Soliton Parameters Research Articles

Do N-soliton breathers exist for the Hirota equation models?

We reveal the method for systematic construction of unexpected bound soliton states (localized breathers on the zero background) and formulate specific conditions for their appearance both in the framework of the canonical and the generalized nonautonomous Hirota models. These conditions show that both all parameters of solitons forming the breather and the dispersion and nonlinearity variations cannot be chosen independently, they are related by the exact integrability of the model. Novel N-soliton breathers derived for the nonautonomous Hirota equation with vanishing boundary conditions differ in that they generally move with varying amplitudes and velocities adapted to variations of the dispersion, nonlinearity, and gain, or losses. We apply the specific conditions for the breather formation to derive novel soliton breather solutions of the nonlinear Schrödinger and the complex modified Korteweg–de Vries equations. Consequently, we demonstrate that the way is opened to generalize and apply the presented method for other evolution equations of the Ablowitz–Kaup–Newell–Segur hierarchy.

Sub pico-second chirp-free optical solitons with Kaup-Newell equation using a couple of strategic algorithms

This paper obtains bright, dark, singular and other combo optical soliton solutions to perturbed Kaup–Newell equation. Two strategic integration schemes are fruitfully applied to obtain these solutions. These solitons appear with constraint conditions on the soliton parameters that are also presented.

Control of interaction between femtosecond dark solitons in inhomogeneous optical fibers

In this paper, we present exact femtosecond one- and two-dark soliton solutions for a variable-coefficient higher-order nonlinear Schrodinger equation via modified Hirota method. The propagation and interaction of femtosecond dark solitons are investigated in inhomogeneous fiber systems. Elastic collision, bound oscillation and parallel propagation can be achieved in both Gaussian distributed parameter system and exponentially periodic distributed parameter system by choosing the appropriate distributed parameters and soliton parameters. The results may be beneficial to the realization of interaction control of femtosecond dark solitons in communication systems.

Variational approach to stationary and pulsating dissipative optical solitons

The trial function approach is a useful technique to modelling soliton solutions for equations that cannot be solved exactly. The authors use the variational approach to find approximate solutions for the dissipative solitons described by the cubic-quintic complex Ginzburg-Landau equation. They find that the evolution equations for the soliton parameters are similar to those derived using the method of moments. The existence of both stationary and pulsating soliton solutions is indicated by both approaches.

Solitons and Josephson-type oscillations in Bose-Einstein condensates with spin-orbit coupling and time-varying Raman frequency

The existence and dynamics of solitons in quasi-one-dimensional Bose-Einstein condensates (BEC) with spin-orbit coupling (SOC) and attractive two-body interactions are described for two coupled atomic pseudo-spin components with slowly and rapidly varying time-dependent Raman frequency. By varying the Raman frequency linearly in time, it was shown that ordinary nonlinear Schr\"odinger-type bright solitons can be converted to striped bright solitons and vice versa. The internal Josephson oscillations between atom-number of the coupled soliton components, and the corresponding center-of-mass motion, are studied for different parameter configurations. In this case, a mechanism to control the soliton parameters is proposed by considering parametric resonances, which can emerge when using time-varying Raman frequencies. Full numerical simulations confirm variational analysis predictions when applied to the region where regular solitons are expected. In the limit of high frequencies, the system is described by a time-averaged Gross-Pitaevskii formalism with renormalized nonlinear and SOC parameters and modified phase-dependent nonlinearities. By comparing full-numerical simulations with averaged results, we have also studied the lower limits for the frequency of Raman oscillations in order to obtain stable soliton solutions.

Decay of Benjamin–Ono solitons under the influence of dissipation

The adiabatic decay of Benjamin–Ono algebraic solitons is studied when the influence of various types of small dissipation and radiative losses due to large scale Coriolis dispersion are taken into consideration. The physically most important dissipations are studied, Rayleigh and Reynolds dissipation, Landau damping, dissipation in a laminar boundary layer and Chezy friction on a rough bottom. The decay laws for the soliton parameters, that is amplitude, velocity and width, are found in analytical form and are compared with the results of direct numerical modelling.

