Mixed Soliton Research Articles

Obtaining new soliton solutions of the fractional generalized perturbed KdV equation

Abstract In this study, the fractional generalized perturbed KdV equation (gpKdV) with beta derivative is considered. The generalized exponential rational function method (GERFM) is applied to this equation for the first time in this study. Thus, dark soliton, bright soliton, singular soliton, mixed soliton, trigonometric function, rational trigonometric function, hyperbolic function and rational exponential function solutions of this fractional equation are obtained for the first time in this study. The 2D, 3D, and density plots, which effectively illustrate the behavior of these solitons, are shown for various values and specific ranges of the solutions.

Double-pole dark-bright mixed solitons for a three-wave-resonant-interaction system

This Letter explores a three-wave-resonant-interaction system that is widely seen in fluid mechanics, plasma physics and nonlinear optics. With the aid of the iterated-form generalized Darboux transformation method, we obtain the second-order analytic solutions that illustrate the double-pole dark-bright mixed solitons. Under certain parameter conditions, the double-pole bright-dark-bright solitons are illustrated through the 3D, density and contour plots. Those plots demonstrate the stable attractive forces between the adjacent soliton components. Moreover, we conduct the asymptotic analysis on the double-pole bright-dark-bright solitons to investigate their physical properties, which are consistent with those in the plots. Our work might contribute to the studies of the dynamic properties of the curved dark-bright solitons and be extended to the double-pole dark-bright mixed solitons within the other integrable systems.

Constructing the soliton wave structure to the nonlinear fractional Kairat-X dynamical equation under computational approach

In this paper, the nonlinear fractional Kairat-X equation is investigated on the basis of computational simulation. The nonlinear fractional Kairat-X equation is an integrable equation and is used to explain the differential geometry of curves and equivalence aspects. Several kinds of solitary wave structures of the nonlinear fractional Kairat-X equation are established successfully via the implantation of the extended simple equation method. Here, we explore the interesting, novel and general solutions in trigonometric, exponential, and rational types, which represent periodic wave solitons, mixed solitons in the shape of bright–dark solutions, kink wave solitons, peakon bright and dark solitons, anti-kink wave solutions, bright solitons, dark solitons, and solitary wave structure. The physical structures of secured results, aided by numerical simulation, have numerous applications in applied sciences such as optical fiber, geophysics, laser optics, mathematical physics, nonlinear optics, nonlinear dynamics, communication system, and engineering. This study explores the physical behavior of models through the visualization of solutions in contour, 2D and 3D plots by revealing that these solutions yield profitable results in the field of mathematical physics. The study demonstrates that the proposed technique is more reliable, efficient, and powerful in analyzing nonlinear evolution equations in various domains of science.

Superposition solitons for the mixed 4-coupled nonlinear Schrödinger equations

In this paper, the mixed 4-coupled nonlinear Schrödinger equations with different nonlinear signs are studied to derive a new type of soliton solutions called the superposition soliton solutions. By using the Hirota method, we obtain the exact one-bright-three-superposition N-soliton solutions analytically. Notably, this kind of soliton solutions have not been researched in prior literature. Under certain conditions, the general mixed (bright-dark) soliton solutions can be obtained from our results such as all bright soliton solutions. In addition, the propagation characteristics, including elastic collision, time periodicity and soliton reaction, are displayed through graphic simulation. On this basis, the influence of various parameters on the phase, direction, and amplitude of soliton propogation is concluded. Finally, the asymptotic behaviors of 2, 3-soliton solutions are analyzed in detail.

Dynamical analysis of exact optical soliton structures of the complex nonlinear Kuralay-II equation through computational simulation

In this paper, we successfully extracted the various types of soliton solutions for the complex nonlinear Kuralay-II equation through the improved F-expansion method with symbolic computational software Mathematica. The extracted soliton solutions for the Kuralay-II equation are interesting, novel and more general such as anti-kink wave solitons, dark solitons, kink wave solitons, bright solitons, periodic wave solitons, mixed solitons in bright-dark soliton shape, peakon solitons, and solitary wave structures. The graphical structure of some extracted solutions is visualized in 2D, 3D and contour plottings with imaginary, real, and absolute values of the functions by using the numerical simulation. The proposed research will contribute to advancing our knowledge about the complex nonlinear Kuralay-II equation and demonstrating the applicability to the proposed approach to investigate other higher-order complex nonlinear equations. The successful investigation demonstrated that the proposed method is effective, simple, more powerful, efficient and can be utilized on a variety of other nonlinear equations. The explored solitary waves and optical solitons will play an important role in the investigation of nonlinear phenomena in various domains of science and engineering.

