New optical stochastic solutions for the Schrödinger equation with multiplicative Wiener process/random variable coefficients using two different methods

Wael W. Mohammed; Farah M. Al-Askar

doi:10.1515/phys-2024-0086

Article Open Access

New optical stochastic solutions for the Schrödinger equation with multiplicative Wiener process/random variable coefficients using two different methods

Wael W. Mohammed and Farah M. Al-Askar

Published/Copyright: October 9, 2024

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From the journal Open Physics Volume 22 Issue 1

Abstract

In this article, we take into consideration the stochastic Schrödinger equation (SSE) perturbed in the Itô sense by the multiplicative Wiener process. We employ an appropriate transformation to turn the SSE into another Schrödinger equation with random variable coefficients (SE-RVCs). We used the generalizing Riccati equation mapping method and the Jacobi elliptic function method to find novel hyperbolic, trigonometric, rational, and elliptic functions solutions for SE-RVCs. After that, we can acquire the SSE solutions. For the first time, in this work, we assume that the solution to the wave equation for the Schrödinger equation is stochastic, whereas all earlier studies assumed it to be deterministic. Furthermore, we give various graphs to display the effect of multiplicative Wiener process on the exact solutions to the SSE. We deduce that the multiplicative Wiener process stabilizes the solutions of the SSE.

Keywords: random variable coefficients; Schrödinger equation; stability by noise; generalizing Riccati equation mapping method

1 Introduction

Partial differential equations (PDEs) are used to model various physical phenomena in fields such as physics, engineering, and biology. Solving PDEs can be challenging due to their complexity, but there are several techniques that can be used to find solutions. Some of these methods are the F-expansion method [1], the Laplace adomian decomposition method [2,3], the generalized Adams-Bashforth-Moulton method [4], the ϕ 6 -model expansion method [5], the modified generalized exponential rational function [6,7], the He’s semi-inverse [8], the bilinear method [9,10], Bernoulli ( G ′ ⁄ G )-expansion method [11,12], the exp ( − ψ ( z ) ) -expansion method [13], the generalized Riccati equation mapping method [14], the Lie symmetry method [15], the multivariate generalized exponential rational integral function [16], and the modified generalized Riccati equation mapping approach [17].

One of the most well-known examples of a PDE is the Schrödinger equation [18], which has a wide range applications in various fields such as nonlinear optics, heat pulses in materials, plasma physics, nonlinear acoustics, hydrodynamics, and several other nonlinear instability phenomenon [19–22]. Also, it has been used to investigate the behavior of quantum fields, for instance, the wave equation for photons in quantum electrodynamics. Scientists can obtain a better knowledge of the fundamental principles of quantum mechanics and change many disciplines of science and industry by solving the Schrödinger equation.

On the other hand, random fluctuations in the Schrödinger equation are a key aspect of quantum mechanics, emphasizing the unpredictability and probabilistic nature of quantum systems. While these fluctuations make it hard to predict a particle’s exact behavior in a quantum system, they also open up new possibilities such as quantum tunneling and superposition. Harnessing these random fluctuations has resulted in technical advances in areas such as quantum computing and cryptography.

Consequently, it is important to consider the following Schrödinger equation with stochastic term:

(1) i Z t − δ 1 Z x x + δ 2 ∣ Z ∣ 2 Z − δ 3 Z = i σ ZW t ,

where Z ( x , t ) is a complex-valued process, x and t denote the spatial coordinate and time, respectively, i = − 1 , δ 1 , δ 2 , and δ 3 are arbitrary constants, W ( t ) is the standard Wiener process, W t = ∂ W ∂ t and σ are the noise amplitude.

Because of the significance of the Schrödinger equation (1), several researchers obtain its solutions with σ = 0 by using diverse approaches such as first integral method [23], ( G ′ ⁄ G , 1 ⁄ G )-expansion and ( 1 ⁄ G ′ ) -expansion methods [24], the generalized exp ( − φ ( ξ ) ) -expansion method [25], mapping and generalized Riccati equation methods [26], mapping method [27], sine–cosine and the tanh methods [28], modified generalized Riccati equation mapping approach [29], and extended generalized Riccati equation mapping method [30]. While the analytical solutions of stochastic Schrödinger equation (SSE) have been achieved by employing different approaches such as Riccati-Bernoulli sub-ODE method and sine–cosine method [31], tanh–coth and He’s semi-inverse techniques [32], unified technique [33], and modified mapping method [34]. Previous studies [31–33] assumed deterministic solutions to the wave equation, but this work assumes stochastic solutions.

Our objective of this article is to find the exact solutions to the SSE (1). To achieve this goal, we convert the SSE into another nonlinear Schrödinger equation with random variable coefficients (SE-RVCs) by using suitable transformation. After that, we obtain the exact solutions for SE-RVCs by using the generalizing Riccati equation mapping method (GREM method) and the Jacobi elliptic function method (JEF method). Finally, by using the used transformation, we can acquire the stochastic solutions for the SSE (1). In this work, we assume for the first time that the solution of the wave equation for the Schrödinger equation is stochastic, whereas all earlier studies assumed that it was deterministic. These acquired solutions are crucial in understanding several difficult physical processes due to the importance of Schrödinger equation (1) in various areas of physics, chemistry, and engineering. To see the influence of stochastic term, we provide some figures by utilizing MATLAB tools. In the end, we address the effect of noise on the obtained solutions.

