Various Types Of Solitons Research Articles

Analytical and dynamical analysis of nonlinear Riemann wave equation in plasma systems

The Riemann wave equation presents appealing nonlinear equations applicable in sea-water and tsunami wave propagation, ion and magneto-sound waves in plasmas, electromagnetic waves in transmission lines, and homogeneous stationary media. This study focuses on deriving soliton solutions in optics and exploring their physical properties. A wave transformation is used to convert a partial differential equation into an ordinary differential equation, from which soliton solutions are obtained using the generalized Riccati equation mapping approach. The solutions encompass various types of solitons, including bright, dark, periodic, and kink solitons. A comparison of solutions from this analytical method enhances the understanding of the nonlinear model’s behavior, with implications in plasma physics, fluid dynamics, optics, and communication technology. Additionally, 2D and 3D graphs illustrate the physical phenomena of the solutions using appropriate constant parameters. The qualitative analysis of the undisturbed planar system involves examining phase portraits in bifurcation theory, followed by introducing an outward force to induce disruption and reveal chaotic phenomena. Chaotic trajectories in the perturbed system are detected through various plots, including 3D, 2D, power spectrum, and chaotic attractor, alongside Lyapunov exponents. Stability analysis under different initial conditions is conducted, and sensitivity assessments are performed using the Runge–Kutta method. The findings are innovative and have not been previously explored for this system, highlighting the reliability, simplicity, and effectiveness of these techniques in analyzing nonlinear models in mathematical physics and engineering.

Impact of high dispersion and cubic-quintic-septic nonlinearity on optical solitons perturbations of the resonant nonlinear Schrödinger equation with multiplicative white noise

This study investigates the effect of high dispersion, perturbations, cubic-quintic-septic nonlinearity, and multiplicative white noise on the resonant nonlinear Schrödinger equation in the Bohm potential. The model being studied is novel and requires investigation into its behavior and characteristics. Two integration schemes, namely the unified Riccati expansion method and the Jacobi elliptic function method, are employed in this study to seek soliton solutions for the governing model. Various types of solitons, such as bright, dark, and singular solitons, are observed. Moreover, solutions shaped like Jacobi's elliptic functions are identified. Parametric constraints characterize the obtained soliton solutions. Furthermore, the results are presented comprehensively, accompanied by visual representations.

Multiwaves, breathers, lump and other solutions for the Heimburg model in biomembranes and nerves

In this manuscript, a mathematical model known as the Heimburg model is investigated analytically to get the soliton solutions. Both biomembranes and nerves can be studied using this model. The cell membrane’s lipid bilayer is regarded by the model as a substance that experiences phase transitions. It implies that the membrane responds to electrical disruptions in a nonlinear way. The importance of ionic conductance in nerve impulse propagation is shown by Heimburg’s model. The dynamics of the electromechanical pulse in a nerve are analytically investigated using the Hirota Bilinear method. The various types of solitons are investigates, such as homoclinic breather waves, interaction via double exponents, lump waves, multi-wave, mixed type solutions, and periodic cross kink solutions. The electromechanical pulse’s ensuing three-dimensional and contour shapes offer crucial insight into how nerves function and may one day be used in medicine and the biological sciences. Our grasp of soliton dynamics is improved by this research, which also opens up new directions for biomedical investigation and medical developments. A few 3D and contour profiles have also been created for new solutions, and interaction behaviors have also been shown.

Resonant collisions among X-type, multi-lump, generalized breathers, N-solitons and rogue waves in plasma

This paper studies the (1+1)-dimensional Kuramoto–Sivashinsky equation (1D-KSE) also known as canonical evolution equation within the context of flame front propagation, plasma instabilities, and phase turbulence in reaction–diffusion systems. We use Hirota’s derivatives to formulate bilinear equations and subsequently compute various types of solitons. We obtain lump solution, lump one strip, lump two strip, X-type, lump periodic solution, generalized breather and rogue wave solutions for our governing model with the help of Hirota bilinear method (HBM) and the ansatz approach. Breathers solutions may be used to improve the efficiency of fiber optic communication systems and solitons in plasma waves. Rogue wave solutions may be used for the safety of oil platforms and ships while lump wave solutions can be utilized in manipulating and controlling laser beams for materials processing or laser surgery. We also discuss one, two and other soliton interactions for 1D-KSE under some constraint conditions. These interactions may be discussed in plasma stability and confinement, which can control fusion as a future energy source.

