Soliton Perturbation Theory Research Articles

Soliton perturbation theory for the generalized fifth-order KdV equation

The soliton perturbation theory is used to study the adiabatic parameter dynamics of solitons due to the generalized fifth-order KdV equation in presence of perturbation terms. The adiabatic change of soliton velocity is also obtained in this paper.

Stochastic Perturbation of Power Law Optical Solitons

The soliton perturbation theory is used to study and analyze the stochastic perturbation of optical solitons, with power law nonlinearity, in addition to deterministic perturbations, that is governed by the nonlinear Schrodinger’s equation. The Langevin equations are derived and analysed. The deterministic perturbations that are considered here are due to filters and nonlinear damping.

Influence of Spectral Filtering and Nonlinear Gain on anOptical Soliton

Using the direct soliton perturbation theory, we investigate theevolution of soliton parameters and the first-order correction of brightsoliton in a system with linear and nonlinear gain (or loss) andspectral filtering in a comprehensive way. The results obtained by meansof our analytic method are consistent with numerical simulations. It isalso found that the stable soliton propagation which has beeninvestigated in a previous report by others is the limit case of ourresults.

Stochastic Perturbation of Parabolic Law Optical Solutions

The stochastic perturbation of optical solitons due to parabolic law nonlinearity is studied in this paper with the aid of soliton perturbation theory. The corresponding Langevin equation is derived and it is proved that the soliton propagates with a fixed mean velocity.

Solitary wave interaction and evolution for a higher-order Hirota equation

Solitary wave interaction and evolution for a higher-order Hirota equation is examined. The higher-order Hirota equation is asymptotically transformed to a higher-order member of the NLS hierarchy of integrable equations, if the higher-order coefficients satisfy a certain algebraic relationship. The transformation is used to derive higher-order one- and two-soliton solutions. It is shown that the interaction is asymptotically elastic and the higher-order corrections to the coordinate and phase shifts are derived. For the higher-order Hirota equation resonance occurs between the solitary waves and linear radiation, so soliton perturbation theory is used to determine the details of the evolving wave and its tail. An analytical expression for the solitary wave tail is derived and it is found that the tail vanishes when the algebraic relationship from the asymptotic theory is satisfied. Hence a two-parameter family of higher-order asymptotic embedded solitons exists. A comparison between the theoretical predictions and numerical solutions shows strong agreement for both solitary wave interaction, where the higher-order coordinate and phase shifts are compared, and solitary wave evolution, with comparisons made of the solitary wave tail.

Soliton perturbation theory for the generalized Benjamin–Bona–Mahoney equation

The soliton perturbation theory is used to study the adiabatic parameter dynamics of solitons due to the Benjamin–Bona–Mahoney equations in presence of perturbation terms. The change in the velocity is also obtained in this paper.

Bright solitons on a continuous wave background for the inhomogeneous nonlinear Schrödinger equation in plasma

In this paper, the inhomogeneous damped nonlinear Schrodinger equation (NLS) which describes wave propagation in plasmas is investigated. Based on Husimi's transformation and lens-type transformation, we reduce the inhomogeneous NLS equation to the standard NLS equation and thus the bright soliton solutions on a continuous wave (cw) background are constructed and discussed. Besides, we also consider the inhomogeneous NLS equation with different damping coefficients. Based on the perturbation theory for bright solitons, the approximate bright soliton solutions for such an equation are obtained, which are in good agreements with direct numerical simulations.

Soliton Perturbation Theory for the Compound KdV Equation

The soliton perturbation theory is used to study the solitons that are governed by the compound Korteweg de-Vries equation in presence of perturbation terms. The adiabatic parameter dynamics of the solitons in presence of the perturbation terms are obtained.

Alfvén soliton and emitted radiation in the presence of perturbations

Soliton perturbation theory based on the inverse scattering transform method is applied to the derivative nonlinear Schrödinger equation, which describes nonlinear Alfvén waves propagating quasiparallel to the external magnetic field. Radiative effects are considered. Spectral distributions of the emitted energy and magnetic helicity rates (in the wave number domain) are calculated analytically. Several forms of perturbations are considered, including the finite electric conductivity, the effect of resonant particles (nonlinear Landau damping), and the influence of the random inhomogeneity of the plasma density. The space structure of the radiative field is determined for a perturbation in the form of the finite electric conductivity.

Soliton perturbation theory for the modified nonlinear Schrödinger’s equation

The soliton perturbation theory is used to study the solitons that are governed by the modified nonlinear Schrödinger’s equation. The adiabatic parameter dynamics of the solitons in presence of the perturbation terms are obtained. In particular, the nonlinear gain (damping) and filters or the coefficient of finite conductivity are treated as perturbation terms for the solitons.

Stochastic Perturbation of Parabolic Law Optical Solitons

The stochastic perturbation of optical solitons due to parabolic law nonlinearity is studied in this paper with the aid of soliton perturbation theory. The corresponding Langevin equation is derived and it is proved that the soliton propagates with a fixed mean velocity. DOI: 10.2529/PIERS060817200007

Direct approach of perturbation theory for kink solitons

A direct approach of perturbation theory for the nonlinear Schrödinger equation is developed based on the separation of variables. The first-order effects of perturbation on a kink soliton, showing the slow time dependence of soliton parameters and the first-order correction.