Here is how the remainder of the article is structured: In Section 2, we derive SE-RVCs from SSE (1) and by utilizing the GREM method and JEF method to find the exact solutions of SE-RVCs. In Section 3, we acquire the solutions of SSE (1). In Section 4, we discuss the results that we obtained and the impacts of noise. Finally, we provide the article’s conclusions.

2 Schrödinger equation with RVCs and its solutions

Here, we obtain the Schrödinger equation with random variable coefficients (SE-RVCs). Using the transformation

(2) Z ( x , t ) = ℱ ( x , t ) exp [ i Y ( x , t ) + σ W ( t ) − 1 2 σ 2 t ] ,

and the Itô derivatives, we obtain SE-RVCs as follows:

(3) i ℱ t − ℱY t − δ 1 [ ℱ x x + 2 i Y x ℱ x + i Y x x ℱ − Y x 2 ℱ ] + A ( t ) ℱ 3 − δ 3 ℱ = 0 ,

where ℱ is a stochastic real function and A ( t ) = δ 2 e σ W ( t ) − 1 2 σ 2 t . Putting imaginary and real parts equal zero, we have

(4) ℱ t − 2 δ 1 Y x ℱ x − δ 1 Y x x ℱ = 0 ,

and

(5) − ℱY t − δ 1 ℱ x x + δ 1 Y x 2 ℱ + A ( t ) ℱ 3 − δ 3 ℱ = 0 .

2.1 GREM method

To find the solutions of the SE-RVCs (3), we assume the solutions of Eqs (4) and (5) in the following special forms:

(6) ℱ ( x , t ) = ∑ k = 0 m α k ( t ) X k ( μ ) , μ = k x + ∫ 0 t λ ( τ ) d τ ,

and

(7) Y ( x , t ) = φ 0 ( t ) + x φ 1 ( t ) ,

where λ , φ 0 , φ 1 , α 0 , α 1 , … , α m − 1 , and α m ≠ 0 are functions of t , and X satisfies

(8) X ′ = s X 2 + r X + p .

First, let us compute the value of m by balancing ℱ ″ with ℱ 3 in Eq. (5) as follows:

m + 2 = 3 m ⇒ m = 1 .

Eq. (6) is rewritten as follows:

(9) ℱ ( x , t ) = α 0 ( t ) + α 1 ( t ) X ( μ ) .

Differentiating Eqs (9) and (7) with respect to t and x , we have

(10) ℱ t = ( α ˙ 0 + p α 1 λ ) + ( α ˙ 1 + α 1 r λ ) X + s λ α 1 X 2 , ℱ x = k α 1 [ s X 2 + r X + p ] , ℱ x x = k 2 α 1 [ 2 s 2 X 3 + 3 s r X 2 + ( 2 s p + r 2 ) X + r p ] , ℱ 3 = α 1 3 X 3 + 3 α 0 α 1 2 X 2 + 3 α 0 2 α 1 X + α 0 3 ,

and

(11) Y t = φ ˙ 0 + x φ ˙ 1 , Y x = φ 1 , Y x x = 0 .

Eqs. (7), (9), (10), and (11) are substituted into Eqs (4) and (5). After that, by equating each coefficient of X k to zero, we have

X 0 : α ˙ 0 + p λ α 1 − 2 p k δ 1 φ 1 α 1 = 0 , X 1 : α ˙ 1 + r λ α 1 − 2 r k δ 1 φ 1 α 1 = 0 , X 2 : r λ α 1 − 2 s δ 1 φ 1 α 1 = 0 ,

and

x X 0 : α 0 φ ⋅ 1 = 0 , x X 1 : α 1 φ ⋅ 1 = 0 , X 0 : α 0 φ ⋅ 0 + r p k 2 δ 1 α 1 − φ 1 2 δ 1 α 0 − A α 0 3 + δ 3 α 0 = 0 , X 1 : α 1 φ ⋅ 0 + ( 2 s p k 2 + r 2 k 2 ) δ 1 α 1 − φ 1 2 δ 1 α 1 − 3 A α 0 2 α 1 + δ 3 α 1 = 0 , X 2 : − 3 s r k 2 δ 1 α 1 + 3 A α 0 α 1 2 = 0 , X 3 : − 2 s 2 k 2 δ 1 α 1 + A α 1 3 = 0 .

We solve these equations to obtain

α 0 ( t ) = r = 0 , α 1 = ℓ , φ 1 = b , λ ( t ) = b ℓ 2 k s 2 A ( t ) ,

and

φ 0 = − δ 3 t + ℓ 2 ( b 2 − 2 s p k 2 ) 2 k 2 s 2 ∫ 0 t A ( τ ) d τ , δ 1 = ℓ 2 A ( t ) 2 k 2 s 2 ,

where b and ℓ are constants. Hence, by utilizing Eq. (9), the solutions of the SE-RVCs (3) are

(12) ℱ ( x , t ) = ℓ X ( μ ) , μ = k x + b ℓ 2 δ 2 k s 2 ∫ 0 t e σ W ( τ ) − 1 2 σ 2 τ d τ .