Kink and breather waves with and without singular solutions to the Zoomeron model

The Zoomeron model is applied to various types of solitons arising in fluid mechanics, laser optics, and nonlinear physics. We derive analytical solutions to the mentioned model using the exp(−λ(ς))-expansion, the generalized Kudryashov, and the generalized tanh schemes. We obtain the kink waves, breather waves, bright soliton, and dark soliton of this model via symbolic computation. We also get both singular kink waves and singular breather waves. The dynamics of the adopted results are shown in density and 3D graphics. The obtained outcomes will play a vital role in further studies of complicated nonlinear models.

Soliton interactions and Yang–Baxter maps for the complex coupled short‐pulse equation

AbstractThe complex coupled short‐pulse equation (ccSPE) describes the propagation of ultrashort optical pulses in nonlinear birefringent fibers. The system admits a variety of vector soliton solutions: fundamental solitons, fundamental breathers, composite breathers (generic or nongeneric), as well as so‐called self‐symmetric composite solitons. In this work, we use the dressing method and the Darboux matrices corresponding to the various types of solitons to investigate soliton interactions in the focusing ccSPE. The study combines refactorization problems on generators of certain rational loop groups, and long‐time asymptotics of these generators, as well as the main refactorization theorem for the dressing factors that leads to the Yang–Baxter property for the refactorization map and the vector soliton interactions. Among the results obtained in this paper, we derive explicit formulas for the polarization shift of fundamental solitons that are the analog of the well‐known formulas for the interaction of vector solitons in the Manakov system. Our study also reveals that upon interacting with a fundamental breather, a fundamental soliton becomes a fundamental breather and, conversely, that the interaction of two fundamental breathers generically yields two fundamental breathers with a polarization shifts, but may also result into a fundamental soliton and a fundamental breather. Explicit formulas for the coefficients that characterize the fundamental breathers, as well as for their polarization vectors are obtained. The interactions of other types of solitons are also derived and discussed in detail and illustrated with plots. New Yang–Baxter maps are obtained in the process.

Solitons in spin-orbit-coupled systems with fractional spatial derivatives

We demonstrate the existence of various types of solitons in the spin-orbit-coupled systems with the fractional dimension based on Lévy random flights, including the systems with or without Zeeman splitting. Specifically, the systems without Zeeman splitting can support families of symmetric solitons, whereas the systems with Zeeman splitting can support families of stable asymmetric solitons. These coupled solitons may come in the form of fundamental single solitons or dipole solitons. The Lévy index, the strength of self- and cross-phase modulation, and the propagation constant strongly affect the waveforms and stability domains of coupled solitons. The stability and instability domains of such single and dipole solitons are calculated by the method of linear stability analysis and are confirmed by the numerical simulation of perturbed propagation. The general conclusion is that for the Lévy index close to 2, corresponding to the normal nonlinear optics, the solitons tend to be stable, while in the opposite case of Lévy index close to 1, corresponding to Cauchy random flights, the solitons tend to become unstable.

Determinant representation of Darboux transformation for the (2+1)-dimensional nonlocal nonlinear Schrödinger-Maxwell-Bloch equations

In this paper, we mainly investigate two-fold Darboux transformation and N-fold Darboux transformation in terms of ratio of ordinary determinant for the (2+1)-dimensional nonlocal nonlinear Schrödinger-Maxwell-Bloch equations with reverse-space. It should be mentioned that the questions of one-fold Darboux transformation addressed, results of one-fold Darboux transformation proved, as well as style of analysis borrow heavily, but the determinant representation involved to cope with higher-order Darboux transformation is nontrivial. By utilizing symbolic computation, some explicit solutions including periodic waves solutions, soliton solutions and complexion solutions are obtained for two-fold Darboux transformation. Choosing particular parameters, figures are plotted to show the propagations and interactions of various types of solitons, such as periodic waves, bright soliton, dark soliton, antidark soliton, two-peak bright soliton, two-peak dark soliton and two-peak antidark soliton.