Analysis of mixed soliton solutions for the nonlinear Fisher and diffusion dynamical equations under explicit approach

Analysis of mixed soliton solutions for the nonlinear Fisher and diffusion dynamical equations under explicit approach

The coupled space-shifted nonlocal nonlinear Schrödinger equation: Multiple bright-dark double-pole solitons, multiple negaton-type solitons, and their associated mixed solitons

We study the multiple bright-dark double-pole solitons, multiple negaton-type solitons, and their associated mixed solitons of the coupled space-shifted nonlocal nonlinear Schrödinger equation. These three distinct classes of soliton waveforms are derived from multiple soliton solutions through three different long-wave limit procedures with specific restrictions of the parameters of the solutions. The multiple bright-dark double-pole soliton solutions are symmetric about the point (x02,0), where x0 is the space-shifting parameter of the coupled space-shifted nonlocal nonlinear Schrödinger equation. The space-shifting parameter x0 only affects half of the multiple bright-dark negaton-type solitons, while it has no impact on the other half. The mixed soliton solutions are composed of multiple double-pole solitons and negaton-type solitons. The unique properties of these three distinct classes of soliton waveforms are examined by performing the long-time asymptotic analysis for them.

Multi-soliton solutions and asymptotic analysis for the coupled variable-coefficient Lakshmanan-Porsezian-Daniel equations via Riemann-Hilbert approach

In this paper, we study multi-soliton solutions and asymptotic analysis for the coupled variable-coefficient Lakshmanan-Porsezian-Daniel equations, which describe the simultaneous propagation of nonlinear waves in the inhomogeneous optical fibers. We analyze the spectrum of the Lax pair to establish the Riemann-Hilbert problem. Using such Riemann-Hilbert problem, we calculate various multi-soliton solutions without reflection, including breather-like and mixed solitons. We illustrate the propagation and interaction dynamics of the solitons through appropriate parameter selection and asymptotic analysis. We find that the interaction between solitons is elastic, the amplitudes of solitons are only determined by the initial velocity and interaction, and the soliton with lower energy always yields a position shift when elastic interaction occurs. In addition, we observe that the existence time of the wave changes with energy and that multiple elastic interactions between solitons can be obtained when we choose appropriate variable coefficients. Then, we investigate the influences of group velocity dispersions and fourth-order dispersions on the interactions of solitons through parameter modulation mode and asymptotic analysis. Furthermore, we present several new types of nonlinear phenomena graphically, including elastic interactions between parabolic solitons and hump-type solitons, elastic interactions between cubic solitons and hump-type solitons, and periodic-changing propagations.

Exact solutions of the nonlocal higher-order Chen–Lee–Liu equation

The subject of this paper is a nonlocal higher-order Chen–Lee–Liu equation, which can be reduced from the generic deformation of generalized Kaup–Newell hierarchy involving the higher-order effects in optical fibers. With the help of the Darboux transformation, soliton molecules, breather molecules, double periodic solutions and various kinds of mixed soliton solutions are displayed, whose dynamic behaviors based on the parameters are graphically demonstrated.

Propagation characteristics of bright and mixed solitons based on the variable coefficient (3+1)-dimensional cubic-quintic complex Ginzburg-Landau equation

In the study of telecommunication system, the variable coefficient (3+1)-dimensional cubic-quintic complex Ginzburg-Landau equation is used as the optical solitons transmission model, which not only explains the physical meaning of the existing model with quintic terms, but also has more nonlinear dynamics characteristics of the higher dimensional system than the lower dimensional system. In this paper, the analytical soliton solutions of the (3+1)-dimensional cubic-quintic CGL equations with variable coefficients are obtained by using the modified Hirota method. By selecting certain parameters of the nonlinear coefficients and spectral filtering terms, a special kind of mixed soliton solution is obtained, which has the characteristics of bright soliton, dark soliton and kinked soliton at the same time. Subsequently, the influence of changing the nonlinear, spectral filtering, linear loss parameters and other parameters on the transmission characteristics of solitons is discussed respectively, so as to realize the control of optical solitons, which can not only control the propagation of optical solitons in different forms, but also can realize the adjustment of the amplitude and pulse width of the pulse and control the propagation direction and energy of the pulse for the mixed solitons of a particular form. The research results of high dimensional CGL system in this paper can be applied to nonlinear optical system, ultra-fast optical digital logic system and other different experiments and application fields.

