W-shaped Soliton Research Articles

Localized wave solutions and their superposition and conversion mechanism for the (2+1)-dimensional Hirota’s system

Via Darboux transformation method, breather and rogue wave solutions for the (2+1)-dimensional Hirota’s system will be derived. Meanwhile, the dynamic features of those solutions, as well as their various special superpositions creating different sequences of rogue wave due to the different simplest ratio of two frequencies, will be analytically and graphically analyzed. In addition, we will study the conversion mechanism, which can convert breathers and rogue waves into solitons and periodic waves. Through this conversion mechanism, we will obtain a range of stationary nonlinear waves such as M-shaped and W-shaped multi-peak solitons, (quasi) periodic waves and (quasi) anti-dark solitons that exhibit the stationary feature. By virtue of the second-order solutions, we will consider all possible semi-transformed waves, interactions between two non-parallel transformed waves and superpositions of two parallel transformed waves.

On assorted soliton wave solutions with the higher-order fractional Boussinesq–Burgers system

The purpose of this study is to highlight the shallow water wave patterns along the ocean shore or in lakes with the higher-order Boussinesq–Burgers system possessing a fractional derivative operator. A generic fractional transformation is used, which turns the proposed model into an nonlinear ordinary differential equations (NLODEs) system. For the construction of new solitons of the mentioned coupled system, the auxiliary equation technique is employed. This approach produced numerous soliton solutions such as bright, singular and w-shaped solitons of the aforesaid model successfully. These results are expressed graphically to exemplify their physical appearance with the help of soft computations in Mathematica. All the solutions yielded by this method are novel and have not been derived yet.

Darboux Transformation and Soliton Solution of the Nonlocal Generalized Sasa–Satsuma Equation

This paper aims to seek soliton solutions for the nonlocal generalized Sasa–Satsuma (gSS) equation by constructing the Darboux transformation (DT). We obtain soliton solutions for the nonlocal gSS equation, including double-periodic wave, breather-like, KM-breather solution, dark-soliton, W-shaped soliton, M-shaped soliton, W-shaped periodic wave, M-shaped periodic wave, double-peak dark-breather, double-peak bright-breather, and M-shaped double-peak breather solutions. Furthermore, interaction of these solitons, as well as their dynamical properties and asymptotic analysis, are analyzed. It will be shown that soliton solutions of the nonlocal gSS equation can be reduced into those of the nonlocal Sasa–Satsuma equation. However, several of these properties for the nonlocal Sasa–Satsuma equation are not found in the literature.

Abundant explicit solutions for the M-fractional coupled nonlinear Schrödinger–KdV equations

In this study, the famous fractional generalized coupled cubic nonlinear Schrödinger–KdV equations arising in many domains of physics and engineering such as depicting the propagation of long waves in dispersive media and the dynamics of short dispersive waves for narrow-bandwidth packet have been investigated. We propose two significant methods named the modified (Gʹ/G, 1/G)-expansion method and the Gʹ/(bGʹ + G + a)-expansion method. After utilizing these two efficient techniques, many types of explicit soliton pulse solutions including the well-known bell-shape bright soliton pulse, the kink and anti-kink dark solitons pulse, the mixed bright-dark soliton pulse, the W-shaped soliton pulse, the periodic waves pulse, and the blow-up soliton pattern solutions are obtained. If we select different values of the frequency, coefficients and orders, the dynamic properties and physical structures of these solutions are discussed, these important results can help us to further understand the inner characteristic of the model.

Phase properties of several nonlinear optical waves described by rational solutions

Phase plays an essential role in both classical and quantum wave dynamics, which has motivated many scientists to study the phases of many different nonlinear waves. Among those nonlinear waves, the ones described by rational solutions have been given much attention, partly because some of them can describe rogue wave dynamics. We revisit the phase of several well-known one-dimensional rational solutions and clarify that sudden phase inversion processes indeed exist for rogue waves, in contrast to the ones reported before. Moreover, we investigate the phase properties of rogue waves and rational W-shaped solitons in the Hirota and Sasa-Satsuma model, for which some high-order physical effects are considered, i.e., the third-order dispersion delayed nonlinear response and self-steeping effects. The quite distinctive phases are uncovered for the rational W-shaped solitons with similar density profiles in the two models, by limitation analysis on the underlying topological vector potentials of other related nonlinear waves. These results indicate that the underlying topological vector potentials can be used to identify the phases of nonlinear waves, especially for localized waves with abrupt phase jumps.