Generation and categories of solitons in various mode-locked fiber lasers

Optical solitons generated from fiber lasers have been extensively investigated for their good stability, narrow pulse width and high peak power, which play an indispensable role in many fields such as mechanical micromachining, optical microscope, imaging technology, optical soliton communication and so on. In this review, we summarize the development of soliton fiber laser, and the current application of soliton pulses. There are many different kinds of mode-locking technologies to generate solitons, such as actively mode-locked fiber lasers, passively mode-locked fiber lasers and hybrid mode-locked fiber lasers. During the long-term research of solitons, various types of solitons have been gradually realized under different group velocity dispersion (GVD) conditions in passively mode-locked fiber lasers, such as conventional solitons (CSs) (GVD < 0), dispersion-managed solitons (DMSs) (GVD≈0), self-similar solitons (GVD ≥ 0), dissipative solitons (DSs) (GVD > 0). And in order to apply solitons to more fields, soliton amplification and compression technologies are widely used to improve the performance of solitons, such as chirped pulse amplification (CPA) technology, self-similar soliton amplification technology, large-mode area (LMA) fiber, etc. Finally, the development prospect of soliton in the future is prospected.

Exact optical solitons to the perturbed nonlinear Schrödinger equation with dual-power law of nonlinearity

In this paper, the perturbed nonlinear Schrodinger equation with dual-power law of nonlinearity is studied by adopting four mathematical methods namely the Modified Kudryashov method, the extended Tanh–Coth method, the modified simple equation method and soliton ansatz method. As a results, a set of various types of solitons that contains dark, singular, dark–singular, bright optical solitons and other form of optical soliton solutions are obtained. Firstly, we solve the perturbed nonlinear Schrodinger equation by considering the dual power law parameter using the first three integration methods and secondly we set this parameter equal to one in order, to solve the problem with the fourth method. The used methods in this paper present various applications in fields of nonlinear wave equations. Comparing our new results with well-known in literature are also given. Moreover, the graphical representations of the modulus of some optical soliton solutions and their 2-D profile are also depicted.

Numerical Consideration of Chen-Lee-Liu Equation through Modification Method for Various Types of Solitons

The purpose of the current study is to assess the effectiveness and exactness of the new Modification of the Adomian Decomposition (MAD) method in providing fast converging numerical solutions for the Chen-Lee-Liu (CLL) equation. In addition, we are able to simulate the scheme and provide a comparative analysis with the help of some exact soliton solutions in optical fibers. Finally, the MAD method uncovered that the strategy is proven to be reliable due to the elevated level of accuracy and less computational advances, as demonstrated by a series of tables and figures.

Interactions of solitons, dromion-like structures and butterfly-shaped pulses for variable coefficient nonlinear Schrödinger equation

Investigation on interactions of various types of solitons plays an important role in the analysis of various physical mechanisms. In this paper, analytic two- and three-soliton solutions for the variable coefficient nonlinear Schrödinger (vcNLS) equation are obtained with the help of the Hirota method. Through choosing the related parameters of solutions, we present various types of soliton interactions, and analyze the influence of corresponding parameters on soliton interactions. Periodic and oscillatory interactions between parabolic solitons are observed firstly. The method to control the butterfly-shaped pulse interaction is provided. Those results have guiding effect on controlling the interaction of solitons in nonlinear optics and Bose–Einstein condensation.

Spatial solitons in photonic lattices with large-scale defects**Project supported by the National Natural Science Foundation of China (Grant No. 10704067) and the Natural Science Foundation of Zhejiang Province, China (Grant No. Y6100381).

We address the existence, stability and propagation dynamics of solitons supported by large-scale defects surrounded by the harmonic photonic lattices imprinted in the defocusing saturable nonlinear medium. Several families of soliton solutions, including flat-topped, dipole-like, and multipole-like solitons, can be supported by the defected lattices with different heights of defects. The width of existence domain of solitons is determined solely by the saturable parameter. The existence domains of various types of solitons can be shifted by the variations of defect size, lattice depth and soliton order. Solitons in the model are stable in a wide parameter window, provided that the propagation constant exceeds a critical value, which is in sharp contrast to the case where the soliton trains is supported by periodic lattices imprinted in defocusing saturable nonlinear medium. We also find stable solitons in the semi-infinite gap which rarely occur in the defocusing media.

