arXiv subjects

Boris A. Malomed

Publications and source records attributed to Boris A. Malomed.

At least 559 records · Page 31Linked to original sources

arXiv2002

Solitons in coupled Ablowitz-Ladik chains

A model of two coupled Ablowitz-Ladik (AL) lattices is introduced. While the system as a whole is not integrable, it admits reduction to the integrable AL model for symmetric states. Stability and evolution of symmetric solitons are studied in detail analytically (by means of a variational approximation) and numerically. It is found that there exists a finite interval of positive values of the coupling constant in which the symmetric soliton is stable, provided that its mass is below a threshold value. Evolution of the unstable symmetric soliton is further studied by means of direct simulations. It is found that the unstable soliton breaks up and decays into radiation, or splits into two counter-propagating asymmetric solitons, or evolves into an asymmetric pulse, depending on the coupling coefficient and the mass of the initial soliton.

nlin.PS↗

arXiv2002

Stable periodic waves in coupled Kuramoto-Sivashinsky - Korteweg-de Vries equations

Periodic waves are investigated in a system composed of a Kuramoto-Sivashinsky - Korteweg-de Vries (KS-KdV) equation, which is linearly coupled to an extra linear dissipative equation. The model describes, e.g., a two-layer liquid film flowing down an inclined plane. It has been recently shown that the system supports stable solitary pulses. We demonstrate that a perturbation analysis, based on the balance equation for the field momentum, predicts the existence of stable cnoidal waves (CnWs) in the same system. It is found that the mean value U of the wave field u in the main subsystem, but not the mean value of the extra field, affects the stability of the periodic waves. Three different areas can be distinguished inside the stability region in the parameter plane (L,U), where L is the wave's period. In these areas, stable are, respectively, CnWs with positive velocity, constant solutions, and CnWs with negative velocity. Multistability, i.e., the coexistence of several attractors, including the waves with several maxima per period, appears at large value of L. The analytical predictions are completely confirmed by direct simulations. Stable waves are also found numerically in the limit of vanishing dispersion, when the KS-KdV equation goes over into the KS one.

nlin.PS↗

arXiv2002

Solitons in a Three-Wave System with Intrinsic Linear Mixing and Walkoff

A modification of the usual Type-II second-harmonic-generation model is proposed, which includes two additional features: linear conversion and walkoff (group-velocity difference) between two components of the fundamental-frequency (FF) wave. Physical interpretations of the model are possible in both temporal and spatial domains. In the absence of the intrinsic walkoff, the linear mixing makes real soliton solutions stable or unstable, depending on the relative sign of the two FF components. Unstable solitons spontaneously re-arrange themselves into stable ones. Fundamental solitons change their shape (in particular, they develop chirp) but remain stable if the intrinsic walkoff is included. In addition, quasi-stable double-humped solitary waves are found in the latter case.

nlin.PS↗

arXiv2002

Two-dimensional solitary pulses in driven diffractive-diffusive complex Ginzburg-Landau equations

Two models of driven optical cavities, based on two-dimensional Ginzburg-Landau equations, are introduced. The models include loss, the Kerr nonlinearity, diffraction in one transverse direction, and a combination of diffusion and dispersion in the other one (which is, actually, a temporal direction). Each model is driven either parametrically or directly by an external field. By means of direct simulations, stable completely localized pulses are found (in the directly driven model, they are built on top of a nonzero flat background). These solitary pulses correspond to spatio-temporal solitons in the optical cavities. Basic results are presented in a compact form as stability regions for the solitons in a full three-dimensional parameter space of either model. The stability region is bounded by two surfaces; beyond the left one, any two-dimensional (2D) pulse decays to zero, while quasi-1D pulses, representing spatial solitons in the optical cavity, are found beyond the right boundary. The spatial solitons are found to be stable both inside the stability region of the 2D pulses (hence, bistability takes place in this region) and beyond the right boundary of this region (although they are not stable everywhere). Unlike the spatial solitons, their quasi-1D counterparts in the form of purely temporal solitons are always subject to modulational instability, which splits them into an array of 2D pulses, that further coalesce into two final pulses. A uniform nonzero state in the parametrically driven model is also modulationally unstable, which leads to formation of many 2D pulses that subsequently merge into few ones.