Propagation of W-shaped and M-shaped solitons with multi-peak interaction for ultrashort light pulse in fibers

Propagation of W-shaped and M-shaped solitons with multi-peak interaction for ultrashort light pulse in fibers

Chirped gap solitons with Kudryashov’s law of self-phase modulation having dispersive reflectivity

The present study is devoted to investigate the chirped gap solitons with Kudryashov’s law of self-phase modulation having dispersive reflectivity. Thus, the mathematical model consists of coupled nonlinear Schrödinger equation (NLSE) that describes pulse propagation in a medium of fiber Bragg gratings (BGs). To reach an integrable form for this intricate model, the phase-matching condition is applied to derive equivalent equations that are handled analytically. By means of auxiliary equation method which possesses Jacobi elliptic function (JEF) solutions, various forms of soliton solutions are extracted when the modulus of JEF approaches 1. The generated chirped gap solitons have different types of structures such as bright, dark, singular, W-shaped, kink, anti-kink and Kink-dark solitons. Further to this, two soliton waves namely chirped bright quasi-soliton and chirped dark quasi-soliton are also created. The dynamic behaviors of chirped gap solitons are illustrated in addition to their corresponding chirp. It is noticed that self-phase modulation and dispersive reflectivity have remarkable influences on the pulse propagation. These detailed results may enhance the engineering applications related to the field of fiber BGs.

Chirped gap solitons in fiber Bragg gratings with polynomial law of nonlinear refractive index

The objective of the present study is to examine the behaviors of chirped optical solitons in fiber Bragg gratings (BGs) with dispersive reflectivity. The form of nonlinear refractive index represents polynomial law nonlinearity. By virtue of phase-matching condition, the discussed model of coupled nonlinear Schrödinger equation is reduced to an integrable form. Consequently, chirped optical solitons having various profiles such as W-shaped, bright, dark, kink and anti-kink solitons are derived. Further to this, the chirp associated with these soliton structures are extracted. The impact of dispersive reflectivity, self-phase modulation and cross-phase modulation on the pulse propagation is investigated and it is induced that the changes of self-phase modulation and cross-phase modulation cause a marked rise in soliton amplitude which is subject to minor variations by dispersive reflectivity. The physical evolutions of chirped optical solitons are described along with the corresponding chirp to pave the way for possible applications in the field of fiber BGs.

W-shaped soliton solutions to the modified Zakharov-Kuznetsov equation of ion-acoustic waves in (3+1)-dimensions arise in a magnetized plasma

<abstract><p>This paper is presented to investigate the exact solutions to the modified Zakharov-Kuznetsov equation that have a critical role to play in mathematical physics. The $ \tan \left(\phi \left(\zeta \right)/2 \right) $-expansion, $ (m+G'(\zeta)/G(\zeta)) $-expansion and He exponential function methods are used to reveal various analytical solutions of the model. The equation regulates the treatment of weakly nonlinear ion-acoustic waves in a plasma consisting of cold ions and hot isothermal electrons throughout the existence of a uniform magnetic field. Solutions in forms of W-shaped, singular, periodic-bright and bright are constructed.</p></abstract>

Interactions of fractional [formula omitted]-solitons with anomalous dispersions for the integrable combined fractional higher-order mKdV hierarchy

In this paper, we investigate the anomalous dispersive relations, inverse scattering transform with a Riemann–Hilbert (RH) problem, and fractional multi-solitons of the integrable combined fractional higher-order mKdV (fhmKdV) hierarchy, including the fractional mKdV (fmKdV), fractional fifth-order mKdV (f5mKdV), fractional combined third–fifth-order mKdV (f35mKdV) equations, etc., which can be featured via completeness of squared scalar eigenfunctions of the ZS spectral problem. We construct a matrix RH problem to present three types of fractional N-solitons illustrating anomalous dispersions of the combined fhmKdV hierarchy for the reflectionless case. As some examples, we analyse the wave velocity of the fractional one-soliton such that we find that the fhmKdV equation predicts a power law relationship between the wave velocity and amplitude, and demonstrates the anomalous dispersion. Furthermore, we illustrate other interesting anomalous dispersive wave phenomena containing the elastic interactions of fractional bright and dark solitons, W-shaped soliton and dark soliton, as well as breather and dark soliton. These obtained fractional multi-solitons will be useful to understand the related nonlinear super-dispersive wave propagations in fractional nonlinear media.

