1. Introduction
Nonlinear waves represent a fascinating area of study, pervading numerous scientific disciplines with their intricate dynamics and practical implications. From fluid dynamics to optics, nonlinear waves appear in a variety of natural phenomena, offering critical insights into the complex behaviors that govern wave propagation. The significance of this research extends to technological applications, where nonlinear wave phenomena are harnessed for advancements in communication, imaging, and energy transfer [1-4]. In addition, having a comprehensive understanding of interactions between nonlinear waves is crucial in climate science, as it plays a key role in the forecasting of climate change and extreme weather events. The motivation for pursuing advanced research in nonlinear waves lies in unraveling the enigmas of intricate systems, enhancing prognostications, and cultivating innovative technologies. As researchers delve deeper into this dynamic field, they strive to expand the limits of existing theoretical frameworks, thereby paving the way for a more profound comprehension of natural phenomena and fostering applications across diverse scientific and technological domains.
Optical soliton propagation is a crucial area in nonlinear optics, where self-stabilizing light waves maintain their shape and intensity by balancing nonlinear and dispersive influences. These solitary waveforms arise from the interplay between the Kerr nonlinearity and dispersive effects. Optical solitons have various applications, including ultrafast laser systems and long-distance telecommunications [5-7]. They are a crucial tool in modern optical communication networks due to their ability to maintain stability while transmitting information. Various techniques have been used to address challenges related to optical solitons. These methods include He’s semi-inverse variational principle [8], generalized exponential differential function method [9], $({G}^{{\prime} }/G)$-expansion method [10], modified simple equation schemes [11], Lie symmetry approach [12], generalized Kudryashov’s approach [13], F-expansion technique [14], generalized Riccati’s equation mapping method [15] and many more.
Stochastic partial differential equations (SPDEs) are mathematical models that integrate stochastic processes with fractional differential equations. These models find applications across diverse fields like materials science, physics, finance, and more. Recently, there have been significant advances in the study of SPDEs, leading to a deeper understanding and broader application of these equations. Nonlinearities can emerge from various sources, such as multiplicative noise, nonlinear drift terms, or interactions between the solution and other variables. Research into the behavior and properties of solutions tdetao nonlinear SPDEs continues to be a dynamic and evolving field. The stochastic effects in nonlinear partial differential equations (NPDEs) play a key role in explaining numerous important phenomena across diverse applied sciences, such as fluid mechanics, electromagnetic wave propagation, magneto-static spin waves and biology [16-18]. A stochastic process involves the evolution of a deterministic variable or system over time and it plays a key role in determining its behavior. One of the most fundamental examples of a stochastic process is Brownian motion, which has significant implications in various scientific and practical fields [19, 20]. Brownian motion, discovered by Robert Brown in the 19th century, refers to the deterministic movement of particles in a fluid. It gave insight into molecular motion and statistical mechanics. It has a Gaussian distribution and no preferential direction. Brownian motion has been widely applied in fields such as finance, physics, biology, and computer science. Understanding it helps predict behavior in systems subject to deterministic forces [21, 22]. The study of stochastic and Brownian motion deepens our insight into the balance between determinism and randomness in both natural and artificial systems.
This study investigates the nonlinear properties of the complex cubic nonlinear Schrödinger’s equation (NLSE) having δ-potential [23] which is given by: 1 ), βt specifies the noise effect that signifies the time derivative of Brownian motion β(t), t ≥ 0, however, Γ is the strength of noise effect, which cannot be negative. In physical systems, negative noise intensity would imply the reversal or cancellation of fluctuations, which contradicts the concept of noise as a random perturbation. The properties of Brownian motion include the following [25]:
Equation (1 ) has been studied from various perspectives: Goodman et al [26] discussed the stability of solitary wave solutions without noise effect, Baskonus et al [27] derived the dark, bright and combo type optical solutions without noise term, Alomair et al [28] studied the new optical solitary solutions, Alkhidhr investigated the optical solutions in optical communications through Brownian process [25].