nlin.PS↗

arXiv2002

Embedded Solitons in Lagrangian and Semi-Lagrangian Systems

We develop the technique of the variational approximation for solitons in two directions. First, one may have a physical model which does not admit the usual Lagrangian representation, as some terms can be discarded for various reasons. For instance, the second-harmonic-generation (SHG) model considered here, which includes the Kerr nonlinearity, lacks the usual Lagrangian representation if one ignores the Kerr nonlinearity of the second harmonic, as compared to that of the fundamental. However, we show that, with a natural modification, one may still apply the variational approximation (VA) to those seemingly flawed systems as efficiently as it applies to their fully Lagrangian counterparts. We call such models, that do not admit the usual Lagrangian representation, \textit{semi-Lagrangian} systems. Second, we show that, upon adding an infinitesimal tail that does not vanish at infinity, to a usual soliton ansatz, one can obtain an analytical criterion which (within the framework of VA) gives a condition for finding \textit{embedded solitons}, i.e., isolated truly localized solutions existing inside the continuous spectrum of the radiation modes. The criterion takes a form of orthogonality of the radiation mode in the infinite tail to the soliton core. To test the criterion, we have applied it to both the semi-Lagrangian truncated version of the SHG model and to the same model in its full form. In the former case, the criterion (combined with VA for the soliton proper) yields an \emph{exact} solution for the embedded soliton. In the latter case, the criterion selects the embedded soliton with a relative error $\approx 1%$.

math-ph↗

arXiv2002

Spatial solitons in a medium composed of self-focusing and self-defocusing layers

We introduce a model combining Kerr nonlinearity with a periodically changing sign ("nonlinearity management") and a Bragg grating (BG). The main result, obtained by means of systematic simulations, is presented in the form of a soliton's stability diagram on the parameter plane of the model; the diagram turns out to be a universal one, as it practically does not depend on the soliton's power. Moreover, simulations of the nonlinear Schroedinger (NLS) model subjected to the same "nonlinearity management" demonstrate that the same diagram determines the stability of the NLS solitons, unless they are very narrow. The stability region of very narrow NLS solitons is much smaller, and soliton splitting is readily observed in that case. The universal diagram shows that a minimum non-zero average value of the Kerr coefficient is necessary for the existence of stable solitons. Interactions between identical solitons with an initial phase difference between them are simulated too in the BG model, resulting in generation of stable moving solitons. A strong spontaneous symmetry breaking is observed in the case when in-phase solitons pass through each other due to attraction between them.

nlin.PS↗

arXiv2002

Scattering of a solitary pulse on a local defect or breather

A model is introduced to describe guided propagation of a linear or nonlinear pulse which encounters a localized nonlinear defect, that may be either static or breather-like one (the model with the static defect applies to an optical pulse in a long fiber link with an inserted additional section of a nonlinear fiber). In the case when the host waveguide is linear, the pulse has a Gaussian shape. In that case, an immediate result of its interaction with the nonlinear defect can be found in an exact analytical form, amounting to transformation of the incoming Gaussian into an infinite array of overlapping Gaussian pulses. Further evolution of the array in the linear host medium is found numerically by means of the Fourier transform. An important ingredient of the linear medium is the third-order dispersion, that eventually splits the array into individual pulses. If the host medium is nonlinear, the input pulse is taken as a fundamental soliton. The soliton is found to be much more resistant to the action of the nonlinear defect than the Gaussian pulse in the linear host medium. In this case, the third-order-dispersion splits the soliton proper and wavepackets generated by the action of the defect.

nlin.PS↗

arXiv2002

Noise-induced dephasing of an ac-driven Josephson junction

We consider phase-locked dynamics of a Josephson junction driven by finite-spectral-linewidth ac current. By means of a transformation, the effect of frequency fluctuations is reduced to an effective additive noise, the corresponding (large) dephasing time being determined, in the logarithmic approximation, by the Kramers' expression for the lifetime. For sufficiently small values of the drive's amplitude, direct numerical simulations show agreement of the dependence of the dephasing activation energy on the ac-drive's spectral linewidth and amplitude with analytical predictions. Solving the corresponding Fokker-Planck equation analytically, we find a universal dependence of a critical value of the effective phase-diffusion parameter on the drive's amplitude at a point of a sharp transition from the phase-locked state to an unlocked one. For large values of the drive amplitude, saturation and subsequent decrease of the activation energy are revealed by simulations, which cannot be accounted for by the perturbative analysis. The same new effect is found for a previously studied case of ac-driven Josephson junctions with intrinsic thermal noise. The work was performed in the framework of a cooperation agreement between Consiglio Nazionale di Ricerca (Italy) and the Israeli Ministry of Science and Technology.

