Rational Solitons Research Articles

New dynamics performances to the established dark solitons in Polariton Condensate

Abstract New diverse enormous soliton solutions to the Gross–Pitaevskii Equation, which describes the dynamics of two dark solitons in a polarization condensate under non-resonant pumping, have been constructed for the first time by using two various Schemes. The two utilized Schemes are the Generalized Kudryashov Scheme, the (G'/G)-Expansion Scheme. Throughout these two suggested schemata we construct new diverse forms solutions that include dark, bright-shaped soliton solutions, combined bright-shaped, dark-shaped soliton solutions, hyperbolic function soliton solution, singular shaped soliton solution and other rational soliton solutions. The two 2D and 3D figures designs have been configured by using the Mathematica-program. In addition, the Haar Wavelet Numerical Scheme has been applied to construct the identical numerical behavior for all soliton solutions achieved by the two suggested Schemes to show existing similarity between the soliton solutions and numerical solutions.

Study of Fifth-Order Nonlinear Caudrey–Dodd–Gibbon Model for Multiwave, M-Shaped Solitons, Lumps, Generalized Breathers, Manifold Periodic, Rogue Wave and Kink Wave Interactions

Numerous novel wave structures, such as lump, lump with one-kink, lump with two-kink soliton, generalized breather, Ma breather, Kuznetsov-Ma breather, Akhmediev breather, manifold periodic, and rogue wave solutions to [Formula: see text]-dimensional fifth-order nonlinear Caudrey–Dodd–Gibbon (FNCDG) model via symbolic computation are studied based on the ansatz function scheme. Studying the lump-type solution involves selecting the function as the generic quadratic polynomial function. The lump with one- and two-kink solutions is created by mixing a quadratic function with one or two exponential functions. By adding hyperbolic function with a quadratic function, the rogue wave solutions are manipulated. In addition, the multiwave, [Formula: see text]-shaped rational solitons and their interactions with kink waves are analytically evaluated. Furthermore, different 3D, 2D, and contour profiles are created in different dimensions by changing the parameter values.

M-shaped rational, homoclinic breather, kink-cross rational, multi-wave and interactional soliton solutions to the fifth-order Sawada-Kotera equation

In this work, we develop multi-wave, homoclinic breathers, M-shaped rational, 1-kink interactions with M-shaped, periodic-cross rational and kink-cross rational solutions for the fifth-order Sawada-Kotera equation, which represents the motion of long waves under gravity in shallow water, using several ansatz transformations. A three-wave approach is used to identify multi-wave solitons. Additionally, novel forms of exact solutions are constructed using the homoclinic breathers technique. The kink-cross rational and periodic-cross rational solitons are investigated using appropriate transformations. We develop M-shaped solitons and demonstrate their behavior by selecting suitable parameter values. Furthermore, the interactions between kink waves and M-shaped solitons are also studied. The obtained solutions are determined in 3D, 2D, and contour profiles, where the free parameters involved in the solutions are assigned specific values. Their physical relevance is explored to highlight the inner context of tangible incidents in the natural domain. Since these recently discovered solutions contain a few arbitrary constants, they can be used to explain the variation in qualitative characteristics of wave phenomena.

Studying exploring paraxial wave equation in Kerr media: Unveiling the governing laws, rational solitons and multi-wave solutions and their stability with applications

Studying exploring paraxial wave equation in Kerr media: Unveiling the governing laws, rational solitons and multi-wave solutions and their stability with applications

Pseudo grid-based physics-informed convolutional-recurrent network solving the integrable nonlinear lattice equations

Traditional discrete learning methods involve discretizing continuous equations using difference schemes, necessitating considerations of stability and convergence. Integrable nonlinear lattice equations possess a profound mathematical structure that enables them to revert to continuous integrable equations in the continuous limit, particularly retaining integrable properties such as conservation laws, Hamiltonian structure, and multiple soliton solutions. The pseudo grid-based physics-informed convolutional-recurrent network (PG-PhyCRNet) is proposed to investigate the localized wave solutions of integrable lattice equations, which significantly enhances the model’s extrapolation capability to lattice points beyond the temporal domain. We conduct a comparative analysis of PG-PhyCRNet with and without pseudo grid by investigating the multi-soliton solutions and rational solitons of the Toda lattice and self-dual network equation. The results indicate that the PG-PhyCRNet excels in capturing long-term evolution and enhances the model’s extrapolation capability for solitons, particularly those with steep waveforms and high wave speeds. Finally, the robustness of the PG-PhyCRNet method and its effect on the prediction of solutions in different scenarios are confirmed through repeated experiments involving pseudo grid partitioning.