$\begin{eqnarray}\begin{array}{l}{\rm{i}}\frac{\partial \phi (x,t)}{\partial t}+\frac{1}{2}\frac{{\partial }^{2}\phi (x,t)}{\partial {x}^{2}}-\varpi \delta \phi (x,t)\\ \quad -\rho {\left|\phi (x,t)\right|}^{2}\phi (x,t)+\tau \phi (x,t){\beta }_{t}=0,\end{array}\end{eqnarray}$
where $\varpi$, δ, and ρ represent real constant coefficients, with δ specifically denoting the Dirac delta function at the origin. The delta potential, controlled by ρ, exhibits the following behavior [24]: it is attractive for ρ < 0 and repulsive for ρ > 0. In equation (In recent studies, there has been a significant transition towards exploring the dynamics of continuous systems via bifurcation and chaos theory [29]. This development has established itself as an effective framework for comprehending complex and nonlinear behaviors in a range of disciplines, such as fluid dynamics, mechanical systems and nonlinear optics. Bifurcation analysis facilitates the identification of critical junctures at which small shifts in parameters can lead to significant transformations in system dynamics, whereas chaos theory investigates outcomes that are unpredictable yet inherently deterministic within these frameworks. The investigation of bifurcation and chaotic dynamics greatly enhances both theoretical understanding and practical applications in various scientific fields. Very recently, Gu et al made a significant contribution to study the chaotic behavior of nonlinear models arising in different nonlinear medium, see [30-32].
This research also explores the dynamical analysis of NLSEs using bifurcation and chaos theory. We aim to derive various stochastic solitary wave solutions through a Brownian process, employing the modified $\exp (-{\rm{\Psi }}(\xi ))$ expansion function method. Additionally, we conduct a multistability and sensitivity analysis of the dynamical system under different initial conditions.
This manuscript is arranged as follows: section 2 outlines the algorithm of the proposed method. Section 3 presents the extraction of stochastic solutions for the complex cubic NLSE using the modified $\exp (-{\rm{\Psi }}(\xi ))$-expansion function method. Section 4 studies the bifurcation analysis of the dynamical system. Section 5 analyzes the chaotic behavior of an autonomous system with a perturbation term. Section 6 conducts the multistabilty analysis of the perturbed dynamical system under varying initial conditions. Section 7 performs the sensitivity analysis of the underlying system at different initial conditions and section 8 provides concluding remarks for the studied equation.
2. General strategy of the method
By introducing the traveling wave variable η = x - vt in (1+1)-dimensional NLPDE form as: 3 ) assumes the following form of solution: 3 ) possesses the following solution families:
$\begin{eqnarray}G(F,{F}_{x},{F}_{y},{F}_{t},{F}_{xx},\ldots \,)=0,\end{eqnarray}$
we get $\begin{eqnarray}O(F,{F}_{\eta },{F}_{\eta \eta },\ldots \,)=0.\end{eqnarray}$
