M-shaped Soliton Research Articles

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.

Novel nonlinear wave transitions and interactions for (2+1)-dimensional generalized fifth-order KdV equation

The (2 + 1)-dimensional generalized fifth-order KdV (2GKdV) equation is revisited via combined physical and mathematical methods. By using the Hirota perturbation expansion technique and via setting the nonzero background wave on the multiple soliton solution of the 2GKdV equation, breather waves are constructed, for which some transformed wave conditions are considered that yield abundant novel nonlinear waves including X/Y-Shaped (XS/YS), asymmetric M-Shaped (MS), W-Shaped (WS), Space-Curved (SC) and Oscillation M-Shaped (OMS) solitons. Furthermore, distinct nonlinear wave molecules and interactional structures involving the asymmetric MS, WS, XS/YS, SC solitons, and breathers, lumps are constructed after considering the corresponding existence conditions. The dynamical properties of the nonlinear molecular waves and interactional structures are revealed via analyzing the trajectory equations along with the change of the phase shifts.

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.

M-shaped, W-shaped and dark soliton propagation in optical fiber for nonlocal fourth order dispersive nonlinear Schrödinger equation under distinct conditions

The formation of soliton in optical fiber governed by nonlocal nonlinear Schrödinger (NLS) equation with fourth order dispersion is studied. The model of nonlocal NLS equation with fourth order dispersion is solved using the new extended auxiliary method and the solutions are obtained. The solutions are in the form of hyperbolic and trigonometric functions which are based on Jacobi elliptic function m. Shape changing soliton in optical fiber for nonlocal fourth order dispersive NLS equation is discussed by suitably choosing the values of kerr and quintic nonlinearities and by varying fourth order dispersion term. The effect of fourth order dispersion on soliton in fiber for different conditions of kerr and quintic nonlinearity is also discussed. In addition, the phase portraits of the system have been investigated and the stability of wave in optical fiber for nonlocal NLS equation is discussed using fourth order Runge-Kutta algorithm. This paper addresses a significant gap in the current literature by examining the impact of fourth order dispersion on the nonlocal NLS equation in optical fiber.

Solitary waves of M-fractional low-pass nonlinear electrical transmission line model arising in network system

In this study, we analyze the solitary wave behavior of a truncated M-fractional low-pass nonlinear electrical transmission line (NLETLs) model. NLETL models are relevant to computer network systems, particularly for high-speed data transmissions. They influence the behavior of signals traveling through network cables. To investigate the dynamics of solitary waves in the model, we applied the modified Sardar sub-equation and extended the sinh-Gordon equation expansion methods. We illustrated the 2D, 3D, and contour shapes of selected solutions for appropriate values of the NLETLs dynamics using Mathematica-14. Kink, anti-kink, bright-dark bell, dark bell, M-shaped periodic soliton, and logarithmic wave solutions were obtained. The results indicate that the proposed techniques may provide valuable, powerful, and efficient insights into the dynamics of nonlinear evolution models. The role of the fractional order derivative in making optical solutions is investigated in detail, which opens up opportunities for the creation of more complex models that can more accurately simulate optical phenomena in the real world.

Nonlinear waves and transitions mechanisms for (2+1)-dimensional Korteweg–de Vries-Sawada-Kotera-Ramani equation

In this paper, state transition waves are investigated in a (2+1)-dimensional Korteweg–de Vries-Sawada-Kotera-Ramani equation by analyzing characteristic lines. Firstly, the N-soliton solutions are given by using the Hirota bilinear method. The breather and lump waves are constructed by applying complex conjugation limits and the long-wave limit method to the parameters. In addition, the transition condition of breather and lump wave are obtained by using characteristic line analysis. The state transition waves consist of quasi-anti-dark soliton, M-shaped soliton, oscillation M-shaped soliton, multi-peak soliton, W-shaped soliton, and quasi-periodic wave soliton. Through analysis, when solitary wave and periodic wave components undergo nonlinear superposition, it leads to the formation of breather waves and transformed wave structures. It can be used to explain the deformable collisions of transformation waves after collision. Furthermore, the time-varying property of transformed waves are studied using characteristic line analysis. Based on the high-order breather solutions, the interactions involving breathers, state transition waves, and solitons are exhibited. Finally, the dynamics of these hybrid solutions are analyzed through symbolic computations and graphical representations.