To determine X , there exist several families for the solutions of Eq. (8) depending on p and s , as follows:

Family I: If p s > 0 , then Eq. (8) has the solutions:

X 1 ( μ ) = p s tan ( p s μ ) , X 2 ( μ ) = − p s cot ( p s μ ) , X 3 ( μ ) = p s ( tan ( 4 p s μ ) ± sec ( 4 p s μ ) ) , X 4 ( μ ) = − p s ( cot ( 4 p s μ ) ± csc ( 4 p s μ ) ) ,

and

X 5 ( μ ) = 1 2 p s tan 1 2 p s μ − cot 1 2 p s μ .

Then, SE-RVCs (3) has the trigonometric functions solutions:

(13) ℱ 1 ( x , t ) = ℓ p s tan p s k x + b ℓ 2 δ 2 k s 2 ∫ 0 t e σ W ( τ ) − 1 2 σ 2 τ d τ ,

(14) ℱ 2 ( x , t ) = − ℓ p s cot p s k x + b ℓ 2 δ 2 k s 2 ∫ 0 t e σ W ( τ ) − 1 2 σ 2 τ d τ ,

(15) ℱ 3 ( x , t ) = ℓ p s tan 4 p s k x + b ℓ 2 δ 2 k s 2 ∫ 0 t e σ W ( τ ) − 1 2 σ 2 τ d τ ± sec 4 p s k x + b ℓ 2 δ 2 k s 2 ∫ 0 t e σ W ( τ ) − 1 2 σ 2 τ d τ ,

(16) ℱ 4 ( x , t ) = ℓ p s cot 4 p s k x + b ℓ 2 δ 2 k s 2 ∫ 0 t e σ W ( τ ) − 1 2 σ 2 τ d τ ± csc 4 p s k x + b ℓ 2 δ 2 k s 2 ∫ 0 t e σ W ( τ ) − 1 2 σ 2 τ d τ ,

and

(17) ℱ 5 ( x , t ) = 1 2 ℓ p s tan 1 2 p s k x + b ℓ 2 δ 2 k s 2 ∫ 0 t e σ W ( τ ) − 1 2 σ 2 τ d τ − cot 1 2 p s k x + b ℓ 2 δ 2 k s 2 ∫ 0 t e σ W ( τ ) − 1 2 σ 2 τ d τ .

Family II: If p s < 0 , then Eq. (8) has the solutions:

X 6 ( μ ) = − − p s tanh ( − p s μ ) , X 7 ( μ ) = − − p s coth ( − p s μ ) , X 8 ( μ ) = − − p s ( coth ( − 4 p s μ ) ± csch ( − 4 p s μ ) ) ,

and

X 9 ( μ ) = − 1 2 − p s tanh 1 2 − p s μ + coth 1 2 − p s μ .

Then, SE-RVCs (3) has the hyperbolic functions solution:

(18) ℱ 6 ( x , t ) = − ℓ − p s tanh [ − p s ( k x + b ℓ 2 δ 2 k s 2 ∫ 0 t e σ W ( τ ) − 1 2 σ 2 τ d τ ,

(19) ℱ 7 ( x , t ) = − ℓ − p s coth [ − p s ( k x + b ℓ 2 δ 2 k s 2 ∫ 0 t e σ W ( τ ) − 1 2 σ 2 τ d τ ,

(20) ℱ 8 ( x , t ) = − ℓ − p s coth − 4 p s k x + b ℓ 2 δ 2 k s 2 ∫ 0 t e σ W ( τ ) − 1 2 σ 2 τ d τ ± csch − 4 p s k x + b ℓ 2 δ 2 k s 2 ∫ 0 t e σ W ( τ ) − 1 2 σ 2 τ d τ ,

and

(21) ℱ 9 ( x , t ) = − 1 2 ℓ − p s tanh 1 2 − p s k x + b ℓ 2 δ 2 k s 2 ∫ 0 t e σ W ( τ ) − 1 2 σ 2 τ d τ + coth 1 2 − p s k x + b ℓ 2 δ 2 k s 2 ∫ 0 t e σ W ( τ ) − 1 2 σ 2 τ d τ .

Family III: If p = 0 and s ≠ 0 , then the solution of Eq. (8) is:

X 10 ( μ ) = − 1 s μ .

Then, the solution of SE-RVCs (3) is

(22) ℱ 10 ( x , t ) = − ℓ s ( k x + b ℓ 2 δ 2 k s 2 ∫ 0 t e σ W ( τ ) − 1 2 σ 2 τ d τ ) .

2.2 JEF method

Supposing the solutions of Eqs (4) and (5), with m = 1 , has the form

(23) ℱ ( x , t ) = a 0 ( t ) + a 1 ( t ) J ( μ ) ,

where J ( μ ) is one of the following elliptic functions s n ( ω μ , n ˇ ) , c n ( ω μ , n ˇ ) , or d n ( ω μ , n ˇ ) . Differentiating Eq. (23) with respect to t and x , we have

(24) ℱ t = a ⋅ 0 + α ⋅ 1 J + ω λ a 1 J ′ , ℱ x = ω k a 1 J ′ , ℱ x x = a 1 k 2 ω 2 J ′ ′ = a 1 B 1 J + a 1 B 2 J 3 , ℱ 3 = a 1 3 J 3 + 3 a 0 a 1 2 J 2 + 3 a 0 2 a 1 J + a 0 3 ,

where B 1 and B 2 are constants depending on ω , n ˇ , k , and they will be defined later. Eqs. (24) and (11) are plugged into Eqs (4) and (5). After that, by equating each coefficient of J k to zero, we have

J 0 : a ⋅ 0 = 0 , J 1 : a ⋅ 1 = 0 , J ′ : ω λ a 1 − 2 δ 1 φ 1 k ω a 1 = 0 ,

and

x J 0 : a 0 φ ⋅ 1 = 0 , x J 1 : a 1 φ ⋅ 1 = 0 , J 0 : − a 0 φ ⋅ 0 + φ 1 2 δ 1 a 0 + A a 0 3 − δ 3 a 0 = 0 , J 1 : − a 1 φ ⋅ 0 + φ 1 2 δ 1 a 1 − δ 1 a 1 B 1 + 3 A a 0 2 a 1 − δ 3 a 1 = 0 , J 2 : 3 A a 0 a 1 2 = 0 , J 3 : − δ 1 B 2 a 1 + A a 1 3 = 0 .

