Soliton Amplitude Research Articles

Propagation of envelope solitons in the presence of generalized (r; q) distribution of electrons

Abstract The study has been made for modulational instability of ion acoustic waves in an electron-ion unmagnetized plasma where ions are dynamic and warm while electrons follows the generalized (r,q) distribution. The Krylov Bogoliubov Mitropolsky (KBM) method is employed to derive the nonlinear Schrödinger equation (NLSE). The dispersive and nonlinear coefficients P and Q are retrieved which depend on σ_{i} (Temperature ratio) and spectral indices (r,q). The bright and dark solitons are excited for the case PQ>0 and PQ<0 respectively. Amplitude of bright solitons is characterized by a localized intensity peak and dark solitons by a localized intensity hole on a homogeneous background. The growth rates and modulationally stable/unstable zones are analyzed numerically. It is established that spectral indices (r,q) of electron distribution function and temperature ratio of ion to electron play a major role in the creation of bright/dark envelope solitons in electron-ion plasmas in the presence of nonthermal distribution.

Sub-pico-second chirped optical solitons in birefringent fibers for space--time fractional Kaup-Newell equation

The present work is devoted to investigate the chirped bright and dark optical solitons of fractional Kaup-Newell equation (KNE) in birefringent fibers. The study is carried out analytically by the traveling wave hypothesis with the conformable derivative which reduces the governing model to an ordinary differential equation (ODE). The obtained equation is handled with the aid of an exotic integration scheme that utilizes the Jacobi elliptic equation in the form of a first-order nonlinear ODE with three-degree terms. Taking the modulus of Jacobi elliptic function to unity, distinct types of bright and dark optical solitons are derived with their corresponding chirping. The fractional order derivative is noted to have a significant influence on the pulse propagation. Additionally, the nonlinearity amount causes also marked variations in the amplitude and width of solitons. The modulation instability of the KNE is reported by implementing the linear stability analysis which confirms that all solutions are stable. The revealed results can be capitalized in improving the relevant physical and engineering applications in the field of birefringent fiber.

Soliton Dynamics over a Disordered Topography.

We report on the dynamics of a soliton propagating on the surface of a fluid in a 4-m-long canal with a random or periodic bottom topography. Using a full space-and-time resolved wave field measurement, we evidence, for the first time experimentally, how the soliton is affected by the disorder, in the context of Anderson localization, and how localization depends on nonlinearity. For weak soliton amplitudes, the localization length is found in quantitative agreement with a linear shallow-water theory. For higher amplitudes, this spatial attenuation of the soliton amplitude is found to be enhanced. Behind the leading soliton slowed down by the topography, different experimentally unreported dynamics occur: fission into backward and forward nondispersive pulses for the periodic case, and scattering into dispersive waves for the random case. Our findings open doors to potential applications regarding ocean coastal protection against large-amplitude waves.

Soliton-comb solutions for fiber-based bottle microresonators.

We predict a stable generation of optical solitons and related low-repetition frequency combs in high-Q fiber-based bottle resonators of different shapes. The modal localization effect is controlled by the shaping of the fiber radius. The solitons consist of axial resonator modes, belong to a single azimuth modal number, and possess anomalously small velocities. They are remarkably similar to each other for different resonator shapes and strongly different from conventional azimuth soliton-comb states: instead of the modal dispersion, the soliton width and amplitude are controlled by the critical self-focusing power.

N-soliton solutions and asymptotic analysis for the massive Thirring model in the laboratory coordinates via the Riemann-Hilbert approach

Abstract In this paper, the N-soliton solutions for massive Thirring model (MTM) in the laboratory coordinates are analyzed via the Riemann-Hilbert approach. The direct scattering including the analyticity, symmetries and asymptotic behaviors of the Jost solutions as |λ| → ∞ and λ → 0 are given. Considering that the scattering coefficients have simple zeros, the matrix Riemann-Hilbert problem, reconstruction formulas and corresponding trace formulas are also derived. Further, the N-soliton solutions in the reflectionless case are obtained explicitly in the form of determinants. The propagation characteristics of one-soliton solutions and interaction properties of two-soliton solutions are discussed. In particular, the asymptotic expressions of two-soliton solutions as |t| → ∞ are obtained, which show that the velocities and amplitudes of the asymptotic solitons don’t change before and after interaction except the position shifts. In addition, three types of bounded states for two-soliton solutions are presented with certain parametric conditions.

