Effects of Brownian noise strength on new chiral solitary structures

Abstract

In this paper, we investigate the nonlinear Chiral Schrödinger equation (CNLSE) in two dimensions where noise term affected randomly with time. This equation characterized some edges states of fractional-Hall Effect features in quantum. The CNLSE with multiplicative noise effects is studied as dynamical system to specify the acceptable solution types and then solved by unified solver method. The presented solutions are periodic envelopes, explosive, dissipative, symmetric solitons, and blow up waves. It was confirmed that the noise factor is dominant on all the wave conversion, growing and damping of envelopes and shocks. The presented technique in this study can be easily utilized for other nonlinear equations in applied science.

Mathematics Subject Classification (2010): 35C08, 65C20, 60H15, 35Q40, 35Q55, 35Q62

Introduction

Mathematical simulation of some vital real life problems typically leads to differential equations and other problems; essential functions of mathematical physics, as well as their extensions and generalizations, are included.^1–8 Recently, the nonlinear stochastic partial differential equations (NSPDEs) have been extensively studied by many scientists due to the fact that these equations are successfully utilized for modeling various physical phenomena such as superfluid, HIV internal virus dynamics, telecommunications, astrophysical dynamics, and growth of biological populations.^9–14 These equations often rely on a noise source such as a white noise and certain probability laws.^15–18 The investigation of optical solitons for the NSPDEs is an important topic as explained in.^19–22

The investigation of chiral solitons is so important in the developments of quantum mechanics, especially in the field of quantum hall effect where chiral excitations are known to become visible. ²³ The governing equations, that are going to be investigated through this study, are (2 + 1)-dimensional CNLSE. Namely, we consider this model in the presence of noise term, which produce various complex phenomena in quantum mechanics, superfluid, molecular biology, statistical mechanics, magneto hydrodynamics, deep water, and many others.^24–28 More and more attention has been paid recently to the effect of a noise on the propagation of the soliton solutions.

In recent years, the generation of fractal solitary structures from many nonlinear equations via fractal variational principles in different media becomes an effective way to describe some new phenomena in our universe.^29–32 The solitary peaks have been examined by unsmooth boundaries effects via fractal KdV equation by using various fractal dimensions of the boundaries. ³² Also, the solitary forms of Boussinesq equation has been examined in fractal space. ²⁹ On the other hand, the nonlinear Chen-Lee-Liu equation in fractal form has been introduced to investigate the optical nonlinear pulses features in optics applications. ³³

We consider the (2 + 1)-dimensional CNLSE as follows^34,35

$i{\mathrm{\Theta }}_t+\alpha \left({\mathrm{\Theta }}_{xx}+{\mathrm{\Theta }}_{yy}\right)+i\left(b_1\left(\mathrm{\Theta }{\mathrm{\Theta }}_x^{*}-{\mathrm{\Theta }}^{*}{\mathrm{\Theta }}_x\right)+b_2\left(\mathrm{\Theta }{\mathrm{\Theta }}_y^{*}-{\mathrm{\Theta }}^{*}{\mathrm{\Theta }}_y\right)\right)\mathrm{\Theta }-i\sigma \mathrm{\Theta }{\mathrm{\Xi }}_t=0,$

(1a)

where   Θ = Θ(x, y, t) is a complex function and the superscript * denotes the complex conjugate. α denotes the coefficient of dispersion. b₁ and b₂ refer to the coefficients of the nonlinear coupling terms, whereas σ is the noise strength. This type of nonlinearity is called current density. The noise Ξ _t is the time derivative of the Brownian motion Ξ(t). Equation (1a) is not Galilean invariant and also is not integrable by the method of inverse scattering transform, because it will fail in the Painleve test of integrability. ³⁵ There are so many studies that consider the (2 + 1)-dimensional CNLSE, but in the absence of stochastic effect.^23,35–38 In the ongoing research, we consider this equation in the presence of noise term in Itô sense. This noise term will properly controlled the growing, damping and conversion influences on the bright/dark envelope and shock forced oscillatory wave amplitudes and frequencies. For many physical phenomena, the Browning motion is used to find suitable models for them if they have a quantity that is constantly in small region with random fluctuations such as noise term in the considered model in equation (1a). Brownian motion is defined as a random movement of small particles in both gases and liquids. In mathematics, we can view the Browning motion as a combination of a Gaussian process and a Markov process that have continuous path happening over continuous time. It is called a standard Wiener process. In summary, the Browning motion is a stochastic process that is continuous in time and it is used to describe random movement in fluids. Now, we introduce the main properties of Brownian motion process Ξ(t) ³⁹ as follows:

• (a) Ξ(t), t ≥ 0 are continuous functions of t, where  t  refers to the time and  Ξ(t) ∼ N(0, t).