We solve these equations to obtain

a 0 ( t ) = 0 , a 1 = ℏ , δ 1 = ℏ 2 A ( t ) B 2 , φ 1 = ε ,

and

φ 0 = − δ 3 t + ℏ 2 ( ε 2 − B 1 ) B 2 ∫ 0 t A ( τ ) d τ , λ ( t ) = 2 ε k ℏ 2 B 2 A ( t ) ,

where ε and ℏ are constants. Hence, by utilizing Eq. (23), the solutions of the SE-RVCs (3) are as follows:

(25) ℱ ( x , t ) = ℏ J ( μ ) , μ = k x + 2 ε k ℏ 2 δ 2 B 2 ∫ 0 t e σ W ( τ ) − 1 2 σ 2 τ d τ .

Let us now define J ( μ ) as follows:

Set 1: If J ( μ ) = s n ( ω μ , n ˇ ) , then Eq. (25) takes the form

(26) ℱ ( x , t ) = ℏ s n ω k x + 2 ε ω k ℏ 2 δ 2 B 2 ∫ 0 t e σ W ( τ ) − 1 2 σ 2 τ d τ , n ˇ ,

where

B 1 = − k 2 ω 2 ( 1 + n ˇ 2 ) and B 2 = 2 k 2 ω 2 n ˇ 2 .

Set 2: If J ( μ ) = c n ( ω μ , n ˇ ) , then Eq. (25) takes the form

(27) ℱ ( x , t ) = ℏ c n ω k x + 2 ε ω k ℏ 2 δ 2 B 2 ∫ 0 t e σ W ( τ ) − 1 2 σ 2 τ d τ , n ˇ ,

where

B 1 = k 2 ω 2 ( 1 − 2 n ˇ 2 ) and B 2 = − 2 k 2 ω 2 n ˇ 2 .

Set 3: If J ( μ ) = d n ( ω μ , n ˇ ) , then Eq. (25) takes the form

(28) ℱ ( x , t ) = ℏ d n ω k x + 2 ε ω k ℏ 2 δ 2 B 2 ∫ 0 t e σ W ( τ ) − 1 2 σ 2 τ d τ , n ˇ ,

where

B 1 = k 2 ω 2 ( 2 − n ˇ 2 ) and B 2 = − 2 k 2 ω 2 .

3 Exact solutions of SSE

By substituting Eq. (12) into Eq. (2), we attain the solution of SSE (1) as follows:

(29) Z ( x , t ) = ℱ ( x , t ) exp i Y ( x , t ) + σ W ( t ) − 1 2 σ 2 t .

3.1 GREM method

Plugging Eqs (13)–(22) into (29), then the SSE has the solutions:

(30) Z 1 ( x , t ) = ℓ p s tan p s k x + b ℓ 2 δ 2 k s 2 ∫ 0 t e σ W ( τ ) − 1 2 σ 2 τ d τ × e [ i Y + σ W ( t ) − 1 2 σ 2 t ] ,

(31) Z 2 ( x , t ) = − ℓ p s cot [ p s ( k x + b ℓ 2 δ 2 k s 2 ∫ 0 t e σ W ( τ ) − 1 2 σ 2 τ d τ e [ i Y + σ W ( t ) − 1 2 σ 2 t ] ,

(32) Z 3 ( x , t ) = ℓ p s tan 4 p s k x + b ℓ 2 δ 2 k s 2 ∫ 0 t e σ W ( τ ) − 1 2 σ 2 τ d τ − sec p s k x + b ℓ 2 δ 2 k s 2 ∫ 0 t e σ W ( τ ) − 1 2 σ 2 τ d τ × e [ i Y + σ W ( t ) − 1 2 σ 2 t ] ,

(33) Z 4 ( x , t ) = ℓ p s cot 4 p s k x + b ℓ 2 δ 2 k s 2 ∫ 0 t e σ W ( τ ) − 1 2 σ 2 τ d τ + csc 4 p s k x + b ℓ 2 δ 2 k s 2 ∫ 0 t e σ W ( τ ) − 1 2 σ 2 τ d τ × e [ i Y + σ W ( t ) − 1 2 σ 2 t ]

(34) Z 5 ( x , t ) = 1 2 ℓ p s tan p s 2 k x + b ℓ 2 δ 2 k s 2 ∫ 0 t e σ W ( τ ) − 1 2 σ 2 τ d τ − cot p s 2 k x + b ℓ 2 δ 2 k s 2 ∫ 0 t e σ W ( τ ) − 1 2 σ 2 τ d τ × e [ i Y + σ W ( t ) − 1 2 σ 2 t ] ,

for p s > 0 ,

(35) Z 6 ( x , t ) = ℓ − p s tanh − p s k x + b ℓ 2 δ 2 k s 2 ∫ 0 t e σ W ( τ ) − 1 2 σ 2 τ d τ e i Y + σ W ( t ) − 1 2 σ 2 t ,

