Showing papers by "Yuri S. Kivshar published in 1994"

Perturbation-induced dynamics of dark solitons

Abstract: We study analytically and numerically the effect of perturbations on (spatial and temporal) dark optical solitons. Our purpose is to elaborate a general analytical approach to describe the dynamics of dark solitons in the presence of physically important effects which break integrability of the primary nonlinear Schr\"odinger equation. We show that the corresponding perturbation theory differs for the cases of constant and varying backgrounds which support the dark solitons. We present a general formalism describing the perturbation-induced dynamics for both cases and also analyze the influence of several physically important effects, such as linear and two-photon absorption, Raman self-induced scattering, gain with saturation, on the propagation of the dark soliton. As we show, the perturbation-induced dynamics of a dark soliton may be treated as a result of the combined effect of the background evolution and internal soliton dynamics, the latter being characterized by the soliton phase angle. A similar approach is applied to the problem of the dark-soliton propagation on a finite-width background. We analyze adiabatic modification of a dark pulse propagating on a dispersively spreading finite-width background, and we prove analytically that a frequency chirp of the background does not affect the soliton motion. As a matter of fact, the results obtained describe the perturbation-induced dynamics of dark solitons in the so-called adiabatic approximation and, as we show for all the cases analyzed, they are in excellent agreement with direct numerical simulations of the corresponding perturbed nonlinear Schr\"odinger equation, provided the effects produced by the emitted radiation are small.

Spatial optical solitons governed by quadratic nonlinearity

Abstract: We demonstrate that a dielectric medium with purely quadratic nonlinearity [the so-called chi((2)) material] can display self-focusing phenomena through a new type of modulational instability of the interacting fundamental and second-harmonic field components and therefore can support propagation of (two-wave) optical solitons. We prove the existence of a family of such solitons, which are found numerically and, in some particular cases, also analytically. The two-wave solitons are stable in the whole parameter region in which they exist.

Gordon–Haus effect on dark solitons

Abstract: The theory of the Gordon–Haus effect with application to dark solitons is presented. It is proved analytically that a random frequency shift of a fundamental dark soliton results in a time jitter 2 times lower than that for bright solitons.

Ring dark solitons

Abstract: Self-defocusing nonlinear media can support dark solitary waves with ring symmetry that are robust but slowly change their parameters. This alternative type of optical dark soliton is investigated numerically and analytically, and it is shown that in the small-amplitude limit such solitons are described by the so-called cylindrical Korteweg--de Vries equation known, e.g., from plasma physics. Ring dark solitons may coexist with other types of dark solitons, displaying almost an elastic interaction.

Dark solitons in discrete lattices.

Abstract: We analyze the effect of discreteness on properties and propagation dynamics of dark solitons in the discrete nonlinear Schr\"odinger equation. We show that for small-amplitude nonlinear waves the lattice discreteness induces novel properties of dark solitons, e.g., such solitons may be transformed into brightlike dark solitons on a modulationally stable background. For large-amplitude dark solitons we demonstrate that discreteness effects may be understood as arising from an effective periodic potential to the soliton's coordinate similar to the Peierls-Nabarro (PN) periodic potential for (topological) kinks in the Frenkel-Kontorova model. We calculate the PN barrier (the height of the PN potential) to a dark soliton numerically and, in the case of strong interparticle coupling, also analytically, and discuss how the existence of the PN barrier may affect the mobility of dark solitons in a discrete lattice. In particular, we predict unexpected types of discreteness-induced instabilities for soliton-bearing models showing that, even being at a bottom of the PN potential well, the dark soliton is unstable and it always starts to move after a series of oscillations around the potential minimum. An intuitive picture for such a discreteness-induced nonlinear instability of dark solitons is presented, and the novelty of this phenomenon in comparison to bright solitons is emphasized.

Modulational instabilities in the discrete deformable nonlinear Schrödinger equation.

