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arXiv1998

Diffusive Mixing of Stable States in the Ginzburg-Landau Equation

For the time-dependent Ginzburg-Landau equation on the real line, we construct solutions which converge, as $x \to \pm\infty$, to periodic stationary states with different wave-numbers $η_\pm$. These solutions are stable with respect to small perturbations, and approach as $t \to +\infty$ a universal diffusive profile depending only on the values of $η_\pm$. This extends a previous result of Bricmont and Kupiainen by removing the assumption that $η_\pm$ should be close to zero. The existence of the diffusive profile is obtained as an application of the theory of monotone operators, and the long-time behavior of our solutions is controlled by rewriting the system in scaling variables and using energy estimates involving an exponentially growing damping term.

patt-sol↗

arXiv1998

Extensive Properties of the Complex Ginzburg-Landau Equation

We study the set of solutions of the complex Ginzburg-Landau equation in $\real^d, d<3$. We consider the global attracting set (i.e., the forward map of the set of bounded initial data), and restrict it to a cube $Q_L$ of side $L$. We cover this set by a (minimal) number $N_{Q_L}(ε)$ of balls of radius $ε$ in $\Linfty(Q_L)$. We show that the Kolmogorov $ε$-entropy per unit length, $H_ε=\lim_{L\to\infty} L^{-d} \log N_{Q_L}(ε)$ exists. In particular, we bound $H_ε$ by $\OO(\log(1/ε)$, which shows that the attracting set is smaller than the set of bounded analytic functions in a strip. We finally give a positive lower bound: $H_ε>\OO(\log(1/ε))$

chao-dyn↗

arXiv1997

Strongly nonlinear convection in binary fluids: Minimal model for extended states using symmetry decomposed modes

Spatially extended stationary and traveling states in the strongly nonlinear regime of convection in layers of binary fluid mixtures heated from below are described by a few-mode-model. It is derived from the proper hydrodynamic balance equations including experimentally relevant boundary conditions with a non-standard Galerkin approximation that uses numerically obtained, symmetry decomposed modes. Properties of the model are elucidated and compared with full numerical solutions of the field equations.

patt-sol↗

arXiv1998

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.

cond-mat↗

arXiv1994

Domain Structures in Fourth-Order Phase and Ginzburg-Landau Equations

In pattern-forming systems, competition between patterns with different wave numbers can lead to domain structures, which consist of regions with differing wave numbers separated by domain walls. For domain structures well above threshold we employ the appropriate phase equation and obtain detailed qualitative agreement with recent experiments. Close to threshold a fourth-order Ginzburg-Landau equation is used which describes a steady bifurcation in systems with two competing critical wave numbers. The existence and stability regime of domain structures is found to be very intricate due to interactions with other modes. In contrast to the phase equation the Ginzburg-Landau equation allows a spatially oscillatory interaction of the domain walls. Thus, close to threshold domain structures need not undergo the coarsening dynamics found in the phase equation far above threshold, and can be stable even without phase conservation. We study their regime of stability as a function of their (quantized) length. Domain structures are related to zig-zags in two-dimensional systems. The latter are therefore expected to be stable only when quenched far enough beyond the zig-zag instability.

patt-sol↗

arXiv1994

Rayleigh-Bénard Convection; Patterns, Chaos, Spatiotemporal Chaos and Turbulence

A coupled map lattice for convection is proposed, which consists of Eulerian and Lagrangian procedures. Simulations of the model not only reproduce a wide-range of phenomena in Rayleigh-Bénard convection experiments but also lead to several prediction of novel phenomena there: For small aspect ratios, the formation of convective rolls, their oscillation, many routes to chaos, and chaotic itinerancy are found, with the increase of the Rayleigh number. For large aspect ratios, the collective oscillation of convective rolls, travelling waves, coherent chaos, and spatiotemporal intermittency are observed. At high Rayleigh numbers, the transition from soft to hard turbulence is confirmed, as is characterized by the change of the temperature distribution from Gaussian to exponential. Roll formation in three-dimensional convection is also simulated, and found to reproduce experiments well.

chao-dyn↗

arXiv1993

Exact Solutions of the One-Dimensional Quintic Complex Ginzburg-Landau Equation

Exact solitary wave solutions of the one-dimensional quintic complex Ginzburg-Landau equation are obtained using a method derived from the Painlevé test for integrability. These solutions are expressed in terms of hyperbolic functions, and include the pulses and fronts found by van Saarloos and Hohenberg. We also find previously unknown sources and sinks. The emphasis is put on the systematic character of the method which breaks away from approaches involving somewhat ad hoc Ansätze.

