Interaction of Ising-Bloch fronts with Dirichlet Boundaries

We study the Ising-Bloch bifurcation in two systems, the Complex Ginzburg Landau equation (CGLE) and a FitzHugh Nagumo (FN) model in the presence of spatial inhomogeneity introduced by Dirichlet boundary conditions. It is seen that the interaction of fronts with boundaries is similar in both systems, establishing the generality of the Ising-Bloch bifurcation. We derive reduced dynamical equations for the FN model that explain front dynamics close to the boundary. We find that front dynamics in a highly non-adiabatic (slow front) limit is controlled by fixed points of the reduced dynamical equations, that occur close to the boundary.Comment: 10 pages, 8 figures, submitted to Phys. Rev.

Front propagation into unstable and metastable states in Smectic C* liquid crystals: linear and nonlinear marginal stability analysis

We discuss the front propagation in ferroelectric chiral smectics (SmC*) subjected to electric and magnetic fields applied parallel to smectic layers. The reversal of the electric field induces the motion of domain walls or fronts that propagate into either an unstable or a metastable state. In both regimes, the front velocity is calculated exactly. Depending on the field, the speed of a front propagating into the unstable state is given either by the so-called linear marginal stability velocity or by the nonlinear marginal stability expression. The cross-over between these two regimes can be tuned by a magnetic field. The influence of initial conditions on the velocity selection problem can also be studied in such experiments. SmC$^*$ therefore offers a unique opportunity to study different aspects of front propagation in an experimental system

"Barber pole turbulence" in large aspect ratio Taylor-Couette flow

Investigations of counter-rotating Taylor-Couette flow (TCF) in the narrow gap limit are conducted in a very large aspect ratio apparatus. The phase diagram is presented and compared to that obtained by Andereck et al. The spiral turbulence regime is studied by varying both internal and external Reynolds numbers. Spiral turbulence is shown to emerge from the fully turbulent regime via a continuous transition appearing first as a modulated turbulent state, which eventually relaxes locally to the laminar flow. The connection with the intermittent regimes of the plane Couette flow (pCf) is discussed

Universal Algebraic Relaxation of Velocity and Phase in Pulled Fronts generating Periodic or Chaotic States

We investigate the asymptotic relaxation of so-called pulled fronts propagating into an unstable state. The ``leading edge representation'' of the equation of motion reveals the universal nature of their propagation mechanism and allows us to generalize the universal algebraic velocity relaxation of uniformly translating fronts to fronts, that generate periodic or even chaotic states. Such fronts in addition exhibit a universal algebraic phase relaxation. We numerically verify our analytical predictions for the Swift-Hohenberg and the Complex Ginzburg Landau equation.Comment: 4 pages Revtex, 2 figures, submitted to Phys. Rev. Let

Fixed-Node Monte Carlo Calculations for the 1d Kondo Lattice Model

The effectiveness of the recently developed Fixed-Node Quantum Monte Carlo method for lattice fermions, developed by van Leeuwen and co-workers, is tested by applying it to the 1D Kondo lattice, an example of a one-dimensional model with a sign problem. The principles of this method and its implementation for the Kondo Lattice Model are discussed in detail. We compare the fixed-node upper bound for the ground state energy at half filling with exact-diagonalization results from the literature, and determine several spin correlation functions. Our `best estimates' for the ground state correlation functions do not depend sensitively on the input trial wave function of the fixed-node projection, and are reasonably close to the exact values. We also calculate the spin gap of the model with the Fixed-Node Monte Carlo method. For this it is necessary to use a many-Slater-determinant trial state. The lowest-energy spin excitation is a running spin soliton with wave number pi, in agreement with earlier calculations.Comment: 19 pages, revtex, contribution to Festschrift for Hans van Leeuwe

On the validity of the linear speed selection mechanism for fronts of the nonlinear diffusion equation

We consider the problem of the speed selection mechanism for the one dimensional nonlinear diffusion equation $u_t = u_{xx} + f(u)$. It has been rigorously shown by Aronson and Weinberger that for a wide class of functions $f$, sufficiently localized initial conditions evolve in time into a monotonic front which propagates with speed $c^*$ such that $2 \sqrt{f'(0)} \leq c^* < 2 \sqrt{\sup(f(u)/u)}$. The lower value $c_L = 2 \sqrt{f'(0)}$ is that predicted by the linear marginal stability speed selection mechanism. We derive a new lower bound on the the speed of the selected front, this bound depends on $f$ and thus enables us to assess the extent to which the linear marginal selection mechanism is valid.Comment: 9 pages, REVTE

Front propagation into unstable states : universal algebraic convergence towards uniformly translating pulled fronts

