Soliton Research Articles

The Lie symmetry analysis, optimal system and exact solutions of the (2+1)-dimensional variable coefficients integrable coupled Burgers equations

The variable coefficients integrable coupled Burgers equations (vcICBs) are researched by the Lie symmetry analysis method. The infinitesimal generators, commutative relations and adjoint relations are given. Optimal system of vcICBs are studied by Olver’s method. Based on the optimal system, the (2+1)-dimensional vcICB equations are reduced to (1+1)-dimensional equations. Some new exact solutions of vcICB equations are derived. Some kink solutions, breathe wave solutions and soliton solutions are got by utilising -expansion method, Riccati equation method, Tanh-expansion method and Painlevé analysis method. By altering different coefficient functions, many different types of exact solutions can be derived. The effect of the coefficient functions on the solution is analysed. By studying the evolution of the solution with time t, the dynamic behaviors of the solutions are analysed.

Dynamical self-trapping of two-dimensional binary solitons in cross-combined linear and nonlinear optical lattices.

Dynamical and self-trapping properties of two-dimensional (2D) binary mixtures of Bose-Einstein condensates in cross-combined lattices, consisting of a one-dimensional (1D) linear optical lattice (LOL) in the x direction for the first component and a 1D nonlinear optical lattice (NOL) in the y direction for the second component, are analytically and numerically investigated. The existence and stability of 2D binary matter wave solitons in these settings are demonstrated both by variational analysis and by direct numerical integration of the coupled Gross-Pitaevskii equations. We find that in the absence of the NOL, binary solitons, stabilized by the action of the 1D LOL and by the attractive intercomponent interaction, can freely move in the y direction. In the presence of the NOL, we find, quite remarkably, the existence of threshold curves in the parameter space separating regions where solitons can move from regions where the solitons become dynamically self-trapped. The mechanism underlying the dynamical self-trapping phenomenon (DSTP) is qualitatively understood in terms of a dynamical barrier induced by the NOL, similar to the Peirls-Nabarro barrier of solitons in discrete lattices. DSTP is numerically demonstrated for binary solitons that are put in motion both by phase imprinting and by the action of external potentials applied in the y direction. In the latter case, we show that the trapping action of the NOL allows one to maintain a 2D binary soliton at rest in a nonequilibrium position of a parabolic trap or to prevent it from falling under the action of gravity. Possible applications of the results are also briefly discussed.

Construction of thin-shell around new wormhole solutions via solitonic quantum wave dark matter

In the present work, we study wormhole geometry in the framework of general relativity and explore the effects of quantum wave dark matter on the dynamical configuration of the shell around the wormhole. Aiming this aspect some specific shape functions have been checked for physically interesting results. Firstly, we develop the basic formalism and the respective field equations. Thereafter, we introduce quantum wave dark matter as an input for getting reasonable wormhole solutions where the physical interpretation of energy conditions within the average pressure scenario has been considered. The positive behavior of quantum wave dark matter density confirms the compatibility for four different parametric values of the model. We also use the specific values of boson mass parameter within the observational data of soliton core density, and soliton core radius for the SgrA∗ candidate. After this we have started to discuss the dynamics of the shell around the wormhole structure and observe the effects of quantum wave dark matter on the stability of the shell by using the equation of motion through radial linear perturbation. The outcomes of the investigation seem to provide several attractive attributes as well as futuristic possibilities to traditional wormhole studies.

Soliton approximation in continuum models of leader-follower behavior.

Complex biological processes involve collective behavior of entities (bacteria, cells, animals) over many length and time scales and can be described by discrete models that track individuals or by continuum models involving densities and fields. We consider hybrid stochastic agent-based models of branching morphogenesis and angiogenesis (new blood vessel creation from preexisting vasculature), which treat cells as individuals that are guided by underlying continuous chemical and/or mechanical fields. In these descriptions, leader (tip) cells emerge from existing branches and follower (stalk) cells build the new sprout in their wake. Vessel branching and fusion (anastomosis) occur as a result of tip and stalk cell dynamics. Coarse graining these hybrid models in appropriate limits produces continuum partial differential equations(PDEs) for endothelial cell densities that are more analytically tractable. While these models differ in nonlinearity, they produce similar equationsat leading order when chemotaxis is dominant. We analyze this leading order system in a simple quasi-one-dimensional geometry and show that the numerical solution of the leading order PDE is well described by a soliton wave that evolves from vessel to source. This wave is an attractor for intermediate times until it arrives at the hypoxic region releasing the growth factor. The mathematical techniques used here thus identify common features of discrete and continuum approaches and provide insight into general biological mechanisms governing their collective dynamics.