Obtaining analytical solutions of (2+1)-dimensional nonlinear Zoomeron equation by using modified F-expansion and modified generalized Kudryashov methods

PurposeThe purpose of the article is to conduct a mathematical and theoretical analysis of soliton solutions for a specific nonlinear evolution equation known as the (2 + 1)-dimensional Zoomeron equation. Solitons are solitary wave solutions that maintain their shape and propagate without changing form in certain nonlinear wave equations. The Zoomeron equation appears to be a special model in this context and is associated with other types of solitons, such as Boomeron and Trappon solitons. In this work, the authors employ two mathematical methods, the modified F-expansion approach with the Riccati equation and the modified generalized Kudryashov’s methods, to derive various types of soliton solutions. These solutions include kink solitons, dark solitons, bright solitons, singular solitons, periodic singular solitons and rational solitons. The authors also present these solutions in different dimensions, including two-dimensional, three-dimensional and contour graphics, which can help visualize and understand the behavior of these solitons in the context of the Zoomeron equation. The primary goal of this article is to contribute to the understanding of soliton solutions in the context of the (2 + 1)-dimensional Zoomeron equation, and it serves as a mathematical and theoretical exploration of the properties and characteristics of these solitons in this specific nonlinear wave equation.Design/methodology/approachThe article’s methodology involves applying specialized mathematical techniques to analyze and derive soliton solutions for the (2 + 1)-dimensional Zoomeron equation and then presenting these solutions graphically. The overall goal is to contribute to the understanding of soliton behavior in this specific nonlinear equation and potentially uncover new insights or applications of these soliton solutions.FindingsAs for the findings of the article, they can be summarized as follows: The article provides a systematic exploration of the (2 + 1)-dimensional Zoomeron equation and its soliton solutions, which include different types of solitons. The key findings of the article are likely to include the derivation of exact mathematical expressions that describe these solitons and the successful visualization of these solutions. These findings contribute to a better understanding of solitons in this specific nonlinear wave equation, potentially shedding light on their behavior and applications within the context of the Zoomeron equation.Originality/valueThe originality of this article is rooted in its exploration of soliton solutions within the (2 + 1)-dimensional Zoomeron equation, its application of specialized mathematical methods and its successful presentation of various soliton types through graphical representations. This research adds to the understanding of solitons in this specific nonlinear equation and potentially offers new insights and applications in this field.

Soliton, breather and rogue wave solutions of the higher-order modified Gerdjikov–Ivanov equation

In this paper, we investigate the dynamics of localized waves for the higher-order modified Gerdjikov–Ivanov equation. Based on Lax pair, the Nth-fold Darboux transformation is constructed. The soliton, breather and rogue wave solutions are obtained and presented graphically. Moreover, the dynamic properties of the above localized waves are analyzed. Specially, the interactions between solitons and bound states are achieved. In addition, the rational W-shaped soliton is derived by the degeneration of breather solutions.

Dimensionless dynamics: Multipeak and envelope solitons in perturbed nonlinear Schrödinger equation with Kerr law nonlinearity

