ac driven sine-Gordon solitons: dynamics and stability

The ac driven sine-Gordon equation is studied analytically and numerically, with the aim of providing a full description of how soliton solutions behave. To date, there is much controversy about when ac driven dc motion is possible. Our work shows that kink solitons exhibit dc or oscillatory motion depending on the relation between their initial velocity and the force parameters. Such motion is proven to be impossible in the presence of damping terms. For breathers, the force amplitude range for which they exist when dissipation is absent is found. All the analytical results are compared with numerical simulations, which in addition exhibit no dc motion at all for breathers, and an excellent agreement is found. In the conclusion, the generality of our results and connections to others systems for which a similar phenomenology may arise are discussed.Comment: 10 pages, latex, PostScript figures included with epsfig, to appear in European Physical Journal B, see GISC homepage at http://valbuena.fis.ucm.es/ for related wor

Stable higher order finite-difference schemes for stellar pulsation calculations

Context: Calculating stellar pulsations requires a sufficient accuracy to match the quality of the observations. Many current pulsation codes apply a second order finite-difference scheme, combined with Richardson extrapolation to reach fourth order accuracy on eigenfunctions. Although this is a simple and robust approach, a number of drawbacks exist thus making fourth order schemes desirable. A robust and simple finite-difference scheme, which can easily be implemented in either 1D or 2D stellar pulsation codes is therefore required. Aims: One of the difficulties in setting up higher order finite-difference schemes for stellar pulsations is the so-called mesh-drift instability. Current ways of dealing with this defect include introducing artificial viscosity or applying a staggered grids approach. However these remedies are not well-suited to eigenvalue problems, especially those involving non-dissipative systems, because they unduly change the spectrum of the operator, introduce supplementary free parameters, or lead to complications when applying boundary conditions. Methods: We propose here a new method, inspired from the staggered grids strategy, which removes this instability while bypassing the above difficulties. Furthermore, this approach lends itself to superconvergence, a process in which the accuracy of the finite differences is boosted by one order. Results: This new approach is shown to be accurate, flexible with respect to the underlying grid, and able to remove mesh-drift.Comment: 15 pages, 11 figures, accepted for publication in A&

Oscillatory relaxation of zonal flows in a multi-species stellarator plasma

The low frequency oscillatory relaxation of zonal potential perturbations is studied numerically in the TJ-II stellarator (where it was experimentally detected for the first time). It is studied in full global gyrokinetic simulations of multi-species plasmas. The oscillation frequency obtained is compared with predictions based on single-species simulations using simplified analytical relations. It is shown that the frequency of this oscillation for a multi-species plasma can be accurately obtained from single-species calculations using extrapolation formulas. The damping of the oscillation and the influence of the different inter-species collisions is studied in detail. It is concluded that taking into account multiple kinetic ions and electrons with impurity concentrations realistic for TJ-II plasmas allows to account for the values of frequency and damping rate in zonal flows relaxations observed experimentally.Comment: 11 figures, 22 page

Markovian versus non-Markovian stochastic quantization of a complex-action model

We analyze the Markovian and non-Markovian stochastic quantization methods for a complex action quantum mechanical model analog to a Maxwell-Chern-Simons eletrodynamics in Weyl gauge. We show through analytical methods convergence to the correct equilibrium state for both methods. Introduction of a memory kernel generates a non-Markovian process which has the effect of slowing down oscillations that arise in the Langevin-time evolution toward equilibrium of complex action problems. This feature of non-Markovian stochastic quantization might be beneficial in large scale numerical simulations of complex action field theories on a lattice.Comment: Accepted for publication in the International Journal of Modern Physics

Chaos of the Relativistic Parametrically Forced van der Pol Oscillator

A manifestly relativistically covariant form of the van der Pol oscillator in 1+1 dimensions is studied. We show that the driven relativistic equations, for which $x$ and $t$ are coupled, relax very quickly to a pair of identical decoupled equations, due to a rapid vanishing of the ``angular momentum'' (the boost in 1+1 dimensions). A similar effect occurs in the damped driven covariant Duffing oscillator previously treated. This effect is an example of entrainment, or synchronization (phase locking), of coupled chaotic systems. The Lyapunov exponents are calculated using the very efficient method of Habib and Ryne. We show a Poincar\'e map that demonstrates this effect and maintains remarkable stability in spite of the inevitable accumulation of computer error in the chaotic region. For our choice of parameters, the positive Lyapunov exponent is about 0.242 almost independently of the integration method.Comment: 8 Latex pages including 12 figures. To be published in Phys. Lett.