According to this method [33], equation ( $\begin{eqnarray}\phi (\eta )=\frac{{\sum }_{i=0}^{n}{A}_{i}{({{\rm{e}}}^{-\chi (\eta )})}^{i}}{{\sum }_{j=0}^{n}{B}_{j}{({{\rm{e}}}^{-\chi (\eta )})}^{j}},\end{eqnarray}$
where Ai, Bj are unknown coefficients and χ(η) satisfies the following equation: $\begin{eqnarray}\chi ^{\prime} (\eta )={{\rm{e}}}^{-\chi (\eta )}+{\alpha }_{1}{{\rm{e}}}^{\chi (\eta )}+{\alpha }_{2}.\end{eqnarray}$
Equation (F1: With α1 ≠ 0, ${\alpha }_{2}^{2}-4{\alpha }_{1}\gt 0$,
$\begin{eqnarray*}\begin{array}{rcl}\chi (\eta ) & = & \mathrm{ln}\left(-\frac{\sqrt{{\alpha }_{2}^{2}-4{\alpha }_{1}}}{2{\alpha }_{1}}\tanh \right.\\ & & \left.\times \left(\frac{\sqrt{{\alpha }_{2}^{2}-4{\alpha }_{1}}}{2}(\eta +d)\right)-\frac{{\alpha }_{2}}{2{\alpha }_{1}}\right).\end{array}\end{eqnarray*}$
F2: With α1 ≠ 0, ${\alpha }_{2}^{2}-4{\alpha }_{1}\lt 0$, $\begin{eqnarray*}\begin{array}{rcl}\chi (\eta ) & = & {\rm{ln}}\left(\frac{\sqrt{-{\alpha }_{2}^{2}+4{\alpha }_{1}}}{2{\alpha }_{1}}\right.\\ & & \left.\times \tanh \left(\frac{\sqrt{-{\alpha }_{2}^{2}+4{\alpha }_{1}}}{2}(\eta +d)\right)-\frac{{\alpha }_{2}}{2{\alpha }_{1}}\right).\end{array}\end{eqnarray*}$
F3: With α1 = 0, α2 ≠ 0, and ${\alpha }_{2}^{2}-4{\alpha }_{1}\gt 0$, $\begin{eqnarray*}\chi (\eta )=-\mathrm{ln}\left(\frac{{\alpha }_{2}}{{{\rm{e}}}^{{\alpha }_{2}(\eta +d)-1}}\right).\end{eqnarray*}$
F4: With α1 ≠ 0, α2 ≠ 0, and ${\alpha }_{2}^{2}-4{\alpha }_{1}=0$, $\begin{eqnarray*}\chi (\eta )=\mathrm{ln}\left(-\frac{2{\alpha }_{2}(\eta +d)+4}{{\alpha }_{2}^{2}(\eta +d)}\right).\end{eqnarray*}$
F5: With α1 = 0, α2 = 0, and ${\alpha }_{2}^{2}-4{\alpha }_{1}=0$, $\begin{eqnarray*}\chi (\eta )=\mathrm{ln}(\eta +d).\end{eqnarray*}$
3. Mathematical setup and stochastic solutions
This section delves into deriving stochastic solutions for equation (1 ) through the application of Brownian motion technique. Applying the following transformation 1 ), the imaginary part leads to s = a, while the real part produces: 7 ), we establish the relationship n = m + 2. Choosing m = 1 yields n = 3. With this specific information, equation (3 ) can be reformulated as: 8 ) into equation (7 ) and following the same procedure as in the previous section yields the result: 9 ):
$\begin{eqnarray}\phi (x,t)=U(\eta ){{\rm{e}}}^{{\rm{i}}(ax+bt+\tau {\beta }_{t})},\quad {where}\quad \eta =r(x-st),\end{eqnarray}$
to considered model ( $\begin{eqnarray}{r}^{2}{U}^{{\prime\prime} }-2\rho {U}^{3}-({a}^{2}+2(b+\varpi \delta ))U=0.\end{eqnarray}$
Through the balancing principle in equation ( $\begin{eqnarray}U(\eta )=\frac{{A}_{0}+{A}_{1}{{\rm{e}}}^{-\chi (\eta )}+{A}_{2}{{\rm{e}}}^{-2\chi (\eta )}+{A}_{3}{{\rm{e}}}^{-3\chi (\eta )}}{{B}_{0}+{B}_{1}{{\rm{e}}}^{-\chi (\eta )}}.\end{eqnarray}$
Plugging equation ( $\begin{eqnarray}\begin{array}{rcl}{A}_{0} & = & \frac{1}{2}\frac{{B}_{0}{\alpha }_{2}r}{\rho \sqrt{\frac{1}{\rho }}},\quad {A}_{1}=\frac{1}{2}\frac{r({B}_{1}{\alpha }_{2}+2{B}_{0})}{\rho \sqrt{\frac{1}{\rho }}},\\ {A}_{2} & = & {B}_{1}r\sqrt{\frac{1}{\rho }},\quad {A}_{3}=0,\\ b & = & ({\alpha }_{1}-\frac{1}{4}{\alpha }_{2}^{2}){r}^{2}-\varpi \delta -\frac{1}{2}{a}^{2}.\end{array}\end{eqnarray}$