Abstract: We study analytically and numerically modulational instability for the discrete deformable nonlinear Schr\"odinger (NLS) equation which represents a natural link between the properties of the integrable Ablowitz-Ladik model and the nonintegrable discrete NLS equation. We show how different discretizations of the nonlinear interaction change modulational instability in the lattice and, correspondingly, conditions for localized modes to exist.

Dynamics of dark solitons

Abstract: Spatial dark solitons are known to be stable nonlinear localized waves in self-defocusing media. We demonstrate how to analyse the dynamics of dark solitons in the presence of small perturbations, and we consider the effect of two-photon absorption on dark solitons as a particular example. We also predict a new type of dark solitons of circular symmetry, ring dark solitons, which are stable against transverse perturbations but slowly change their parameters. Ring dark solitons may coexist with dark strips or optical vortex solitons displaying almost elastic interactions.

Compactons in Discrete Lattices

Abstract: As is well-known, solitons appear in a result of a balance between weak nonlinearity and dispersion. However, when the wave dispersion is purely nonlinear, some novel features may be observed and the most remarkable one is the existence of the so-called compactons, i. e. solitons with finite wavelength recently discovered by Rosenau and Hyman1 for a special class of the Korteweg-de Vries (KdV) type equations with nonlinear dispersion. These travelling-wave solutions have a remarkable property: Unlike the standard KdV soliton which narrows as the amplitude increases, the compacton’s width is independent of the amplitude. Having the constant width, such solutions can not be obtained, however, in a result of a proper continuum limit to discrete models. Indeed, soliton-bearing partial differential equations may be derived from discrete models of solids in a result of expansions in the wave amplitude and inverse pulse width which normally need a scaling procedure. In other words, the continuum limit approach yields the condition of the slowly varying wave envelope which is consistent with the effect of weak nonlinearity balanced by a weak dispersion. As soon as we deal with compactons instead of standard solitons, the continuum limit approximation cannot be properly justified because higher-order derivatives will be only numerically small.

Standing localized modes in nonlinear lattices.

Abstract: The theory of standing localized modes in discrete nonlinear lattices is presented. We start from a rather general model describing a chain of particles subjected to an external (on-site) potential with cubic and quartic nonlinearities (the so-called Klein-Gordon model), and, using the approximation based on the discrete nonlinear Schro$iuml---dinger equation, derive a system of two coupled nonlinear equations for slowly varying envelopes of two counterpropagating waves of the same frequency. We show that spatially localized modes exist in the frequency--wave number domain where the lattice displays modulational instability; two families of localized modes are found for this case as separatrix solutions of the effective equations for the wave envelopes. When the nonlinear plane wave in the lattice is stable to small modulations of its amplitude, nonlinear localized modes appear as dark solitons associated with the so-called extended modulational instability. These localized modes may be treated as domain walls or kinks connecting two standing plane-wave modes of the similar structure. We investigate analytically and numerically the special family of such localized solutions that, in the vicinity of the zero-dispersion point, cover exactly the case of the so-called self-induced gap solitons recently introduced by Kivshar [Phys. Rev. Lett. 70, 3055 (1993)]. Application of the theory to the case of parametrically driven damped lattices is also briefly discussed, and it is mentioned that some of the solutions considered in the present paper may be extended to include the case of localized modes in driven damped lattices, provided the mode frequency and amplitude are fixed by the parameters of the external parameters of the external parametric ac force.