patt-sol↗

arXiv1994

Pattern Dynamics of a Coupled Map Lattice for Open Flow

The pattern dynamics of the one-way coupled logistic lattice which can serve as a phenomenological model for open flow is investigated and shown to be extremely rich. For medium and large coupling strengths, we find spatially periodic, quasiperiodic and chaotic patterns with temporal periodicity and analyze their stability with the help of a newly introduced spatial map. Criteria are established for predicting the down-flow bifurcation behavior of the coupled map lattice, and for selecting attractors through modulation of the boundary. For smaller coupling strengths we find a novel regime in which chaotic defects form a periodic lattice that is shown to be the result of a boundary crisis.

chao-dyn↗

arXiv1994

Bound Pairs of Fronts in a Real Ginzburg-Landau Equation Coupled to a Mean Field

Motivated by the observation of localized traveling-wave states (`pulses') in convection in binary liquid mixtures, the interaction of fronts is investigated in a real Ginzburg-Landau equation which is coupled to a mean field. In that system the Ginzburg-Landau equation describes the traveling-wave amplitude and the mean field corrsponds to a concentration mode which arises due to the slowness of mass diffusion. For single fronts the mean field can lead to a hysteretic transition between slow and fast fronts. Its contribution to the interaction between fronts can be attractive as well as repulsive and depends strongly on their direction of propagation. Thus, the concentration mode leads to a new localization mechanism, which does not require any dispersion in contrast to that operating in the nonlinear Schrödinger equation. Based on this mechanism alone, pairs of fronts in binary-mixture convection are expected to form {\it stable} pulses if they travel {\it backward}, i.e. opposite to the phase velocity. For positive velocities the interaction becomes attractive and destabilizes the pulses. These results are in qualitative agreement with recent experiments. Since the new mechanism is very robust it is expected to be relevant in other systems as well in which a wave is coupled to a mean field.

patt-sol↗

arXiv1994

Hole Solutions in the 1d Complex Ginzburg-Landau Equation

The cubic Complex Ginzburg-Landau Equation (CGLE) has a one parameter family of traveling localized source solutions. These so called 'Nozaki-Bekki holes' are (dynamically) stable in some parameter range, but always structually unstable: A perturbation of the equation in general leads to a (positive or negative) monotonic acceleration or an oscillation of the holes. This confirms that the cubic CGLE has an inner symmetry. As a consequence small perturbations change some of the qualitative dynamics of the cubic CGLE and enhance or suppress spatio-temporal intermittency in some parameter range. An analytic stability analysis of holes in the cubic CGLE and a semianalytical treatment of the acceleration instability in the perturbed equation is performed by using matching and perturbation methods. Furthermore we treat the asymptotic hole-shock interaction. The results, which can be obtained fully analytically in the nonlinear Schroedinger limit, are also used for the quantitative description of modulated solutions made up of periodic arrangements of traveling holes and shocks.

patt-sol↗

arXiv1994

From the defocusing nonlinear Schroedinger to the complex Ginzburg-Landau equation

Perturbation approaches developed so far for the dark soliton solutions of the (fully integrable) defocusing nonlinear Schroedinger equation cannot describe the dynamics resulting from dissipative perturbations of the Ginzburg-Landau type. Here spatially slowly decaying changes of the background wavenumber occur which requires the use of matching technics. It is shown how the perturbation selects a 1 or 2-parameter subfamily from the 3-parameter family of dark solitons of the nonlinear Schroedinger equation. The dynamics of the perturbed system can then be described analytically as motion within this selected subfamily yielding interesting scenarios. Interaction with shocks occurring in the complex Ginzburg-Landau equation can be included in a straight forward way.

patt-sol↗

arXiv1995

Tangent Bifurcation of Band Edge Plane Waves, Dynamical Symmetry Breaking and Vibrational Localization

We study tangent bifurcation of band edge plane waves in nonlinear Hamiltonian lattices. The lattice is translationally invariant. We argue for the breaking of permutational symmetry by the new bifurcated periodic orbits. The case of two coupled oscillators is considered as an example for the perturbation analysis, where the symmetry breaking can be traced using Poincare maps. Next we consider a lattice and derive the dependence of the bifurcation energy on the parameters of the Hamiltonian function in the limit of large system sizes. A necessary condition for the occurence of the bifurcation is the repelling of the band edge plane wave's frequency from the linear spectrum with increasing energy. We conclude that the bifurcated orbits will consequently exponentially localize in the configurational space.

