Soliton Solutions Research Articles

Optical soliton solutions and modulation instability for unstable conformable Schrödinger model

The unstable time fractional Schrödinger model (UTFSM) is studied through the development of disturbances in marginally stable or unstable media. A modified Sardar-sub equation technique (MSSE) for a conformable fractional-order nonlinear evolution model is presented in this paper. The objective here is to construct new wave solutions for UTFSM. These solutions have particular relevance in quantum physics and assume several forms such as rational, exponential, trigonometric and hyperbolic functions, as well as combo solutions. This technique produces various shapes of dark, kink-type solitons and periodic solitary waves by setting proper parametric values. These discrete physical frameworks contribute to an understanding from analysis of unstable dynamical models. Additionally, we investigate modulation instability and stability analysis to ensure that obtained solutions are highly stable. The multiplicity of waves and solutions emphasizes how this technique can be used for different nonlinear fractional models in quantum physics and other areas.

Lump solitions, fractal soliton solutions, superposed periodic wave solutions and bright-dark soliton solutions of the generalized (3+1)-dimensional KP equation via BNNM

Lump solitions, fractal soliton solutions, superposed periodic wave solutions and bright-dark soliton solutions of the generalized (3+1)-dimensional KP equation via BNNM

Dynamics of novel soliton and periodic solutions to the coupled fractional nonlinear model

This study secures the soliton solutions of the (2+1)-dimensional Davey–Stewartson equation (DSE) incorporating the properties of the truncated M-fractional derivative. The DSE and its coupling with other systems have extensive applications in many fields, including physics, applied mathematics, engineering, hydrodynamics, plasma physics, and nonlinear optics. Various solutions, such as dark, singular, bright-dark, bright, complex, and combined solitons, are derived. In addition, exponential, periodic, and hyperbolic solutions are also generated. The newly designed integration method, known as the modified Sardar subequation method (MSSEM), has been applied in this study for extracting the solutions. The approach is efficient in explaining fractional nonlinear partial differential equations (FNLPDEs) by confirming pre-existing solutions and producing new ones. Furthermore, we plot the density, 2D, and 3D graphs with the associated parameter values to visualize the solutions. The outcomes of this work indicate the effectiveness of the method utilized to improve nonlinear dynamical behavior. We anticipate that our work will be helpful for a large number of engineering models and other related problems.

Vector multipole solitons of fractional-order coupled saturable nonlinear Schrödinger equation

Three kinds of vector multipole solitons of fractional coupled saturable nonlinear Schrödinger equation are reported, including fractional dipole-dipole, dipole-tripole and tripole-dipole vector soliton solutions. Firstly, their existence domains, which are modulated by potential function parameters, are constructed in a certain interval. Secondly, the stable regions of three kinds of vector multipole solitons, which are modulated by the soliton power of each component, are found. The properties of solitons are explored through these existence and stability domains. Finally, the stability of three kinds of fractional vector multipole solitons is verified by the numerical evolution. Compared with the integer-order results, there are differences in the existence and stable regions of soliton solutions, and the Lévy index affects the existence and stability of vector multipole solitons.

Riemann–Hilbert problem for a (3+1)-dimensional nonlinear evolution equation

This paper concentrates on a (3+1)-dimensional nonlinear evolution equation. By introducing a transformation, the (3+1)-dimensional nonlinear evolution equation is decomposed into three integrable (1+1)-dimensional models. On the basis of a quartet Lax pair, we build the associated matrix Riemann–Hilbert problem. As a consequence, solving the obtained matrix Riemann–Hilbert problem with the identity jump matrix, corresponding to the reflectionless, the soliton solution to the (3+1)-dimensional nonlinear evolution equation is acquired. Specially, the one-soliton solutions are worked out and analyzed graphically.

Analytic wave solutions to the beta-time fractional modified equal width equation based on two efficient approaches

By using the two distinct methods known as the exp(-ϕ(η))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\exp (-\\phi (\\eta ))$$\\end{document} and the Expa\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Exp_a$$\\end{document}-function methods, various forms of soliton solutions of the modified equal width wave equation (MEWE) with beta time derivative (BTD) are produced in this study. This model is used as a model in partial differential equations for the simulation of one-dimensional wave transmission in nonlinear media with dispersion processes. The obtained solutions are in the form of rational, trigonometry and hyperbolic trigonometry functions. Using Mathematica software, the resulting solitons are validated. Graphs are also used at the conclusion to explain the findings. These soliton solutions imply that these two methods are more dependable, simple and efficient than other methods. The findings can be used to explain how studious structures and other comparable non-linear physical structures are substantially understood. The obtained results are very helpful in the fields of optics, kinetics solid-state physics and hydro-magnetic waves in cold plasma.

