Biomedical subjects

Boris A Malomed

Publications and source records attributed to Boris A Malomed.

At least 19 recordsLinked to original sources

PubMed2006

Gap solitons in quasiperiodic optical lattices.

Families of solitons in one- and two-dimensional (1D and 2D) Gross-Pitaevskii equations with the repulsive nonlinearity and a potential of the quasicrystallic type are constructed (in the 2D case, the potential corresponds to a fivefold optical lattice). Stable 1D solitons in the weak potential are explicitly found in three band gaps. These solitons are mobile, and they collide elastically. Many species of tightly bound 1D solitons are found in the strong potential, both stable and unstable (unstable ones transform themselves into asymmetric breathers). In the 2D model, families of both fundamental and vortical solitons are found and are shown to be stable.

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PubMed2006

Exact soliton-on-plane-wave solutions for two-component Bose-Einstein condensates.

By means of the Darboux transformation, we obtain analytical solutions for a soliton set on top of a plane-wave background in coupled Gross-Pitaevskii equations describing a binary Bose-Einstein condensate. We consider basic properties of the solutions with and without the cross interaction [cross phase modulation (XPM)] between the two components of the background. In the absence of the XPM, this solutions maintain properties of one-component condensates, such as the modulation instability (MI); in the presence of the cross interaction, the solutions exhibit different properties, such as restriction of the MI and soliton splitting.

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PubMed2006

Semidiscrete composite solitons in arrays of quadratically nonlinear waveguides.

We demonstrate that an array of discrete waveguides on a slab substrate, both featuring chi2 nonlinearity, supports stable solitons composed of discrete and continuous components. Two classes of fundamental composite soliton are identified: ones consisting of a discrete fundamental-frequency (FF) component in the waveguide array, coupled to a continuous second-harmonic (SH) component in the slab waveguide, and solitons with an inverted FF/SH structure. Twisted bound states of the fundamental solitons are found, too. In contrast with the usual systems, the intersite-centered fundamental solitons and bound states with the twisted continuous components are stable over almost the entire domain of their existence.

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PubMed2006

Two-dimensional solitons in saturable media with a quasi-one-dimensional lattice potential.

We study families of solitons in a two-dimensional model of the light transmission through a photorefractive medium equipped with a (quasi-)one-dimensional photonic lattice. The soliton families are bounded from below by finite minimum values of the peak and total power. Narrow solitons have a single maximum, while broader ones feature side lobes. Stability of the solitons is checked by direct simulations. The solitons can be set in motion across the lattice (actually, made tilted in the spatial domain), provided that the respective boost parameter does not exceed a critical value. Collisions between moving solitons are studied too. Collisions destroy the solitons, unless their velocities are sufficiently small. In the latter case, the colliding solitons merge into a single stable pulse.

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PubMed2006

Two-dimensional solitons in the Gross-Pitaevskii equation with spatially modulated nonlinearity.

We introduce a dynamical model of a Bose-Einstein condensate based on the two-dimensional Gross-Pitaevskii equation, in which the nonlinear coefficient is a function of radius. The model describes a situation with spatial modulation of the negative atomic scattering length, via the Feshbach resonance controlled by a properly shaped magnetic of optical field. We focus on the configuration with the nonlinear coefficient different from zero in a circle or annulus, including the case of a narrow ring. Two-dimensional axisymmetric solitons are found in a numerical form, and also by means of a variational approximation; for an infinitely narrow ring, the soliton is found in an exact form (in the latter case, exact solitons are also found in a two-component model). A stability region for the solitons is identified by means of numerical and analytical methods. In particular, if the nonlinearity is supported on the annulus, the upper stability border is determined by azimuthal perturbations; the stability region disappears if the ratio of the inner and outer radii of the annulus exceeds a critical value . The model gives rise to bistability, as the stationary solitons coexist with stable axisymmetric breathers, whose stability region extends to higher values of the norm than that of the static solitons. The collapse threshold strongly increases with the radius of the inner hole of the annulus. Vortex solitons are found too, but they are unstable.