Optical soliton solutions of time-fractional coupled nonlinear Schrödinger system via Kudryashov-based methods

Purpose:In this study, we investigated the time-fractional coupled nonlinear Schrödinger (TF-CNLS) system in conformable fractional sense. TF-CNLS system models the circularly-polarized waves in the optic branch of physics frequently. Methodology:To obtain optical solitons of the model via new Kudryashov method and Kudryashov auxiliary equation methods have been applied. Findings:After implementing the mentioned methods on the model, we obtained bright, singular, W-shaped and bright soliton solutions. According to obtained soliton types, it is clearly seen that the methods are efficient and have ease of use. We depicted the results obtained by 3D and 2D graphs and investigated the wave motion according to the parameter and fractional order changes. Originality:We have applied a modulation instability to the TF-CNLS system for the first time. Due to literature research, we propose that this study is innovative and has novel inference.

Exact solutions for the conformable fractional coupled nonlinear Schrödinger equations with variable coefficients

In this work, the general algebraic method is proposed for finding some solutions of a conformable fractional coupled nonlinear Schrödinger equation with variable coefficients arising in inhomogeneous fibers with two orthogonal polarization states. With the aid of symbolic computations, many types of new soliton pulse and periodic pulse solutions including complex doubly periodic solutions, solitary wave solutions, and trigonometric function solutions are obtained. Some 3D and 2D numerical simulations about these solutions are portraited, which show the novelty and visibility of the dynamical structure and propagation behavior of the corresponding model. Moreover, we found the W-shaped soliton pulse and the periodic wave pulse can be effectively controlled by changing the relative parameters of the frequency coefficients and orders, which will help us to have a better understanding about the internal structure of this model.

The mixed solutions and nonlinear wave transitions for the (2 + 1)-dimensional Sawada-Kotera equation

In this paper, we will obtain the mixed solutions of the -dimensional Sawada-Kotera equation by the Hirota bilinear method and the long wave limit method. Through applying the long wave limit method to the N-soliton solutions, we will obtain the corresponding lump solutions. Besides, we will get the mixed solutions of lump, breather and soliton solutions. Under the transition condition, we will obtain different types of nonlinear waves which include quasi-anti-dark soliton, multi-peak soliton, M-shaped soliton, quasi-periodic soliton and W-shaped soliton from the breath wave and the lump solutions. In particular, we will derive the inelastic collision modes, the elastic collision modes and the collision-free modes under specific conditions. Finally, we will display the mixed solutions and the nonlinear waves through illustrations.

Application of the unified method to solve the Biswas–Arshed model

We apply the unified method to retrieve optical soliton solutions of the Biswas–Arshed model (BAM) with the Kerr law nonlinearity in this paper. We first derive the ordinary differential form of this model from its partial differential form via a variable transformation. Then we add many new dynamical solitons of combo trigonometric, hyperbolic, and rational function solutions by Maple-18 package in this literature. The derived solutions exhibit some dynamics as singular solitons with exponentially increasing and decreasing amplitudes, singular solitons with gradually increasing and decreasing amplitudes, and singular solitons with constant amplitudes. Moreover, W-shaped solitons with multiple bright and dark waves, double periodic optical solitons with exponentially increasing and decreasing amplitudes, and double periods with symmetric and asymmetric amplitudes are obtained. Besides this, various dynamical properties of the obtained solutions are presented graphically.

Data-driven solutions and parameter discovery of the Sasa–Satsuma equation via the physics-informed neural networks method

Physics-informed neural networks (PINNs) can be used to predict various solutions of nonlinear partial differential equations in diverse physical fields. We use the PINNs to predict the (single-) double-hump soliton, multi-hump breather, anti-dark soliton, Mexican-hat soliton, twisted rogue-wave pairs and rational W-shaped soliton of the Sasa–Satsuma equation in optical fibers and deep water wave. The L2 relative error of the prediction results can achieve high accuracy even with only less training data and training time while the results show that the prediction error will increase with the increase of prediction time. In addition, we employ the PINNs to investigate the parameter discovery of the Sasa–Satsuma equation via the Mexican-hat soliton solution. For the anti-dark soliton, we compare the predicted results and the amount of data for the PINNs method and the traditional split-step Fourier technique, and show that the PINNs method requires less data with the same accuracy of prediction results.