Application of the G’/G Expansion Method for Solving New Form of Nonlinear Schrödinger Equation in Bi-Isotropic Fiber

Bi-isotropic media (chiral and non-reciprocal) present an outstanding challenge for the scientific community. Their characteristics have facilitated the emergence of new and remarkable applications. In this paper, we focus on the novel effect of chirality, characterized through a newly proposed formalism, to highlight the nonlinear effect induced by the magnetization vector under the influence of a strong electric field. This research work is concerned with a new formulation of constitutive relations. We delve into the analysis and discussion of the family of solutions of the nonlinear Schrödinger equation, describing the pulse propagation in nonlinear bi-isotropic media, with a novel approach to constitutive equations. We apply the extended -expansion method with varying dispersion and nonlinearity to define certain families of solutions of the nonlinear Schrödinger equation in bi-isotropic (chiral and non-reciprocal) optical fibers. This clarification aids in understanding the propagation of light with two modes of propagation: a right circular polarized wave (RCP) and a left circular polarized wave (LCP), each having two different wave vectors in nonlinear bi-isotropic media. Various novel exact solutions of bi-isotropic optical solitons are reported in this study. Introduction: The investigation of exact solutions for nonlinear partial differential equations (PDEs) holds significant importance in understanding nonlinear physical phenomena. Nonlinear waves manifest across various scientific domains, notably in optical fibers and solid-state physics. In recent years, several potent methodologies have emerged for identifying solitons and periodic wave solutions of nonlinear PDEs. These include the -expansion method [1-6], the new mapping method [9-10], the method of generalized projective Riccati equations [11-16], and the expansion method [17]. Consequently, an original mathematical approach is proposed to evaluate nonlinear effects in bi-isotropic optical fibers, stemming from magnetization under the influence of a strong electric field [19-20]. The extended -expansion method emerges as a potent technique for deriving solution families of the nonlinear Schrödinger equation in bi-isotropic optical fibers. This method employs a perturbation expansion in powers of the dimensionless parameter and is applicable for both weak and strong nonlinearities. It accommodates varying dispersion and nonlinearity, rendering it suitable for modeling a wide array of optical fibers. Results and Conclusion: This investigate is concerned with a new formulation of constitutive relation linking to the magnetic effect, to understand rigorously the physical nature of biisotropic effects and to generalize the main macroscopic models. We inferred the nonlinear Schrodinger equation for a bi-isotropic medium term with a nonlinear term of magnetizing. In this article, the extended -expansion method is a powerful technique for determining a family of solutions of the nonlinear Schrödinger equation in bi-isotropic optical fibers. This method is based on the use of a perturbation expansion in powers of the dimensionless parameter, and it is valid for both weak and strong nonlinearities. The method allows for the inclusion of varying dispersion and nonlinearity, making it well-suited for modeling a wide range of optical fibres. Overall, the extended -expansion method is a valuable tool for understanding the dynamics of nonlinear optical systems, and it is expected to have a wide range of applications in the field of nonlinear optics.

Study of multi solitons, breather soliton structures in the earth's magnetotail region

The investigation of wave packets with varying magnitudes, which predominantly carry energy across different physical mediums, proves to be highly intriguing and unveils numerous nonlinear characteristics within Earth's magnetotail region and magnetosphere, as confirmed by extensive space observations. Nonlinear features in wave dynamics are elucidated through diverse nonlinear structures, among which breather structures hold prominence and manifest across various wave fields. We have considered an unmagnetized collisionless homogeneous hydrodynamical governing equations set in an electron-ion plasma comprising hot and cold ions. In this study, we have explored 1-soliton, 2-soliton, and breather structures in the framework of the Gardner equation (GE). We have derived the GE from the normalized set of governing equations employing the reductive perturbation technique (RPT). The GE, an extension of the Korteweg-de Vries (KdV) equation, encompasses the combined effects of quadratic and cubic nonlinearities. By employing the Hirota bilinear method (HBM), we have obtained multi-soliton solutions and breather soliton solutions. We have noticed the variations in electron temperature ratios and density concentration ratios substantially impact and reshape breather soliton structures. Such alterations in breather structures, influenced by different electron temperatures and concentration ratios, hold potential significance in understanding energy transport within Earth's magnetotail region. The alteration of shapes and soliton structures may also be investigated in the dynamics of wave fields upon interaction such as hydrodynamics, optics, optic fibers, signal processing, etc.