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PubMed2005

Photon-number fluctuation and correlation of bound soliton pairs in mode-locked fiber lasers.

Quantum photon-number fluctuation and correlation of bound soliton pairs in mode-locked fiber lasers are studied on the basis of the complex Ginzburg-Landau equation model. We find that, depending on their phase difference, the total photon-number noise of the bound soliton pair can be larger or smaller than that of a single soliton, and the two solitons in the soliton pair have a corresponding positive or negative photon-number correlation. It is predicted for the first time to our knowledge that out-of-phase soliton pairs can exhibit less noise as a result of negative correlation.

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PubMed2005

Matter-wave solitons in nonlinear optical lattices.

We introduce a dynamical model of a Bose-Einstein condensate based on the one-dimensional (1D) Gross-Pitaevskii equation (GPE) with a nonlinear optical lattice (NOL), which is represented by the cubic term whose coefficient is periodically modulated in the coordinate. The model describes a situation when the atomic scattering length is spatially modulated, via the optically controlled Feshbach resonance, in an optical lattice created by interference of two laser beams. Relatively narrow solitons supported by the NOL are predicted by means of the variational approximation (VA), and an averaging method is applied to broad solitons. A different feature is a minimum norm (number of atoms), N = N(min), necessary for the existence of solitons. The VA predicts N(min) very accurately. Numerical results are chiefly presented for the NOL with the zero spatial average value of the nonlinearity coefficient. Solitons with values of the amplitude A larger than at N = N(min) are stable. Unstable solitons with smaller, but not too small, A rearrange themselves into persistent breathers. For still smaller A, the soliton slowly decays into radiation without forming a breather. Broad solitons with very small A are practically stable, as their decay is extremely slow. These broad solitons may freely move across the lattice, featuring quasielastic collisions. Narrow solitons, which are strongly pinned to the NOL, can easily form stable complexes. Finally, the weakly unstable low-amplitude solitons are stabilized if a cubic term with a constant coefficient, corresponding to weak attraction, is included in the GPE.

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PubMed2005

Gap solitons in a medium with third-harmonic generation.

We find two-component optical solitons in a nonlinear waveguide with a Bragg grating, including Kerr effects and third-harmonic generation (THG). The model may be realized in temporal and in spatial domains. Two species of fundamental gap solitons (GSs) are found. The first ("THG-gap soliton") has the bulk of its energy at the fundamental frequency (FF) and a lesser part in the third-harmonic (TH) band. The FF part of the soliton is always single humped; the TH part may be single or double humped. Stability domains for quiescent and moving THG-gap solitons strongly shrink with increase of velocity. The second species is the usual ("simple") GS, sitting entirely in the TH band. More complex solutions are also found, in the form of a bound state of a THG-gap soliton and two simple GSs, with a finite binding energy. When a THG-gap soliton is unstable, the instability is oscillatory. It may ultimately cause the THG-gap soliton to throw off some radiation and evolve into a localized structure with the FF and TH components out of phase, with or without internal oscillations. Stable solitons feature an excited state (i.e., they support a localized eigenmode).

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PubMed2005

Motion of discrete solitons assisted by nonlinearity management.

We demonstrate that time-periodic modulation of the nonlinearity coefficient in the discrete nonlinear Schrödinger equation strongly facilitates creation of traveling solitons in the lattice. We predict this possibility in a semi-qualitative form analytically, and test it in direct numerical simulations. Systematic computations reveal several generic dynamical regimes, depending on the amplitude and frequency of the time modulation, and on the initial thrust which sets the soliton in motion. These regimes include irregular motion of the soliton, regular motion of a decaying one, and regular motion of a stable soliton. The motion may occur in both the straight and reverse directions, relative to the initial thrust. In the case of stable motion, extremely long simulations in a lattice with periodic boundary conditions demonstrate that the soliton keeps moving indefinitely long without any visible loss. Velocities of moving stable solitons are in good agreement with the analytical prediction, which is based on requiring a resonance between the ac drive and motion of the soliton through the periodic lattice. The generic dynamical regimes are mapped in the model's parameter space. Collisions between moving stable solitons are briefly investigated too, with a conclusion that two different outcomes are possible: elastic bounce, or bounce with mass transfer from one soliton to the other. The model can be realized experimentally in a Bose-Einstein condensate trapped in a deep optical lattice.

