Stochastic Soliton Research Articles

Stochastic soliton and rogue wave solutions of the nonlinear Schrödinger equation with white Gaussian noise

In this letter, we propose a novel integrable nonlinear Schrödinger equation and its Lax pair, influenced by Brownian motion and white Gaussian noise. We aim to construct and solve new integrable systems affected by the white Gaussian noise. Utilising the classical and generalised Darboux transformations, the stochastic soliton solutions and the stochastic rogue wave solutions of this novel integrable nonlinear Schrödinger equation are obtained and expressed in determinant form. Studies of stochastic soliton and rogue wave solutions of the NLS equation are essential for complex physical and mathematical phenomena where nonlinear interactions and randomness play crucial roles.

Characterizing stochastic solitons behavior in (3+1)-dimensional Schrödinger equation with Cubic–Quintic nonlinearity using improved modified extended tanh-function scheme

An extended version of (3+1)-dimensional non-linear Schrödinger equation that has a cubic–quintic nonlinear component under the stochastic effects is examined in this investigation. Several stochastic exact solutions of this model is acquired through the application of the improved modified extended tanh-function scheme (IMETFS). This method offers a practical and effective approach to finding precise solutions to several kinds of nonlinear partial differential equations. In addition, these solutions include stochastic soliton solutions (bright, singular, combo dark-singular), and exact solution such as singular periodic, Jacobi elliptic function, Weierstrass elliptic doubly periodic solution, rational, and exponential functions. Since it is the first study of its sort to examine multiplicative white noise’s impacts in this particular setting, it offers fresh insights and innovative research approaches for the field’s future studies. The work adds much to our understanding of soliton theory and how it relates to optical fiber technology while illuminating hitherto unknown facets of multiplicative white noise. To illustrate the impact of the noise, a few recovered solutions with varying noise strengths are given graphically as examples.

Coherence-controlled chaotic soliton bunch

Controlling the coherence of chaotic soliton bunch holds the promise to explore novel light-matter interactions and manipulate dynamic events such as rogue waves. However, the coherence control of chaotic soliton bunch remains challenging, as there is a lack of dynamic equilibrium mechanism for stochastic soliton interactions. Here, we develop a strategy to effectively control the coherence of chaotic soliton bunch in a laser. We show that by introducing a lumped fourth-order-dispersion (FOD), the soliton oscillating tails can be formed and generate the potential barriers among the chaotic solitons. The repulsive force between neighboring solitons enabled by the potential barriers gives rise to an alleviation of the soliton fusion/annihilation from stochastic interactions, endowing the capability to control the coherence in chaotic soliton bunch. We envision that this result provides a promising test-bed for a variety of dynamical complexity science and brings new insights into the nonlinear behavior of chaotic laser sources.

Exploring advanced non-linear effects on highly dispersive optical solitons with multiplicative white noise

This research investigates the dynamics of highly stochastic optical solitons governed by an eighth-order nonlinear Schrödinger equation. The study considers spatio-temporal dispersion effects, higher-order nonlinearity, and multiplicative white noise in the Itô sense. Two robust methods, the singular manifold method and the new generalized exp(-ϕ(ζ)) expansion method, are employed to derive novel closed-form optical soliton solutions. Our exploration of stochastic soliton behavior using Itô calculus sheds light on the influence of multiplicative white noise on the model. Notably, the phase component of the solitons incorporates the white noise, leading to a spectrum of soliton solutions including singular, periodic, singular periodic, combined bright-dark solitons, and various solitary waves. The research provides physical interpretations and visual representations of these solutions through 3D and 2D graphs, using reliable parameter values. Our approach stands out due to the novelty of the problem and the application of untested methods in this context, resulting in numerous new and original optical soliton solutions. These outcomes demonstrate the efficacy of our approach in addressing nonlinear challenges in engineering and the natural sciences, surpassing previous efforts in the literature.

The mKdV equation under the Gaussian white noise and Wiener process: Darboux transformation and stochastic soliton solutions

In this paper, we propose a novel integrable system named the stochastic mKdV equation, along with its corresponding Lax pair. We aim to extend the methodology of deterministic integrable systems to construct and solve stochastic integrable systems. The Darboux transformation effectively obtains analytic solutions for the integrable stochastic mKdV equation. Using the Darboux transformation, soliton solutions incorporating stochastic terms are obtained as Wronskian determinants. Furthermore, we conduct an in-depth analysis of the dynamics exhibited by the stochastic one-soliton and the two-soliton solutions.