Symmetry-Optimized Dynamical Analysis of Optical Soliton Patterns in the Flexibly Supported Euler–Bernoulli Beam Equation: A Semi-Analytical Solution Approach

This study employs spatial optimization principles to investigate the nonlinear vibration of a flexibly supported Euler–Bernoulli beam, a (1 + 1)-dimensional system subjected to axial loads. The modified Khater method, a crucial tool in mechanical engineering, is utilized to analyze analytical solutions, which include a symmetric spatial representation of the waveform as an integral part of each solution. Notably, periodic soliton solutions for the nonlinear model closely align with numerical and approximate analytical solutions, demonstrating the accuracy of our modeling approach. Density diagrams, contour diagrams, and Poincaré maps depicting the obtained analytical solutions are presented to elucidate their accuracy and provide visual confirmation of the optimized engineering model’s physical significance. The planar dynamical system is derived through the Galilean transformation by employing mathematical models and appropriate parameter values, thereby further refining problem understanding. Sensitivity analysis is conducted, and phase portraits with equilibrium points are illustrated by analyzing a special case of the investigated dynamical system, emphasizing its symmetrical properties. Lastly, we perform a global analysis to identify periodic, quasi-periodic, and chaotic behaviors, with an extra weak forcing term confirmed by Poincaré maps and a two-dimensional symmetric basin of the largest Lyapunov exponent.

Abundant Soliton Solutions to the Generalized Reaction Duffing Model and Their Applications

The main aim of this study is to obtain soliton solutions of the generalized reaction Duffing model, which is a generalization for a collection of prominent models describing various key phenomena in science and engineering. The equation models the motion of a damped oscillator with a more complex potential than in basic harmonic motion. Two effective techniques, the mapping method and Bernoulli sub-ODE technique, are used for the first time to obtain the soliton solutions of the proposed model. Initially, the traveling wave transform, which comes from Lie symmetry infinitesimals, is applied, and a nonlinear ordinary differential equation form is derived. These approaches effectively retrieve a hyperbolic, Jacobi function as well as trigonometric solutions while the appropriate conditions are applied to the parameters. Numerous innovative solutions, including the kink wave, anti-kink wave, bell shape, anti-bell shape, W-shape, bright, dark and singular shape soliton solutions, were produced via the mapping and Bernoulli sub-ODE approaches. The research includes comprehensive 2D and 3D graphical representations of the solutions, facilitating a better understanding of their physical attributes and proving the effectiveness of the proposed methods in solving complex nonlinear equations. It is important to note that the proposed methods are competent, credible and interesting analytical tools for solving nonlinear partial differential equations.

The localized excitation on the Jacobi elliptic function periodic background for the Gross–Pitaevskii equation

In this paper, the nonlinear wave solutions for Gross–Pitaevskii equation on the periodic wave background are investigated by Darboux-Bäcklund transformation, from which the soliton and breather wave solutions on the Jacobi elliptic cn and dn functions backgrounds are derived. The corresponding evolutions and dynamical properties of nonlinear wave solutions under different parameters are discussed. These results reported in this paper may raise the possibility of related experiments and potential applications in nonlinear science fields, such as nonlinear optics, oceanography and so on.

PT\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathscr{P}\\mathscr{T}$$\\end{document}-symmetric KdV solutions and their algebraic extension with zero-width resonances

A class of complex breather and soliton solutions to both KdV and mKdV equations are identified with a Pöschl-Teller type PT\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathscr{P}\\mathscr{T}$$\\end{document}-symmetric potential. However, these solutions represent only the unbroken-PT\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathscr{P}\\mathscr{T}$$\\end{document} phase owing to their isospectrality to an infinite potential well in the complex plane having real spectra. To obtain the broken-PT\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathscr{P}\\mathscr{T}$$\\end{document} phase, an extension of the potential satisfying the sl2,R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$sl\\left( 2,\\mathbb {R}\\right)$$\\end{document} potential algebra is mandatory that additionally supports non-trivial zero-width resonances.

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.