• (b) For s < t < u < k, $\mathrm{\Xi }\left(s\right)-\mathrm{\Xi }\left(t\right))$ and Ξ(k) − Ξ(u) are independent.

• (c) Ξ(t) − Ξ(s) follows a normal distribution with zero mean and variance t − s, that is, $\mathrm{\Xi }\left(t\right)-\mathrm{\Xi }\left(s\right)\sim \sqrt{t-s}N\left(\mathrm{0,1}\right)$ , where N(0, 1)  is a standard normal distribution.

We note that the probability density function of the normal distribution is displayed as

$f\left(x\right)=\frac{1}{\sigma \sqrt{2\pi }}{e}^{-\frac{{\left(x-\mu \right)}^2}{2 {\sigma }^2}},-\infty <x<\infty$

where  μ   refers to the mean and  σ²  represents the variance. The standard normal distribution is a special case of normal distribution. It has zero mean and unit variance. Its corresponding probability distribution function is given by

$f\left(z\right)=\frac{1}{\sqrt{2\pi }}{e}^{-\frac{z^2}{2}},-\infty <z<\infty .$

The stochastic integral ${\int }_0^t\mathrm{\Theta }\left(s\right)d\mathrm{\Xi }\left(s\right)$ can be interpreted in distinct ways. The most widely used explications of a stochastic integral are the Itô and Stratonovich. When the stochastic integral is evaluated at the left-end point, the integral is called an Itô stochastic integral and if the stochastic integral is evaluated at the midpoint, the integral is called a Stratonovich stochastic integral. ⁴⁰

We aim to produce some new stochastic solutions for the (2 + 1)-dimensional CNLSE forced by multiplicative noise in Itô sense, using the unified solver approach. The proposed solver reveals various advantages, such as it averts intricate and tedious computations, and provides us with exact solutions in an explicit form via free physical parameters. This solver is simple, sturdily built and vigorous. One can easily use this solver as a box solver. The presented solver can be applied for the huge classes of NSPDEs. Indeed, this solver can be easily extended for solving stochastic fractional NPDEs. One of the more interesting points is to clarify the influence of the noise term on the acquired solutions. We also illustrate the nonlinear dynamical behavior of some selected stochastic solutions.

The rest of the framework of this paper is structured as follows: Section 2 considers the (2 + 1)-dimensional stochastic CNLSE via Itô sense. Namely, CNLSE with multiplicative noise effects is studied as dynamical system to specify the acceptable solution types and then solved by the unified solver method. Section 3 provides the physical interpretation for the phenomena arising through the solutions of considered model. Conclusions appear in Section 4.

The stochastic ODE for equation (1a) and its solutions

Using the traveling wave transformation, ³⁴ we obtain

$\mathrm{\Theta }\left(x,y,t\right)=\theta \left(\zeta \right)e^{i\eta +\sigma \mathrm{\Xi }\left(t\right)-{\sigma }^{2}t},\hspace{0.17em}\zeta =r_{1}x+r_{2}y-r_{3}t,\hspace{0.17em}\eta ={\eta }_{1}x+{\eta }_{2}y+{\eta }_{3}t+\varpi ,$

(2a)

where η₁, η₂ represent the frequencies in x and y directions; η₃ and ϖ denote soliton frequency and phase constant, whereas σ is the noise strength.

Putting equations (2a) into (1a) yields

$-{\sigma }^2\theta +\left(-r_3+2\alpha \left(r_1{\eta }_1+r_2{\eta }_2\right)\right){\theta }^{\prime }=0$

(2b)

from the imaginary part and

$\alpha \left(r_1^2+r_2^2\right){\theta }^{″}+2\left(b_1{\eta }_1+b_2{\eta }_2\right){\theta }^3{e}^{2\sigma \hspace{0.17em}\mathrm{\Xi }\left(t\right)-2{\sigma }^{2}t}-\left(\alpha \left({\eta }_1^2+{\eta }_2^2\right)+{\eta }_3\right)\theta =0$

(2c)

from the real part. Equation (2c) is used to inspect the soliton solutions under the condition