Long-timescale soliton dynamics in the Korteweg–de Vries equation with multiplicative translation-invariant noise

This paper studies the behavior of solitons in the Korteweg–de Vries equation under the influence of multiplicative noise. We introduce stochastic processes that track the amplitude and position of solitons based on a rescaled frame formulation and stability properties of the soliton family. We furthermore construct tractable approximations to the stochastic soliton amplitude and position which reveal their leading-order drift. We find that the statistical properties predicted by our method agree well with numerical evidence.

Noise effect on soliton phenomena in fractional stochastic Kraenkel-Manna-Merle system arising in ferromagnetic materials

This work dives into the Conformable Stochastic Kraenkel-Manna-Merle System (CSKMMS), an important mathematical model for exploring phenomena in ferromagnetic materials. A wide spectrum of stochastic soliton solutions that include hyperbolic, trigonometric and rational functions, is generated using a modified version of Extended Direct Algebraic Method (EDAM) namely r+mEDAM. These stochastic soliton solutions have practical relevance for describing magnetic field behaviour in zero-conductivity ferromagnets. By using Maple to generate 2D and 3D graphical representations, the study analyses how stochastic terms and noise impact these soliton solutions. Finally, this study adds to our knowledge of magnetic field behaviour in ferromagnetic materials by shedding light on the effect of noise on soliton processes inside the CSKMMS.

Optical solitons of stochastic perturbed Radhakrishnan–Kundu–Lakshmanan model with Kerr law of self-phase-modulation

This study aims to obtain the optical soliton solutions of the stochastic perturbed Radhakrishnan–Kundu–Lakshmanan equation which models the optical solitons in optical waveguides such as silica fibers with Kerr law in the presence of chromatic and third-order dispersions by multiplicative white noise in Itô calculus. The study constitutes one of the works aimed at adapting the model in accordance with the importance of the noise effect in nonlinear optics. In the first step, the nonlinear ordinary differential form of the investigated problem has been obtained by using an appropriate complex wave transformation. The enhanced Kudryashov technique, which is the combination of Kudryashov and the new Kudryashov methods, is applied to the nonlinear ordinary differential form. As the result, bright, dark and singular stochastic soliton solutions of the analyzed problem have been carried out by assigning appropriate parameter values and the impact of noise effect on the soliton behavior has been investigated. Lastly, the graphical representations and required comments have been presented in the related sections.

Unraveling the Dynamics of Singular Stochastic Solitons in Stochastic Fractional Kuramoto–Sivashinsky Equation

This work investigates the complex dynamics of the stochastic fractional Kuramoto–Sivashinsky equation (SFKSE) with conformable fractional derivatives. The research begins with the creation of singular stochastic soliton solutions utilizing the modified extended direct algebraic method (mEDAM). Comprehensive contour, 3D, and 2D visual representations clearly depict the categorization of these stochastic soliton solutions as kink waves or shock waves, offering a clear description of these soliton behaviors within the context of the SFKSE framework. The paper also illustrates the flexibility of the transformation-based approach mEDAM for investigating soliton occurrence not only in SFKSE but also in a wide range of nonlinear fractional partial differential equations (FPDEs). Furthermore, the analysis considers the effect of noise, specifically Brownian motion, on soliton solutions and wave dynamics, revealing the significant influence of randomness on the propagation, generation, and stability of soliton in complex stochastic systems and advancing our understanding of extreme behaviors in scientific and engineering domains.

Optical soliton solutions of the stochastic perturbed Radhakrishnan-Kundu-Lakshmanan equation via Itô Calculus

This study presents, for the first time, optical solitons of the stochastic perturbed Radhakrishnan-Kundu-Lakshmanan equation with Kerr law nonlinearity in the presence of chromatic and spatio-temporal dispersions. The stochastic form involves multiplicative white noise in the Ito sense, besides; the Kudryashov and the new Kudryashov methods are picked to analyze. The analysis of the stochastic soliton solutions of the Radhakrishnan-Kundu-Lakshmanan equation and the impact of noise on these solitons are the primary motivations for choosing both of these techniques rather than obtaining many solitons. Therefore, the first goal is to obtain the most basic soliton types, bright and dark solitons, and the second goal is to observe the white noise effect on these solitons. By applying the proposed methods, bright and dark solitons are obtained, and the noise effect on these solitons is illustrated using both 3D and 2D graphic presentations, along with necessary comments. The presentation of the examined model for the first time in this article reflects its originality in terms of contributing both the study and the obtained results to the literature.