In this work, the dimensionless form of the improved perturbed nonlinear Schrödinger equation with Kerr law of fiber nonlinearity is solved for distinct exact soliton solutions. We examined the multi-wave solitons and rational solitons of the governing equation using the logarithmic transformation and symbolic computation using an ansatz functions approach. Multi-wave solitons in fluid dynamics describe the situation in which a fluid flow shows several different regions (or peaks) of high concentration or intensity of a particular variable (e.g., velocity, pressure, or vorticity). Multi-wave solitons in turbulent flows might indicate the existence of several coherent structures, like eddies or vortices. These formations are areas of concentrated energy or vorticity in the turbulent flow. Understanding how these peaks interact and change is essential to comprehending the energy cascade and dissipation in turbulent systems. Furthermore, a sub-ordinary differential equation approach is used to create solutions for the Weierstrass elliptic function, periodic function, hyperbolic function, Chirped free, dark-bright (envelope solitons), and rational solitons, as well as the Jacobian elliptic function, periodic function, and rational solitons. Also, as the Jacobian elliptic function's' modulus m approaches values of 1 and 0, we find trigonometric function solutions, solitons-like solutions, and computed chirp free-solitons. Envelope solitons can arise in stratified fluids and spread over the interface between layers, such as layers in the ocean with varying densities. Their research aids in the management and prediction of wave events in artificial and natural fluid settings. In fluids, periodic solitons are persistent, confined wave structures that repeat on a regular basis, retaining their form and velocity over extended distances. These structures occur in a variety of settings, including internal waves in stratified fluids, shallow water waves, and even plasma physics.

Exploring travelling wave solutions, bifurcation, chaos, and sensitivity analysis in the (3+1)-dimensional gKdV-ZK model: A comprehensive study using Lie symmetry methodology

This article presents a study on the generalized Korteweg-de Vries-Zakharov-Kuznetsov (gKdV-ZK) model, which is a nonlinear system that demonstrates the effect of magnetic fields on weak ion-acoustic waves in plasma consisting of cold and hot electrons. The research entails investigating the reduction of symmetry through Lie group analysis, scrutinizing the characteristics of the dynamic structure using bifurcation phase diagrams, and examining the dynamic behaviour of the perturbed dynamical system employing chaos theory. Methods such as 3D and 2D phase portraits, time series analysis, Poincaré maps, exploration of multistability in the autonomous structure across various initial conditions, Lyapunov exponents, and bifurcation diagrams are exercised to demonstrate chaotic behaviour. Additionally, the research establishes general forms of solitary wave solutions, encompassing hyperbolic, trigonometric, and rational soliton solutions, through the utilization of a modified auxiliary equation approach to analytically address the examined problem. These findings are visually depicted as 2D and 3D graphs with carefully selected parameters, accompanied by their corresponding constraint conditions. Furthermore, the sensitivity analysis of the studied equation is deliberated upon and visually illustrated. The uncovered findings are captivating, innovative, and potentially beneficial for comprehending various physical phenomena in engineering and science.

Dynamical study and diverse soliton collisions for the Kraenkel–Manna–Merle system in ferrites

We investigate the rational solitons of the Kraenkel–Manna–Merle (KMM) system in ferrites via symbolic computation with the ansatz functions technique and logarithmic transformation. The KMM system describes the propagations of the ferromagnetic particles in ferrite materials. Multiwave solitons are reported by using the three-wave technique. Homocentric breathers (HBs) approach is also applied to build new forms of exact solutions. We construct M-shaped solitons and their dynamics are shown via the appropriate values of parameters. Two types of interactions for rational soliton with kink wave are investigated. Additionally, kink cross-rational solitons (KCRSs) and periodic cross-rational solitons (PCRSs) are examined via suitable transformation.

Mathematical modeling and component generalization of (2+1)-dimensional Schwarz–Kortweg–de Vries model in shallow water waves

This work employs the sub-ODE method to compute new soliton solutions for the component generalization of [Formula: see text]-dimensional Schwarz–Kortweg–de Vries (SKdV) equation in shallow water waves. The rational soliton solutions (RSS), Jacobian elliptic function (JEF), periodic solutions (PS), hyperbolic function solutions (HFS), positive solitons, dark soliton solutions (DSS) and bright soliton solutions (BSS) are constructed with this technique. Solitons-like solutions, HFS and trigonometric-function solutions (TFS) are also found as the JEF’s parameter m approaches values of 1 and 0, respectively. The suggested approach for the nonlinear equations in mathematical physics is concise, simple, effective, and promising.

Exploration conversations laws, different rational solitons and vibrant type breather wave solutions of the modify unstable nonlinear Schrödinger equation with stability and its multidisciplinary applications

Exploration conversations laws, different rational solitons and vibrant type breather wave solutions of the modify unstable nonlinear Schrödinger equation with stability and its multidisciplinary applications

Various solitons induced by relative phase in the nonlinear Schrödinger Maxwell-Bloch system.

