Serge F. Mingaleev

Self-Trapping and Stable Localized Modes in Nonlinear Photonic Crystals

We predict the existence of stable nonlinear localized modes near the band edge of a two-dimensional reduced-symmetry photonic crystal with a Kerr nonlinearity. Employing the technique based on the Green function, we reveal a physical mechanism of the mode stabilization associated with the effective nonlinear dispersion and long-range interaction in the photonic crystals.

Long-range interaction and nonlinear localized modes in photonic crystal waveguides

We develop the theory of nonlinear localized modes (intrinsic localized modes or discrete breathers) in two-dimensional (2D) photonic crystal waveguides. We consider different geometries of the waveguides created by an array of nonlinear dielectric rods embedded into an otherwise perfect linear 2D photonic crystal, and demonstrate that the effective interaction in such waveguides is nonlocal, being described by a nonlinear lattice model with long-range coupling and nonlocal nonlinearity. We reveal the existence of different types of nonlinear guided mode that are also localized in the waveguide direction, and describe their unique properties, including bistability.

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 (LRIs) between base pairs of DNA is offered for the description of nonlinear dynamics of the DNA molecule. We show that LRIs 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.

Localized excitations in discrete nonlinear Schro¨dinger systems: effects of nonlocal dispersive interactions and noise

A one-dimensional discrete nonlinear Schr ¨ (DNLS) model with the power dependence,r s on the distance r ,o f dispersive interactions is proposed. The stationary states of the system are studied both analytically and numerically. Two kinds of trial functions, exp-like and sech-like are exploited and the results of both approaches are compared. Both on-site and inter-site stationary states are investigated. It is shown that for s sufficiently large all features of the model are qualitatively the same as in the DNLS model with nearest-neighbor interaction. Fors less than some critical value, scr, there is an interval of bistability where two stable stationary states exist at each excitation number. The bistability of on-site solitons may occur for dipole‐dipole dispersive interaction .s D 3/, whilescr for inter-site solitons is close to 2.1. In the framework of the DNLS equation with nearest-neighbor coupling we discuss the stability of highly localized, “breather-like”, excitations under the influence of thermal fluctuations. Numerical analysis shows that the lifetime of the breather is always finite and in a large parameter region inversely proportional to the noise variance for fixed damping and nonlinearity. We also find that the decay rate of the breather decreases with increasing nonlinearity and with increasing damping. Copyright

Effect of nonlocal dispersion on self-interacting excitations

Abstract The dynamics of self-interacting quasiparticles in 1D systems with long-range dispersive interactions is expressed in terms of a nonlocal nonlinear Schrodinger equation. Two branches of stationary solutions are found. The new branch which contains a cusp soliton is shown to be unstable and blowup is observed. Moving solitons radiate with a wavelength proportional to the velocity.

Nonlinearity-induced conformational instability and dynamics of biopolymers

We propose a simple phenomenological model for describing the conformational dynamics of biopolymers via the nonlinearity-induced buckling and collapse instability. We describe the buckling instability analytically, and then demonstrate its role in the folding dynamics of macromolecules through the three-dimensional numerical simulations of long semiflexible chains.

Effects of finite curvature on soliton dynamics in a chain of non-linear oscillators

We consider a curved chain of non-linear oscillators and show that the interplay of curvature and non-linearity leads to a number of qualitative effects. In particular, the energy of non-linear localized excitations centred on the bending decreases when curvature increases, i.e. bending manifests itself as a trap for excitations. Moreover, the potential of this trap is a double-well one, thus leading to a symmetry-breaking phenomenon: a symmetric stationary state may become unstable and transform into an energetically favourable asymmetric stationary state. The essentials of symmetry breaking are examined analytically for a simplified model. We also demonstrate a threshold character of the scattering process, i.e. transmission, trapping, or reflection of the moving non-linear excitation passing through the bending.

Multi-component structure of nonlinear excitations in systems with length-scale competition

Abstract:We investigate the properties of nonlinear excitations in different types of soliton carrying systems with long-range dispersive interactions. We show that length-scale competition in such systems, universally results in a multi-component structure of nonlinear excitations which may lead to a new type of multistability: coexistence of different nonlinear excitations at the same value of the spectral parameter (i.e., velocity in the case of anharmonic lattices or frequency in nonlinear Schrödinger models).

BOUND STATES OF ENVELOPE AND BOUSSINESQ SOLITONS IN ANHARMONIC LATTICES

We investigate a soliton charge and energy transport in anharmonic molecular systems and show that at large enough anharmonicity parameter there are two kinds of envelope solitons, one of which is a Davydov soliton. It has the usual one-bell shape and may exist at any anharmonicity. The other kind has a two-bell shape. The two-bell shape is a bound state of Davydov and Boussinesq solitons. It is caused by excitation (electron) tunnelling in the effective lattice potential.

Numerical study of breathers in a bent chain of oscillators with long-range interaction

Most of the studies of breathers in networks of oscillators are limited to nextneighbour interaction. However, long-range interaction becomes critical when the geometry of the chain is taken into account, as the distance between oscillators and, therefore, the coupling, depends on the shape of the system. In this paper we analyse the existence and stability of breathers, i.e. localized oscillations in a simple model for a bent chain of oscillators with long-range interaction.