Timoléon C. Kofané

Non-Lagrangian collective variable approach for optical solitons in fibres

We use a non-Lagrangian method to express the generalized nonlinear Schrodinger equation, for pulse propagation in optical fibres, in terms of the pulse parameters, called collective variables, such as the pulse width, amplitude, chirp and frequency. The collective variable equations of motion, which include the important effects due to fibre losses, third-order dispersion, stimulated Raman scattering and self-steepening are derived using a direct averaging method without the help of any Lagrangian.

Nonlinear excitations in a compressible quantum Heisenberg chain

Abstract We investigate, both analytically and numerically, nonlinearly coupled magnetic and elastic excitations of compressible Heisenberg chains. From a shallow water wave treatment of perturbation terms, one can derive two types of coupled equations which are coupled Boussinesq and nonlinear Schrodinger (NLS) equations and coupled Boussinesq and NLS-like equations. We also simulate collisions between magnetic and elastic solitons in the compressible Heisenberg chain when a nonlinearized approach is performed to deal with the magnetic modes in the presence of harmonic as well as anharmonic interactions. Finally, from a fast Fourier transform (FFT) algorithm, the dynamical structure factor is computed for all the numerical solutions of the elastic mode for the model under consideration.

Chaotic behavior in deformable models: the double-well doubly periodic oscillators

Abstract The motion of a particle in a one-dimensional perturbed double-well doubly periodic potential is investigated analytically and numerically. A simple physical model for calculating analytically the Melnikov function is proposed. The onset of chaos is studied through an analysis of the phase space, a construction of the bifurcation diagram and a computation of the Lyapunov exponent. The parameter regions of chaotic behavior predicted by the theoretical analysis agree very well with numerical simulations.

Note on collective variable theory of nonlinear Schrödinger solitons

Inconsistencies arise in a recently developed collective variable theory of nonlinear Schrodinger solitons, as a result of a particular formulation of the energy-conservation principle in terms of the time derivative of the phase of the original field. We show that the inconsistencies are resolved either by correctly reformulating the energy-conservation principle or by directly averaging the nonlinear Schrodinger equation.

Subharmonic and homoclinic bifurcations in the driven and damped sine-Gordon system

Abstract Chaotic responses induced by an applied biharmonic driven signal on the sine-Gordon (sG) system influenced by a constant dc-driven and the damping fields are investigated using a collective coordinate approach for the motion of the breather in the system. For this biharmonic signal, one term has a large amplitude at low frequency. Thus, the classical Melnikov method does not apply to such a system; however, we use the modified version of the Melnikov method to homoclinic bifurcations of the perturbed sG system. Additionally resonant breathers are studied using the modified subharmonic Melnikov theory. This dynamic behavior is illustrated by some numerical computations.

Chaotic behaviour in deformable models: the asymmetric doubly periodic oscillators

Abstract The motion of a particle in a one-dimensional perturbed asymmetric doubly periodic (ASDP) potential is investigated analytically and numerically. A simple physical model for calculating analytically the Melnikov function is proposed. The onset of chaos is studied through an analysis of the phase space, a construction of the bifurcation diagram and a computation of the Lyapunov exponent. Theory predicts the regions of chaotic behaviour of orbits in a good agreement with computer calculations.

Horseshoe-shaped maps in chaotic dynamics of long Josephson junction driven by biharmonic signals

Abstract A collective coordinate approach is applied to study chaotic responses induced by an applied biharmonic driven signal on the long Josephson junction influenced by a constant dc-driven field with breather initial conditions. We derive a nonlinear equation for the collective variable of the breather and a new version of the Melnikov method is then used to demonstrate the existence of Smale horseshoe-shaped maps in its dynamics. Additionally, numerical simulations show that the theoretical predictions are well reproduced. The subharmonic Melnikov theory is applied to study the resonant breathers. Results obtained using this approach are in good agreement with numerical simulations of the dynamics of the Poincare islands.

Long-range effects on the periodic deformable sine-Gordon chains

The model of long-range interatomic interactions is found to reveal a number of new features, closely connected with the substrate potential shape parameter s. The phase trajectories, as well as an analytical analysis, provide information on a disintegration of solitons upon reaching some critical values of the lattice parameters. An implicit form for two classes of these topological solitons (kink) is calculated exactly.