New and effective solitary applications in Schrödinger equation via Brownian motion process with physical coefficients of fiber optics

1.   Introduction

Nonlinear complex phenomena in a superfluid, optical fiber communications, solid-state physics, epidemiology, plasma physics can be expressed in deterministic or stochastic nonlinear partial differential equations (SNPDEs) ^{[1,2,3,4,5,6,7,8]}. There are several advantages for using stochastic rather than deterministic equations, including in finance, biology, chemistry, mechanics, microelectronics and economics ^[9,10,11,12]. A complete comprehension of SNPDEs theory requires expertise in advanced probability and stochastic processes. Recently, many researchers have proposed and developed various numerical and analytical methods for solving NPDEs ^{[13,14,15,16]}.

The NLSE depicts how waves move across mediums with dispersive and nonlinear effects. This equation turns into the basic ingredient for describing the wave behaviors in so many vital applications of applied science such as Bose-Einstein condensations (BEC), bimolecular dynamical modes, deep water, coastal water motions and semiconductors ^{[17,18,19,20,21,22]}. The NLSE has localized solutions (soliton), which are especially sturdy and propagate without changing form. The solitonic features in the molecular chain model that presented by NLSE with saturated nonlinear coefficient and discrete coupled NLSE have been investigated ^[23,24]. Alkhidhr et al. ^[9] investigated the stochastic unstable NLSE and higher-order dispersive NLSE. Alharbi et al. ^[3] studied the perturbed NLSE with Kerr law nonlinearity in the presence of random dispersion and nonlinear effects. Kumar et al. ^[25] presented some innovative solutions to the generalized Schrödinger-Boussinesq equation, which illustrates the interaction of complex short-wave and real long-wave envelopes. Houwe et al. investigated the effects of modulated self-Kerr nonlinearities on discrete solitons propagated in optomechanical arrays. It was noted that nonlinear terms minimize the transient regimes of temporal solitonic structures ^[26]. The effect of noise on the propagation of these soliton solutions has received increased attention in recent years ^[9,25,27]. On the other hand, propagating solitons in materials having saturation impacts cannot be modeled by the NLS equation ^[28,29]. To study these effects, the WKI equation must be taken into account ^[28,29]. Li et al. investigated long time asymptotic solutions for finite density WKI equation ^[29]. Riemann–Hilbert (RH) problems is developed to study the focusing NLS equation with multiple high-order poles under nonzero boundary conditions. The solutions behaviours in various discrete spectra have been obtained ^[30].

Recently, the development of fractal solitary structures from a large number of nonlinear equations using fractal variational principles in various media turns into an effective technique to describe certain new occurrences in our universe ^{[31,32,33,34,35,36]}. Abdelrahman et al. ^[27] extracted new stochastic solutions for the conformable fractional nonlinear NLSE. The Adomian decomposition technique was defined and used to solve the newly constructed nonlinear Schrödinger equation with spatiotemporal dispersion ^[37]. Also, various fractional formulations have been presented to extract exact optical soliton solutions for modified NLSE with spatiotemporal dispersion ^[38,39].

The Wiener process is often called Brownian motion (BM) due to its connection with the physical process called Brownian motion. Actually, this stochastic process serves as a foundation for a variety of models in applied science ^[40,41]. Since a BM process is non-differentiable, Itô's formula is often utilized to get the explicit solution of SNPDEs with a BM process ^[42,43]. A 1D NLSE with an additive space-time white noise is investigated in ^[44]. This paper considers the NLSE with various physical optical fiber coefficients forced by multiplicative noise in Itô sense ^[45]:

$\psi = \psi (x, t)$ depicts the slowly pulse amplitude and $\sigma$ is the noise strength. Indeed, $\alpha, \delta$ & $\lambda$ denote fiber dispersion, nonlinear and fiber loss effects. The noise $W_{t}$ denotes derivative of the Brownian motion process $W(t)$ in time ^[46]. This noise term will properly manage the increasing, damping, and conversion affects on the amplitudes and frequencies of the bright/dark envelope and shock forced oscillatory wave.