(36) Z 7 ( x , t ) = ℓ − p s coth − p s k x + b ℓ 2 δ 2 k s 2 ∫ 0 t e σ W ( τ ) − 1 2 σ 2 τ d τ e i Y + σ W ( t ) − 1 2 σ 2 t ,

(37) Z 8 ( x , t ) = ℓ − p s coth − 4 p s k x + b ℓ 2 δ 2 k s 2 ∫ 0 t e σ W ( τ ) − 1 2 σ 2 τ d τ + csch − 4 p s k x + b ℓ 2 δ 2 k s 2 ∫ 0 t e σ W ( τ ) − 1 2 σ 2 τ d τ e i Y + σ W ( t ) − 1 2 σ 2 t ,

(38) Z 9 ( x , t ) = − 1 2 ℓ − p s tanh 1 2 − p s k x + b ℓ 2 δ 2 k s 2 ∫ 0 t e σ W ( τ ) − 1 2 σ 2 τ d τ + coth 1 2 − p s k x + b ℓ 2 δ 2 k s 2 ∫ 0 t e σ W ( τ ) − 1 2 σ 2 τ d τ e i Y + σ W ( t ) − 1 2 σ 2 t ,

for p s < 0 , and

(39) Z 10 ( x , t ) = − 1 s k x + b ℓ 2 δ 2 k s 2 ∫ 0 t e σ W ( τ ) − 1 2 σ 2 τ d τ e i Y + σ W ( t ) − 1 2 σ 2 t for s ≠ 0 ,

where

Y ( x , t ) = b x − δ 3 t + δ 2 ℓ 2 ( b 2 − 2 s p k 2 ) 2 k 2 s 2 ∫ 0 t e σ W ( τ ) − 1 2 σ 2 τ d τ .

Remark 1

Putting σ = 0 (i.e., no noise) and δ 3 = 0 in Eqs (32), (33), and (37), we have the same results that stated in [24].

Remark 2

Putting σ = 0 and δ 3 = 0 in Eqs (30) and (35), we have the same results that stated in [23].

3.2 JEF method

Substituting Eqs (26)–(28) into (29), then the SSE has the solutions:

(40) Z ( x , t ) = ℏ s n ω k x + ε ℏ 2 δ 2 k ω n ˇ 2 ∫ 0 t e σ W ( τ ) − 1 2 σ 2 τ d τ , n ˇ × e [ i Y ( x , t ) + σ W ( t ) − 1 2 σ 2 t ] ,

(41) Z ( x , t ) = ℏ c n ω k x − ε ℏ 2 δ 2 k ω n ˇ 2 ∫ 0 t e σ W ( τ ) − 1 2 σ 2 τ d τ , n ˇ × e [ i Y ( x , t ) + σ W ( t ) − 1 2 σ 2 t ] ,

and

(42) Z ( x , t ) = ℏ d n ω k x − ε ℏ 2 δ 2 k ω ∫ 0 t e σ W ( τ ) − 1 2 σ 2 τ d τ , n ˇ × e [ i Y ( x , t ) + σ W ( t ) − 1 2 σ 2 t ] ,

where

Y = ε x − δ 3 t + ℏ 2 ( ε 2 − B 1 ) B 2 ∫ 0 t A ( τ ) d τ .

When n ˇ → 1 , then Eqs (40)–(42) tend to

(43) Z ( x , t ) = ℏ tanh ω k x + ε ℏ 2 δ 2 k ω ∫ 0 t e σ W ( τ ) − 1 2 σ 2 τ d τ × e [ i Y ( x , t ) + σ W ( t ) − 1 2 σ 2 t ] ,

and

(44) Z ( x , t ) = ℏ sech ω k x − ε ℏ 2 δ 2 k ω ∫ 0 t e σ W ( τ ) − 1 2 σ 2 τ d τ × e [ i Y ( x , t ) + σ W ( t ) − 1 2 σ 2 t ] .

4 Physical meaning and effect of noise

Physical meaning: The SSE provides a powerful tool for studying the behavior of quantum systems in the presence of stochastic influences. By incorporating random fluctuations into the evolution of quantum states, this equation offers a more realistic and nuanced understanding of quantum phenomena. Through its ability to capture the physical implications of noise and uncertainty in quantum systems, the stochastic Schrödinger equation helps to deepen our understanding of the inherently probabilistic nature of the quantum world. Here, we obtained the exact stochastic solutions of the SSE (1). We utilized two methods including the GREM method and the JEF method. There are many kinds of solutions including singular periodic solutions, dark solutions, and light solutions. Singular solitons are important because of their special properties that allow for effective information transfer and the investigation of complex wave dynamics in many different fields of science. In nonlinear optics, singular solitons play an essential role in light transmission over optical fibers. Solitons, unlike regular light waves, do not scatter over long distances due to their self-focusing qualities, allowing for the effective transport of information at high rates. This makes them crucial in telecommunications, where data transfer over vast distances is critical for preserving network connectivity and speed.