$\alpha {\sigma }^4\left(r_1^2+r_2^2\right)e^{-\frac{2\zeta {\sigma }^2}{2\alpha {\eta }_1{r}_1+2\alpha {\eta }_2{\rho }_2-{\rho }_3}}+2(b_1{\eta }_1+b_2{\eta }_2)(r_3-2\alpha ({\eta }_1{r}_1+{\eta }_2{r}_2){)}^2=0$

(2d)

Expectation on both side for equation (2c) gives

$\alpha \left(r_1^2+r_2^2\right){\theta }^{″}+2\left(b_1{\eta }_1+b_2{\eta }_2\right){\theta }^3{e}^{-2{\sigma }^{2}t}E\left(e^{2\sigma \hspace{0.17em}\mathrm{\Xi }\left(t\right)}\right)-\left(\alpha \left({\eta }_1^2+{\eta }_2^2\right)+{\eta }_3\right)\theta =0,$

(2e)

where $E\left(e^{2\sigma \hspace{0.17em}\mathrm{\Xi }\left(t\right))}\right)=e^{2{\sigma }^{2}t}$ . Therefore, equation (2e) becomes

$\alpha \left(r_1^2+r_2^2\right){\theta }^{″}+2\left(b_1{\eta }_1+b_2{\eta }_2\right){\theta }^3-\left(\alpha \left({\eta }_1^2+{\eta }_2^2\right)+{\eta }_3\right)\theta =0.$

(2f)

Chiral NLSE dynamical mode

The investigated Chiral nonlinear mode in two dimensions can be reduced to the formal classical energy integral of particles in potential well. The particle kinetic energy and quasi-potential represented in the energy equation reads

$\frac{1}{2}{\left(\frac{\partial \theta }{\partial \zeta }\right)}^2+V\left(\theta \right)=0.$

(2g)

Equation (2f) is a dynamical system of a particle in the potential given by

$\mathrm{V}=\frac{{\theta }^4\left(b_1{\eta }_1+b_2{\eta }_2\right)}{2\alpha \left(r_1^2+r_2^2\right)}-\frac{{\theta }^2\left(\alpha \left({\eta }_1^2+{\eta }_2^2\right)+{\eta }_3\right)}{2\alpha \left(r_1^2+r_2^2\right)}.$

(2h)

The graphical dynamical behaviors of V(θ) and the phase portrait are numerically simulated in Figures 1 and 2.

Figure 1.

Change of V and $\frac{d\theta }{d\zeta }$ with θ for α = 0.

Figure 2.

Change of V and $\frac{d\theta }{d\zeta }$ with θ for α = −1.5.

Stochastic solutions

Exact and unified solver technique solutions

The exact solution of equation (2f) is

$\theta \left(\zeta \right)=\frac{2\left(\alpha \left({\eta }_1^2+{\eta }_2^2\right)+{\eta }_3\right)\sqrt{\frac{1}{\left(\alpha \left({\eta }_1^2+{\eta }_2^2\right)+{\eta }_3\right)\left(b_1{\eta }_1+b_2{\eta }_2\right)}}\exp \left(\frac{\zeta \sqrt{\alpha \left({\eta }_1^2+{\eta }_2^2\right)+{\eta }_3}}{\sqrt{\alpha \left(r_1^2+r_2^2\right)}}\right)}{\exp \left(\frac{2\zeta \sqrt{\alpha \left({\eta }_1^2+{\eta }_2^2\right)+{\eta }_3}}{\sqrt{\alpha \left(r_1^2+r_2^2\right)}}\right)+1}.$

(2i)

Consequently, the solutions of equation (1a) is given by

$\begin{array}{c}\mathrm{\Theta }\left(x,y,t\right)=2\left(\alpha \left({\eta }_1^2+{\eta }_2^2\right)+{\eta }_3\right)\sqrt{\frac{1}{\left(\alpha \left({\eta }_1^2+{\eta }_2^2\right)+{\eta }_3\right)\left(b_1{\eta }_1+b_2{\eta }_2\right)}}\\ \frac{\exp \left(\frac{\zeta \sqrt{\alpha \left({\eta }_1^2+{\eta }_2^2\right)+{\eta }_3}}{\sqrt{\alpha \left(r_1^2+r_2^2\right)}}\right)}{\exp \left(\frac{2\zeta \sqrt{\alpha \left({\eta }_1^2+{\eta }_2^2\right)+{\eta }_3}}{\sqrt{\alpha \left(r_1^2+r_2^2\right)}}\right)+1}{e}^{i\left({\eta }_{1}x+{\eta }_{2}y+{\eta }_{3}t+\varpi \right)+\sigma \hspace{0.17em}\mathrm{\Xi }\left(t\right)-{\sigma }^{2}t}.\end{array}$