nlin.PS↗

arXiv2002

Progressive motion of an ac-driven kink in an annular damped system

A novel dynamical effect is presented: systematic drift of a topological soliton in ac-driven weakly damped systems with periodic boundary conditions. The effect is demonstrated in detail for a long annular Josephson junction. Unlike earlier considered cases of the ac-driven motion of fluxons (kinks), in the present case the long junction is_spatially uniform_. Numerical simulations reveal that progressive motion of the fluxon commences if the amplitude of the ac drive exceeds a threshold value. The direction of the motion is randomly selected by initial conditions, and a strong hysteresis is observed. An analytical approach to the problem is based on consideration of the interaction between plasma waves emitted by the fluxon under the action of the ac drive and the fluxon itself, after the waves complete round trip in the annular junction. The analysis predicts instability of the zero-average-velocity state of the fluxon interacting with its own radiation tails, provided that the drive's amplitude exceeds an explicitly found threshold. The predicted threshold amplitude strongly depends on the phase shift gained by the wave after the round trip. A very similar dependence is found in the simulations, testifying to the relevance of the analytical consideration.

nlin.PS↗

arXiv2002

Cuspons, peakons and regular gap solitons between three dispersion curves

A general wave model with the cubic nonlinearity is introduced to describe a situation when the linear dispersion relation has three branches, which would intersect in the absence of linear couplings between the three waves. Actually, the system contains two waves with a strong linear coupling between them, to which a third wave is then coupled. This model has two gaps in its linear spectrum. Realizations of this model can be made in terms of temporal or spatial evolution of optical fields in, respectively, a planar waveguide or a bulk-layered medium resembling a photonic-crystal fiber. Another physical system described by the same model is a set of three internal wave modes in a density-stratified fluid. A nonlinear analysis is performed for solitons which have zero velocity in the reference frame in which the group velocity of the third wave vanishes. Disregarding the self-phase modulation (SPM) term in the equation for the third wave, we find two coexisting families of solitons: regular ones, which may be regarded as a smooth deformation of the usual gap solitons in a two-wave system, and cuspons with a singularity in the first derivative at their center. Even in the limit when the linear coupling of the third wave to the first two vanishes, the soliton family remains drastically different from that in the linearly uncoupled system; in this limit, regular solitons whose amplitude exceeds a certain critical value are replaced by peakons. While the regular solitons, cuspons, and peakons are found in an exact analytical form, their stability is tested numerically, which shows that they all may be stable. If the SPM terms are retained, we find that there again coexist two different families of generic stable soliton solutions, namely, regular ones and peakons.

nlin.PS↗

arXiv2001

Stable and chaotic solutions of the complex Ginzburg-Landau equation with periodic boundary conditions

We study, analytically and numerically, the dynamical behavior of the solutions of the complex Ginzburg-Landau equation with diffraction but without diffusion, which governs the spatial evolution of the field in an active nonlinear laser cavity. Accordingly, the solutions are subject to periodic boundary conditions. The analysis reveals regions of stable stationary solutions in the model’s parameter space, and a wide range of oscillatory and chaotic behaviors. Close to the first bifurcation destabilizing the spatially uniform solution, a stationary single-humped solution is found in an asymptotic analytical form, which turns out to be in very good agreement with the numerical results. Simulations reveal a series of stable stationary multi-humped solutions

nlin.PS↗

arXiv2001

"Dispersion management" for solitons in a Korteweg-de Vries system

The existence of ``dispersion-managed solitons'', i.e., stable pulsating solitary-wave solutions to the nonlinear Schrödinger equation with periodically modulated and sign-variable dispersion is now well known in nonlinear optics. Our purpose here is to investigate whether similar structures exist for other well-known nonlinear wave models. Hence, here we consider as a basic model the variable-coefficient Korteweg-de Vries equation; this has the form of a Korteweg-de Vries equation with a periodically varying third-order dispersion coefficient, that can take both positive and negative values. More generally, this model may be extended to include fifth-order dispersion. Such models may describe, for instance, periodically modulated waveguides for long gravity-capillary waves. We develop an analytical approximation for solitary waves in the weakly nonlinear case, from which it is possible to obtain a reduction to a relatively simple integral equation, which is readily solved numerically. Then, we describe some systematic direct simulations of the full equation, which use the soliton shape produced by the integral equation as an initial condition. These simulations reveal regions of stable and unstable pulsating solitary waves in the corresponding parametric space. Finally, we consider the effects of fifth-order dispersion.

nlin.SI↗

arXiv2001

Stationary states of Bose-Einstein condensates in single- and multi-well trapping potentials

The stationary solutions of the Gross-Pitaevskii equation can be divided in two classes: those which reduce, in the limit of vanishing nonlinearity, to the eigenfunctions of the associated Schrödinger equation and those which do not have linear counterpart. Analytical and numerical results support an existence condition for the solutions of the first class in terms of the ratio between their proper frequency and the corresponding linear eigenvalue. For one-dimensional confined systems, we show that solutions without linear counterpart do exist in presence of a multi-well external potential. These solutions, which in the limit of strong nonlinearity have the form of chains of dark or bright solitons located near the extrema of the potential, represent macroscopically excited states of a Bose-Einstein condensate and are in principle experimentally observable.