Effect of noise: Now, we address the impact of multiplicative white noise on the exact solutions of the SSE (1). Numerous numerical simulations of various solutions with different intensity of noise are shown. Figures 1, 2, 3 display the solutions Z ( x , t ) described in Eqs (41), (43) and (44) for different intensity of noise σ as follows:

$Figure 1 (a)–(e) Exhibit 3D-profile of ∣ Z ( x , t ) ∣ | {\mathcal{Z}}\left(x,t)| described in Eq. (41) with ℏ = n ˇ = 0.5 \hslash =\check{n}=0.5 , δ 2 = 2 {\delta }_{2}=2 , ε = − 1 \varepsilon =-1 , ω = k = 1 \omega =k=1 , x ∈ [ − 4 , 4 ] x\in \left[-4,4] , and t ∈ [ 0 , 4 ] t\in \left[0,4] , (f) 2D-profile of Eq. (41) with σ = 0 \sigma =0 , 0.1, 0.3, 1, and 2. (a) σ = 0 \sigma =0 , (b) σ = 0.1 \sigma =0.1 , (c) σ = 0.3 \sigma =0.3 , (d) σ = 1 \sigma =1 , (e) σ = 2 \sigma =2 , (f) σ = 0 \sigma =0 , 0.1, 0.3, 1, 2.$

Figure 1

(a)–(e) Exhibit 3D-profile of ∣ Z ( x , t ) ∣ described in Eq. (41) with ℏ = n ˇ = 0.5 , δ 2 = 2 , ε = − 1 , ω = k = 1 , x ∈ [ − 4 , 4 ] , and t ∈ [ 0 , 4 ] , (f) 2D-profile of Eq. (41) with σ = 0 , 0.1, 0.3, 1, and 2. (a) σ = 0 , (b) σ = 0.1 , (c) σ = 0.3 , (d) σ = 1 , (e) σ = 2 , (f) σ = 0 , 0.1, 0.3, 1, 2.

$Figure 2 (a)–(e) Exhibit 3D-profile of ∣ Z ( x , t ) ∣ | {\mathcal{Z}}\left(x,t)| introduced in Eq. (43) with ℏ = n ˇ = 1 \hslash =\check{n}=1 , δ 2 = 2 {\delta }_{2}=2 , ε = 1 \varepsilon =1 , ω = k = 1 \omega =k=1 , x ∈ [ − 4 , 4 ] x\in \left[-4,4] , and t ∈ [ 0 , 4 ] t\in \left[0,4] (f) displays 2D-profile of Eq. (43) with σ = 0 \sigma =0 , 0.1, 0.3, 1, and 2. (a) σ = 0 \sigma =0 , (b) σ = 1 \sigma =1 , (c) σ = 0.3 \sigma =0.3 , (d) σ = 1 \sigma =1 , (e) σ = 2 \sigma =2 , and (f) σ = 0 \sigma =0 , 0.1, 0.3, 1, 2.$

Figure 2

(a)–(e) Exhibit 3D-profile of ∣ Z ( x , t ) ∣ introduced in Eq. (43) with ℏ = n ˇ = 1 , δ 2 = 2 , ε = 1 , ω = k = 1 , x ∈ [ − 4 , 4 ] , and t ∈ [ 0 , 4 ] (f) displays 2D-profile of Eq. (43) with σ = 0 , 0.1, 0.3, 1, and 2. (a) σ = 0 , (b) σ = 1 , (c) σ = 0.3 , (d) σ = 1 , (e) σ = 2 , and (f) σ = 0 , 0.1, 0.3, 1, 2.

$Figure 3 (a)–(e) Exhibit 3D-profile of ∣ Z ( x , t ) ∣ | {\mathcal{Z}}\left(x,t)| introduced in Eq. (44) with ℏ = n ˇ = 0.5 \hslash =\check{n}=0.5 , δ 2 = 2 , ε = 1 {\delta }_{2}=2,\hspace{1em}\varepsilon =1 , ω = k = 1 \omega =k=1 , x ∈ [ − 4 , 4 ] x\in \left[-4,4] , and t ∈ [ 0 , 3 ] t\in \left[0,3] (f) displays 2D-profile of Eq. (44) with σ = 0 \sigma =0 , 0.1, 0.3, 1, and 2. (a) σ = 0 \sigma =0 , (b) σ = 0.1 \sigma =0.1 , (c) σ = 0 \sigma =0 , (d) σ = 1 \sigma =1 , (e) σ = 2 \sigma =2 , and (f) σ = 0 \sigma =0 , 0.1, 0.3, 1, 2.$

Figure 3

(a)–(e) Exhibit 3D-profile of ∣ Z ( x , t ) ∣ introduced in Eq. (44) with ℏ = n ˇ = 0.5 , δ 2 = 2 , ε = 1 , ω = k = 1 , x ∈ [ − 4 , 4 ] , and t ∈ [ 0 , 3 ] (f) displays 2D-profile of Eq. (44) with σ = 0 , 0.1, 0.3, 1, and 2. (a) σ = 0 , (b) σ = 0.1 , (c) σ = 0 , (d) σ = 1 , (e) σ = 2 , and (f) σ = 0 , 0.1, 0.3, 1, 2.

Figures 1, 2, 3 show that when noise is ignored (i.e., σ = 0 ), different kinds of solutions appear, including periodic solutions, singular solutions, optical light solutions, dark solutions, and so on. When noise is introduced at σ = 0.1 , 0.3, 1, 2, the surface flattens after some transit patterns. This result shows how multiplicative white noise affects the SSE (1) solutions, stabilizing them around zero.