Nonlinear Ion-Acoustic Solitary Waves in a Weakly Relativistic Electron-Positron-Ion Plasma with Relativistic Electron and Positron Beams

In this investigation, compressive and rarefactive solitons are demonstrated to exist in a plasma model that includes unmagnetized weak-relativistic positive ions, negative ions, electrons, electron beam and positron beam. For these weakly relativistic non-linear ion-acoustic waves in unmagnetized plasma with electron inertia and relativistic beam, the existence of compressive and rarefactive soliton is investigated by deriving the Korteweg-de Vries (KdV) equation. It has been observed that the amplitude and width of compressive and rarefactive solitons vary differently in response to pressure variation and the presence of electron inertia. The research determines the requirements that must be met for the existence of the nonlinear ion-acoustic solitons. The fluid equations of motion governing the one-dimensional plasma serve as the foundation for the analysis. Various relational forms of the strength parameter (ε) are chosen to stretch the space and time variables, leading to a variety of nonlinearities. The findings can have implications not only for astrophysical plasmas but also for inertial confinement fusion plasmas.

Electron acoustic solitary wave with finite cold electron temperature

Abstract This study investigates the nonlinear propagation of Electron Acoustic Waves (EAWs) in a homogeneous, collisionless, dispersive plasma medium comprising two thermodynamically distinct electron species and a static charge-neutral ion background. Utilizing the properties of the Sagdeev pseudopotential, we established the necessary conditions for the existence of solitary waves in the presence of a finite cold electron temperature with the appropriate boundary conditions.
Next, we derived the conditions for a soliton solution when the cold electron temperature is absent, employing the same pseudopotential method. It was demonstrated that incorporating finite cold electron temperature significantly alters the conditions required to achieve a soliton solution. We derived a small amplitude soliton solution by expanding the Sagdeev potential up to the fourth order, and we discuss the implications of this solution.

Painlevé Analysis and Soliton Solutions for an Inhomogeneous Heisenberg Ferromagnetic System

In this paper we investigate an inhomogeneous higher-order nonlinear Schrödinger equation generated from the deformation of an inhomogeneous Heisenberg ferromagnetic system. Considering the variable coefficients from the inhomogeneities in the ferromagnetic medium, we present the Painlevé-integrable condition. Introducing an auxiliary function, we obtain the one-, two- and N-soliton solutions via the Hirota method. Furthermore, we graphically analyze the influence of inhomogeneities on the soliton propagation and interaction. It is found that the inhomogeneities will cause an increasing-decreasing process of soliton amplitude, and the change of the propagation direction. The perturbation terms can affect the propagation region of soliton but have no influence on the soliton amplitude. We consider the interactions between/among two and three solitons with the same or different initial propagation directions, and observe that the inhomogeneities can influence the soliton amplitudes and propagation directions, while the perturbation parameter can only affect the propagation direction of the soliton. In addition, bound state of solitons and its interaction with soliton are analyzed under the given excitation condition.

Analytical solutions to the (2+1)-dimensional cubic Klein–Gordon equation in the presence of fractional derivatives: A comparative study

The study seeks to obtain new analytical solutions for the (2+1)-dimensional cubic Klein–Gordon (cKG) equation using the beta derivative. By applying the unified method to the equation, various types of solitons have been generated, including periodic solitons, periodic solitons with equal and unequal wavelengths, bright solitons, and periodic singular solitons with unequal wavelengths. To demonstrate the fundamental dynamics of the soliton family, three-dimensional and two-dimensional graphs showcasing various novel solutions that satisfy the relevant equations are provided. In relation to fractionality, the bright waveform retains its overall shape, but its smoothness improves as the fractional parameters increase. Conversely, periodic wave solutions show enhanced periodicity as the fractional parameters rise. Additionally, the study provides a comprehensive comparison of solutions derived from models utilizing conformable, M-truncated, and beta derivatives. The investigation explores the effect of the fractional parameter on soliton amplitude, using graphs to illustrate this impact by assigning specific values to the fractional parameter. The properties of the waves can be modified through changes to the model's parameters to produce the appropriate wave profiles. Consequently, the solutions we obtained could be particularly valuable for analyzing physical problems associated with nonlinear complex dynamical systems.

Dynamical Study with Exact Travelling Waves with High Amplitude Solitons to Clannish Random Walker’s Parabolic Equation

Dynamical Study with Exact Travelling Waves with High Amplitude Solitons to Clannish Random Walker’s Parabolic Equation

Wave propagation and soliton behavior in biomechanical tissues: A mathematical approach