Employing the family of solutions from F1-F5 characterized by the modified $\exp (-{\rm{\Psi }}(\xi ))$ expansion function method, we derive the following set of solutions with expressions (F1: For α1 ≠ 0, ${\alpha }_{2}^{2}-4{\alpha }_{1}\gt 0$, we get
$\begin{eqnarray}\begin{array}{rcl}\phi (x,t) & = & \frac{{M}_{1}}{{B}_{0}{M}_{1}+{B}_{1}}\left(\frac{{B}_{0}r{\alpha }_{2}}{2\rho \sqrt{\frac{1}{\rho }}}+\frac{r({B}_{1}{\alpha }_{2}+2{B}_{0})}{2\rho \sqrt{\frac{1}{\rho }}{M}_{1}}\right.\\ & & \left.+\frac{{B}_{1}r\sqrt{\frac{1}{\rho }}}{{M}_{1}^{2}}\right){{\rm{e}}}^{{\rm{i}}(ax+bt+\tau \beta (t))},\quad \rho \gt 0,\end{array}\end{eqnarray}$
where $\begin{eqnarray*}\begin{array}{rcl}{M}_{1} & = & -\frac{\sqrt{{\alpha }_{2}^{2}-4{\alpha }_{1}}{\rm{\tanh }}\left(\right.\frac{1}{2}\sqrt{{\alpha }_{2}^{2}-4{\alpha }_{1}}(\eta +k)\left)\right.+{\alpha }_{1}}{2{\alpha }_{1}},\\ a & = & ({\alpha }_{1}-\frac{1}{4}{\alpha }_{2}^{2}){r}^{2}-\varpi \delta -\frac{1}{2}{a}^{2},\quad \eta =r(x-at),\end{array}\end{eqnarray*}$
and k is constant of integration. The physical significance of Brownian process for solution (
Figure 1. Graphical representation of solutions ( |
F2: For α1 ≠ 0, ${\alpha }_{2}^{2}-4{\alpha }_{1}\lt 0$, we get
$\begin{eqnarray}\begin{array}{rcl}\phi (x,t) & = & \frac{{M}_{2}}{{B}_{0}{M}_{2}+{B}_{1}}\left(\frac{{B}_{0}r{\alpha }_{2}}{2\rho \sqrt{\frac{1}{\rho }}}+\frac{r({B}_{1}{\alpha }_{2}+2{B}_{0})}{2\rho \sqrt{\frac{1}{\rho }}{M}_{2}}\right.\\ & & \left.+\frac{{B}_{1}r\sqrt{\frac{1}{\rho }}}{{M}_{2}^{2}}\right){{\rm{e}}}^{{\rm{i}}(ax+bt+\tau \beta (t))},\quad \rho \gt 0,\end{array}\end{eqnarray}$
where $\begin{eqnarray*}{M}_{2}=\frac{\sqrt{-{\alpha }_{2}^{2}+4{\alpha }_{1}}\tanh \left(\right.\frac{1}{2}\sqrt{-{\alpha }_{2}^{2}+4{\alpha }_{1}}(\eta +k)\left)\right.-{\alpha }_{2}}{2{\alpha }_{1}}.\end{eqnarray*}$
The physical significance of Brownian process for solution (
Figure 2. Graphical representation of solutions ( |
F3: For α1 = 0, α2 ≠ 0 and ${\alpha }_{2}^{2}-4{\alpha }_{1}\gt 0$, we get
$\begin{eqnarray}\phi (x,t)=\frac{{\alpha }_{2}r({e}^{\eta +k}+1)}{2\rho \sqrt{\frac{1}{\rho }}({e}^{\eta +k}-1)}{{\rm{e}}}^{{\rm{i}}(ax+bt+\tau \beta (t))},\quad \rho \gt 0.\end{eqnarray}$
The physical significance of Brownian process for solution (
Figure 3. Graphical representation of solutions ( |
F4: For α1 ≠ 0, α2 ≠ 0 and ${\alpha }_{2}^{2}-4{\alpha }_{1}=0$, we get
$\begin{eqnarray}\phi (x,t)=\frac{{\alpha }_{2}r}{\rho \sqrt{\frac{1}{\rho }}({\alpha }_{2}(k+\eta )+2)}{{\rm{e}}}^{{\rm{i}}(ax+bt+\tau \beta (t))},\quad \rho \gt 0.\end{eqnarray}$
The physical significance of Brownian process for solution (
Figure 4. Graphical representation of solutions ( |
F5: For α1 = 0, α2 = 0 and ${\alpha }_{2}^{2}-4{\alpha }_{1}=0$, we get
$\begin{eqnarray}\phi (x,t)=\frac{2r}{2\rho \sqrt{\frac{1}{\rho }}(k+\eta )}{{\rm{e}}}^{{\rm{i}}(ax+bt+\tau \beta (t))},\quad \rho \gt 0.\end{eqnarray}$
The physical significance of Brownian process for solution (
Figure 5. Graphical representation of solutions ( |
4. Bifurcation analysis