(2j)

In view of the unified solver technique, ⁴¹ the stochastic solutions for equation (2c) are:

Family I:

The first stochastic solutions of equation (2c) are written as

${\theta }_{\mathrm{1,2}}\left(x,y,t\right)=\pm \sqrt{\frac{\alpha \left({\eta }_1^2+{\eta }_2^2\right)+{\eta }_3}{b_1{\eta }_1+b_2{\eta }_2}}\hspace{1em}\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{h}\left(\sqrt{\frac{\alpha \left({\eta }_1^2+{\eta }_2^2\right)+{\eta }_3}{\alpha \left(r_1^2+r_2^2\right)}}\left(r_{1}x+r_{2}y-r_{3}t\right)\right).$

(2k)

Thus, based on equation (2k), the stochastic solutions of the equation (1a) are

${\mathrm{\Theta }}_{\mathrm{1,2}}\left(x,y,t\right)=\pm \sqrt{\frac{\alpha \left({\eta }_1^2+{\eta }_2^2\right)+{\eta }_3}{b_1{\eta }_1+b_2{\eta }_2}}\hspace{1em}\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{h}\left(\sqrt{\frac{\alpha \left({\eta }_1^2+{\eta }_2^2\right)+{\eta }_3}{\alpha \left(r_1^2+r_2^2\right)}}\left(r_{1}x+r_{2}y-r_{3}t\right)\right)\hspace{1em}{e}^{i\left({\eta }_{1}x+{\eta }_{2}y+{\eta }_{3}t+\varpi \right)+\sigma \hspace{0.17em}\mathrm{\Xi }\left(t\right)-{\sigma }^{2}t}.$

(2l)

• Family II:

The second stochastic solutions of equation (2c) are presented as

${\theta }_{\mathrm{3,4}}\left(x,y,t\right)=\pm \sqrt{\frac{35\hspace{1em}\left(\alpha \left({\eta }_1^2+{\eta }_2^2\right)+{\eta }_3\right)}{36\hspace{1em}\left(b_1{\eta }_1+b_2{\eta }_2\right)}}{\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{h}}^2\left(\sqrt{\frac{5\hspace{1em}\left(\alpha \left({\eta }_1^2+{\eta }_2^2\right)+{\eta }_3\right)}{12\hspace{1em}\alpha \left(r_1^2+r_2^2\right)}}\hspace{1em}\left(r_{1}x+r_{2}y-r_{3}t\right)\right).$

(2m)

So, using equation (2m), the stochastic solutions of the equation (1a) are displayed as

$\begin{array}{c}{\mathrm{\Theta }}_{\mathrm{3,4}}\left(x,y,t\right)=\sqrt{\frac{35\left(\alpha \left({\eta }_1^2+{\eta }_2^2\right)+{\eta }_3\right)}{36\left(b_1{\eta }_1+b_2{\eta }_2\right)}}{\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{h}}^2\left(\sqrt{\frac{5\left(\alpha \left({\eta }_1^2+{\eta }_2^2\right)+{\eta }_3\right)}{12\alpha \left(r_1^2+r_2^2\right)}}\hspace{1em}\left(r_{1}x+r_{2}y-r_{3}t\right)\right)\\ e^{i\left({\eta }_{1}x+{\eta }_{2}y+{\eta }_{3}t+\varpi \right)+\sigma \hspace{0.17em}\mathrm{\Xi }\left(t\right)-{\sigma }^{2}t}.\end{array}$

(2n)

• Family III:

The third stochastic solutions of equation (2c) are demonstrated as

${\theta }_{\mathrm{5,6}}\left(x,y,t\right)=\pm \sqrt{\frac{\alpha \left({\eta }_1^2+{\eta }_2^2\right)+{\eta }_3}{2\left(b_1{\eta }_1+b_2{\eta }_2\right)}}\tanh \left(\sqrt{\frac{-\left(\alpha \left({\eta }_1^2+{\eta }_2^2\right)+{\eta }_3\right)}{2\alpha \left(r_1^2+r_2^2\right)}}\left(r_{1}x+r_{2}y-r_{3}t\right)\right).$

(2o)

Therefore, based on equation (2o), the stochastic solutions of the equation (1a) are

