C. Polymilis

STABLE ANTI-DARK LIGHT BULLETS SUPPORTED BY THE THIRD-ORDER DISPERSION

Abstract We consider transverse dynamics of quasi-one-dimensional (Q1D) spatio-temporal solitons governed by the nonlinear Schrodinger equation with the higher-order dispersion. We obtain stable Q1D dark solitons, unstable Q1D anti-dark solitons and stable anti-dark lump solitons in the form of light bullets.

Exact travelling wave solutions for a generalized nonlinear Schrödinger equation

A wide class of exact travelling wave solutions of a generalized nonlinear Schrodinger equation (GNLS) is obtained and analysed in detail. This class of solutions incorporates bright and dark solitary waves, periodic waves, unbounded waves and other solitary waves as asymptotic limits of the periodic or unbounded modes. The method of analysis adopted is based on reducing the GNLS to an ordinary differential equation and studying the phase plane of the resulting dynamical system. Application of the obtained results to the problem of propagation of femtosecond duration pulses in nonlinear optical fibres is also discussed.

Ultrashort solitary-wave propagation in dielectric media with resonance-dominated chromatic dispersion

Nonlinear pulse propagation in single-mode inhomogeneous dielectric waveguides is analyzed by means of the reductive perturbation method. The chromatic dispersion of the fiber takes impurity-related resonance phenomena into account, while the nonlinear properties are described by means of a time- (frequency-) dependent dielectric constant with cubic nonlinearity. For the case of short-envelope propagation, a perturbed nonlinear Schrodinger equation, reflecting higher-order linear and nonlinear effects, is derived and then transformed into a generalized higher-order nonlinear Schrodinger (GHONLS) equation that is valid for both the anomalous- and the normal-dispersion regimes. In the search for quasi-stationary-wave solutions the GHONLS equation is then reduced to a nonlinear ordinary differential equation, which is analyzed by phase-space analysis. The latter leads to bright- and dark soliton solutions that can be analytically derived and correspond to separatrices on the phase plane of the associated dynamical system. Emphasis is given to the connections among the initial spatiotemporal pulse information and the types of mode (bright or dark solitons) that can be excited.

Bifurcations of beam-beam like maps

The bifurcations of a class of mappings including the beam-beam map are examined. These maps are asymptotically linear at infinity where they exhibit invariant curves and elliptic periodic points. The dynamical behaviour is radically different with respect to the Henon-like polynomial maps whose stability boundary (dynamic aperture) is at a finite distance. Rather than the period-doubling bifurcations exhibited by the Henon-like maps, we observe a systematic appearance of tangent bifurcations and in phase space one observes the disappearance of chains of islands born from the origin and coming from infinity. This behaviour has relevant consequences on the transport process.

Optical stripes and bullets for a modified nonlinear Schrödinger equation

We study the existence, formation, and stability of quasi-one-dimensional (stripes) and two-dimensional (bullets) spatio-temporal soliton solutions for a (2+1)-dimensional modified nonlinear Schrodinger equation, in the presence of the self-steepening effect. These solutions, which are on top of a continuous-wave background, are either dark or antidark, the latter being supported by the self-steepening effect. We show that there exist stable small-amplitude lump solitons, which, in the context of nonlinear optics, constitute novel light bullets.

Conditions for soliton trapping in random potentials using Lyapunov exponents of stochastic ODEs

Abstract The conditions for trapping of Schrodinger solitons in a random potential are analysed. A stochastic ODE is derived for the position of the soliton centre and the behaviour of its solution is studied for different types of stochasticity. The problem of trapping or propagation of the solitons in the potential is connected with the calculation of the Lyapunov exponents for stochastic ODEs and rigorous criteria for trapping are provided. Possible applications to the problem of strong dispersion management are discussed.

The homoclinic tangle of a 1:2 resonance in a 2-D Hamiltonian system

We study in great detail the geometry of the homoclinic tangle, with respect to the energy, corresponding to an unstable periodic orbit of type 1:2, on a surface of section representing a 2-D Hamiltonian system. The tangle consists of two resonance areas, in contrast with the tangles of type-l or -{l, m, k, x = 0} considered in previous studies, that consist of only one resonance area. We study the intersections of the inner and outer lobes of the same resonance area and of the two resonance areas. The intersections of the lobes follow certain rules. The detailed study of these rules allows us to derive quantitative relations about the number of intersections and to understand the complex behavior of the higher order lobes by studying the lower order lobes. We find 1st, 2nd, 3rd, etc. order intersections formed by lobes making 1, 2, 3, etc. turns around an island. After a sufficiently high order of iterations a lobe may intersect its image and thus produce a Poincaré recurrence. Numerical results for a wide interval of energies are presented. The number of intersections changes through tangencies. In any finite interval of the energy between two tangencies of 1st order, an infinite number of higher order tangencies occur and thus, according to the Newhouse theorem, there exist nearby islands of stability.

Numerical Study of the Phase Space of a four Dimensional Symplectic Map

We study the structure of non periodic orbits in a 4-D symplectic map, composed of two coupled 2-D maps. Such maps correspond to 3 degrees of freedom Hamiltonian systems.

A Method for locating Singularities in the Complex Time Domain for Integrable Dynamical Systems

A simple method for the determination of the position of singularities in the complex time domain for dynamical systems which are described by ordinary differential equations is presented. The method is designed for integrable separable systems whose solutions are not expressible in closed form. A direct consequence of this method is that it ‘closes’ the phase space. Simple physical meaning is given to the singularity position.