Bifurcations of dynamical system determines that either candidate system depends on underlying physical parameters of the system [34]. Through Galilean transformation equation (7 ) converts into: 15 ) has the following
$\begin{eqnarray}\left\{\begin{array}{l}\frac{{\rm{d}}U}{{\rm{d}}\xi }=V,\quad \\ \frac{{\rm{d}}V}{{\rm{d}}\xi }={k}_{1}U+{k}_{2}{U}^{3},\quad \end{array}\right.\end{eqnarray}$
where ${k}_{1}=\frac{{a}^{2}+2(b+\varpi \delta )}{{r}^{2}}$ and ${k}_{2}=\frac{2\rho }{{r}^{2}}$. System ( $\begin{eqnarray}{{ \mathcal R }}_{1}=(0,0),\quad {{ \mathcal R }}_{2}=\left(\sqrt{-\frac{{k}_{1}}{{k}_{2}}},0\right),\quad {{ \mathcal R }}_{3}=\left(-\sqrt{-\frac{{k}_{1}}{{k}_{2}}},0\right).\end{eqnarray}$
having fixed points with Jacobian: $\begin{eqnarray}{ \mathcal J }(U,V)=\left|\begin{array}{cc}0 & 1\\ {k}_{1}+3{k}_{2}{U}^{2} & 0\end{array}\right|=-({k}_{1}+3{k}_{2}{U}^{2}).\end{eqnarray}$
As we know, if ${ \mathcal J }(U,V)\lt 0$, then ${ \mathcal J }(U,V)$ represents a saddle point; if ${ \mathcal J }(U,V)\gt 0$, it indicates a center point; and if ${ \mathcal J }(U,V)=0$, it denotes a cuspidal point. The stability of unperturbed system (
Figure 6. Phase portraits of bifurcation for system ( |
5. Chaotic analysis
In this study, the chaotic response of the candidate system is induced by introducing a perturbation term. Accordingly, by adding the perturbation to the dynamical system (15 ), we obtain:
$\begin{eqnarray}\left\{\begin{array}{l}\frac{{\rm{d}}U}{{\rm{d}}\xi }=q,\quad \\ \frac{{\rm{d}}V}{{\rm{d}}\xi }={k}_{1}U+{k}_{2}{U}^{3}+{g}_{0}\cos (W),\quad \\ \frac{{\rm{d}}W}{{\rm{d}}\xi }=\varphi ,\quad \end{array}\right.\end{eqnarray}$
where W = φξ and g0, φ represent the strength and frequency of the external force. This system captures the underlying dynamics and acts as a strong candidate for exploring chaotic behavior. Numerous tools are available for predicting chaos in autonomous systems. In this study, we employ several of these tools, which are detailed as [35]:5.1. Numerical simulation
This subsection provides an in-depth analysis of numerical simulations employing the previously mentioned chaos detection tools. It identifies regions of chaos and offers greater insights into the system’s behavior and its connection to quantum chaos.
6. Multistability analysis
Multistability analysis investigates the existence of several stable states within a dynamical system when subjected to perturbations, whereby the system may attain distinct equilibrium points contingent upon initial conditions or external influences. This analysis explores how variations in parameters or disturbances affect stability and instigate transitions among these states. This inquiry is essential for comprehending intricate behaviors such as the cohabitation of multiple solutions or sudden alterations in system dynamics. The multistability analysis of the system (18 ) is simulated in figure 13.
7. Sensitivity analysis
This section is used to examine the sensitivity analysis of the following Hamiltonian system: 19 ) under varying initial conditions is illustrated in figure 14.