Exploring the fractional Hirota Maccari system for its soliton solutions via impressive analytical strategies

In the field of marine engineering, the characteristics of wave propagation play an imperative character. In many geographical regions, the key source of environmental effects on artificial floating or stationary structures or seashores is waves. This article deals with the Hirota Maccari system (HMs) via a new fractional derivative operator known as the beta derivative. This aforesaid equation is a significant model which deals with a variety of nonlinear phenomenons in the fields of optical fibers, physics and other scientific fields. A complex wave hypothesis is applied to transform the aforesaid system with beta derivative into an ODE system. The modified Kudryashov’s (New version) and the Auxiliary equation methods are used to build a variety of soliton solutions of HMs with beta derivative. The singular, periodic, bright, dark solitons and mixed solitons are constructed with the above-mentioned approaches via soft computation in Mathematica. Some of them are numerically simulated for 2D and 3D representation. All the solutions of fractional HMs produced by the above two techniques are novel and have not been derived yet.

Dynamics of higher‐order rational and semi‐rational soliton solutions of the coupled modified KdV lattice equation

The coupled modified KdV lattice equation is a discrete version of the coupled modified KdV equation. This paper presents the general formulae for the higher‐order rational, semi‐rational, and mixed soliton solutions in terms of determinants by constructing the generalized perturbation ‐fold Darboux transformation. Particularly, we exhibit the first‐, second‐, and third‐order rational and semi‐rational soliton solutions. Finally, numerical simulations are carried out to explore the dynamical behaviors and instability of those discrete rational and semi‐rational soliton solutions, respectively. The results would enrich our understanding on the two‐layered fluid waves near ocean shores described by the coupled modified KdV lattice equation.

Mixed soliton solutions for the ultradiscrete BKP equation and its reduction

Mixed soliton solutions for the ultradiscrete BKP equation and its reduction

Darboux transformation, exact solutions and conservation laws for the reverse space-time Fokas–Lenells equation

In this paper, we firstly deduce a reverse space-time Fokas–Lenells equation which can be derived from a rather simple but extremely important symmetry reduction of corresponding local equation. Next, the determinant representations of one-fold Darboux transformation and N-fold Darboux transformation are expressed in detail by special eigenfunctions of spectral problem. Depending on zero seed solution and nonzero seed solution, exact solutions, including bright soliton solutions, kink solutions, periodic solutions, breather solutions, rogue wave solutions and several types of mixed soliton solutions, can be presented. Furthermore, the dynamical behaviors are discussed through some figures. It should be mentioned that the solutions of nonlocal Fokas–Lenells equation possess new characteristics different from the ones of local case. Besides, we also demonstrate the integrability by providing infinitely many conservation laws. The above results provide an alternative possibility to understand physical phenomena in the field of nonlinear optics and related fields.

Mixed soliton solutions for the (2+1)-dimensional generalized breaking soliton system via new analytical mathematical method

In this research, we study the dispersive phenomena for the (2+1)-dimensional generalized breaking soliton system (GBSS), which describe the interaction phenomena between Riemann wave and long wave via two space variable in nonlinear media. The constructed solitary wave results for the (2+1)-dimensional GBSS via extended modified rational expansion (EMRE) method in the form of rational, elliptic, and trigonometric functions including mixed solitons, single bright and dark solitions, combined bright and dark solitons, kink and anti-kink wave solitions, traveling waves, and periodic waves. The physical phenomena of demonstrated results represented by higher dimensional and contour plots graphically with the help of symbolic computation. The obtained results are very helpful in the study of interaction phenomena in fiber optics, mathematical physics, fluid dynamics, geophysics, engineering and many other various areas of scientific fields.