Stochastic soliton solutions of conformable nonlinear stochastic systems processed with multiplicative noise

Stochastic equations are powerful mathematical tools used to study systems having random effects. The current manuscript studies the two distinct systems: the stochastic conformable Zakharov system (SCZs) and stochastic conformable Hirota-Macari system (SCHMs) with multiplicative noises. Both systems are investigated using the conformable derivative in terms of stratonovich formulation. These systems are chosen for study due to their relevance and significance in understanding the behavior of complex physical phenomena influenced by stochastic effect. The novelty lies in the applications of the unified solver method, which is a succinct, direct and efficient technique for obtaining solutions in rational, trigonometric and hyperbolic forms. To demonstrate the effect of multiplicative noise and conformable derivative on the solutions of SCZs and SCHMs, 3D and 2D graphs are illustrated by using Mathematica 11. Our findings highlight the sustaining role of the multiplicative noise, which stabilizes the solutions of systems around zero. Overall, this study deepens the understanding of the interaction between stochastic effects and conformable derivatives in the context of these complex systems.

Solitons and other nonlinear waves for stochastic Schrödinger‐Hirota model using improved modified extended tanh‐function approach

The improved modified extended tanh‐function approach was used to study optical stochastic soliton solutions and other exact stochastic solutions for the nonlinear Schrödinger‐Hirota equation with multiplicative white noise. The derived solutions include stochastic bright solitons, stochastic singular solitons, stochastic periodic solutions, stochastic singular periodic solutions, stochastic exponential solutions, stochastic rational solutions, and stochastic Jacobi elliptic doubly periodic solutions. Constraints on the parameters were taken into account to ensure the existence of the obtained stochastic soliton solutions. Additionally, selected solutions were presented graphically to illustrate the physical characteristics of the stochastic solutions. In this paper, we used Mathematica (11.3) packages to find the coefficients and Matlab (R2015a) packages to plot the graphs.

Soliton Shielding of the Focusing Nonlinear Schrödinger Equation.

We first consider a deterministic gas of N solitons for the focusing nonlinear Schrödinger (FNLS) equation in the limit N→∞ with a point spectrum chosen to interpolate a given spectral soliton density over a bounded domain of the complex spectral plane. We show that when the domain is a disk and the soliton density is an analytic function, then the corresponding deterministic soliton gas surprisingly yields the one-soliton solution with the point spectrum the center of the disk. We call this effect soliton shielding. We show that this behavior is robust and survives also for a stochastic soliton gas: indeed, when the N-soliton spectrum is chosen as random variables either uniformly distributed on the circle, or chosen according to the statistics of the eigenvalues of the Ginibre random matrix the phenomenon of soliton shielding persists in the limit N→∞. When the domain is an ellipse, the soliton shielding reduces the spectral data to the soliton density concentrating between the foci of the ellipse. The physical solution is asymptotically steplike oscillatory, namely, the initial profile is a periodic elliptic function in the negative x direction while it vanishes exponentially fast in the opposite direction.

Program-Controlled Single Soliton Generation Driven by the Thermal-Compensated Avoided Mode Crossing

Single soliton microcombs enable the chip-scaled comb source with an extremely high repetition rate suitable for many applications. However, it is challenging to deterministically generate a single soliton due to the thermal effect and the stochastic soliton formation. Meanwhile, mode coupling induced avoided mode crossings (AMXs) also affect the soliton generation. Here, we analyze the thermal compensation effect of an auxiliary laser on the single soliton generation based on a Si <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sub> N <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</sub> micro-resonator with weak AMXs. Both simulation and experiments demonstrate that only in a proper blue-detuned range of the auxiliary laser can achieve the high-probable single soliton generation. Based on the above observation, we present a program-controlled scheme for reliable single soliton generation based on the red-detuned pump entrance forward tuning method. A single soliton is successfully generated in 100 trials with a 100% success rate. We also stabilize the single soliton generation by pump back tuning and feedback control of the auxiliary wavelength and achieve more than 5 hours of the soliton duration time in the experiments.

Investigation of nonlinear problems governed by stochastic phi-4 type equations in nuclear and particle physics

In this article, the stochastic Phi-4 model is under consideration analytically. The stochastic Phi-4 equation has been crucial to nuclear and particle physics. The randomness of the nuclear particle at the micro level is represented by the stochastic phenomena which are shown by the white noise. The stochastic version of the Phi-4 model is investigated for the first time analytically to gain the soliton solutions by using the two techniques such as Sardar sub-equation and modified exponential rational function. The stochastic soliton solutions are successfully constructed in the form of exponential, trigonometric, and hyperbolic forms. Additionally, the stability analysis of this model is observed. Lastly, the stochastic behavior of some solutions is shown in the form of 2D, and 3D by choosing the different values of parameters.

