Lorenz Kramer

The world of the complex Ginzburg-Landau equation

The cubic complex Ginzburg-Landau equation is one of the most-studied nonlinear equations in the physics community. It describes a vast variety of phenomena from nonlinear waves to second-order phase transitions, from superconductivity, superfluidity, and Bose-Einstein condensation to liquid crystals and strings in field theory. The authors give an overview of various phenomena described by the complex Ginzburg-Landau equation in one, two, and three dimensions from the point of view of condensed-matter physicists. Their aim is to study the relevant solutions in order to gain insight into nonequilibrium phenomena in spatially extended systems.

Pattern propagation in nonlinear dissipative systems

Abstract We discuss the problem of pattern selection in situations where a stable, nonuniform state of a nonlinear dissipative system propagates into an initially unstable, homogeneous region. Our strategy is to consider this process as a generalization of front propagation in a nonlinear diffusion problem for which rigorous results are known; and we point out that these known properties are consistent with a marginal-stability hypothesis that has been suggested in the theory of dendritic crystal growth. We then describe a more general interpretation of the marginal-stability hypothesis and, finally, present numerical evidence for its validity from three different pattern-forming models.

Theory of Light Scattering from Fluctuations of Membranes and Monolayers

General linear viscoelastic response relations are used to obtain the long‐wavelength fluctuations of a thin membrane separating two viscous fluids. The intensity and spectrum of coherent light scattered inelastically from thermal fluctuations are calculated. It is found that the intensity is large enough to allow measurements on black membranes. Our theory includes clean surfaces, interfaces, and insoluble monolayers as special cases.

Core structure and low-energy spectrum of isolated vortex lines in clean superconductors atT ≪T c

We present analytic solutions of the Eilenberger equations for the low-frequency Greens functions corresponding to those quasiparticles whose trajectories pass near the center of an isolated vortex. Using these results we find that for type II superconductors in the temperature rangeTc≫T≫Tc2/εF the order parameter and the supercurrent near the vortex center increase over a lengthξ1∼ξBCST/Tc (ξBCS=BCS coherence length) and that the density of states at the Fermi surface isN0 2π3ξBCS2/3 In (ξBCS/ξ1). The results can be reproduced with the Bogoliubov equations for the elementary excitations. It is shown that this peculiar behavior is connected with the low-lying bound states in the vortex core. ForTťc2/εF one hasξ1∼kF−1.

The Eckhaus instability for traveling waves

Abstract The Eckhaus instability for traveling waves is studied experimentally, theoretically and numerically. The existence of stable compression pulses is established in a certain domain of parameters. For the experimental parameters, these pulses are transients and are predicted to develop finite time phase slips, corresponding to the annihilation of two cells.

On the Eckhaus instability for spatially periodic patterns

Abstract The range of stable wavevectors is near the threshold for appearance of periodic patterns in quasi-one-dimensional systems limited by the long-wavelength Eckhaus instability. At this instability saddle-point solutions characterizing the wavelength-changing processes inside the stable range merge with the periodic solutions. We first analyse this bifurcation near threshold using the amplitude expansion in lowest order. Then a nonlinear equation for the evolution of slow modulations of the periodic pattern far from threshold but near the Eckhaus instability is derived and used to analyse the universal properties of the Eckhaus bifurcation. More detailed information concerning the spatial symmetry of saddle-point solutions is obtained by numerical integration of simple model systems.

Nonequilibrium theory of dirty, current-carrying superconductors: phase-slip oscillators in narrow filaments near T c

General equations for the dynamic behavior of dirty superconductors in the Ginzburg-Landau regime Tc-T ≪ Tcare derived from microscopic theory. In the immediate vicinity of Tca local equilibrium approximation leads to a simple generalized time-dependent Ginzburg-Landau equation. The oscillatory phase-slip solutions presented previously are discussed in greater detail.

Structure and dynamics of dislocations in anisotropic pattern-forming systems

Abstract The motion of dislocations in convective roll patterns provides an important wavevector selection mechanism. In this work the structure and velocity of dislocations is calculated near threshold using amplitude equations appropriate for systems with an axial anisotropy. The fact that then the roll pattern has a preferred direction leads to characteristic differences to isotropic systems like Rayleigh-Benard convection in simple fluids. Furthermore the nucleation process of dislocation pairs is discussed by analyzing the threshold solution that describes the nucleation barrier.

Symmetry-breaking Hopf bifurcation in anisotropic systems

Abstract Symmetry-breaking Hopf bifurcation from a spatially uniform steady state of a spatially extended anisotropic system is considered. This work is motivated by the experimental observation of a Hopf bifurcation to oblique traveling rolls in electrohydrodynamic convection in planarly aligned nematic liquid crystals. Symmetry forces four traveling rolls to lose stability simultaneously. Four coupled complex ordinary differential equations describing the nonlinear interaction of the traveling rolls are analyzed using methods of equivariant bifurcation theory. Six branches of periodic solutions always bifurcate from the trivial state at the Hopf bifurcation. These correspond to traveling and standing wave patterns. In an open region of coefficient space there is a primary bifurcation to a quasiperiodic standing wave solution. The Hopf bifurcation can also lead directly to an aperiodic attractor in the form of an asymptotically stable, structurally stable heteroclinic cycle. The theory is applied to a model for the transition from normal to oblique traveling rolls.

Amplitude equations for description of chemical reaction–diffusion systems

Abstract We consider a new method for modeling waves in complex chemical systems close to bifurcation points. The method overcomes numerical problems connected with the high dimensional configuration phase space of realistic chemical systems without sacrificing the quantitative accuracy of the calculations. The efficiency is obtained by replacing the conventional use of kinetic equations considering just a few species by the use of amplitude equations for determining the evolution of the state. Coupled with calculation of an explicit function connecting the amplitude space and the concentration space this method permits the quantitative determination of the concentrations of all species. We also introduce a new method for calculating the boundaries of convective and absolute stability of waves for a chemical model at an operating point close to a supercritical Hopf bifurcation and with a slow stable mode.