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PubMed2005

Trapping Bragg solitons by a pair of defects.

We study collisions of moving solitons in a fiber Bragg grating with a structure composed of two local defects of the grating, attractive or repulsive. Results are summarized in the form of diagrams showing the share of the trapped energy as a function of the soliton's velocity and defects' strength. The moving soliton can be trapped by a cavity bounded by repulsive defects; a well-defined region of the most efficient trapping is identified. The trapped soliton performs persistent oscillations in the cavity, with the frequency in the GHz range. For attractive defects, essential differences are found from the earlier studied case of the collision of a soliton with a single defect: in this case, too, there appears a well-defined region of the most efficient trapping, and the largest velocity, up to which the soliton can be captured, increases. The findings may be significant for experiments aimed at the creation of "standing-light" pulses in the fiber gratings and for related applications. Collisions between identical solitons moving across the two-defect structure are also studied. On the attractive set, soliton-soliton collisions may give rise to symmetric capture of the solitons by both defects or merger into a single pulse trapped at one defect.

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PubMed2005

Coupled-mode theory for spatial gap solitons in optically induced lattices.

We derive two systems of coupled-mode equations for spatial gap solitons in one-dimensional (1D) and quasi-one-dimensional (Q1D) photonic lattices induced by two interfering optical beams in a nonlinear photorefractive crystal. The models differ from the ordinary coupled-mode system (e.g., for the fiber Bragg grating) by saturable nonlinearity and, if expanded to cubic terms, by the presence of four-wave-mixing terms. In the 1D system, solutions for stationary gap solitons are obtained in an implicit analytical form. For the Q1D model and for tilted ("moving") solitons in both models, solutions are found in a numerical form. The existence of stable tilted solitons in the full underlying model of the photonic lattice in the photorefractive medium is also shown. The stability of gap solitons is systematically investigated in direct simulations, revealing a nontrivial border of instability against oscillatory perturbations. In the Q1D model, two disjointed stability regions are found. The stability border of tilted solitons does not depend on the tilt. Interactions between stable tilted solitons are investigated too. The collisions are, chiefly, elastic, but they may be inelastic close to the instability border.

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PubMed2005

Stability of double-peaked solitons in Bragg gratings with the quadratic nonlinearity.

We report systematic results for the existence and stability of double-peaked (DP) solitons in the known model including the Bragg grating, which acts on the fundamental- and second-harmonic waves, and the quadratic (chi(2)) nonlinearity, which accounts for the parametric interaction between the harmonics. We identify existence and stability regions for the DP solitons in the plane of relevant parameters (the relative Bragg reflectivity at the two harmonics, and phase mismatch q between them). We conclude that the existence region considerably expands with the soliton's velocity v, while the stability area remains nearly constant up to a critical value of v. The stability region quickly vanishes as one crosses the critical value, while the region of the existence of unstable DP solitons does not disappear. The stability is confined to negative soliton frequencies, and almost entirely to q<0. Collisions between stable moving solitons are investigated too, with a conclusion that they are always destructive.

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PubMed2005

Gap solitons in a model of a hollow optical fiber.