5 Conclusion

In this article, we considered the SSE (1) driven by multiplicative Wiener process in the Itô sense. By using appropriate transformation, we converted the SSE to another Schrödinger equation with random variable coefficients (3) (SE-RVCs). By using the GREM method and JEF method, we obtained a new stochastic exact solutions for SE-RVCs in the type of trigonometric, hyperbolic, rational, and elliptic functions. After that, we acquired the obtained solutions of SSE (1). Moreover, we generalized some previous solutions such as the results reported in previous studies [23,24]. Because of the importance of Schrödinger equation in nonlinear waves in a liquid-filled elastic tube, solitary wave and nonlinear instability problems, plasma waves and hydromagnetic, heat transfer in a solid, nonlinear optics, and propagation in the piezoelectric semi- conductors, the acquired solutions are crucial in understanding several difficult physical processes. Finally, some graphics were included to demonstrate the effect of stochastic term on the stochastic exact solutions of the SSE. We deduced that the multiplicative Wiener process stabilized the solutions of SSE (1). In the future work, we can address the Schrödinger equation (1) with additive noise.

Acknowledgments

Princess Nourah bint Abdulrahman University Researcher Supporting Project number (PNURSP2024R 273), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.

Funding information: Princess Nourah bint Abdulrahman University Researcher Supporting Project number (PNURSP2024R 273), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors state no conflict of interest.

References

[1] Zhang JL, Wang ML, Wang YM, Fang ZD. The improved F-expansion method and its applications. Phys Lett A. 2006;350:103–9. 10.1016/j.physleta.2005.10.099Search in Google Scholar

[2] Baleanu D, Aydogn SM, Mohammadi H, Rezapour S. On modelling of epidemic childhood diseases with the Caputo-Fabrizio derivative by using the Laplace Adomian decomposition method. Alex Eng J. 2020;59:3029–39. 10.1016/j.aej.2020.05.007Search in Google Scholar

[3] Aydogan SM, Baleanu D, Mohammadi H, Rezapour S. On the mathematical model of Rabies by using the fractional Caputo-Fabrizio derivative. Adv Differ Equ. 2020;2020:382. 10.1186/s13662-020-02798-4Search in Google Scholar

[4] Tuan NH, Mohammadi H, Rezapour S. A mathematical model for COVID-19 transmission by using the Caputo fractional derivative. Chaos Solitons Fractals. 2020;140:110107. 10.1016/j.chaos.2020.110107Search in Google Scholar PubMed PubMed Central

[5] Iqbal MS, Inc M, Yasin MW, Ahmed N, Tawfiq FM, Bilal M, et al. Soliton solutions of nonlinear stochastic Fitz-Hugh Nagumo equation. Opt Quant Electron. 2024;56:1047. 10.1007/s11082-024-06819-4Search in Google Scholar

[6] Zhu C, Al-Dossari M, Rezapour S, Shatey S. On the exact soliton solutions and different wave structures to the modified Schrödingeras equation. Results Phys. 2023;54:107037. 10.1016/j.rinp.2023.107037Search in Google Scholar

[7] Günay B, Alsallami SAM, Rezapour S, Shateyi S. Analytical soliton solutions for the generalized Schrödingeras equation in optical fiber communication systems. Results Phys. 2023;54:106792. 10.1016/j.rinp.2023.106792Search in Google Scholar

[8] Albosaily S, Elsayed EM, Albalwi MD, Alesemi M, Mohammed WW. The analytical stochastic solutions for the stochastic potential Yu-Toda-Sasa-Fukuyama equation with conformable derivative using different methods. Fractal Fract. 2023;7:787. 10.3390/fractalfract7110787Search in Google Scholar

[9] Gu Y, Peng L, Huang Z, Lai Y. Soliton, breather, lump, interaction solutions and chaotic behavior for the (2+1)-dimensional KPSKR equation. Chaos Solitons Fractals. 2024;187:115351. 10.1016/j.chaos.2024.115351Search in Google Scholar

[10] Gu Y, Manafian J, Malmir S, Eslami B, Ilhan OA. Lump, lump-trigonometric, breather waves, periodic wave and multi-waves solutionsfor a Konopelchenko-Dubrovsky equation arising in fluid dynamics. Int J Mod Phys B. 2023;37(15):2350141. 10.1142/S0217979223501412Search in Google Scholar

[11] Gu Y, Aminakbari N. Bernoulli GGasffsa-expansion method for nonlinear Schrödinger equation with third-order dispersion. Mod Phys Lett B. 2022;36:2250028. 10.1142/S0217984922500282Search in Google Scholar

[12] Gu Y, Aminakbari N. New optical soliton solutions for the variable coefficients nonlinear Schrödinger equation. Opt Quant Electron. 2022;54:255. 10.1007/s11082-022-03645-4Search in Google Scholar

[13] Gu Y, Meng F. Searching for analytical solutions of the (2+1)-dimensional KP equation by two different systematic methods. Complexity 2019;2019:9314693. 10.1155/2019/5162038Search in Google Scholar

[14] Kumar S, Mann N. A variety of newly formed soliton solutions and patterns of dynamic waveforms for the generalized complex coupled Schrödinger-Boussinesq equations. Opt Quant Electron. 2023;55:723. 10.1007/s11082-023-04869-8Search in Google Scholar