Nonlinear Spatial Localized Strain Waves

A possible way of study of single waves in solids is discussed. The soliton is one of these waves without shape and parameters varying. Soliton deformation parameters are connected with the elastic moduli of the third order that allows defining values of these moduli by means of the measured solitondeformation parameters in various type waveguides made of the same material. The conditions under which a soliton can exist in a rod are analytically determined. For simultaneous excitation of loading in several wave guides two new energetic photosensitive structures (the mixtures are given) initiated by means of short light impulses of noncoherent light sources are proposed. Conditions of excitation of the waves on the basis of multipoint optical initiation loading impulses are described. As a technique for registration the shadowgraph visualization is proposed. It is discussed, how the problem connected to the use of energetic initiation structures consisting in the power background illumination can be solved. The shadow scheme with the use of a tiny dot explosive light source (Tbr ~41 kK) allows to carry out modelling experiments on research of slabbing actions, jet formations, fluffings, hydrodynamic instability during shock-wave loading of investigated samples, which makes it attractive for determination of parameters in equations-of-state for investigated materials, creation of numerical models and their validation. Some examples showing basic possibility of application of the declared techniques are included.

Dark and combined optical solitons, and modulation instability analysis in dispersive metamaterial

This paper obtains the dark and dark-bright or combined optical solitons to the nonlinear schrödinger equation (NLSE) describing propagation in dispersive metamaterial in optical fibers. The integration algorithm is the complex envelope function ansatz. This naturally lead to some constraint conditions placed on the soliton parameters which must hold for the solitons to exist. The intensities and the nonlinear phase shifts of the solitons are reported. Furthermore, the modulation instability analysis (MI) is studied based on the standard linear-stability analysis and the MI gain spectrum is got. Numerical simulation of some obtained results are analyzed with interesting figures showing the physical meaning of the solutions.

Fraction‐Dimensional Accessible Solitons in a Parity‐Time Symmetric Potential

AbstractBy using the modified Snyder‐Mitchell (MSM) model, which can describe the propagation of a paraxial beam in fractional dimensions (FDs), we find the exact "accessible soliton” solutions in the strongly nonlocal nonlinear media with a self‐consistent parity‐time (PT) symmetric complex potential. The exact solutions are constructed with the help of two special functions: the complex Gegenbauer and the generalized Laguerre polynomials in polar coordinates, parametrized by two nonnegative integer indices ‐ the radial and azimuthal mode numbers (n,m), and the beam modulation depth. By the choice of different soliton parameters, the intensity and angular profiles display symmetric and asymmetric structures. We believe that it is important to explore the MSM model in FDs and PT‐symmetric potentials, for a better understanding of nonlinear FD physical phenomena. Different physical systems in which the model might be of relevance are briefly discussed.

Modification of the discrete spectral parameters of optical solitons in fibers with variable dispersion

The spectral parameters of solitons result from the nonlinear Fourier transform for a pulse propagating in a fiber. A new approach for controlling the discrete component of spectral parameters is proposed. In the presence of a resonance between field oscillations and fiber dispersion, the real part of spectral parameters change, which leads to multisoliton pulse separation into fundamental solitons. It is proposed to use this effect for signal decoding in communication lines operating based on the nonlinear Fourier transform.

Dispersion managed dissipative solitons with higher order nonlinearity and frequency-selective feedback

We present the generation and interaction dynamics of dispersion managed soliton in a dissipative cubic-quintic nonlinear optical fiber coupled with frequency-selective feedback. Analytically, perturbative variational method is used to obtain evolution equations for different soliton parameters that are subsequently solved to study the propagation characteristics of the dispersion managed dissipative solitons (DMDS). The DMDSs show breather like characteristics and are found to be robust against certain level of initial noise. Both the standard Gaussian and more generic super-Gaussian pulse profile are shown to yield stable DMDS. Interaction of two DMDSs of same phase may lead to a bound state. By varying the temporal separation and phase difference between the DMDSs the bound state can be switched between low to high speed regime.

Dark optical and other soliton solutions for the three different nonlinear Schrödinger equations

In this work, the dark optical and singular soliton solutions of three different nonlinear Schrodinger equations (NLSE) are analyzed using Ricatti-Bernoulli (RB) sub-ODE method. We obtain new solutions that, in our opinion, are not known yet. The constraint conditions between the soliton parameters are determined. The obtained solitons may be used to explain and understand the physical nature of the wave spreads in the most dispersive optical medium.

Planar light bullets under conditions of second-harmonic generation.

We study solutions to second-harmonic-generation equations in two-dimensional media with anomalous dispersion. The analytical solution is obtained in an approximate form of the planar spatiotemporal two-component soliton by means of the averaged Lagrangian method. It is shown that a decrease in the amplitudes of both soliton components and an increase in the value of the transverse coordinate are accompanied by an increase in their temporal duration. Within this variational approach, we have managed to find a stability criterion for the light bullet and a period of oscillations of soliton parameters. Then, we use the obtained form as an initial configuration to carry out the direct numerical simulation of soliton dynamics. We demonstrate stable propagation of spatiotemporal solitons undergoing small oscillations predicted analytically for a long distance. The formation of a two-component light bullet is shown when we launch a pulse only at the fundamental frequency. In addition, we investigate the phase and group-velocity mismatch effects on the propagation of pulses.