We introduce a model for two coupled waves propagating in a hollow-core fiber: a linear dispersionless core mode and a dispersive nonlinear surface mode. The linear coupling between them may open a bandgap through the mechanism of avoidance of crossing between dispersion curves. The third-order dispersion of the surface mode is necessary for the existence of the gap. Numerical investigation reveals that the entire bandgap is filled with solitons, and they are stable in direct simulations. The gap-soliton (GS) family includes stable pulses moving relative to the given reference frame up to limit values of the corresponding boost delta, beyond which they do not exist. The limit values are asymmetric for delta > or < 0. Recently observed solitons in hollow-core photonic crystal fibers may belong to this GS family.

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PubMed2005

Quasistable two-dimensional solitons with hidden and explicit vorticity in a medium with competing nonlinearities.

We consider basic types of two-dimensional (2D) vortex solitons in a three-wave model combining quadratic chi((2)) and self-defocusing cubic chi((3))(-) nonlinearities. The system involves two fundamental-frequency (FF) waves with orthogonal polarizations and a single second-harmonic (SH) one. The model makes it possible to introduce a 2D soliton, with hidden vorticity (HV). Its vorticities in the two FF components are S(1,2) = +/-1 , whereas the SH carries no vorticity, S(3) = 0 . We also consider an ordinary compound vortex, with 2S(1) = 2S(2) = S(3) = 2 . Without the chi((3))(-) terms, the HV soliton and the ordinary vortex are moderately unstable. Within the propagation distance z approximately 15 diffraction lengths, Z(diffr), the former one turns itself into a usual zero-vorticity (ZV) soliton, while the latter splits into three ZV solitons (the splinters form a necklace pattern, with its own intrinsic dynamics). To gain analytical insight into the azimuthal instability of the HV solitons, we also consider its one-dimensional counterpart, viz., the modulational instability (MI) of a one-dimensional CW (continuous-wave) state with "hidden momentum," i.e., opposite wave numbers in its two components, concluding that such wave numbers may partly suppress the MI. As concerns analytical results, we also find exact solutions for spreading localized vortices in the 2D linear model; in terms of quantum mechanics, these are coherent states with angular momentum (we need these solutions to accurately define the diffraction length of the true solitons). The addition of the chi((3))(-) interaction strongly stabilizes both the HV solitons and the ordinary vortices, helping them to persist over z up to 50 Z(diffr). In terms of the possible experiment, they are completely stable objects. After very long propagation, the HV soliton splits into two ZV solitons, while the vortex with S(3) = 2S(1,2) = 2 splits into a set of three or four ZV solitons.

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PubMed2005

Finite-band solitons in the Kronig-Penney model with the cubic-quintic nonlinearity.

We present a model combining a periodic array of rectangular potential wells [the Kronig-Penney (KP) potential] and the cubic-quintic (CQ) nonlinearity. A plethora of soliton states is found in the system: fundamental single-humped solitons, symmetric and antisymmetric double-humped ones, three-peak solitons with and without the phase shift pi between the peaks, etc. If the potential profile is shallow, the solitons belong to the semi-infinite gap beneath the band structure of the linear KP model, while finite gaps between the Bloch bands remain empty. However, in contrast with the situation known in the model combining a periodic potential and the self-focusing Kerr nonlinearity, the solitons fill only a finite zone near the top of the semi-infinite gap, which is a consequence of the saturable character of the CQ nonlinearity. If the potential structure is much deeper, then fundamental and double (both symmetric and antisymmetric) solitons with a flat-top shape are found in the finite gaps. Computation of stability eigenvalues for small perturbations and direct simulations show that all the solitons are stable. In the shallow KP potential, the soliton characteristics, in the form of the integral power Q (or width w) versus the propagation constant k, reveal strong bistability, with two and, sometimes, four different solutions found for a given k (the bistability disappears with the increase of the depth of the potential). Disobeying the Vakhitov-Kolokolov criterion, the solution branches with both dQ/dk > 0 and dQ/dk < 0 are stable. The curve Q(k) corresponding to each particular type of the solution (with a given number of local peaks and definite symmetry) ends at a finite maximum value of Q (breathers are found past the end points). The increase of the integral power gives rise to additional peaks in the soliton's shape, each corresponding to a subpulse trapped in a local channel of the KP structure (a beam-splitting property). It is plausible that these features are shared by other models combining saturable nonlinearity and a periodic substrate.