Construction of Soliton Solutions of Time-Fractional Caudrey–Dodd–Gibbon–Sawada–Kotera Equation with Painlevé Analysis in Plasma Physics

Fractional calculus with symmetric kernels is a fast-growing field of mathematics with many applications in all branches of science and engineering, notably electromagnetic, biology, optics, viscoelasticity, fluid mechanics, electrochemistry, and signals processing. With the use of the Sardar sub-equation and the Bernoulli sub-ODE methods, new trigonometric and hyperbolic solutions to the time-fractional Caudrey–Dodd–Gibbon–Sawada–Kotera equation have been constructed in this paper. Notably, the definition of our fractional derivative is based on the Jumarie’s modified Riemann–Liouville derivative, which offers a strong basis for our mathematical explorations. This equation is widely utilized to report a variety of fascinating physical events in the domains of classical mechanics, plasma physics, fluid dynamics, heat transfer, and acoustics. It is presumed that the acquired outcomes have not been documented in earlier research. Numerous standard wave profiles, such as kink, smooth bell-shaped and anti-bell-shaped soliton, W-shaped, M-shaped, multi-wave, periodic, bright singular and dark singular soliton, and combined dark and bright soliton, are illustrated in order to thoroughly analyze the wave nature of the solutions. Painlevé analysis of the proposed study is also part of this work. To illustrate how the fractional derivative affects the precise solutions of the equation via 2D and 3D plots.

Study of lump and periodic waves along with rogue waves and breathers for mathematical modeling of biological dynamical system

Globally, infectious diseases pose a significant threat. Notable cases include influenza, Hepatitis B and HIV. COVID-19, caused by the coronavirus, has been the subject of recent discussion due to its great transmissibility. This study explores the Susceptible–Infectious–Recovered (SIR) epidemic model, explaining several analytical approaches to comprehend the nonlinear incident rates and geographic dispersion. The model investigates phenomena such as M-shaped solitons, homoclinic breather-like solitons, lump waves (LWs), lump solution (LS) with one or two kinks, rogue waves (RWs), periodic waves (PWs) and periodic cross-kink waves. These solutions aid in understanding the disease spread and informing containment strategies by identifying optimal outbreak control methods. This work studies localized wave solutions in nonlinear wave equations, known as soliton solutions including the LS. RWs are characterized by unexpectedly large amplitude and rapid profile changes, drawing attention for their erratic features. PWs exhibit periodic repetitions over time and space. Besides, some type of solitons with special waveforms are the periodic cross-kink, M-shaped and homoclinic breather-like solitons. Graphical representation visualizes the behavior of these effective waves, providing insights into virus spread in specific regions over time. SIR models help identify optimal strategies for controlling outbreaks. The work adds to our understanding of epidemic dynamics by illuminating how the movement of susceptible and infected individuals affects the spread of disease.

Multiple solitons structures in optical fibers via PNLSE with a novel truncated M-derivative: modulated wave gain

This study introduces a novel truncated Mittage–Leffler (M)- proportional derivative (TMPD) and examines its impact on the perturbed nonlinear Schrödinger equation (PNLSE) that includes fourth-order dispersion and cubic-quintic nonlinearity. The TMPD-PNLSE is used to model light signals in nanofibers. In addition to dispersion and Kerr nonlinearity, which are characteristics of the NLSE, the PNLSE also exhibits self-steepening and self-phase modulation effects. The unified method is implemented to derive exact solutions for the model equation. These solutions provide a variety of phenomena; including breathers, geometric chaos, and complex solitons. The solutions also exhibit numerous structures, such as geometric chaos, where undulated M-shaped and M-shaped solitons are embedded. The modulation instability is analyzed, finding that it is triggered when the coefficient of the fourth-order dispersion surpasses a critical value.

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.

M-shaped solitons in cubic nonlinear media with a composite linear potential

M-shaped solitons in cubic nonlinear media with a composite linear potential

Construction of M-shaped solitons for a modified regularized long-wave equation via Hirota's bilinear method

Construction of M-shaped solitons for a modified regularized long-wave equation <i>via</i> Hirota's bilinear method

Rhombus-shaped, M-shaped- solitons, 2D-lumps vector and tunneling in perturbed Gerdjikov–Ivanov equation with fourth-order dispersion: Modulation instability and global bifurcation

The Perturbed Gerdjikov–Ivanov Equation (PGIE) describes optical solitons in nonlinear fiber optics and photonic crystal fibers. It was currently studied in the literature. Here, we consider a new complex field equation, the PGIE with fourth-order dispersion. The exact solutions of the aforementioned equation are obtained by implementing the unified method. These solutions are evaluated numerically and they are shown in graphs. Our objectives, here, are to inspect the onset of self- steepening and self-phase modulation and investigate the structures of solitons produced. It is found that these phenomena are launched for high values of the relevant coefficients, and they may be overlapped. Multiple solitons shapes are observed; M-shaped solitons, rhombus shapes, solitons with tunneling, 2D-lumps vector, and solitons- cascade. Further, the analysis of modulation instability (MI) and global bifurcation are carried out. It is established that the MI triggers when the coefficient of Raman scattering exceeds a critical value. It is worth mentioning that two kinds of bifurcations can be studied, namely local and global bifurcations. Here, we are concerned with studying global bifurcation. This is based on constructing the phase portrait via Hamiltonian function.