Y-shaped soliton solutions for the (2+1)-dimensional bidirectional Sawada–Kotera equation

On the basis of the Hirota bilinear method, resonance Y-shaped soliton and its interaction with other localized waves of (2+1)-dimensional bidirectional Sawada–Kotera equation are derived by introducing the constraint conditions. These types of mixed soliton solutions exhibit complex interaction phenomenon between the resonance Y-shaped solitons and line waves, breather waves, and high-order lump waves. The dynamic behaviors of the interaction solutions are analyzed and illustrated.

Various Soliton Solutions and Asymptotic State Analysis for the Discrete Modified Korteweg-de Vries Equation

Under investigation is the discrete modified Korteweg-de Vries (mKdV) equation, which is an integrable discretization of the continuous mKdV equation that can describe some physical phenomena such as dynamics of anharmonic lattices, solitary waves in dusty plasmas, and fluctuations in nonlinear optics. Through constructing the discrete generalized m , N − m -fold Darboux transformation for this discrete system, the various discrete soliton solutions such as the usual soliton, rational soliton, and their mixed soliton solutions are derived. The elastic interaction phenomena and physical characteristics are discussed and illustrated graphically. The limit states of diverse soliton solutions are analyzed via the asymptotic analysis technique. Numerical simulations are used to display the dynamical behaviors of some soliton solutions. The results given in this paper might be helpful for better understanding the physical phenomena in plasma and nonlinear optics.

Highly dispersive optical soliton perturbation with cubic–quintic–septic law via two methods

The main consideration of this paper is to discuss cubic optical solitons in a polarization-preserving fiber modeled by nonlinear Schrödinger equation (NLSE). We extract the solutions in the forms of hyperbolic, trigonometric including a class of solitary wave solutions like dark, bright–dark, singular, singular periodic, multiple-optical soliton and mixed complex soliton solutions. A recently developed integration tool known as new extended direct algebraic method (NEDAM) is applied to analyze the governing model. Moreover, the studied equation is discussed with two types of nonlinearity. The constraint conditions are explicitly presented for the resulting solutions. The accomplished results show that the applied computational system is direct, productive, reliable and can be carried out in more complicated phenomena.

Multi-fold binary Darboux transformation and mixed solitons of a three-component Gross-Pitaevskii system in the spinor Bose-Einstein condensate

The Bose-Einstein-condensation applications give rise to the superfluidity in the liquid helium and superconductivity in the metals. In this paper, we work on a three-component Gross-Pitaevskii system, which describes the matter waves in an spin-1 spinor Bose-Einstein condensate. We construct a multi-fold binary Darboux transformation with the zero seed solutions to describe the three vertical spin projection of the spin-1 spinor BEC, which is different from all the existing Darboux-type ones for the same system, and derive three types of the exponential-and-rational mixed soliton solutions associated with two conjugate complex eigenvalues. For such mixed solitons, we give their asymptotic expressions, indicating that they consist of the Ieda-Miyakawa-Wadati (IMW)-polar-state or IMW-ferromagnetic solitons but possess the time-dependent velocities. Asymptotically and graphically, interaction mechanisms between the mixed and exponential solitons are classified in six cases, and we exhibit the inelastic and elastic interactions through calculating the modifications of the polarization matrices and phase shifts for the two interacting solitons. We find that both the IMW-polar-state solitons, including the mixed and exponential solitons, can not alter the other soliton’s intensity distribution during the interaction, while the mixed or exponential soliton in the IMW-ferromagnetic state does.

Construction of analytical wave solutions to the conformable fractional dynamical system of ion sound and Langmuir waves

ABSTRACT In this article, the exact wave structures are discussed to conformable time-fractional system of ion sound under influence of ponderomotive force. It is caused by non-linear force and is experienced by a charged particle in an oscillating electromagnetic field (EMF) of inhomogeneity. It is investigated that system exhibits the trigonometric, hyperbolic and exponential function solutions including some special known solitary, shock, singular, single and mixed solitons. The strategy that is used to extract the wave structures for this ionic system of Langmuir waves is known as generalized exponential rational function method. Moreover, the constraint conditions are also reported which validate the existence of solutions. Furthermore, 3-dimensional, 2-dimensional and corresponding contour graphs are presented for the physical movement of the earned solutions under the appropriate selection of the parameters. The obtained results are helpful for understanding of the study of wave propagation and are also vital for numerical and experimental verification in plasma physics.