Studies of Phase Turbulence in the One Dimensional Complex Ginzburg-Landau Equation

The phase-turbulent (PT) regime for the one dimensional complex Ginzburg-Landau equation (CGLE) is carefully studied, in the limit of large systems and long integration times, using an efficient new integration scheme. Particular attention is paid to solutions with a non-zero phase gradient. For fixed control parameters, solutions with conserved average phase gradient $\nu$ exist only for $|\nu|$ less than some upper limit. The transition from phase to defect-turbulence happens when this limit becomes zero. A Lyapunov analysis shows that the system becomes less and less chaotic for increasing values of the phase gradient. For high values of the phase gradient a family of non-chaotic solutions of the CGLE is found. These solutions consist of spatially periodic or aperiodic waves travelling with constant velocity. They typically have incommensurate velocities for phase and amplitude propagation, showing thereby a novel type of quasiperiodic behavior. The main features of these travelling wave solutions can be explained through a modified Kuramoto-Sivashinsky equation that rules the phase dynamics of the CGLE in the PT phase. The latter explains also the behavior of the maximal Lyapunov exponents of chaotic solutions.Comment: 16 pages, LaTeX (Version 2.09), 10 Postscript-figures included, submitted to Phys. Rev.

A New Monte Carlo Algorithm for Protein Folding

We demonstrate that the recently proposed pruned-enriched Rosenbluth method (P. Grassberger, Phys. Rev. E 56 (1997) 3682) leads to extremely efficient algorithms for the folding of simple model proteins. We test them on several models for lattice heteropolymers, and compare to published Monte Carlo studies. In all cases our algorithms are faster than all previous ones, and in several cases we find new minimal energy states. In addition to ground states, our algorithms give estimates for the partition sum at finite temperatures.Comment: 4 pages, Latex incl. 3 eps-figs., submitted to Phys. Rev. Lett., revised version with changes in the tex

Nonlinear Schrödinger solitons scattering off an interface

We integrate the one-dimensional nonlinear Schrodinger equation numerically for solitons moving in external potentials. In particular, we study the scattering off an interface separating two regions of constant potential modeled by a linear ramp. Transmission coefficients and inelasticities are computed as functions of the potential difference and the slope of the ramp. Our data show that the ramp's slope has a strong influence revealing unexpected windows of reflection in a transmission regime. The transmission coefficients for very small potential differences are compared with the theoretical predictions derived by perturbation theory. Also the time evolution of the solitary waves after the scattering is studied. We observed that they in general behave like solitons with an amplified amplitude. Due to this, they oscillate. The oscillation period is measured and compared with theoretical predictions

Interaction of Nonlinear Schrödinger Solitons with an External Potential

Employing a particularly suitable higher order symplectic integration algorithm, we integrate the 1-d nonlinear Schrodinger equation numerically for solitons moving in external potentials. In particular, we study the scattering off an interface separating two regions of constant potential. We find that the soliton can break up into two solitons, eventually accompanied by radiation of non-solitary waves. Reflection coefficients and inelasticities are computed as functions of the height of the potential step and of its steepness

Higher Order Unitary Integrators for the Schrödinger Equation

For classical Hamiltonian dynamics, much progress has recently been made in manifestly symplectic integrators. In this work some of these integrators will be generalized to the Schrodinger equation. There they correspond to algorithms which preserve unitarity and time reversal invariance exactly. In particular, we apply them to the one dimensional harmonic and anharmonic oscillators. The accuracies of three different algorithms are compared. 1 Introduction: Symplectic integrators In the present paper we shall discuss some new integration algorithms for Schrodinger equations with coordinate independent kinetic energies. These algorithms are rather straightforward modifications of recently proposed algorithms for classical mechanical systems. Thus we shall first discuss these integrators which are known as "symplectic integrators" since they preserve the symplectic structure of classical mechanics. Symplectic integrators were introduced by De Vogelaere in 1956, in a series of unpublis..