below:

where $\rho$ is an arbitrary constants.

$\bullet$$N$ is control between the nonlinear terms and the highest order derivatives in Eq (2.2).

Some special solutions of Eq (2.5) are as below ^[40];

${\bf{1)}}$$\ell < 0,$

${\bf{2)}}$$\ell > 0,$

${\bf{3)}}$$\ell = 0,$ $\rho = const.,$

Note. The generalized hyperbolic and trigonometric functions are as below^[2];

where ${E_\vartheta }(\varsigma) = \mathop {\mathop {\sum }\limits^N }\limits_{i = 0} \frac{{{\varsigma ^i}}}{{\Gamma (1 + i\vartheta)}}$ is the Mittag-Leffler function.

From Eq (2.2), is found $N = 1$. Then we can choose the solution of Eq (2.2) are given by:

where $G(\varsigma)$ satisfied Eq (2.5).

Now, replacing (2.6) and (2.5) into (2.2), by equating the all coefficients of $G(\varsigma)$, we can solve equations. Then we obtain the following some groups of solutions:

One of the obtained these groups is given by;

The exact solutions of Eq (2.1) are given by;

${\bf{1)}}$ Hyperbolic function solutions (when $\ell < 0$)$,$

${\bf{2)}}$ Trigonometric function solutions (when $\ell > 0$)$,$

${\bf{3)}}$ Wave solutions (when $\ell = 0,$ $\rho = const.$)$,$

3.   Stochastic case applications

In this part, we use the inverse Hermite transform and Theorem 4.1.1 in ^[41] to analyse white noise functional solutions of Eq (1.2). The properties of generalized hyperbolic, trigonometric and exponential functions express that there is a limited open station $\Omega \subset {\rm{R}} \times {{\rm{R}}_ + },$ $a < \infty,$ $b > 0$ such that the solutions $q(x, \tau, z)$ and $r(x, \tau, z)$ of Eq (2.1) and whole its fractional derivatives which are contained in Eq (2.1) are uniformly limited for $(x, \tau, z) \in \Omega \times {K_a}(b),$ continuous with respect to $(x, \tau) \in \Omega$ for whole $z \in \Omega \times {K_a}(b)$ and analytic with respect to $z \in {K_a}(b)$, for whole $(x, \tau) \in \Omega$. From Theorem 4.1.1 in ^[42], there is $P(x, \tau) \in {(S)_{ - 1}}$ such that $q(x, \tau, z) = Q(x, \tau)(z)$ and $r(x, \tau, z) = R(x, \tau)(z)$ for whole $(x, \tau, z) \in \Omega \times {K_a}(b)$ and $Q(x, \tau),$ $R(x, \tau)$ solves Eq (1.5) in ${(S)_{ - 1}}$. Thus, we will investigate the white noise functional solutions of Eq (1.5) for $a(\tau) > 0,$ ${c_1}(\tau) > 0$ and ${c_2}(\tau) > 0$ wit the aid of inverse Hermite transform for Eqs (2.8)–(2.12) as below.

${\bf{1)}}$ Hyperbolic function random solutions (when $\ell < 0$)$,$

${\bf{2)}}$ Trigonometric function random solutions (when $\ell > 0$)$,$

${\bf{3)}}$ Wave random solutions (when $\ell = 0,$ $\rho = const.$)$,$

4.   Examples

In this part, we analyse special application example to show the accessibility of our consequences and to support the real help of these results. We interpret that the solutions of Eq (1.5) are powerfully count on the type of the given functions ${c_1}(\tau),$ ${c_2}(\tau)\;$and $a(\tau)$. Thus, for different types of ${c_1}(\tau),$ ${c_2}(\tau)\;$and $a(\tau)$, we can obtain different solutions of Eq (1.5) that come from Eqs (3.1)–(3.5). We demonstrate this by giving the following example.

When $\vartheta = 1$,

Suppose ${c_1}(\tau) = \partial$$a(\tau),$ ${c_2}(\tau) =$$\rho$$a(\tau),$ $a(\tau) = h(\tau) + \lambda {W_\tau }$ and ${d_1} = {d_2} = 1.$Where $\partial,$ $\rho$ and $\lambda$ are arbitrary constants, $h(\tau)$ is a limited mensurable function on ${{\rm{R}}_ + }$ and ${W_\tau }$ is the Gaussian white noise that is the time derivative (in the strong sense in ${(S)_{ - 1}}$) of the Brownian motion ${B_\tau }$.

The Hermite transform of ${W_\tau }$ is expressed by ${\tilde W_\tau }(z) = \mathop {\mathop {\sum }\limits^\infty }\limits_{i = 0} {z_i}\int\limits_0^t {{\Psi _i}(\tau)d\tau }$ ^[42]. Using the describtion of ${\tilde W_\tau }(z),$Eqs (3.1)–(3.5) given the white noise functional solution of Eq (1.5) as below:

where we have already used the following relation ^[32]

5.   Physical reviews

In this section, we shown some figures to analyse the action of the obtained solutions of Eq (1.3).

In Figures 1 and 2, we show the evolutional effects of stochastic equation (1.5) for Brownian motion ${B_\tau } = random\; [0, 1]{\rm{sinh}}2\tau$. In Figures 3 and 4, we consider and we considered the effects of stochastic equation (1.5) without effect of stochastic term ${W_\tau }.$

6.   conclusions

In this article, we investigated the coupled nonlinear Schrödinger equations with the aid of conformable derivative for deterministic and stochastic forms. In addition to, we studied Wick-model stochastic coupled nonlinear Schrödinger equations with time conformable derivatives. We analysed some exact solutions with the aid of the modified fractional sub-equation method, Hermite transform and White noise theory. We obtained stochastic hiperbolic and trigonometric wave solutions via inverse Hermite transform. Furthermore, we investigate an example, to show the stochastic solutions can be found as Brownian motion functional solutions. In addition to, if $\vartheta = 1$, then the stochastic solutions (4.1)–(4.5) express a new set of stochastic solutions for the Wick-model stochastic coupled nonlinear Schrödinger equations by using integer derivatives.

This study emphasize that the modified fractional sub-equation method is adequate to solve the stochastic nonlinear equations in mathematical physics. The studied method in this paper is normal, direct and computerized method, which lets us to do confused and boring algebraic computation. It is expressed that the method can be also applied to other nonlinear stochastic differential equations in mathematical physics.

Acknowledgments

We would like to express our thanks to the anonymous referees who help us improved this paper.

Conflict of interest

All authors declare no conflict of interest in this paper.