This study presents a mathematical model for understanding wave propagation and soliton behavior in biomechanical tissues, explicitly focusing on the Achilles tendon. Utilizing the Korteweg-de Vries (KdV) equation, the research incorporates the Achilles tendons’ nonlinear elastic and viscoelastic properties to explore how mechanical waves propagate through this complex tissue. The tendon’s nonlinear elasticity leads to wave steepening, while its viscoelasticity introduces dispersive effects that counteract this steepening, resulting in the formation of solitons—stable, localized waves that maintain their shape as they propagate. Key findings from this study reveal that the formation and propagation of solitons are strongly influenced by the tendon’s mechanical properties. Numerical simulations show that stiffer tendons, characterized by a higher elasticity modulus, support faster soliton propagation, with wave speeds ranging from 18.9 m/s in damaged tendons to 28.6 m/s in stiffened tendons. Additionally, soliton amplitude increases with tissue stiffness, with the highest amplitude observed in stiffened tendons (5.1 mm) and the lowest in damaged tendons (3.2 mm). The study also demonstrates that energy dissipation due to the tendon’s viscoelasticity plays a critical role in soliton behavior. Damaged tendons exhibit the highest energy loss (18.6%), leading to shorter soliton propagation distances, while stiffer tendons retain more energy (96.1%) and allow solitons to travel further distances (up to 180 mm). Moreover, the balance between nonlinearity and dispersion is crucial for maintaining soliton stability. Excessive nonlinearity leads to unstable solitons, while higher levels of dispersion contribute to more stable waveforms.

Auto-resonance process under the interaction of solitons with external force and dissipation

Algebraic soliton interactions with an external force in the presence of Reynolds viscosity is investigated. In the absence of an external force, the soliton amplitude decays over time. However, when an external force is introduced, it acts as a restoring force, and in some cases, the soliton’s amplitude is preserved. A dynamical system that governs the soliton amplitude and its crest position is obtained assuming a weak force and weak viscosity. For an external force with a Gaussian shape, the dynamical system has two equilibrium points, namely, a saddle and a stable spiral. Asymptotic results are compared with direct numerical simulations, and a strong qualitative agreement is observed. The stable spiral predicted by the asymptotic theory is stable in the sense that soliton solutions with a chosen amplitude and crest position near the spiral point are attracted to it, preserving their amplitude and location.

Multiple solitons and breathers on periodic backgrounds in the complex modified Korteweg–de Vries equation

This study explores multiple soliton and breather solutions on periodic backgrounds in the complex modified Korteweg–de Vries equation. The compact determinant formulas and their detailed derivation process for these solutions are provided via the bilinear method. We confirm that on periodic backgrounds, soliton amplitudes exhibit regular periodic behaviors, while breather amplitudes display quasi-periodic behaviors, as is expected for a breather with one period propagating over a periodic wave with another period. The asymptotic expressions for the solitons and breathers, which establish the high accuracy of the derived solutions, are provided to reveal the soliton and breather dynamics on the periodic backgrounds.

Effective regulation of the interaction process among three optical solitons

Abstract The interaction between three optical solitons is a complex and valuable research direction, which is of practical application for promoting the development of optical communication and all-optical information processing technology. In this paper, we start from the study of the variable-coefficient coupled higher-order nonlinear Schrödinger equation (VCHNLSE), and obtain an analytical three-soliton solution of this equation. Based on the obtained solution, the interaction of the three optical solitons is explored when they are incident from different initial velocities and phases. When the higher-order dispersion and nonlinear functions are sinusoidal, hyperbolic secant, and hyperbolic tangent functions, the transmission properties of three optical solitons before and after interactions are discussed. Besides, this paper achieves effective regulation of amplitude and velocity of optical solitons as well as of the local state of interaction process, and interaction-free transmission of the three optical solitons is obtained with a small spacing. The relevant conclusions of the paper are of great significance in promoting the development of high-speed and large-capacity optical communication, optical signal processing, and optical computing.

The multiple bright soliton pairs of the fully PT-symmetric nonlocal Davey-Stewartson I equation

This article investigates the dynamics of multiple bright soliton pair interactions in the fully PT -symmetric nonlocal Davey–Stewartson I equation. The bright soliton pair solutions are derived by employing the bilinear KP-hierarchy reduction method, and are expressed in terms of determinants. To study the interactions of the multiple soliton pairs, the long-time asymptotic analysis for these soliton solutions is performed by using the analysis of determinants, and the asymptotic expressions of the N individual soliton pair solutions are given as the sum of expressions for the 2N single soliton solutions. The asymptotics shows that the soliton pairs only exhibit elastic collisions and the two solitons in each soliton pair share equal amplitudes

Higher-order beyond-band discrete solitons in binary waveguide arrays

We study higher-order beyond-band discrete solitons (HOBBDSs) and quasi-HOBBDSs, which can be constructed by multiplying the solutions of fundamental single-peaked beyond-band discrete solitons by a soliton order parameter larger than unity. In the quasi-continuous regime when the HOBBDS peak amplitude is low (thus its width is large) and the soliton order parameter is a small integer number, HOBBDSs periodically evolve during propagation and their dynamics are similar to those of higher-order solitons governed by the nonlinear Schrödinger equation in an optical fiber, including the periodicity, pattern evolution, and independence of the period length on the soliton order parameter. If the soliton order parameter is still small but not an integer, then one can obtain the quasi-HOBBDSs whose profiles almost periodically evolve during propagation. The breathing length of quasi-HOBBDSs decreases if the soliton order parameter increases. Moreover, the breathing length of quasi-HOBBDSs is approximately inversely proportional to the square values of the peak amplitude of the fundamental beyond-band discrete solitons, just like what happens with the period length of the higher-order solitons governed by the nonlinear Schrödinger equation. If the fundamental beyond-band discrete solitons are intense enough and/or the soliton order parameter is large enough, then most of the energy of the beams is eventually trapped in a single waveguide.