[formula omitted]-soliton solutions of coupled Schrödinger–Boussinesq equation with variable coefficients

In this paper, soliton solutions of the coupled variable coefficients Schrödinger–Boussinesq equation, which describes the stationary propagation of coupled upper-hybrid waves and magnetoacoustic waves in a magnetized plasma, are investigated. Based on the Hirota bilinear method, the bilinear form, one, two, three and N-soliton solutions of Schrödinger–Boussinesq equation are derived. By means of numerical calculation, the specific figures of different soliton solutions are obtained by selecting different coefficients. The propagation and interaction of the soliton solutions are analyzed graphically.

On traveling wave solutions with stability and phase plane analysis for the modified Benjamin-Bona-Mahony equation.

The modified Benjamin-Bona-Mahony (mBBM) model is utilized in the optical illusion field to describe the propagation of long waves in a nonlinear dispersive medium during a visual illusion (Khater 2021). This article investigates the mBBM equation through the utilization of the rational [Formula: see text]-expansion technique to derive new analytical wave solutions. The analytical solutions we have obtained comprise hyperbolic, trigonometric, and rational functions. Some of these exact solutions closely align with previously published results in specific cases, affirming the validity of our other solutions. To provide insights into diverse wave propagation characteristics, we have conducted an in-depth analysis of these solutions using 2D, 3D, and density plots. We also investigated the effects of various parameters on the characteristics of the obtained wave solutions of the model. Moreover, we employed the techniques of linear stability to perform stability analysis of the considered model. Additionally, we have explored the stability of the associated dynamical system through the application of phase plane theory. This study also demonstrates the efficacy and capabilities of the rational [Formula: see text]-expansion approach in analyzing and extracting soliton solutions from nonlinear partial differential equations.

Bifurcation analysis and soliton structures of the truncated [formula omitted]-fractional Kuralay-II equation with two analytical techniques

The truncated M-fractional Kuralay (TMFK)-II equation is prevalent in the exploration of specific complex nonlinear wave phenomena. Such types of wave phenomena are more applicable in science and engineering. These equations could potentially provide insights into understanding the intricate dynamics of optical phenomena, encompassing solitons, nonlinear effects, and wave interactions. This study aims to uncover a diverse range of soliton solutions to the model, spanning trigonometric, hyperbolic, exponential, and rational expressions. These solutions are unveiled through the application of extended hyperbolic functions and improved F-expansion techniques, representing the primary objective of this research. The three-dimensional (3D) and two-dimensional (2D) combined charts are plotted for some of these solutions. The impact of the fractional parameters and time variations is also illustrated. Moreover, the models are converted into a planar dynamical system using a Galilean transformation, and the analysis of bifurcation is examined. This research underscores the versatility of the aforementioned techniques for exploring complex nonlinear phenomena across various engineering and scientific disciplines. Finally, the findings of this study hold significant implications for advancing our understanding and analysis of nonlinear wave dynamics in diverse physical systems.

Extraction of soliton solutions for the fractional Kaup-Boussinesq system: A comparative study

This paper is based on finding soliton solutions to fractional Kaup-Boussinesq (FKB) system. The fractional derivatives such as β-derivative and truncated M-fractional derivative are used in this study. The unified approach, generalized projective riccati equations method (GPREM) and improved tan (φ(ζ)/2)-expansion approaches are efficiently used for obtaining bright soliton, dark soliton, singular soliton, periodic soliton, dark-singular combo soliton and dark-bright combo soliton. The numerical simulations are also carried out by 3D and 2D, graphs of some of the obtained solutions to discuss the fractional effects.

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.

Beam propagation in cubic-quintic nonlinear optical system supported by modified parity-time symmetric Rosen-Morse complex potential.

The present study explores the stability and persistence of nonlinear waves in self-focusing cubic-quintic media employing couplings of nonlinearity, spatial diffraction, and the parity-time symmetric Rosen-Morse complex field. Here, we discover that a system supported by Rosen-Morse potential can develop eigenmodes but does not accommodate stable soliton solutions for any potential parameters due to the non-vanishing terms in the imaginary component of potential. The study expands by modifying Rosen-Morse potential and discovers that the region of sustained PT-symmetry in the self-focusing material is enhanced and influenced by the order of the nonlinear function, spectral filtering, and gain/loss in the system. Stable soliton conditions for both broken PT-symmetric and PT-symmetric regions are established by the linear stability analysis using numerical simulations. Nonlinear propagation of the beam in the modified PT system is explored and identifies that stable beam propagation is possible only if the system is supported by the field, which is below the threshold imaginary potential.

Dualism in the Theory of Soliton Solutions

Dualism in the Theory of Soliton Solutions