M. P. Soerensen

Perturbation theories for sine-Gordon soliton dynamics

Abstract Three recent perturbation assumptions for soliton dynamics on Josephson junctions are compared with direct numerical integrations of the perturbed sine-Gordon equation. The McLaughlin-Scott theory yields the better prediction of soliton transmission or pinning at a microshort in presence of loss and bias without or with surface resistance loss.

Numerical evidence for global bifurcations leading to switching phenomena in long Josephson junctions

Abstract Fluxons in long Josephson junctions are physical manifestations of travelling waves that connect rest states of the model partial differential equation (p.d.e.), which is a perturbed sine-Gordon equation. In the reduced traavelling wave ordinary differential equation (o.d.e.), fluxons correspond to heteroclinic connections between fixed points. In the absence of surface impedence effects, fluxons persist in parameter regimes until the fixed points disappear, after which the system ‘switches’ to another configuration. It is known that the presence of surface impedence produces a singular perturbation of the model equation, together with a new phenomenon: the fluxons switch in parameter regimes before the fixed points are lost. Why this occurs is unknown, and is the focus of this paper. Two disjoint possibilities are: (1) instability: fluxons still exist, but they become unstable in the p.d.e. due to surface impedance effects; (2) nonexistence: the fluxons fail to exist, even though the fixed points remain. Here, we provide compelling numerical evidence for the second scenario, characterized by a global bifurcation in the travelling wave phase space: a breakdown of heteroclinicorbits, undetected at the local linearized level. Moreover, this global o.d.e. bifurcation occurs at parameter values consistent with the p.d.e. switching phenomenon.

Nonlinear fiber external cavity mode locking of erbium-doped fiber lasers

A detailed numerical study is performed of the dynamics of optical pulse compression in an additive pulse mode-locking configuration consisting of an erbium-doped fiber laser coupled to a nonlinear optical fiber. Pulse evolution is treated by the use of an extended nonlinear Schrodinger equation incorporating a gain-and-loss term in the active fiber laser and a loss term for the nonlinear fiber, which forms the nonlinear auxilliary cavity. To treat ultrashort optical pulses with durations of less than 100 fs, we have considered third-order dispersion and self-frequency shift. On the basis of extensive numerical simulations of the all-fiber coupled-cavity configuration, a stable operating range for pulse compression is identified. It is shown that a variety of less-desirable dynamical evolutions, including pulse splitting, are obtained under other operating conditions.

Solitary Waves on Nonlinear Elastic Rods

Acoustic waves on elastic rods with circular cross section are governed by so-called improved Boussinesq equations when transverse motion and nonlinearity in the medium are taken into account. Solitary waves to these equations are shown to possess soliton-like properties in agreement with the fact that the impro ed Boussinesq equations are nearly integrable. Numerical investigations of blow up (in finite time), reflection and fission of solitary waves are presented in the cases of ending rods and rods with varying cross-section. The results are applied in a model for DNA-molecules in aqueous solution which is capable to predict the influence of anharmonicity on the spectral properties of this important molecule.

On low-dimensional chaos in RF SQUIDs

Abstract Superconducting quantum interference devices (SQUIDs) respond chaotically to external oscillating fluxes. The small deviations from one-dimensional return maps are investigated. Within a narrow region of parameter space a sequence of period doublings, windows with odd periods and chaotic behaviour, intermittency and bifurcation between coexisting attractors of low dimension are found.

Soliton propagation in three coupled nonlinear Schrödinger equations

Abstract Soliton propagation in a system described by three coupled nonlinear Schrodinger equations is investigated. Different initial conditions are considered. The existence of bound states of solitons and soliton separation is demonstrated analytically by perturbation theory and by numerical simulations.

Chaotic behaviour of a pendulum with variable length

SummaryThe Melnikov function for the prediction of Smale horseshoe chaos is applied to a driven damped pendulum with variable length. Depending on the parameters, it is shown that this dynamical system undertakes heteroclinic bifurcations which are the source of the unstable chaotic motion. The analytical results are illustrated by new numerical simulations. Furthermore, using the averaging theorem, the stability of the subharmonics is studied.RiassuntoIn questo articolo si applica la teoria di Melnikov per predire analiticamente la presenza di caos (Smale-horseshoe) in un pendolo con lunghezza variabile in presenza di dissipazione e di un termine forzante. Si mostra che tale sistema dinamico presenta una cascata di biforcazioni eterocliniche quando i parametri che entrano nell’equazione differenziale che lo descrive sono variati. La presenza di queste biforcazioni è la sorgente del moto caotico. Si studia inoltre la stabilità delle subarmoniche facendo uso del teorema della media temporale.

Fluxon dynamics on the circular Josephson oscillator

Abstract A hamiltonian perturbation theory is developed for the perturbed sine-Gordon equation with periodic boundary conditions modelling the Josephson ring oscillator. Stationary fluxon velocities are determined as function of length, loss and bias parameters.

Subharmonic generation in Josephson junction fluxon oscillators biased on Fiske steps

Numerical integration of the perturbed sine‐Gordon equation describing a long overlap‐geometry Josephson junction in a magnetic field indicates a branched structure of the first Fiske step. The major portion of the step corresponds to a simply periodic fluxon oscillation whereas the branches are characterized by subharmonic generation.

Numerical Study of Bifurcations in the BCS Gap Equation

The classical BCS strategy is employed to study layered high Tc superconductors. Starting from a tight binding description and a phenomenological pairing interaction between electrons an anisotropic BCS gap equation is derived. It will be shown that asymmetric pairing interaction within the layers leads to perturbed pitchfork bifurcations separating smoothly different phases of the gap parameter. If the hopping probabilities in the x and y direction differ, local minima of the Gibbs’ free energy can turn into global minima and vice versa, leading to phase transitions which do not result from bifurcations.