Construction of special soliton solutions to the stochastic Riccati equation

Abstract A scheme for the analytical stochastization of ordinary differential equations (ODEs) is presented in this article. Using Itô calculus, an ODE is transformed into a stochastic differential equation (SDE) in such a way that the analytical solutions of the obtained equation can be constructed. Furthermore, the constructed stochastic trajectories remain bounded in the same interval as the deterministic solutions. The proposed approach is in a stark contrast to methods based on the randomization of solution trajectories and is not focused on the analysis of martingales. This article extends the theory of Itô calculus by directly implementing it into analytical schemes for the solution of differential equations based on the generalized operator of differentiation. The efficacy of the presented analytical stochastization techniques is demonstrated by deriving stochastic soliton solutions to the Riccati differential equation. The presented semi-analytical stochastization scheme is relevant for the investigation of the global dynamics of different biological and biomedical processes where the variation interval of the stochastic solution is predetermined by the rationale of the model.

Synthesized soliton crystals

Dissipative Kerr soliton (DKS) featuring broadband coherent frequency comb with compact size and low power consumption, provides an unparalleled tool for nonlinear physics investigation and precise measurement applications. However, the complex nonlinear dynamics generally leads to stochastic soliton formation process and makes it highly challenging to manipulate soliton number and temporal distribution in the microcavity. Here, synthesized and reconfigurable soliton crystals (SCs) are demonstrated by constructing a periodic intra-cavity potential field, which allows deterministic SCs synthesis with soliton numbers from 1 to 32 in a monolithic integrated microcavity. The ordered temporal distribution coherently enhanced the soliton crystal comb lines power up to 3 orders of magnitude in comparison to the single-soliton state. The interaction between the traveling potential field and the soliton crystals creates periodic forces on soliton and results in forced soliton oscillation. Our work paves the way to effectively manipulate cavity solitons. The demonstrated synthesized SCs offer reconfigurable temporal and spectral profiles, which provide compelling advantages for practical applications such as photonic radar, satellite communication and radio-frequency filter.

Analytical manner for abundant stochastic wave solutions of extended KdV equation with conformable differential operators

This work investigates the Wick‐type stochastic and extended Korteweg‐de Vries (EKdV) equation under conformable differential operators. Novel wave solutions with soliton and periodic types are produced to the deterministic EKdV under conformable differential operators. Abundant stochastic wave solutions are explored for the Wick‐type stochastic EKdV equation by using conformable differential operators and the inverse Hermite transform. The stochastic soliton and periodic solutions are denoted as functional solutions with Brownian motion environment. For the acquired solutions, comparisons and graphical representations with three‐dimensional graphs are shown for peculiar values of the existing parameters. According to the existing literature, utilization of the conformable differential operators is a novel contribution for solving the stochastic EKDV and other stochastic nonlinear problems.

Exact solutions of stochastic fractional Korteweg de–Vries equation with conformable derivatives

We deal with the Wick-type stochastic fractional Korteweg de–Vries (KdV) equation with conformable derivatives. With the aid of the Exp-function method, white noise theory, and Hermite transform, we produce a novel set of exact soliton and periodic wave solutions to the fractional KdV equation with conformable derivatives. With the help of inverse Hermite transform, we get stochastic soliton and periodic wave solutions of the Wick-type stochastic fractional KdV equation with conformable derivatives. Eventually, by an application example, we show how the stochastic solutions can be given as Brownian motion functional solutions.

Variational principles for stochastic soliton dynamics.

We develop a variational method of deriving stochastic partial differential equations whose solutions follow the flow of a stochastic vector field. As an example in one spatial dimension, we numerically simulate singular solutions (peakons) of the stochastically perturbed Camassa-Holm (CH) equation derived using this method. These numerical simulations show that peakon soliton solutions of the stochastically perturbed CH equation persist and provide an interesting laboratory for investigating the sensitivity and accuracy of adding stochasticity to finite dimensional solutions of stochastic partial differential equations. In particular, some choices of stochastic perturbations of the peakon dynamics by Wiener noise (canonical Hamiltonian stochastic deformations, CH-SD) allow peakons to interpenetrate and exchange order on the real line in overtaking collisions, although this behaviour does not occur for other choices of stochastic perturbations which preserve the Euler-Poincaré structure of the CH equation (parametric stochastic deformations, P-SD), and it also does not occur for peakon solutions of the unperturbed deterministic CH equation. The discussion raises issues about the science of stochastic deformations of finite-dimensional approximations of evolutionary partial differential equation and the sensitivity of the resulting solutions to the choices made in stochastic modelling.