We study the effect of relative phase on the characteristics of rogue waves and solitons described by rational solutions in the nonlinear Schrödinger Maxwell-Bloch system. We derived the rational rogue wave and soliton solutions with adjustable relative phase and present the parameter range of different types of rogue waves and solitons. Our findings show that the relative phase can alter the distribution of rational solitons and even change the type of rational solitons, leading to a rich array of rational soliton types by adjusting the relative phase. However, the relative phase does not affect the structure of the rogue wave, because the relative phase of the rogue wave changes during evolution. In particular, we confirm that the rational solitons with varying relative phases and the rogue waves at corresponding different evolution positions share the same distribution mode. This relationship holds true for rogue waves or breathers and their stable counterparts solitons or periodic waves in different nonlinear systems. The implications of our study are significant for exploring fundamental excitation elements in nonlinear systems.

Exploration of solitons and analytical solutions by sub-ODE and variational integrators to Klein-Gordon model

In this paper, we use the sub-ODE method to analyze soliton solutions for the renowned nonlinear Klein-Gordon model (NLKGM). This method provides a variety of soliton solutions, including three positive solitons, three Jacobian elliptic function solutions, bright solitons, dark solitons, periodic solitons, rational solitons and hyperbolic function solutions. Applications for these solitons can be found in optical communication, fiber optic sensors, plasma physics, Bose-Einstein condensation and other areas. We also study some numerical solutions by using forward, backward, and central difference techniques. Moreover, we discuss variational integrators (VIs) using the projection technique for NLKGM. We develop a numerical solution for NLKGM using the discrete Euler lagrange equation, the Lagrangian and the Euler lagrange equation. At the end, in various dimensions, covering 3D, 2D, and contour, we will also plot several graphs for the obtained NLKGM solutions. A contour plot is a type of graphic representation that displays a three-dimensional surface on a two-dimensional plane by using contour lines. Each contour line in the plotted function represents one of the function's constant values, mapping the function's value across the plane. This model has been studied across multiple soliton solutions using various methods in the open literature, but this model for VIs and finite deference scheme (FDS) is the first time it has been studied. Within the various numerical techniques accessible for solving Hamiltonian systems, variational integrators distinguish themselves because of their symplectic quality. Here are some of the symplectic properties: symplectic orthogonality, energy conservation, area preservation, and structure preservation.

Various types of discrete exact localized wave solutions and dynamical analysis for the discrete complex modified Korteweg-de Vries equation

Under consideration is the second-order integrable discretization of complex modified Korteweg-de Vries (mKdV) equation which is regarded as the discrete counterpart of the mKdV equation having an essential role in describing the propagation behavior of water waves and acoustic waves in nonlinear media. First of all, based on the known linear spectral problem, the discrete generalized (n,N−n)-fold Darboux transformation is constructed to derive various types of discrete exact localized wave solutions, including soliton and semi-rational soliton solutions on vanishing background, breather, rogue wave and hybrid interaction solutions on plane wave background, and rational soliton solution on constant background, and the relevant evolution structures are studied graphically. Secondly, the asymptotic analysis is used to discuss the elastic interaction for two-soliton solutions and limit states for rational soliton solutions. Finally, the numerical simulation is utilized to investigate the dynamical behavior and propagation stability of some exact solutions. The findings presented in this paper may contribute to explaining physical phenomena described by the mKdV equation.

W-chirped solitons and modulated waves patterns in parabolic law medium with anti-cubic nonlinearity

In this work, investigation of the W-chirped solitons and modulation instability are addressed in nonlinear optic fibers where cubic-quintic and parabolic law parameters are used. For this reason, the modified cubic-quintic complex Ginzburg–Landau equation with parabolic law nonlinearity is employed. Using the sub-ODE method, the chirp component and combined chirped soliton solutions, including rational soliton solutions, are obtained. Using specific parameters of the nonlinear structure, one can identify dark and bright solitons. For a suitable constant parameter of the chirp component, the W-chirped bright as well as the combined bright-dark solitons are displayed via the 3D plots. The linear stability analysis of the plane wave is used to derive an expression of the modulation instability spectrum that is employed to show the effects of the nonlinear parameters in normal and anomalous dispersion regimes. It is pointed out that both cubic and quintic parameters can enlarge or reduce the bandwidth of the modulation instability. Through the numerical simulation of the continuous wave, it becomes apparent that for suitable optical fiber parameter values, modulated wave objects are formed to exhibit mild oscillations. For specific wavenumber excitation, another modulated wave object is depicted to show how the unstable mode can switch to the stable mode for the long-time evolution of the continuous wave. The obtained results could certainly open a new future for signal treatment in optical fiber, where parabolic law is considered.