Kinks in the presence of rapidly varying perturbations

Abstract: The dynamics of sine-Gordon kinks in the presence of rapidly varying periodic perturbations of different physical origins is described analytically and numerically. The analytical approach is based on asymptotic expansions, and it allows one to derive, in a rigorous way, an effective nonlinear equation for the slowly varying field component in any order of the asymptotic procedure as expansions in the small parameter ${\mathrm{\ensuremath{\omega}}}^{\mathrm{\ensuremath{-}}1}$, \ensuremath{\omega} being the frequency of the rapidly varying ac driving force. Three physically important examples of such a dynamics, i.e., kinks driven by a direct or parametric ac force, and kinks on a rotating and oscillating background, are analyzed in detail. It is shown that in the main order of the asymptotic procedure the effective equation for the slowly varying field component is a renormalized sine-Gordon equation in the case of the direct driving force or rotating (but phase locked to an external ac force) background, and it is the double sine-Gordon equation for the parametric driving force. The properties of the kinks described by the renormalized nonlinear equations are analyzed, and it is demonstrated analytically and numerically which kinds of physical phenomena may be expected in dealing with the renormalized, rather than the unrenormalized, nonlinear dynamics. In particular, we predict several qualitatively new effects which include, e.g., the perturbation-induced internal oscillations of the 2\ensuremath{\pi} kink in a parametrically driven sine-Gordon model, and the generation of kink motion by a pure ac driving force on a rotating background.

Diffusion in the frenkel–kontorova model with anharmonic interatomic interactions

Abstract: Low-temperature diffusion and transport properties of the generalized Frenkel–Kontorova model are investigated analytically in the framework of a phenomenological approach which treats a system of strongly interacting atoms as a system of weaklyinteracting quasiparticles (kinks). The model takes into account realistic (anharmonic) interaction of particles subjected into a periodic substrate potential, and such a generalization leads to a series of novel effects which we expect are related to the experimentally-observed phenomena in several quasi-one-dimensional systems. Analysing the concentration dependences in the framework of the kink phenomenology, we use the renormalization procedure when the atomic structure with a complex unit cell is treated as (more simple) periodic structure of kinks. Using phenomenology of the ideal kink gas, the low-temperature states of the chain are described as those consisting of "residual" kinks supplemented by thermally-excited kinks. This approach allows us to describe the ground states of the chain as a hierarchy of "melted" kink lattices. Dynamical and diffusion properties of the system are then described in terms of the kink dynamics and kink diffusion. The motion equation for a single kink is reduced to a Langevin-type equation which is investigated with the help of the Kramers theory. Susceptibility, conductivity, self-diffusion and chemical diffusion coefficients of the chain are calculated as functions of the kink diffusion coefficient. In this way, we qualitatively analyze, for the first time to our knowledge, dependence of the different diffusion coefficients on the concentration of atoms in the chain. The results are applied to describe peculiarities in conductivity and diffusion coefficients of quasi-one-dimensional systems, in particular, superionic conductors and anisotropic layers of atoms adsorbed on crystal surfaces which were earlier investigated experimentally.

Is two-photon absorption a limitation to dark soliton switching?

Abstract: It is shown that two-photon absorption does not always place a fundamental limitation on soliton devices: The steering angle, the key characteristic of switching devices based on the propagation of dark spatial solitons, is almost preserved as the intensity of the background wave decays in the presence of weak two-photon absorption.

Internal-mode-induced resonances in soliton-impurity interactions

Abstract: Sotiton-impurity interactions in the systems where the topological solitons (kinks) possess internal oscillatory modes are studied analytically and numerically. Taking the well-known φ 4 model as a particular but rather general example, we demonstrate that the kink can resonantly pass two (or more) localized impurities due to excitation and deexcitation of its internal mode.

Dynamics of Parametrically Driven Sine-Gordon Breathers

Abstract: We consider the damped and parametrically driven sine-Gordon equation $$ {u_{tt}} - {u_{xx}} + \sin u = \alpha {u_t} + \Gamma \sin (\omega t)\sin \left( {\frac{u}{n}} \right), $$ (1) with periodic boundary conditions u (x − L/2, t) = u(x + L/2, t). Equation (1) with n = 2 may be derived, for example, as an effective equation of motion for the magnetization vector in several magnetic models, u being an angle describing the orientation of the magnetic vector in a selected (easy anisotropy) plane. The perturbation from the r. h. s. of Eq. (1) appears if one considers a variable magnetic field1, 2. Another physically relevant example of Eq. (1) with n = 1 is a long Josephson junction with parametrically varying critical current3.