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PubMed2005

Nonlinear lattice dynamics of Bose-Einstein condensates.

The Fermi-Pasta-Ulam (FPU) model, which was proposed 50 years ago to examine thermalization in nonmetallic solids and develop "experimental" techniques for studying nonlinear problems, continues to yield a wealth of results in the theory and applications of nonlinear Hamiltonian systems with many degrees of freedom. Inspired by the studies of this seminal model, solitary-wave dynamics in lattice dynamical systems have proven vitally important in a diverse range of physical problems-including energy relaxation in solids, denaturation of the DNA double strand, self-trapping of light in arrays of optical waveguides, and Bose-Einstein condensates (BECs) in optical lattices. BECs, in particular, due to their widely ranging and easily manipulated dynamical apparatuses-with one to three spatial dimensions, positive-to-negative tuning of the nonlinearity, one to multiple components, and numerous experimentally accessible external trapping potentials-provide one of the most fertile grounds for the analysis of solitary waves and their interactions. In this paper, we review recent research on BECs in the presence of deep periodic potentials, which can be reduced to nonlinear chains in appropriate circumstances. These reductions, in turn, exhibit many of the remarkable nonlinear structures (including solitons, intrinsic localized modes, and vortices) that lie at the heart of the nonlinear science research seeded by the FPU paradigm.

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PubMed2005

Solitons in a linearly coupled system with separated dispersion and nonlinearity.

We introduce a model of dual-core waveguide with the cubic nonlinearity and group-velocity dispersion (GVD) confined to different cores, with the linear coupling between them. The model can be realized in terms of photonic-crystal fibers. It opens a way to understand how solitons are sustained by the interplay between the nonlinearity and GVD which are not "mixed" in a single nonlinear Schrodinger (NLS) equation, but are instead separated and mix indirectly, through the linear coupling between the two cores. The spectrum of the system contains two gaps, semi-infinite and finite ones. In the case of anomalous GVD in the dispersive core, the solitons fill the semi-infinite gap, leaving the finite one empty. This soliton family is entirely stable, and is qualitatively similar to the ordinary NLS solitons, although shapes of the soliton's components in the nonlinear and dispersive cores are very different, the latter one being much weaker and broader. In the case of the normal GVD, the situation is completely different: the semi-infinite gap is empty, but the finite one is filled with a family of stable gap solitons featuring a two-tier shape, with a sharp peak on top of a broad "pedestal." This case has no counterpart in the usual NLS model. An extended system, including weak GVD in the nonlinear core, is analyzed too. In either case, when the solitons reside in the semi-infinite or finite gap, they persist if the extra GVD is anomalous, and completely disappear if it is normal.

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PubMed2004

Resonant nonlinearity management for nonlinear Schrödinger solitons.

We consider effects of a periodic modulation of the nonlinearity coefficient on fundamental and higher-order solitons in the one-dimensional NLS equation, which is an issue of direct interest to Bose-Einstein condensates in the context of the Feshbach-resonance control, and fiber-optic telecommunications as concerns periodic compensation of the nonlinearity. We find from simulations, and explain by means of a straightforward analysis, that the response of a fundamental soliton to the weak perturbation is resonant, if the modulation frequency omega is close to the intrinsic frequency of the soliton. For higher-order n-solitons with n=2 and 3, the response to an extremely weak perturbation is also resonant, if omega is close to the corresponding intrinsic frequency. More importantly, a slightly stronger drive splits the 2- or 3-soliton, respectively, into a set of two or three moving fundamental solitons. The dependence of the threshold perturbation amplitude, necessary for the splitting, on omega has a resonant character too. Amplitudes and velocities of the emerging fundamental solitons are accurately predicted, using exact and approximate conservation laws of the perturbed NLS equation.

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