Rational soliton solutions in the nonlocal coupled complex modified Korteweg–de Vries equations

Abstract In this article, our work oversees with the nonlocal coupled complex modified Korteweg–de Vries equations (cmKdV), which is a nonlocal generalization for coupled cmKdV equations. The n-fold Darboux transformation (DT) is constructed in the form of determinants for the nonlocal coupled cmKdV equations. Via generalized DT method, we obtain the rational soliton solutions describing M-shaped soliton, W-shaped soliton, and the interactions on the plane wave and periodic background. The results can be useful to study the dynamical behaviors of soliton solutions in nonlocal wave models.

A novel approach to study generalized coupled cubic Schrödinger–Korteweg-de Vries equations

The Kortewegde Vries (KdV) equation represents the propagation of long waves in dispersive media, whereas the cubic nonlinear Schrödinger (CNLS) equation depicts the dynamics of narrow-bandwidth wave packets consisting of short dispersive waves. A model that couples these two equations seems intriguing for simulating the interaction of long and short waves, which is important in many domains of applied sciences and engineering, and such a system has been investigated in recent decades. This work uses a modified Sardar sub-equation procedure to secure the soliton-type solutions of the generalized cubic nonlinear Schrödinger–Korteweg-de Vries system of equations. For various selections of arbitrary parameters in these solutions, the dynamic properties of some acquired solutions are represented graphically and analyzed. In particular, the dynamics of the bright solitons, dark solitons, mixed bright-dark solitons, W-shaped solitons, M-shaped solitons, periodic waves, and other soliton-type solutions. Our results demonstrated that the proposed technique is highly efficient and effective for the aforementioned problems, as well as other nonlinear problems that may arise in the fields of mathematical physics and engineering.

Comparison between three distinct perceptions to the new solitary solutions of the generalized Hirota–Satsuma coupling KDV system

In this paper, we will implement a comparison between three different perceptions to the behavior of the exact solution to the generalized Hirota–Satsuma coupling Korteweg–de Vries System (GHSCKDVS). The proposed model plays a vital role in different branches of physics and applied mathematics. Specially, this model describes dispersive waves as the shallow water waves in hydrodynamics and it has various types of solutions like soliton solutions and traveling wave solutions. These three various visions to the behavior of the exact solution are established via three various techniques that have three different mathematical analyses. These three important different techniques are the Paul–Painlevé approach method (PPAM) examined previously to extract the exact solutions for many nonlinear partial differential equations (NLPDE) and achieved good results, the Ricatti–Bernolli Sub-ODE method (RBSODM) which is one of the famous ansatze approaches that doesn’t surrender to the balance rule, treats the balance rule fails and realizes impressive forms of the exact solutions. Many new various types of soliton solutions as hyperbolic and trigonometric function solutions, represented by the M-shaped, W-shaped, kink, anti-kink and dark soliton solutions have been established via these two methods. Moreover, we will document the identical numerical solutions for all achieved exact solutions realized by the two methods mentioned above via the famous numerical variational iteration method (VIM) that usually gives good results. The powerful of our achieved numerical solutions related to that their initial conditions emerged from the extracted exact solutions.

The nonlinear wave solutions and parameters discovery of the Lakshmanan-Porsezian-Daniel based on deep learning

We apply the deep learning approach to learn some nonlinear wave solutions of the Lakshmanan-Porsezian-Daniel (LPD) model characterizing the evolution of ultrashort optical pulse in optical fibers. Based on the strong universal approximation theorem, we give the initial-boundary value data and residual collocation points, choose the parameters initialization Xavier method and parameters optimization Adam and L-BFGS algorithms to construct the optimal neural network model. Then, we derive the data-driven solutions of the rogue wave, anti-dark soliton, multi-peak soliton, non-rational W-shaped soliton, rational W-shaped soliton as well as periodic-wave solutions for the LPD model. Finally, we study the parameters discovery of such model via the anti-dark soliton solution with 1% perturbation (or without perturbation).

Application of Hirota operators for controlling soliton interactions for Bose-Einstien condensate and quintic derivative nonlinear Schrödinger equation

With the help of the Hirota bilinear method (HBM), we study soliton interactions of quasi-1D Bose-Einstein Condensate system (BECs) with dipole-dipole attraction and repulsion. BEC is an extended form of nonlinear Schrödinger equation (NLSE) and it consists of quadratic-cubic nonlinearities, linear gain or loss and time modulated dispersion. Due to its spatially varying coefficients property it has significance in the field of fluid dynamics, classical and quantum field theories, nonlinear optics and physics etc. We will also discuss soliton interactions with graphically descriptions for QDNLSE. We obtain some parabolic, anti parabolic, M-shaped, W-shaped, butterflies, bright, anti dark, V-shaped, S-shaped and other solitons for our governing models.