cond-mat.stat-mech↗

arXiv2001

Nonlinear parametric instability in double-well lattices

A possibility of a nonlinear resonant instability of uniform oscillations in dynamical lattices with harmonic intersite coupling and onsite nonlinearity is predicted. Numerical simulations of a lattice with a double-well onsite anharmonic potential confirm the existence of the nonlinear instability with an anomalous value of the corresponding power index, 1.57, which is intermediate between the values 1 and 2 characterizing the linear and nonlinear (quadratic) instabilities. The anomalous power index may be a result of competition between the resonant quadratic instability and nonresonant linear instabilities. The observed instability triggers transition of the lattice into a chaotic dynamical state.

nlin.PS↗

arXiv2001

Families of Bragg-grating solitons in a cubic-quintic medium

We investigate the existence and stability of solitons in an optical waveguide equipped with a Bragg grating (BG) in which nonlinearity contains both cubic and quintic terms. The model has straightforward realizations in both temporal and spatial domains, the latter being most realistic. Two different families of zero-velocity solitons, which are separated by a border at which solitons do not exist, are found in an exact analytical form. One family may be regarded as a generalization of the usual BG solitons supported by the cubic nonlinearity, while the other family, dominated by the quintic nonlinearity, includes novel ``two-tier'' solitons with a sharp (but nonsingular) peak. These soliton families also differ in the parities of their real and imaginary parts. A stability region is identified within each family by means of direct numerical simulations. The addition of the quintic term to the model makes the solitons very robust: simulating evolution of a strongly deformed pulse, we find that a larger part of its energy is \emph{retained} in the process of its evolution into a soliton shape, only a small share of the energy being lost into radiation, which is opposite to what occurs in the usual BG model with cubic nonlinearity.

nlin.PS↗

arXiv2001

Quadratic solitons in cubic crystals

Starting from the Maxwell's equations and without resort to the paraxial approximation, we derive equations describing stationary (1+1)-dimensional beams propagating at an arbitrary direction in an optical crystal with cubic symmetry and purely quadratic nonlinearity. The equations are derived separately for beams with the TE and TM polarizations. In both cases, they contain and cubic nonlinear terms, the latter ones generated via the cascading mechanism. The final TE equations and soliton solutions to them are quite similar to those in previously known models with mixed quadratic-cubic nonlinearities. On the contrary to this, the TM model is very different from previously known ones. It consists of four first-order equations for transverse and longitudinal components of the electric field at the fundamental and second harmonics. Fundamental-soliton solutions of the TM model are also drastically different from the usual "quadratic" solitons, in terms of the parity of their components. In particular, the transverse and longitudinal components of the electric field at the fundamental harmonic in the fundamental TM solitons are described, respectively, by odd and single-humped even functions of the transverse coordinate. Amplitudes of the longitudinal and transverse fields become comparable for very narrow solitons, whose width is commensurate to the carrier wavelength.

nlin.PS↗

arXiv2001

One- and two-dimensional solitons in saturable media

Very narrow spatial bright solitons in (1+1)D and (2+1)D versions of cubic-quintic and full saturable models are studied, starting from the full system of the Maxwell's equations, rather than from the paraxial (NLS) approximation. For the solitons with both TE and TM polarizations, it is shown that there always exists a finite minimum width, and they cease to exist at a critical value of the propagation constant, at which their width diverges. Full similarity of the results obtained for both nonlinearities suggests that the same general conclusions apply to narrow solitons in any non-Kerr model.

nlin.PS↗

arXiv2000

Propagation of an optical pulse in a fiber link with random dispersion management

A model of a long optical communication line consisting of alternating segments with anomalous and normal dispersion, whose lengths are picked up randomly from a certain interval, is considered. At the first stage of the analysis, we calculate small changes of parameters of a quasi-Gaussian pulse passing a two-segment cell by means of the variational approach (VA), and approximate the evolution of the pulse passing many cells by smoothed ODEs with random coefficients, which are then solved numerically. Next, we perform systematic direct simulations of the model. Results are presented as dependences of the pulse's mean width, and standard deviation of the width from its mean value, on the propagation distance. The results produced by VA and direct simulations are similar. Averaging over 200 different realizations of the random-length set reveals slow long-scale dynamics of the pulse, frequently in the form of long-period oscillations of its width. It is thus found that the soliton is most stable in the case of the zero path-average dispersion (PAD), less stable in the case of anomalous PAD, and least stable in the case of normal PAD. The soliton's stability also strongly depends on its energy, the soliton with small energy being much more robust than its large-energy counterpart.

nlin.PS↗