Resonant optical solitons with quadratic-cubic nonlinearity by semi-inverse variational principle

This paper obtains resonant soliton solutions to the nonlinear Schrödinger's equation that is studied in quadratic-cubic nonlinear media. The semi-inverse variational principle is applied to retrieve the soliton solution. The soliton parameters appear with restrictions that are also presented.

KPII: Cauchy–Jost function, Darboux transformations and totally nonnegative matrices**To the memory of our friend and colleague Peter P Kulish

Direct definition of the Cauchy–Jost (known also as Cauchy–Baker–Akhiezer) function is given in the case of a pure solitonic solution. Properties of this function are discussed in detail using the Kadomtsev–Petviashvili II equation as an example. This enables formulation of the Darboux transformations in terms of the Cauchy–Jost function and classification of these transformations. Action of Darboux transformations on Grassmanians—i.e. on the space of soliton parameters—is derived and the relation of the Darboux transformations with the property of total nonnegativity of elements of corresponding Grassmanians is discussed.

Nonlinear dynamics of damped DNA systems with long-range interactions

We investigate the nonlinear dynamics of a damped Peyrard-Bishop DNA model taking into account long-range interactions with distance dependence |l|−s on the elastic coupling constant between different DNA base pairs. Considering both Stokes and long-range hydrodynamical damping forces, we use the discrete difference operator technique and show in the short wavelength modes that the lattice equation can be governed by the complex Ginzburg-Landau equation. We found analytically that the technique leads to the correct expression for the breather soliton parameters. We found that the viscosity makes the amplitude of the breather to damp out. We compare the approximate analytic results with numerical simulations for the value s=3 (dipole-dipole interactions).

Discrete bright–dark soliton solutions and parameters controlling for the coupled Ablowitz–Ladik equation

We consider the nonautonomous discrete vector bright–dark solutions and their controllable behaviors in the coupled Ablowitz–Ladik equation with variable coefficients, which possesses complicated wave propagation in time. Based on the differential–difference symmetry transformation and the Lame polynomial solutions, we use the Jacobi elliptic functions $$\hbox {sn}(n, m), \hbox {cn}(n, m)$$ , $$\hbox {dn}(n, m)$$ and present the nonautonomous discrete vector bright–dark solutions, which are localized in space and keep the localization longer in time. Moreover, we also exhibit the wave propagation of nonautonomous Lame polynomial solutions of higher order and their dynamics for some chosen parameters and functions. And the managements and dynamic behaviors of these solutions are investigated analytically.

Approximate solutions of bright and dark optical solitons in birefrigent fibers

In this article, the residual power series method (RPSM) for solving the nonlinear Schrödinger equation is introduced. Residual power series algorithm gets Maclaurin expansion of the numerical soliton solutions to birefringent fibers. Dark and singular numerical optical soliton solutions are listed for these nonlinearities. We studied these cases as an application for the nonlinear Schrödinger equation. Then we have provided a numerical study of the effect of changing the bright and dark soliton parameters. The solutions of our equation are computed in the form of rabidly convergent series with easily calculable components by using mathematica software package. Reliability of the method is given graphical consequens and series solutions are made use of to illustrate the solution. The found consequens show that the method is a power and efficient method in determination of solution the nonlinear Schrödinger equation.

Optical and other solitons for the fourth-order dispersive nonlinear Schrödinger equation with dual-power law nonlinearity

In this work, the soliton solutions of the fourth-order nonlinear Schrödinger equation (NLSE) with dual-power law nonlinearity is analyzed using Ricatti-Bernoulli (RB) sub-ODE and modified Tanh-Coth methods. We obtain new solutions that are not in existence in previous time. The constraint conditions between the soliton parameters are determined. The solutions we obtained may be used to explain and understand the physical nature of the wave spreads in the most dispersive optical medium.

Efficient particle filtering for stochastic Korteweg–de Vries equations

We propose an efficient algorithm to perform nonlinear data assimilation for Korteweg–de Vries solitons. In particular we develop a reduced particle filtering method to reduce the dimension of the problem. The method decomposes a solitonic pulse into a clean soliton and small radiative noise, and instead of inferring the complete pulse profile, we only infer the two soliton parameters with particle filter. Numerical examples are provided to demonstrate that the proposed method can provide rather accurate results, while being much more computationally affordable than a standard particle filter.