MetaPINNs: Predicting soliton and rogue wave of nonlinear PDEs via the improved physics-informed neural networks based on meta-learned optimization

Efficiently solving partial differential equations (PDEs) is a long-standing challenge in mathematics and physics research. In recent years, the rapid development of artificial intelligence technology has brought deep learning-based methods to the forefront of research on numerical methods for partial differential equations. Among them, physics-informed neural networks (PINNs) are a new class of deep learning methods that show great potential in solving PDEs and predicting complex physical phenomena. In the field of nonlinear science, solitary waves and rogue waves have been important research topics. In this paper, we propose an improved PINN that enhances the physical constraints of the neural network model by adding gradient information constraints. In addition, we employ meta-learning optimization to speed up the training process. We apply the improved PINNs to the numerical simulation and prediction of solitary and rogue waves. We evaluate the accuracy of the prediction results by error analysis. The experimental results show that the improved PINNs can make more accurate predictions in less time than that of the original PINNs.

Interaction and manipulation for non-autonomous bright soliton solution of the coupled derivative nonlinear Schrödinger equations with Riemann–Hilbert method

Some bright wave solutions of the coupled derivative nonlinear Schrödinger(CDNLS) equations are derived with Riemann–Hilbert(RH) method. We observe some novel non-autonomous bright soliton structures by choosing the different nonlinear perturbation function g(t). It is shown that there are some possibilities to manipulate the interactions of non-autonomous wave solution through choosing nonlinear and gain/loss functions. Some dynamic behaviors of interactions between two bright solitons have been asymptotically analyzed, the two parabolic-type bright solitons propagate with the opposite directions after the interaction. And some interactions of the linear-, parabolic- and periodic-type bright two-solitons are elastic. A general approach is presented to manipulate soliton waves, which has the potential application for the optical self-routing, non-Kerr media and Bose–Einstein condensates(BEC).

Abundant dynamical solitary waves through Kelvin-Voigt fluid via the truncated M-fractional Oskolkov model

This effort explores the dynamic behavior of solitary wave fronts using the Oskolkov model in a truncated M-fractional derivative form. This effort has been performed through the application of modified extended tanh and a novel form of modified Kudryashov techniques. All obtained solutions manifest in the form of sinusoidal, hyperbolic, and constant basis function solutions within the fractional model. Within this framework, the applied techniques yield various solitary wave structures, including bright and dark bell solitons, kinks, anti-kinks, linked lumps, and collisions of kinks. Additionally, they produce singular kink solitons and lump waves. The effects of fractionality and the coefficient of the highest-order derivative are depicted in Figs. 1 through 11. Moreover, all figures are organized to illustrate the properties of the innovative soliton wave propagations. The studies have also revealed that the fractional Oskolkov model can support both fundamental and higher-order solitons, each characterized by distinct properties and highly efficient behaviors. Various dynamical solitary waves obtained here can be described transmission waves through Kelvin-Voigt fluid with real happenings as it represented by fractional form.

The Fractional Soliton Wave Propagation of Non-Linear Volatility and Option Pricing Systems with a Sensitive Demonstration

In this study, we explore a fractional non-linear coupled option pricing and volatility system. The model under consideration can be viewed as a fractional non-linear coupled wave alternative to the Black–Scholes option pricing governing system, introducing a leveraging effect where stock volatility corresponds to stock returns. Employing the inverse scattering transformation, we find that the Cauchy problem for this model is insolvable. Consequently, we utilize the Φ6-expansion algorithm to generate generalized novel solitonic analytical wave structures within the system. We present graphical representations in contour, 3D, and 2D formats to illustrate how the system’s behavior responds to the propagation of pulses, enabling us to predict suitable parameter values that align with the data. Finally, a conclusion is given.