[15] Kumar S, Dhiman SS. Exploring cone-shaped solitons, breather, and lump-forms solutions using the Lie symmetry method and unified approach to a coupled Breaking soliton model. Phys Scr. 2024;99:025243. 10.1088/1402-4896/ad1d9eSearch in Google Scholar

[16] Niwas M, Kumar S, Rajput R, Chadha D. Exploring localized waves and different dynamics of solitons in (2+1)-dimensional Hirota bilinear equation:a multivariate generalized exponential rational integral function approach. Nonlinear Dyn. 2024;112:9431–44. 10.1007/s11071-024-09555-2Search in Google Scholar

[17] Kumar S, Hamid I. New interactions between various soliton solutions, including bell, kink, and multiple soliton profiles, for the (2+1)-dimensional nonlinear electrical transmission line equation. Opt Quant Electron. 2024;56:1173. 10.1007/s11082-024-06960-0Search in Google Scholar

[18] Schrödinger E. An undulatory theory of the mechanics of atoms and molecules. Phys Rev. 1926;28(6):1049–70. 10.1103/PhysRev.28.1049Search in Google Scholar

[19] Ashcroft NW, Mermin ND. Solid state physics. New York: Holt, Rinehart and Winstor; 1976. Search in Google Scholar

[20] Chu HT. Eigen energies and eigen states of conduction electrons in pure bismithunder size and magnetic fields quatizations. J Physics Chem Solids. 1989;50(3):319–24. 10.1016/0022-3697(89)90494-0Search in Google Scholar

[21] Kelley PI. Self-focusing of optical beams. Phys Rev Lett. 1965;15:1005–8. 10.1103/PhysRevLett.15.1005Search in Google Scholar

[22] Blencowe M. Quantum electromechanical systems. Phys Rep. 2004;395:159–222. 10.1016/j.physrep.2003.12.005Search in Google Scholar

[23] Taghizadeh N, Mirzazadeh M, Farahrooz F. Exact solutions of the nonlinear Schrödinger equation by the first integral method. J Math Anal Appl. 2011;374(2):549–53. 10.1016/j.jmaa.2010.08.050Search in Google Scholar

[24] Kaplan M, Ünsal Ö, Bekir A. Exact solutions of nonlinear Schrödinger equation by using symbolic computation. Math Methods Appl Sci. 2016;39(8):2093–9. 10.1002/mma.3626Search in Google Scholar

[25] Zhimin MA, Yuhuai S. New exact solutions of cubic nonlinear Schrodinger equations. Chin J Quant Electron. 2019;36:416. Search in Google Scholar

[26] Ahmed AI, Algolam MS, Cesarano C, Rizk D, Gassem F, Mohammed WW. Dynamical behavior of the fractional generalized nonlinear Schrödinger equation of third-order. Opt Quant Electron. 2024;56:843. 10.1007/s11082-024-06626-xSearch in Google Scholar

[27] Mohammed WW, Iqbal N, Bourazza S, Elsayed EM. The optical structures for the fractional chiral nonlinear Schrödinger equation with time-dependent coefficients. Opt Quant Electron. 2024;56:1476. 10.1007/s11082-024-06800-1Search in Google Scholar

[28] Wazwaz A-M Reliable analysis for nonlinear Schrödinger equations with a cubic nonlinearity and a power law nonlinearity. Math Comput Model. 2006;43:178–84. 10.1016/j.mcm.2005.06.013Search in Google Scholar

[29] Hamid I, Kumar S. Symbolic computation and Novel solitons, traveling waves and soliton-like solutions for the highly nonlinear (2+1)-dimensional Schrödinger equation in the anomalous dispersion regime via newly proposed modified approach. Opt Quant Electron. 2023;55:755. 10.1007/s11082-023-04903-9Search in Google Scholar

[30] Kumar S, Kukkar A. Dynamics of several optical soliton solutions of a (3+1)-dimensional nonlinear Schrödinger equation with parabolic law in optical fibers. Modern Phys Lett B. 2024;2450453. 10.1142/S0217984924504530Search in Google Scholar

[31] Abdelrahman M, Mohammed WW, Alesemi M, Albosaily S. The effect of multiplicative noise on the exact solutions of nonlinear Schrödinger equation. AIMS Math. 2021;6:2970–80. 10.3934/math.2021180Search in Google Scholar

[32] Mohammed WW, Boulares H, Khelifi F, Elabbasy EM, Aly AS. The exact solutions for the nonlinear Schrödinger equation forced by multiplicative noise on Itô sense. Comput Sci. 2021;16:1147–59. Search in Google Scholar

[33] Alkhidhr HA. The new stochastic solutions for three models of non-linear Schrödingeras equations in optical fiber communications via Itô sense. Front Phys. 2023;11:144704. 10.3389/fphy.2023.1144704Search in Google Scholar

[34] Mohammed WW, Cesarano C, Alqsair NI, Sidaoui R. The impact of Brownian motion on the optical solutions of the stochastic ultra-short pulses mathematical model. Alex Eng J. 2024;101:186–92. 10.1016/j.aej.2024.05.054Search in Google Scholar

Received: 2024-07-09

Revised: 2024-08-27

Accepted: 2024-09-02

Published Online: 2024-10-09

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/phys-2024-0086

Keywords for this article

random variable coefficients; Schrödinger equation; stability by noise; generalizing Riccati equation mapping method

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