Stable operating region in a harmonically actively mode-locked fiber laser

We theoretically study the recovery of a harmonically actively mode-locked soliton fiber laser from pulse dropout. In such lasers, a large number of pulses propagate simultaneously in the cavity. In order to obtain stable operation, pulses that are dropped due to changes in environmental conditions should recover, while other pulses that propagate in the cavity should remain stable. Soliton perturbation theory is used to find stability conditions for the noise in a time slot where a steady state pulse exists and in a time slot where a pulse is dropped. In the stable operating region of the laser, noise is stable in the presence of a pulse while noise becomes unstable in time slots where a pulse is dropped. Such a stability condition ensures that the laser can recover from accidental pulse dropout. A good agreement between the results of a reduced model and the results of a comprehensive numerical simulation was obtained. The results of this paper may enable to improve the stability of actively mode-locked fiber lasers.

Dynamics of solitons in the model of nonlinear Schrödinger equation with an external harmonic potential: I. Bright solitons

The dynamics of nonlinear solitary waves is studied by using the model of nonlinear Schrödinger equation (NSE) with an external harmonic potential. The model allows one to analyse on the general basis a variety of nonlinear phenomena appearing both in a Bose—Einstein condensate in a magnetic trap, whose profile is described by a quadratic function of coordinates, and in nonlinear optics, physics of lasers, and biophysics. It is shown that exact solutions for a quantum-mechanical particle in a harmonic potential and solutions obtained within the framework of the adiabatic perturbation theory for bright solitons in a parabolic trap are completely identical. This fact not only proves once more that solitons behave like particles but also that they can preserve such properties in different traps for which the parabolic approximation is valid near potential energy minima. The conditions are found for formation of stable stationary states of antiphase solitons in a harmonic potential. The interaction dynamics of solitons in nonstationary potentials is studied and the possibility of the appearance of a soliton parametric resonance at which the amplitude of soliton oscillations in a trap exponentially increases with time is shown. It is shown that exact solutions of the problem found using the Miura transformation open up the possibility to control the dynamics of solitons. New effects are predicted, which are called the reversible and irreversible denaturation of solitons in a nonstationary harmonic potential.

Dynamics of ultraslow optical solitons in a cold three-state atomic system

We present a systematic study on the dynamics of a ultraslow optical soliton in a cold, highly resonant three-state atomic system under Raman excitation. Using a method of multiple scales we derive a modified nonlinear Schrödinger equation with high-order corrections that describe effects of linear and differential absorption, nonlinear dispersion, delay response of nonlinear refractive index, diffraction, and third-order dispersion. Taking these effects as perturbations we investigate in detail the evolution of the ultraslow optical soliton using a standard soliton perturbation theory. We show that due to these high-order corrections the ultraslow optical soliton undergoes deformation, change of propagating velocity, and shift of oscillating frequency. In addition, a small radiation superposed by dispersive waves is also generated from the soliton. The results of the present work may provide a guidance that is useful for experimental demonstration of ultraslow optical soliton in cold atomic systems.

Analytical approach to soliton ratchets in asymmetric potentials

We use soliton perturbation theory and collective coordinate ansatz to investigate the mechanism of soliton ratchets in a driven and damped asymmetric double sine-Gordon equation. We show that, at the second order of the perturbation scheme, the soliton internal vibrations can couple effectively, in presence of damping, to the motion of the center of mass, giving rise to transport. An analytical expression for the mean velocity of the soliton is derived. The results of our analysis confirm the internal mode mechanism of soliton ratchets proposed in [Phys. Rev. E 65, 025602(R) (2002)].

Stochastic perturbation of dispersion-managed optical solitons

The stochastic perturbation of dispersion-managed optical solitons is studied in this paper, in addition to deterministic preturbation terms, by the aid of soliton perturbation theory. The super-Gaussian pulses are considered and the corresponding Langevin equations are derived and analyzed. It is shown that in presence of the perturbation terms, the soliton propagates down the fiber with a fixed mean energy.

Soliton propagation in the presence of intrapulse Raman scattering and nonlinear gain

The propagation of ultrashort solitons in fiber-optic systems in the presence of intrapulse Raman scattering is investigated. Narrowband filtering and nonlinear gain are used to control the self-frequency shift and the background instability. The soliton perturbation theory is used to obtain the steady-state solutions as well as their existence conditions. Linear stability analysis is employed to study the steady-state solutions and the results are verified by numerically solving the propagation equation. We show that, using narrowband filtering and nonlinear gain, stable soliton propagation over long distances can be realized in fiber-optic systems in the presence of intrapulse Raman scattering.

Exploring the connection between quasistationary and squared eigenfunction expansion techniques in soliton perturbation theory

We give a short review of the techniques that have been used to study soliton perturbations for nonlinear integrable systems in 1 + 1 dimensions. In particular, we discuss the connection between direct perturbation methods using squared eigenfunction expansions and quasistationary perturbation theory. We show for the KdV (Korteweg–de Vries) how the results for quasistationary approximations are related to those for the standard adiabatic perturbation theory.

A novel analytic technique for the disintegration of coherent structure solutions of non-integrable and integrable nonlinear PDEs under forcing

We consider a technique for deriving exact analytic coherent structure (pulse/front/domain wall) solutions of general NLPDEs via the use of truncated invariant Painlevé expansions. Coupling this to Melnikov theory, we then consider the breakdown to chaos of such analytic coherent structure solutions of the Zakharov–Kuznetsov and reaction-diffusion equations under forcing. We also demonstrate that similar treatments are possible for integrable systems (using the well-studied forced KdV equation as an example) where the soliton/kink solutions represent the homoclinic/heteroclinic structures of the reduced ODEs. A method of treating the dynamics of the system prior to the onset of chaos by the use of intrinsic harmonic balance, multiscale or direct soliton perturbation theory is briefly discussed. It is conceivable that resummation of such perturbation series via the use of Fade approximants or other techniques may enable one to analytically follow the homoclinic or heteroclinic tangling beyond the first transversal intersection of the stable and unstable manifolds and into the chaotic regime.