Lump wave solutions, lump-stripe soliton inelastic collision phenomena and rogue-type wave solutions for a generalized breaking soliton system in (3+1)-dimensions

Based on five types of structures of the functional multi-layer neural network model, some explicit exact solutions for a (3+1)-dimensional generalized breaking soliton system including lump waves, lump-stripe solitons and rogue-type waves are explored by using bilinear neural network method without the datasets training. The 3-dimensional plots, 2-dimensional curves and density plots of these results obtained are provided to display their dynamic characteristics. In particular, the complete processes of lump wave and stripe soliton fusion phenomena are presented. We hope the formulations established in this paper could be applied to study the exact traveling wave solutions of other integrable nonlinear evolution equations.

Bifurcations, chaotic dynamics, sensitivity analysis and some novel optical solitons of the perturbed non-linear Schrödinger equation with Kerr law non-linearity

This study introduces an analysis of bifurcations, chaotic dynamics, and sensitivity using the Galilean transformation applied to the perturbed non-linear Schrödinger equation (NLSE) with Kerr-law nonlinearity. Additionally, we employ the efficient Sardar sub-equation (Sse) method to derive various optical soliton solutions. Initially, we outline the general framework of the Sse method proposed. Next, we utilize a traveling wave transform on the NLSE, transforming it into a system of nonlinear ordinary differential equations (NLODEs) which are further separated into their real and imaginary components. Furthermore, we apply this method to derive new optical solitons for the NLSE equation with Kerr law. We perform exact analytical simulations of these optical soliton solutions and investigate the effects of different parameters on the soliton waves.

The modulations of higher order solitonic pressure and energy of fluid filled elastic tubes

In this article, we investigate the weak higher order nonlinear solitonic pressure waves in elastic, incompressible, nonviscous fluid-filled tubes. The higher order Korteweg–de Vries equation has been developed from the perturbed nonlinear equation. The dependence of higher-order solitonic pressure, higher order fluid velocity, and higher order solitonic energy on the nonlinear coefficient and the perturbation parameter has been established. The presented solutions will be more beneficial in the several disciplines of blood flow and many other related fields of applied science.

Asymptotic Theory of Solitons Generated from an Intense Wave Pulse

Asymptotic Theory of Solitons Generated from an Intense Wave Pulse

The new wave structures to the perturbed NLSE via Wiener process with its wide-ranging applications

This article extracts stochastic soliton waves for the perturbed nonlinear Schödinger’s equation (PNLSE) forced by multiplicative noise through the Itô sense by utilizing two unified solver methods. The presented solutions involve three types: rational function, trigonometric function, and hyperbolic function solutions. These stochastic solutions are critical for studying numerous complicated phenomena in heat transfer, new physics, and many other fields of applied science. We demonstrate the effect of multiplicative noise on the solution of the stochastic PNLSE, which have never been studied before. The study and acquired solutions clarify that the unified solver technique is sturdy and efficient. Finally, several 2D and 3D graphs for selected solutions are shown.

Dynamical forms of various optical soliton solutions and other solitons for the new Schrödinger equation in optical fibers using two distinct efficient approaches

In this work, the new nonlinear [Formula: see text]-dimensional Schrödinger (NLS) equation, which describes wave propagation in optical fibers, has been successfully studied using the [Formula: see text]-expansion method and the Unified method (UM). These techniques have been applied for the first time in extracting new analytic solutions for the mentioned NLS equation. The obtained solutions exhibit various types of optical solitons, including singular solitons, solitary waves, periodic solitons, and wave collisions. These solutions have wide-ranging applications in various fields such as fiber optics, telecommunication systems, plasma physics, hydrodynamics, and nonlinear optics. The behavior of the derived solutions has been visually represented using three-dimensional, two-dimensional, and contour plots, considering appropriate choices of the involved parameters within specific time intervals. These plots provide insights into the dynamics and characteristics of the solutions. The effectiveness, reliability, and ease of application of the [Formula: see text]-expansion method and the UM are demonstrated through the obtained solutions. These mathematical tools have proven to be powerful in solving nonlinear partial differential equations, offering valuable contributions to the field of mathematical physics, soliton theory and nonlinear sciences.

