Thermal diffusion of solitons on anharmonic chains with long-range coupling.

We extend our studies of thermal diffusion of nontopological solitons to anharmonic Fermi-Pasta-Ulam-type chains with additional long-range couplings. The observed superdiffusive behavior in the case of nearest-neighbor interaction turns out to be the dominating mechanism for the soliton diffusion on chains with long-range interactions. Using a collective variable technique in the framework of a variational analysis for the continuum approximation of the chain, we derive a set of stochastic integrodifferential equations for the collective variables (CVs) soliton position and the inverse soliton width. This set can be reduced to a statistically equivalent set of Langevin-type equations for the CV, which shares the same Fokker-Planck equation. The solution of the Langevin set and the Langevin dynamics simulations of the discrete system agree well and demonstrate that the variance of the soliton increases stronger than linearly with time (superdiffusion). This result for the soliton diffusion on anharmonic chains with long-range interactions reinforces the conjecture that superdiffusion is a generic feature of nontopological solitons.

Envelope solitons on anharmonic damped atomic chains.

We investigate the influence of dissipation on envelope solitons on anharmonic chains. We consider both Stokes and hydrodynamical damping and derive the evolution equations for the envelope in both the continuum and the quasi-continuum approximation of the chain. We introduce an appropriate collective variable ansatz for the envelope in order to describe the effect of damping on the soliton shape. We derive ordinary differential equations for the evolution of the three collective variables amplitude, width, and chirp which describe the spatial modulation of the envelope. The analytical results are in good agreement with the simulations of the discrete system for high-energy excitations on the chain. Our results derived from the quasi-continuum approximation show significant improvements compared to the continuum approximation.

Persistent breathers in long-ranged discrete nonlinear Schrödinger models.

The DNLS model including Kac-Baker long-range interactions and nonlinear damping exhibits prominent effects in computer simulations. The combination of long-range forces and damping yields a periodic pattern of stationary breathers from an originally uniformly distributed background. The inverse interaction radius determines the periodicity which can be understood in the quasicontinuum approximation of the system. For the undamped system, we investigate the impact of the long-range interactions on the transition to the persistent-breather phase, which only depends on the energy and the norm of the DNLS. Using Monte Carlo techniques, we can monitor the localization strength as a function of the the long-range radius and the system temperature, which is formally negative in the persistent-breather phase.

Effects of competing short- and long-range dispersive interactions on discrete breathers.

The discrete nonlinear Schrödinger equation with competing short-range and long-range interactions is considered in spatial dimensions d> or =2. This model equation is derived for a situation of two linearly coupled excitations (independently of dimension), and we analytically and numerically study its properties in 2+1 dimensions. We analyze theoretically and demonstrate numerically the dependence of the discrete breather solutions on the amplitude and range of the interactions. We find that complete suppression of the existence thresholds obtained recently for short-range interactions can be achieved beyond a critical value of the amplitude or of the range of the long-range kernel. For supercritical values of the corresponding parameters, staggered branches of solutions are obtained both in theory as well as in the numerical experiment.

Collapse arresting in an inhomogeneous two-dimensional nonlinear Schrödinger model.

Collapse of (2+1)-dimensional beams in the inhomogeneous two-dimensional cubic nonlinear Schrödinger equation is analyzed numerically and analytically. It is shown that in the vicinity of a narrow attractive inhomogeneity, the collapse of beams that in a homogeneous medium would collapse may be arrested under certain circumstances.

Spatiotemporal collapse in a nonlinear waveguide with a randomly fluctuating refractive index.

Analytical results, based on the virial theorem and the Furutsu-Novikov theorem, of the spatiotemporal evolution of a pulse in a nonlinear waveguide with a randomly fluctuating refractive index are presented. For initial conditions in which total collapse occurs in a homogeneous waveguide, random fluctuations postpone the collapse. Sufficiently large-amplitude and short-wavelength fluctuations can cause an initially localized pulse to spread instead of contracting. We show that the disorder can be applied to induce a high degree of controllability of the spatiotemporal extent of the pulses in the nonlinear waveguide.

Collapse arresting in an inhomogeneous quintic nonlinear Schrödinger model.

Collapse of (1+1)-dimensional beams in the inhomogeneous one-dimensional quintic nonlinear Schrödinger equation is analyzed both numerically and analytically. It is shown that in the vicinity of a narrow attractive inhomogeneity, the collapse of beams in which the homogeneous medium would blow up may be delayed and even arrested.

Engineered nonlinear lattices.

We show that with the quasi-phase-matching technique it is possible to fabricate stripes of nonlinearity that trap and guide light like waveguides. We investigate an array of such stripes and find that when the stripes are sufficiently narrow, the beam dynamics is governed by a quadratic nonlinear discrete equation. The proposed structure therefore provides an experimental setting for exploring discrete effects in a controlled manner. In particular, we show propagation of breathers that are eventually trapped by discreteness. When the stripes are wide the beams evolve in a structure we term a quasilattice, which interpolates between a lattice system and a continuous system.

Models for energy and charge transport and storage in biomolecules.

Two models for energy and charge transport and storage in biomolecules are considered. A model based on the discrete nonlinear Schrödinger equation with long-range dispersive interactions (LRI's) between base pairs of DNA is offered for the description of nonlinear dynamics of the DNA molecule. We show that LRI's are responsible for the existence of an interval of bistability where two stable stationary states, a narrow, pinned state and a broad, mobile state, coexist at each value of the total energy. The possibility of controlled switching between pinned and mobile states is demonstrated. The mechanism could be important for controlling energy storage and transport in DNA molecules. Another model is offered for the description of nonlinear excitations in proteins and other anharmonic biomolecules. We show that in the highly anharmonic systems a bound state of Davydov and Boussinesq solitons can exist.

Self-guiding light in layered nonlinear media.

We study the propagation of intense optical beams in layered Kerr media. With appropriate shapes, beams with a power close to the self-focusing threshold are shown to propagate over long distances as quasi-stationary waveguides in cubic media supporting a periodic nonlinear refractive index.