patt-sol↗

arXiv1995

Solitary Waves under the Influence of a Long-Wave Mode

The dynamics of solitons of the nonlinear Schrödinger equation under the influence of dissipative and dispersive perturbations is investigated. In particular a coupling to a long-wave mode is considered using extended Ginzburg-Landau equations. The study is motivated by the experimental observation of localized wave trains (`pulses') in binary-liquid convection. These pulses have been found to drift exceedingly slowly. The perturbation analysis reveals two distinct mechanisms which can lead to a `trapping' of the pulses by the long-wave concentration mode. The are given by the effect of the concentration mode on the local growth rate and on the frequency of the wave. The latter, dispersive mechanism has not been recognized previously, despite the fact that it dominates over the dissipative contribution within the perturbation theory. A second unexpected result is that the pulse can be accelerated by the concentration mode despite the reduced growth rate ahead of the pulse. The dependence of the pulse velocity on the Rayleigh number is discussed, and the hysteretic `trapping' transitions suggested by the perturbation theory are confirmed by numerical simulations, which also reveal oscillatory behavior of the pulse velocity in the vicinity of the transition. The derivation and reconstitution of the extended Ginzburg-Landau equations is discussed in detail.

patt-sol↗

arXiv1997

Influence of boundaries on pattern selection in through-flow

The problem of pattern selection in absolutely unstable open flow systems is investigated by considering the example of Rayleigh-Bénard convection. The spatiotemporal structure of convection rolls propagating downstream in an externally imposed flow is determined for six different inlet/outlet boundary conditions. Results are obtained by numerical simulations of the Navier-Stokes equations and by comparison with the corresponding Ginzburg-Landau amplitude equation. A unique selection process is observed being a function of the control parameters and the boundary conditions but independent of the history and the system length. The problem can be formulated in terms of a nonlinear eigen/boundary value problem where the frequency of the propagating pattern is the eigenvalue. PACS: 47.54.+r, 47.20.Bp, 47.27.Te, 47.20.Ky

patt-sol↗

arXiv1996

The Spatio-Temporal Structure of Spiral-Defect Chaos

We present a study of the recently discovered spatially-extended chaotic state known as spiral-defect chaos, which occurs in low-Prandtl-number, large-aspect-ratio Rayleigh-Benard convection. We employ the modulus squared of the space-time Fourier transform of time series of two-dimensional shadowgraph images to construct the structure factor ${S}({\vec k},ω)$. This analysis is used to characterize the average spatial and temporal scales of the chaotic state. We find that the correlation length and time can be described by power-law dependences on the reduced Rayleigh number $ε$. These power laws have as yet no theoretical explanation.

chao-dyn↗

arXiv1996

Non-Monotonic Dispersion of Surface Waves in Magnetic Fluids

The dispersion relation of surface waves of a magnetic fluid in a magnetic field is studied experimentally. We verify the theoretically predicted existence of a non-monotonic dispersion relation. In particular, we demonstrate the existence of two different wave numbers occuring at the same frequency in an annular geometry.

patt-sol↗

arXiv1994

SUSY-Based Variational Method for the Anharmonic Oscillator

Using a newly suggested algorithm of Gozzi, Reuter, and Thacker for calculating the excited states of one dimensional systems, we determine approximately the eigenvalues and eigenfunctions of the anharmonic oscillator, described by the Hamiltonian $H= p^2/2 + g x^4$. We use ground state post-gaussian trial wave functions of the form $Ψ(x) = N{\rm{exp}}[-b |x|^{2n}]$, where $n$ and $b$ are continuous variational parameters. This algorithm is based on the hierarchy of Hamiltonians related by supersymmetry (SUSY) and the factorization method. We find that our two parameter family of trial wave functions yields excellent energy eigenvalues and wave functions for the first few levels of the anharmonic oscillator.

patt-sol↗

arXiv1994

Normal Form, Symmetry and Infinite Dimensional Lie Algebra for System of Ode's

The normal form for a system of ode's is constructed from its polynomial symmetries of the linear part of the system, which is assumed to be semi-simple. The symmetries are shown to have a simple structure such as invariant function times symmetries of degree one called basic symmetries. We also show that the set of symmetries naturally forms an infinite dimensional Lie algebra graded by the degree of invariant polynomials. This implies that if this algebra is non-commutative then the method of multiple scales with more than two scaling variables fails to apply.

patt-sol↗