Modulation instability and localized wave excitations for a higher-order modified self-steepening nonlinear Schrödinger equation in nonlinear optics

This paper investigates a higher-order modified nonlinear Schrödinger equation with higher-order dispersion and self-steepening effects, which can be used to study the dynamics of asymmetric and steepened optical pulse transmission in optical fibres. The modulation instability of the plane-wave solution has been analysed, and the state transition of the rogue waves under the higher-order dispersion effect is presented. The findings demonstrate that the self-steepening effect may also produce an asymmetric unstable frequency band. Combined with the modulation instability, the rogue wave, rational soliton and mixed interaction of localized waves have a quantitative relationship with plane-wave parameters. Various exact solutions can be accurately located and obtained according to the generalized Darboux transformation. The asymptotic analysis of rational solutions demonstrates the state transition of higher-order rogue waves. This paper illustrates the significance of modulation instability for studying the integrable systems by the Darboux transformation. It is a new guidance to solve the difficulty of exact excitation of asymmetric localized waves in complex integrable systems. The results have prospective applications and references for the emergence, amplification and compression of asymmetric pulses in optical experiments.

Deep neural networks learning forward and inverse problems of two-dimensional nonlinear wave equations with rational solitons

In this paper, we investigate the forward problems on the data-driven rational solitons for the (2+1)-dimensional Kadomtsev–Petviashvili-I (KP-I) equation and spin-nonlinear Schrödinger (spin-NLS) equation via the deep neural networks learning. Moreover, the inverse problems of the (2+1)-dimensional KP-I equation and spin-NLS equation are studied via deep learning. The main idea of the data-driven forward and inverse problems is to use the deep neural networks with the activation function to approximate the solutions of the considered (2+1)-dimensional nonlinear wave equations by optimizing the chosen loss functions related to the considered nonlinear wave equations.

Rational solitons and rogue waves for the integrable nonlocal Lakshmanan-Porsezian-Daniel equation

In this paper, we investigate rational solitons and rogue wave (RW) solutions of the nonlocal Lakshmanan-Porsezian-Daniel (LPD) equation by the generalized Darboux transformation method. The exact rational solutions are presented in terms of determinants whose entries are given by the initial eigenfunction and continuous-wave solution. Their dynamics investigated reveals the interesting patterns such as rational solitons, singular and nonsingular RWs depending on the selection of the free parameters. The compression effects of the RWs with respect to the strength of the higher-order nonlinear effects parameter γ are discussed and graphically illustrated.

Diverse soliton solutions and dynamical analysis of the discrete coupled mKdV equation with 4×4 Lax pair

Under consideration in this study is the discrete coupled modified Korteweg–de Vries (mKdV) equation with 4 × 4 Lax pair. Firstly, through using continuous limit technique, this discrete equation can be mapped to the coupled KdV and mKdV equations, which may depict the development of shallow water waves, the optical soliton propagation in cubic nonlinear media and the Alfven wave in a cold collision-free plasma. Secondly, the discrete generalized (r, N–r)-fold Darboux transformation is constructed and extended to solve this discrete coupled equation with the fourth-order linear spectral problem, from which diverse exact solutions including usual multi-soliton and semi-rational soliton solutions on the vanishing background, higher-order rational soliton and mixed hyperbolic-rational soliton solutions on the non-vanishing background are derived, and the limit states of some soliton and rational soliton solutions are analyzed by the asymptotic analysis technique. Finally, the numerical simulations are used to explore the dynamical behaviors of some exact soliton solutions. These results may be helpful for understanding some physical phenomena in fields of shallow water wave, optics, and plasma physics.