$\begin{eqnarray}\left\{\begin{array}{l}\frac{{\rm{d}}U}{{\rm{d}}\xi }=V,\quad \\ \frac{{\rm{d}}V}{{\rm{d}}\xi }={k}_{1}U+{k}_{2}{U}^{3}.\quad \end{array}\right.\end{eqnarray}$
The sensitivity analysis of the system (8. Results and discussion
This section is vital for interpreting the obtained results and explaining their physical significance in nonlinear optics and mathematical physics. It provides a deeper understanding of how the findings advance these fields, particularly in terms of the system dynamics and their broader implications. The pioneer contribution of this study is the investigation of optical stochastic solutions using the Brownian motion process and dynamical systems analysis for the complex cubic NLSE. The use of optical stochastic solutions through the Brownian motion process is important as it captures the effects of random fluctuations in nonlinear optical systems. These solutions provide a more accurate representation of system dynamics under noise, which is crucial for understanding real-world phenomena in optics and related disciplines. By using the modified $\exp (-{\rm{\Psi }}(\xi ))$ expansion method, we derive optical solitary wave solutions influenced by noise at varying intensity levels of Γ, as shown in figures 1-5. It is important to remark that intensity of noise Γ cannot be negative. It typically represents the strength or magnitude of randomness or fluctuations in a system, and a negative value would not make physical sense. In physical systems, negative noise intensity would imply the reversal or cancellation of fluctuations, which contradicts the concept of noise as a random perturbation. Therefore, Γ ≥ 0 is required for consistency with the physical meaning of noise intensity.
Another key aspect of this work is the dynamical analysis, incorporating bifurcation and chaos theory, multistability and sensitivity analysis. In the bifurcation analysis, we examine the stability of the unperturbed system near the fixed points for k1k2 < 0, as illustrated in figure 6. For the chaotic analysis, we introduce the perturbed dynamical system and identify the chaotic behavior in the solution of system using various chaos detecting tools, as shown in figures 7-12. The multistability analysis in this work, a key characteristic of chaos, reveals multiple stable states within the perturbed dynamical system that contribute to chaotic behavior, as shown in figure 13. This indicates that even small changes in initial conditions can cause significant changes in the system’s behavior. Additionally, the sensitivity analysis of the studied equation is analyzed at different initial conditions, as shown in figure 14. Meanwhile, the approaches used offer new insights into the interaction between stochastic effects and the nonlinear qualitative analysis of the equation, making valuable contributions to both nonlinear optics and mathematical physics.
Figure 7. Phase portraits of the system ( |
Figure 8. Poincaré map of the system ( |
Figure 9. Time series analysis of the system ( |
Figure 10. Bifurcation diagram of the system ( |
Figure 11. LEs of the system ( |
Figure 12. Power spectrum of the system ( |
Figure 13. Multistability analysis of the system ( |
Figure 14. Sensitivity analysis of system ( |
9. Conclusion
The primary aim of this study was to investigate the unresolved nonlinear dynamics and optical stochastic solutions of the complex cubic NLSE with δ-potential, using a Brownian process. The research commenced by employing the modified $\exp (-{\rm{\Psi }}(\xi ))$ expansion function method to derive optical stochastic solutions, incorporating the influence of noise. To show the physical noise effect on the stochastic solutions, we display these solutions in 3D and 2D plots for various noise strengths Γ = 0, 0.5, 1, as shown in figures 1-5. Additionally, we examined the dynamics of the candidate dynamical system with and without the perturbation term, employing bifurcation and chaos theory. For bifurcation analysis, we present phase portraits of bifurcation depending on the parameter values k1 and k2, as shown in figure 6. However, to show the presence of chaos in perturbed dynamical system, a couple of chaos detecting tools were used to predict chaotic behavior in the candidate system, as shown in figures 7-12. Through multistability analysis, we revealed multiple stable states within the perturbed dynamical system that contribute to chaos, as shown in figure 13. In the end, we conducted a sensitivity analysis of the Hamiltonian system under varying initial conditions, as shown in figure 14. These results provide deeper insights into noise, stability and complexity in nonlinear optical systems. The novelty of this work lay in uncovering new results that had not been previously explored for the given equation. These findings offered valuable insights into the behavior of the complex cubic NLSE with δ-potential and its applications in nonlinear optics, quantum mechanics and Bose-Einstein condensates.
Future directions include extending the analysis to higher-dimensional NLSE models, investigating the impact of non-Gaussian noise, and exploring how noise influences soliton collisions and interactions. This involves studying how Brownian motion affects soliton stability, energy transfer and scattering in nonlinear optical systems.