Depending on the nonlinear equation of motion and on the initial conditions, different regions of a front may dominate the propagation mechanism. The most familiar case is the so-called pushed front, whose speed is determined by the nonlinearities in the front region itself. Pushed dynamics is always found for fronts invading a linearly stable state. A pushed front relaxes exponentially in time towards its asymptotic shape and velocity, as can be derived by linear stability analysis. To calculate its response to perturbations, solvability analysis can be used. We discuss, why these methods and results in general do not apply to fronts, whose dynamics is dominated by the leading edge of the front. This can happen, if the invaded state is unstable. Leading edge dominated dynamics can occur in two cases: The first possibility is that the initial conditions are 'flat', i.e., decaying slower in space than e^{-lambda^* x for $xto infty$ with $lambda^*$ defined below. The second and more important case is the one in which the initial conditions are 'steep', i.e., decay faster then e^{-lambda^* x. In this case, which is known as ``pulling'' or ``linear marginal stability'', it is as if the spreading leading edge is pulling the front along. In the central part of this paper, we analyze the convergence towards uniformly translating pulled fronts. We show, that when such fronts evolve from steep initial conditions, they have a universal relaxation behavior as time $ttoinfty$, which can be viewed as a general center manifold result for pulled front propagation. In particular, the velocity of a pulled front always relaxes algebraically like v(t)=v^*-3/(2lambda^*t); left(1-sqrt{pi/big((lambda^*)^2Dtbig)right)+O(1/t^2), where the parameters $v^*$, $lambda^*$, and $D$ are determined through a saddle point analysis from the equation of motion linearized about the unstable invaded state. This front velocity is independent of the precise value of the amplitude which one tracks to measure the front velocity. The interior of the front is essentially slaved to the leading edge, and develops universally as phi(x,t)=Phi_{v(t)left(x-int^t dtau ;v(tau)right)+O(1/t^2), where Phi_{v(x-vt) is a uniformly translating front solution with velocity $v$. We first derive our results in detail for the well known nonlinear diffusion equation of type $partial_t phi =partial_x^2phi+phi-phi^3$, where the invaded unstable state is $phi=0$, and then generalize our results to more general (sets of) partial differential equations with higher spatial or temporal derivatives, to {em p.d.e.'s with memory kernels, and also to difference equations occuring, e.g., in numerical finite difference codes. Our {it universal result for pulled fronts thus also implies independence of the precise nonlinearities, independence of the precise form of the dynamical equation, and independence of the precise initial conditions, as long as they are sufficiently steep. The only remaind

Shift in the velocity of a front due to a cut-off

We consider the effect of a small cut-off epsilon on the velocity of a traveling wave in one dimension. Simulations done over more than ten orders of magnitude as well as a simple theoretical argument indicate that the effect of the cut-off epsilon is to select a single velocity which converges when epsilon tends to 0 to the one predicted by the marginal stability argument. For small epsilon, the shift in velocity has the form K(log epsilon)^(-2) and our prediction for the constant K agrees very well with the results of our simulations. A very similar logarithmic shift appears in more complicated situations, in particular in finite size effects of some microscopic stochastic systems. Our theoretical approach can also be extended to give a simple way of deriving the shift in position due to initial conditions in the Fisher-Kolmogorov or similar equations.Comment: 12 pages, 3 figure

Critical jamming of frictional grains in the generalized isostaticity picture

While frictionless spheres at jamming are isostatic, frictional spheres at jamming are not. As a result, frictional spheres near jamming do not necessarily exhibit an excess of soft modes. However, a generalized form of isostaticity can be introduced if fully mobilized contacts at the Coulomb friction threshold are considered as slipping contacts. We show here that, in this framework, the vibrational density of states (DOS) of frictional discs exhibits a plateau when the generalized isostaticity line is approached. The crossover frequency to elastic behavior scales linearly with the distance from this line. Moreover, we show that the frictionless limit, which appears singular when fully mobilized contacts are treated elastically, becomes smooth when fully mobilized contacts are allowed to slip.Comment: 4 pages, 4 figures, submitted to PR

Universal algebraic relaxation of velocity and phase in pulled fronts generating periodic or chaotic states

We investigate the asymptotic relaxation of so-called pulled fronts propagating into an unstable state, and generalize the universal algebraic velocity relaxation of uniformly translating fronts to fronts that generate periodic or even chaotic states. A surprising feature is that such fronts also exhibit a universal algebraic phase relaxation. For fronts that generate a periodic state, like those in the Swift-Hohenberg equation or in a Rayleigh-Bénard experiment, this implies an algebraically slow relaxation of the pattern wavelength just behind the front, which should be experimentally testable