$\begin{array}{c}{\mathrm{\Theta }}_{\mathrm{5,6}}\left(x,y,t\right)=\pm \sqrt{\frac{\alpha \left({\eta }_1^2+{\eta }_2^2\right)+{\eta }_3}{2\left(b_1{\eta }_1+b_2{\eta }_2\right)}}\tanh \left(\sqrt{\frac{-\left(\alpha \left({\eta }_1^2+{\eta }_2^2\right)+{\eta }_3\right)}{2\alpha \left(r_1^2+r_2^2\right)}}\left(r_{1}x+r_{2}y-r_{3}t\right)\right)\\ e^{i\left({\eta }_{1}x+{\eta }_{2}y+{\eta }_{3}t+\varpi \right)+\sigma \hspace{0.17em}\mathrm{\Xi }\left(t\right)-{\sigma }^{2}t}.\end{array}$

(2p)

Physical interpretation

The (2+1)-dimension stochastic solutions of CNLSE have been released by exact and unified solver techniques. The CNLSE is of interest of many researchers to describe physical phenomena in the Quantum field of Fractional-Hall Effect edges state.^24,27,28 The studied CNLSE in two dimensions can be treated as a particle motion with classical energy equation (2g), which has potential in equation (2h). The potential-phase portrait were numerically deliberated in Figures 1 and 2. It was represented that, there are three points corresponds to a center saddle point and two equilibriums in the phase portrait as in Figure 1. On the other hand, three points correspond to a center equilibrium point and two saddles in the phase portrait as in Figure 2. One can expect all solutions information from the heteroclinic orbits, saddle connections, and periodic orbit obtained in the studied phase portrait. Under noise term and time-dependent Brownian-function Ξ(t), the expectation equation with $E\left(e^{2\sigma \hspace{0.17em}\mathrm{\Xi }\left(t\right)}\right)=e^{2{\sigma }^{2}t}$ reduces CNLSE to ordinary differential equation, which solved to give some solutions that expected in phase portrait. In absence of noise effects (σ = 0), equations (2j), (2l), (2n), and (2p) reduces to

$\begin{array}{c}\mathrm{\Theta }\left(x,y,t\right)=2\left(\alpha \left({\eta }_1^2+{\eta }_2^2\right)+{\eta }_3\right)\sqrt{\frac{1}{\left(\alpha \left({\eta }_1^2+{\eta }_2^2\right)+{\eta }_3\right)\left(b_1{\eta }_1+b_2{\eta }_2\right)}}\\ \frac{\exp \left(\frac{\zeta \sqrt{\alpha \left({\eta }_1^2+{\eta }_2^2\right)+{\eta }_3}}{\sqrt{\alpha \left(r_1^2+r_2^2\right)}}\right)}{\exp \left(\frac{2\zeta \sqrt{\alpha \left({\eta }_1^2+{\eta }_2^2\right)+{\eta }_3}}{\sqrt{\alpha \left(r_1^2+r_2^2\right)}}\right)+1}{e}^{i\left({\eta }_{1}x+{\eta }_{2}y+{\eta }_{3}t+\varpi \right)}\end{array}$

(3a)

${\mathrm{\Theta }}_{\mathrm{7,8}}\left(x,y,t\right)=\pm \sqrt{\frac{\alpha \left({\eta }_1^2+{\eta }_2^2\right)+{\eta }_3}{b_1{\eta }_1+b_2{\eta }_2}}\hspace{1em}\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{h}\left(\sqrt{\frac{\alpha \left({\eta }_1^2+{\eta }_2^2\right)+{\eta }_3}{\alpha \left(r_1^2+r_2^2\right)}}\left(r_{1}x+r_{2}y-r_{3}t\right)\right)\hspace{1em}{e}^{i\left({\eta }_{1}x+{\eta }_{2}y+{\eta }_{3}t+\varpi \right)},$

(3b)

${\mathrm{\Theta }}_{\mathrm{9,10}}\left(x,y,t\right)=\pm \sqrt{\frac{35\hspace{1em}\left(\alpha \left({\eta }_1^2+{\eta }_2^2\right)+{\eta }_3\right)}{36\hspace{1em}\left(b_1{\eta }_1+b_2{\eta }_2\right)}}{\hspace{1em}\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{h}}^2\left(\sqrt{\frac{5\hspace{1em}\left(\alpha \left({\eta }_1^2+{\eta }_2^2\right)+{\eta }_3\right)}{12\hspace{1em}\alpha \left(r_1^2+r_2^2\right)}}\hspace{1em}\left(r_{1}x+r_{2}y-r_{3}t\right)\right)\hspace{1em}{e}^{i\left({\eta }_{1}x+{\eta }_{2}y+{\eta }_{3}t+\varpi \right).}$