Dynamics of ultrashort optical solitons in a system of nonsymmetric quantum objects

This paper theoretically discusses the propagation of an ultrashort laser pulse consisting of ordinary and extraordinary components in a system of anisotropic quantum centers. The treatment does not use the assumption that the density of quantum centers is small and makes no limitations on the transition dipole moments. It is shown that, if the spectral pulse width is much greater than the frequency of the quantum transitions, the dynamics of its ordinary component obeys a modified sine-Gordon equation that belongs to the class of integrable models. Soliton solutions of the given equation are constructed, and they are classified. Depending on the duration of the soliton, solutions are possible in the form of a 2π pulses (kinks and antikinks) or 0π steady-state traveling pulses (neutral kinks), which have no counterpart in the isotropic case. The behavior of the medium is studied when various types of solitons are transmitted.

On the current conversion problem in charge density wave crystals. II. Dislocations

The theory of solitons in crystallographically ordered charge density waves (CDW) at low temperatures is developed. Various types of solitons, which appear in a course of electron's self-trapping, are described. The adaptation of CDW media to the formation of grand 2 grsolitons is considered. A model independent approach is suggested to describe interactions of solitons with the Coulomb field and with phase deformations of CDW. The effects of screening, self-screening and commensurability are considered. Interactions between solitons and with impurities are described. Cases of attraction among solitons are found, which may lead to their aggregation into microscopical phase-slip centers.

Solitons in charge density wave crystals

The theory of solitons in crystallografically ordered charge density waves (CDW) at low temperatures is developed. Various types of solitons, which appear in a course of electron's selftrapping are described. The adaptation of CDW media to the formation of π- and 2π-solitons is considered. A model independent approach is suggested to describe interactions of solitons with the Coulomb field and with phase deformations of CDW. The effects of screening, selfscreening and commensurability are considered. Interactions between solitons are described. Cases of attraction among solitons are found, which may lead to their aggregation into microscopical phase-slip centers.

On the current conversion problem in charge density wave crystals. I. Solitons

The theory of solitons in crystallographically ordered charge density waves (CDW) at low temperatures is developed. Various types of solitons, which appear in a course of electron's self-trapping, are described. The adaptation of CDW media to the formation of grand 2 grsolitons is considered. A model independent approach is suggested to describe interactions of solitons with the Coulomb field and with phase deformations of CDW. The effects of screening, self-screening and commensurability are considered. Interactions between solitons and with impurities are described. Cases of attraction among solitons are found, which may lead to their aggregation into microscopical phase-slip centers.

Soliton generation in stimulated Raman scattering and excitation of polar optical phonons

A theoretical analysis is made of the soliton regime in transient stimulated Raman scattering of light by polaritons. Conditions for soliton generation at the frequencies of the interacting waves are determined. It is found that the soliton regime is achieved for in-phase and antiphase configurations of the mechanical vibrations and oscillations of the infrared electromagnetic field. Destructive spatial-temporal modulation of the relative phase occurs in other cases. Analytic equations are derived to describe the main soliton parameters (amplitude, profile, dispersion law, velocity, and duration). It is found that the solitons have a Lorentzian profile and are π pulses, and their dispersion law corresponds to the center of the scattering line of stationary polaritons. An investigation is made of the frequency dependence of the soliton velocity near the lower and upper polariton branches and the ranges of polariton frequencies for which various types of solitons may be generated for given parameters of the problem are determined.

Phase Hamiltonian and Solitons in Dimerized and Trimerized Systems

Simple descriptions in terms of the phase variables are made for various types of solitons in dimerized and trimerized states of one-dimensional interacting electron system in the continuum limit. Effect of mutual interactions between electrons on such characteristics of a soliton as the formation energy, the charge and spin distributions, and the lattice displacement pattern are examined in the classical approximation.

Phase Hamiltonian and Solitons in Dimerized and Trimerized Systems

Simple descriptions are made for various types of solitons in dimerized and trimerized states of one-dimensional interacting electron system in terms of the phase variables. Formation energies of these solitons are evaluated classically.