In the ongoing research, we aim to extract some new stochastic solutions of Eq (1.1) utilizing the unified solver approach ^[47]. This solver is straightforward, sturdy, burly and averts tedious computations. Moreover, this solver produces various effective solutions for describing physical phenomena. This solver can simply be utilized as a box solver. The presented solver is applicable to a wide range of NSPDE classes. Furthermore, it is simple to expand for solutions of stochastic fractional NPDEs. We also explain how the noise term affects the solutions that have been provided. Some suggested stochastic solutions' nonlinear dynamical behavior is shown. To the best of our knowledge, the proposed solver for resolving the stochastic NLSE with various physical optical fiber coefficients has never been used before.

This article is organized as follows. Section 2 presents the notions of the Brownian motion process and its prosperities. Section 3 offers closed-form solutions for a variety of NPDEs. Section 4 presents the new stochastic solutions for Eq (1.1) via Itô sense. Section 5 introduces the interpretation of the presented solutions. We also show that the influence of the noise parameter plays an essential role in changing solution trajectories. Conclusions are summarized in Section 6.

2.   Brownian motion and normal distribution

Brownian motion is used to describe the random movement that happened by small particles that are suspended in liquid or gas. The Brownian motion prohibits the particles from stabilizing in fluids because it affects on them to be in constant motion. This leads to the stability of colloidal solutions. Therefore, it assists to differentiate between true and colloid solutions. Furthermore, we note that the size of particles and the viscosity of the fluid are inversely proportional to the speed of the motion.

Before we speak about the Browning motion, we display a brief description of the normal distribution. One of the main properties of the normal distribution is that it is a symmetric and continuous distribution. Also, the data around its mean is more occurrence than when they are far from it. Furthermore, its mean, median and mode are equal and it is symmetric around the mean. So, half of the values are to the right of the mean and the other half on the left with the total area under the curve equals one.

The normal distribution's probability density function is given by

$\mu$ and $\sigma$ denote the mean and standard deviation, which controls the spread of the data from the mean. One of the important special cases of the normal distribution is the standard normal distribution. It has zero mean and unit variance. The corresponding probability distribution function of it is

Now, we note that the Wiener process is defined as a stochastic process, which is continuous in time. A Brownian motion process $\{ W(t)\}_{t\geq 0}$ has the following properties:

$({\rm{a}})$ $W(t),$ $t\geq 0$ is a continuous function of time $t$ and $W(t) \sim N (0, t)$.

$({\rm{b}})$ For $s < t < u < k,$ $W(s)- W(t)$ & $W(k)- W(u)$ are independent.

$({\rm{c}})$ $W(t)- W(s)$ follows a normal distribution with zero mean and variance $t-s$, i.e. $W(t)- W(s) \sim \sqrt{t-s}\, N (0, 1)$, $\, N (0, 1)$ is a standard normal distribution.

3.   Unified solver approach

Consider the following NPDEs:

Using the wave transformation:

Equation (3.1) simplified to the following ODE:

where $w$ is the wave speed. In applied science, several NPDEs can be reduced to:

where $L$, $M$ and $N$ are constants rely on the proposed equation's constants and the wave transformations' speed. The solutions of Eq (3.4) are based on the unified solver approach ^[47], given by:

ⅰ) Rational solutions: (when $N = 0$)

ⅱ) Trigonometric solutions: (when $\frac{N}{L} < 0$)

and

ⅲ) Hyperbolic solutions: (when $\frac{N}{L} > 0$)

and

Here, the constant $\varsigma$ is arbitrary.

4.   Stochastic solutions

Using the transformation

Equation (1.1) was transformed into the following ODE

where $\omega, p, c$ and $\mu$ are constants. Taking expectation of both sides, we have

Since $E(e^{2\sigma W (t)}) = e^{2\sigma ^{2}t}$, Eq (4.3) reduced to

with dispersion relation

Equation (4.4) depicts an energy equation of particle dynamical motion in the form

The model has an exact solution as follows

In light of the presented solver, Eq (4.4) has the following solutions:

First: Rational solutions are:

Hence, the stochastic solutions of Eq (1.1) are

Second: Trigonometric solutions are

and

Hence, the stochastic solutions of Eq (1.1) are

and

Third: Hyperbolic solutions are:

and

Thus, the stochastic solutions for Eq (1.1) are

and

Here, $p, c, \mu, \varrho, \lambda, \alpha$ are constants.