Highly controllable stabilization and switching of multiple colliding soliton sequences with generic Ginzburg–Landau gain–loss

We investigate propagation of J soliton sequences in a nonlinear optical waveguide array with generic weak Ginzburg–Landau (GL) gain–loss and nearest-neighbor (NN) interaction. The propagation is described by a system of J perturbed coupled nonlinear Schrödinger (NLS) equations. The NN interaction property leads to the elimination of collisional three-pulse interaction effects, which prevented the observation of stable multisequence soliton propagation with J>2 sequences in the presence of generic GL gain–loss in all previous studies. We show that the dynamics of soliton amplitudes can be described by a generalized J-dimensional Lotka–Volterra (LV) model. Stability and bifurcation analysis for the equilibrium points of the LV model, which is augmented by an application of the Lyapunov function method, is used to develop setups that lead to robust and scalable transmission stabilization and switching for a general J value. The predictions of the LV model are confirmed by extensive numerical simulations with the perturbed coupled-NLS model with J=3, 4, and 5 soliton sequences. Furthermore, soliton stability and the agreement between the LV model’s predictions and the simulations are not reduced with increasing value of J. Therefore, our study provides the first demonstration of robust control of multiple colliding sequences of NLS solitons in the presence of generic weak GL gain–loss with an arbitrary number of sequences. Due to the robustness and scalability of the results, they can have important applications in stabilization and switching of broadband soliton-based optical waveguide transmission.

Riemann–Hilbert approach, dark solitons and double‐pole solutions for Lakshmanan–Porsezian–Daniel equation in an optical fiber, a ferromagnetic spin or a protein

AbstractInverse scattering transform for the defocusing Lakshmanan–Porsezian–Daniel equation with nonzero boundary condition is constructed via the Riemann–Hilbert approach. Since poles of the associated reflection coefficient are simple, ‐dark soliton solutions corresponding to simple poles are constructed. For ‐dark soliton solutions, results show that the soliton amplitude and width are not affected by the strength of the higher‐order linear and nonlinear effects , but soliton velocity has a linear correlation with ; the interactions between the two‐dark solitons and among the three‐dark solitons are elastic and experience phase and position shifts. Besides, asymptotic analysis of the double‐pole solutions for the focusing Lakshmanan–Porsezian–Daniel equation with nonzero boundary condition is presented. Different from ‐dark soliton solutions which locate in the straight lines and experience position shift after the interaction, the double‐pole solutions diverge from each other logarithmically and experience no position shift after the interaction.

Modified Korteweg–de Vries solitons with quartic nonlinearity in a dusty plasma

The present multicomponent dusty plasma with ions, Cairns distributed electrons and immobile dusts has been investigated first time through modified Korteweg–de Vries (mKdV) equation of quartic nonlinearity derived by reductive perturbation technique. In this new investigation, it is found that the dust ion-acoustic (DIA) solitary waves have smaller amplitudes compared to the amplitudes of mKdV-DIA compressive solitons of our previous investigation []. Roles of non-thermal parameter (β), dust to ion density ratio (σ), number of dust charge (Z d ) and initial streaming speed of ion (u i0) in the growth of amplitudes and widths of this new mKdV-DIA solitons are investigated.

Dynamical and statistical features of soliton interactions in the focusing Gardner equation.

In this paper, the dynamical properties of soliton interactions in the focusing Gardner equation are analyzed by the conventional two-soliton solution and its degenerate cases. Using the asymptotic expressions of interacting solitons, it is shown that the soliton polarities depend on the signs of phase parameters, and that the degenerate solitons in the mixed and rational forms have variable velocities with the time dependence of attenuation. By means of extreme value analysis, the interaction points in different interaction scenarios are presented with exact determination of positions and occurrence times of high transient waves generated in the bipolar soliton interactions. Next, with all types of two-soliton interaction scenarios considered, the interactions of two solitons with different polarities are quantitatively shown to have a greater contribution to the skewness and kurtosis than those with the same polarity. Specifically, the ratios of spectral parameters (or soliton amplitudes) are determined when the bipolar soliton interactions have the strongest effects on the skewness and kurtosis. In addition, numerical simulations are conducted to examine the properties of multi-soliton interactions and their influence on higher statistical moments, especially confirming the emergence of the soliton interactions described by the mixed and rational solutions in a denser soliton ensemble.