(3c)

${\mathrm{\Theta }}_{\mathrm{11,12}}\left(x,y,t\right)=\pm \sqrt{\frac{\alpha \left({\eta }_1^2+{\eta }_2^2\right)+{\eta }_3}{2\left(b_1{\eta }_1+b_2{\eta }_2\right)}}\hspace{1em}\tanh \left(\sqrt{\frac{-\left(\alpha \left({\eta }_1^2+{\eta }_2^2\right)+{\eta }_3\right)}{2\alpha \left(r_1^2+r_2^2\right)}}\left(r_{1}x+r_{2}y-r_{3}t\right)\right)e^{i\left({\eta }_{1}x+{\eta }_{2}y+{\eta }_{3}t+\varpi \right).}$

(3d)

Here, 3D CNLSE solution profiles have been inspected with x, y and t for σ = 0 in Figures 3–13. Figures 3 and 4 described periodic envelopes and solitonic solutions for equation (3a). It was noted that solitonic form is directed to negative x axis is given in Figure 4. In Figures 5–8 some plots for equation (3b) solutions are introduced to show pairs of symmetric solitons as in Figures 5 and 7 and positive soliton directed to negative t axis as in Figure 6 and xy plan as in Figure 8. Equation for solutions (3c) represents three types of solutions as in Figures 9–11, two of them are bright solutions, namely, envelope bright soliton and stationary soliton as sketched in Figures 9 and 10 and the third one is a blow up solution as seen in Figure 11. Furthermore, solution (3d) denotes a dissipative behaviors with blow up and symmetric explosive forma as in Figures 12 and 13.

Figure 3.

Change of real part of equation (3a) with x, t, α = 0.5.

Figure 4.

Change of |Θ| of equation (3a) with x, t, α = 0.5.

Figure 5.

Change of real part of equation (3b) with x, t, α = 0.5.

Figure 6.

Change of |Θ| of equation (3b) with x, t, α = 0.5.

Figure 7.

Change of imaginary part of equation (3b) with x, y, α = 0.5.

Figure 8.

Change of |Θ| of equation (3b) with x, y, α = 0.5.

Figure 9.

Change of real part of equation (3c) with x, y, α = 0.5.

Figure 10.

Change of |Θ| of equation (3c) with x, y, α = 0.5.

Figure 11.

Change of |Θ| of equation (3c) with x, y, α = −1.5.

Figure 12.

Change of |Θ| of equation (3d) with x, t, α = 0.5.

Figure 13.

Change of |Θ| of equation (3d) with x, y, α = 0.5.

On the other hand, we examine the noise strength effects on the solitonic solutions properties, which may grow or damp with time. Solution (2l) is varying with time t and noise parameter σ as depicted in Figure 14. It was performed that by raising σ, the amplitudes of envelope profile contracting with t up to σ = 0.6 and then it began converting to shock oscillatory wave with deformed beaks. Finally, the variation of solution (2n) with time t and σ is given in Figure 15. By increasing σ, the stationary soliton amplitude damped with small movement of left tail up to σ = 0.6, and then it began converting to shock wave with stochastic amplitude. The stochastic properties of (2+1)-dimension CNLSE solutions were modulated by noise in Itô sense by varying solitary properties and fluctuated under stochastic σ parameter.

Figure 14.

Stochastic process for real part of equation (2l).

Figure 15.

Stochastic process for |Θ| of equation (2l).

Our stochastic solutions may be valid with the modern investigations in extraordinary materials as topological insulators and strong correlated systems in quantum chromo dynamical studies, turbulences, and superconductor.

Conclusions

The Brownian stochastic two dimensional CNLSE have been investigated by dynamical system plots and then solved to obtain efficient and dynamic solitary structures. These specific solitary structures may be useful and important in new and modern quantum Hall Effect studies in topological materials. The Brownian noises influences on the structural properties have been examined. The parameter σ modulated the obtained structures, which may be useful and important in quantum Hall Effect studies. The Brownian noises strength influences on the features of solitonic forms which may growing or damping by time. The parameter σ modulated the envelope to behave as shock oscillatory with deformed beaks and stationary soliton to shock wave with stochastic amplitude. On the other ward, the envelope amplitudes may contract or converting to oscillatory shocks with distorted beaks. This study may be leading in quantum physics applications.