Numerical simulation of coherent mid-infrared supercontinuum generation based on carbon chloride core tapered photonic crystal fibers

We numerically studied coherent mid-infrared (MIR) supercontinuum (SC) generated by filling carbon chloride (CCl4, C2Cl4) into the tapered photonic crystal fiber (PCF). The mechanism involves nonlinear soliton compression, Raman soliton self-frequency shift and dispersive wave generated in a carbon chloride core tapered PCF pumped by a 1.98 μm femtosecond laser. Numerical solution of the generalized nonlinear Schrodinger equation shows that 1.15–5.5 μm (CCl4-core tapered PCF) and 1.15–5.7 μm (C2Cl4-core tapered PCF) coherent MIR-SC sources can be realized. Our results show that the above liquid-core tapered PCFs are promising nonlinear media for constructing broadband coherent MIR-SC light sources.

Quasi-Grammian solutions of the coupled Gerdjikov–Ivanov equation

We construct the N-fold standard binary Darboux transformation (bDT) for the coupled Gerdjikov–Ivanov (cGI) equation. Then, using the bDT presented in this article, we explicitly construct the exact solutions of the cGI equation in terms of quasi-Grammians. As applications of the bDT, we present various particular solutions for the cGI equation, including soliton, breather, and rogue wave solutions.

Multiple Soliton Solutions for Coupled Modified Korteweg–de Vries (mkdV) with a Time-Dependent Variable Coefficient

In this manuscript, we implement analytical multiple soliton wave and singular soliton wave solutions for coupled mKdV with a time-dependent variable coefficient. Based on the similarity transformation and Hirota bilinear technique, we construct both multiple wave kink and wave singular kink solutions for coupled mKdV with a time-dependent variable coefficient. We implement the Hirota bilinear technique to compute analytical solutions for the coupled mKdV system. Such calculations are made by using a software with symbolic computation software, for instance, Maple. Recently some researchers used Maple in order to show that the bilinear method of Hirota is a straightforward technique which can be used in the approach of differential, nonlinear models. We analyzed whether the experiments proved that the procedure is effective and can be successfully used for many other mathematical models used in physics and engineering. The results of this study display that the profiles of multiple-kink and singular-kink soliton types can be efficiently controlled by selecting the particular form of a similar time variable. The changes in the solitons based on the changes in the arbitrary function of time allows for more applications of them in applied sciences.

Traveling wave solutions to the nonlinear space–time fractional extended KdV equation via efficient analytical approaches

In this paper, the nonlinear space-time fractional extended Korteweg-de Vries (F-eKdV) equation involving the conformable fractional derivative (CFD) is considered. Bernoulli and Riccati equations as a simplest equation method together with applying modified traveling wave transformation are used to obtain the exact solution of F-eKdV equation. Also, by exp⁡(−Φ(ζ))-expansion method, different kind of exact traveling wave solutions are gained, including periodic-singular, dark-singular and plane-wave solutions. As a result, several soliton solutions of the proposed equation are revealed. These novel soliton waves are constructed using the software package Maple with some given parametric values and displayed graphically in 2-dimensional (2D) and 3-dimensional (3D) plots by using MATLAB. To illustrate how the fractional operator affects the results, some of the acquired solutions are presented for various values of the fractional order α and also compared to their exact solutions in classical case (i.e. α=1). In the given graphical representations, the physical meaning of the reported solutions is illustrated. From the obtained results, it is observed that the methods are effective and dependable to solve nonlinear fractional partial differential equations (NFPDEs).

Spatial self-bending soliton phenomenon of (2+1) dimensional bidirectional Sawada-Kotera equation

In this study, we delve into the phenomenon of spatial self-bending solitons within the context of the (2+1)-dimensional bidirectional Sawada-Kotera equation. By employing the Hirota bilinear method, the bilinear form of the (2+1)-dimensional bidirectional Sawada-Kotera equation can be obtained. Through specific parameter constraints (eAjs=0), we establish the N-soliton solution and subsequently derive the spatial self-bending soliton. The curvature of this spatial self-bending soliton is also elucidated. Additionally, we explore the interaction between the spatial self-bending soliton and respiratory waves, along with the interaction between the spatial self-bending soliton and higher-order lump waves through the method of the long wave limit.

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.