5.   Results and discussion

Mathematical treatments of the stochastic nonlinear Eq (1.1), which contains nonlinear, dispersive, losing and noise terms, produce a dynamical energy Eq (4.6) with restricted Eq (4.5) and various structurally important solutions that can be as characteristics of wave behaviors in optical communications. Using Matlab release 18 and Mathematica release 13, we introduce some 2D and 3D graphs for some selected solutions of Eq (1.1) for appropriate parametric choices. Eq (4.5) gives all physical relations between Schrödinger equation coefficients and the transformation parameters as depicted in Figure 1. In this figure, we show the relationship between $\omega$, $\alpha$, $\lambda$ and $\sigma$, which is a conditional constraint on the energy equation. It was found that $\omega$ increased by increasing both $\lambda$ and $\sigma$. Moreover, it was concluded that $\omega$ increases with the increase of $\alpha$ until a specific value and then begins to gradually rebate. By examining Eq (4.6) with potential phase portrait variations as shown in Figures 2 and 3. Both potential and phase portraits represent the regions of stability (instability) of a particle in a potential well with a change in $\delta$ parameter. Phase portrait is a relation plotted between $dq/d\vartheta$ and $q$ at different values of energy constant for $p = 0.3, \lambda = 0.3, \; \alpha = -3,$ $c = 0.5$ and $\mu = 1$. It was noted that there are two stable and one unstable point for $\delta = -2$ as in Figure 2 and an unstable state for $\delta = 2$ as in Figure 3. To discuss some of the obtained solutions in this work for $\sigma = 0$, Equation (4.7) described bright solitary solutions in the form of breather and bell-shaped solitons as in Figures 4 and 5. The rational leading form Eq (4.9) is an effective solution for the production of periodic blow-up solution and explosive solitary behavior as given in Figures 6 and 7.

On the other hand, many trigonometric and hyperbolic dissipative solutions have been illustrated in Eqs (4.12), (4.13), (4.16) and (4.17). In Figure 8, a blow-up dissipative structure has been obtained. One of the priorities of this work is to explore the effects of the random pulses on the state and characteristics of the resulting waves through the change of stochastic time and the consequent changes of wave shapes and phase differences. Figure 9 shows the variation of Eq (4.7) with time and $\sigma$. We found that the increase of $\sigma$ reduces the amplitude and affected the soliton tails until the soliton shape converts to soliton-like dissipative stochastic wave. Variations of Eq (4.16) with time and $\sigma$ is depicted in Figures 10 and 11. It was determined to that the shock wave tails were deformed with $\sigma$ in form of damping in positive tail and forcing in negative tail. Furthermore, the random parameter damped the envelope soliton in the negative $t$-axis and forced the envelope in the positive $t$-axis.

In summary, the stochastic random effects control the properties of solitary and other forms in this model in the form of damping and forcing wave tails.

6.   Conclusions

The random NLSE equation with losing and noise terms has been solved by a mathematical solver. The stability of the dynamical system corresponding to the random NLSE equation has been examined by phase plane to identify the solutions appropriate for this model. Several wave profiles have been derived in the form of blow-up, periodical, soliton, breather and shock-like structures. The noise effects in stochastic behaviors play an improvements controller for wave features. The parameter of random noise modified the shape and features of the model's solutions. It causes some physical effects as damping, fluctuations and forcing for amplitudes and wave tails. The results obtained here may be profitable in physical models with losing terms in stochastic dispersive models.

The unified solver technique in this work via stochastic sense can be applied in studying nonlinear equations using some recently proposed definitions of fractional calculus such as the generalized fractional derivative (GFD) definition ^[48,49]. Actually, this definition with its special functions ^[50] is so powerful because it overcomes some issues associated with conformable derivatives and other fractional derivatives.

Conflict of interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.