Study of a nonlinear Schrodinger equation with truncated M proportional derivative

In this work, we introduce a novel truncated M proportional (T-MP) derivative and consider a T-MP perturbed derivative nonlinear Schrodinger equation (PDNLSE). The PDNLSE is a nonlinear model which arises in nano optical fibers (photonic nanowires). While the DNLSE exhibits self-steepening (SS), the PDNLSE exhibits self-phase modulation (SPM) and Raman scattering (RS) effects, Here, the exact solutions of the PDENLSE are derived, here, by implementing the unified method. These solutions are displayed in graphs. It is found that a sufficient condition for a hyper-chaotic solution to hold is that an elliptic function solution exists. Non elliptic solutions may be, also, hyper-chaotic. It is worth mentioning that hyper chaotic may occur in economic and financial mathematics. Nonchaotic solutions exhibit many geometric structures, chirped solitons, and rhombus-shaped and M-shaped solitons. It is found that modulation instability triggers when the coefficient of the third-order dispersion exceeds a critical value. Further, the global bifurcation is investigated via phase portrait by constructing the Hamiltonian function.

Dynamical Study of Coupled Riemann Wave Equation Involving Conformable, Beta, and M-Truncated Derivatives via Two Efficient Analytical Methods

In this study, the Jacobi elliptic function method (JEFM) and modified auxiliary equation method (MAEM) are used to investigate the solitary wave solutions of the nonlinear coupled Riemann wave (RW) equation. Nonlinear coupled partial differential equations (NLPDEs) can be transformed into a collection of algebraic equations by utilising a travelling wave transformation. This study’s objective is to learn more about the non-linear coupled RW equation, which accounts for tidal waves, tsunamis, and static uniform media. The variance in the governing model’s travelling wave behavior is investigated using the conformable, beta, and M-truncated derivatives (M-TD). The aforementioned methods can be used to derive solitary wave solutions for trigonometric, hyperbolic, and jacobi functions. We may produce periodic solutions, bell-form soliton, anti-bell-shape soliton, M-shaped, and W-shaped solitons by altering specific parameter values. The mathematical form of each pair of travelling wave solutions is symmetric. Lastly, in order to emphasise the impact of conformable, beta, and M-TD on the behaviour and symmetric solutions for the presented problem, the 2D and 3D representations of the analytical soliton solutions can be produced using Mathematica 10.

An extended Hirota bilinear method and new wave structures of (2+1)-dimensional Sawada–Kotera equation

An extended Hirota bilinear method is proposed via changing the summation rules of τ function in the multiple soliton solution expressions while investigating the (2+1)-dimensional Sawada–Kotera equation using the classical Hirota bilinear method. Multi-soliton solutions possessing n arbitrary molecule parameters σi(i=1,…,n),n(n−1)/2 interaction parameters bij(i<j=2,3,…,n), and showing a series of complicated dynamic behaviors are obtained. Asymptotic analysis unveils the dynamical evolution characteristics of the new two-soliton solutions. Abundant wave structures involving single peak soliton, M-shaped soliton, molecular soliton, web-like soliton and various interaction structures are presented and discussed simultaneously.

Multi-peak and rational soliton propagations for (3 + 1)-dimensional generalized Konopelchenko–Dubrovsky–Kaup Kupershmidt model in fluid mechanics, ocean dynamics and plasma physics

This paper retrieves the investigation of rational solitons via symbolic computation with logarithmic transformation and ansatz functions approach for the [Formula: see text]-dimensional generalized Konopelchenko–Dubrovsky–Kaup-Kupershmidt (GKDKK) equation in fluid mechanics, ocean dynamics and plasma physics. We find two categories of M-shaped rational solitons and their dynamics will be revealed through graphs by choosing the suitable values of involved parameters. In addition, two categories of M-shaped rational solitons and their interactions with kink waves will be analyzed. Furthermore, homoclinic breathers, multi-wave and kink cross rational solitons will be investigated. The periodic, rational, dark, bright, Weierstrass elliptic function and positive soliton solutions will also be retrieved with the aid of Sub-ODE approach. Moreover, stability characteristics of solutions will be evaluated.

Breathers, Transformation Mechanisms and Their Molecular State of a (3+1)-Dimensional Generalized Yu–Toda–Sasa–Fukuyama Equation

A (3+1)-dimensional generalized Yu–Toda–Sasa–Fukuyama equation is considered systematically. N-soliton solutions are obtained using Hirota’s bilinear method. The employment of the complex conjugate condition of parameters of N-soliton solutions leads to the construction of breather solutions. Then, the lump solution is obtained with the aid of the long-wave limit method. Based on the transformation mechanism of nonlinear waves, a series of nonlinear localized waves can be transformed from breathers, which include the quasi-kink soliton, M-shaped kink soliton, oscillation M-shaped kink soliton, multi-peak kink soliton, and quasi-periodic wave by analyzing the characteristic lines. Furthermore, the molecular state of the transformed two-breather is studied using velocity resonance, which is divided into three aspects, namely the modes of non-, semi-, and full transformation. The analytical method discussed in this paper can be further applied to the investigation of other complex high-dimensional nonlinear integrable systems.

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.

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.