Charles R. Doering

Low-dimensional behaviour in the complex Ginzburg-Landau equation

The complex Ginzburg-Landau equation in one spatial dimension with periodic boundary conditions is studied from the viewpoint of effective low-dimensional behaviour by three distinct methods. Linear stability analysis of a class of exact solutions establishes lower bounds on the dimension of the universal, or global, attractor and the Fourier spanning dimension, defined as the number of Fourier modes required to span the universal attractor. The authors use concepts from the theory of inertial manifolds to determine rigorous upper bounds on the Fourier spanning dimension, which also establishes the finite dimensionality of the universal attractor. Upper bounds on the dimension of the attractor itself are obtained by bounding (or, for some parameter values, computing exactly) the Lyapunov dimension and invoking a recent theorem that asserts that the Lyapunov dimension, defined by the Kaplan-Yorke formula with the universal (global) Lyapunov exponents, is an upper bound on the Hausdorff dimension. This study of low dimensionality in the complex Ginzburg-Landau equation allows for an examination of the current techniques used in the rigorous investigation of finite-dimensional behaviour. Contact is made with some recent results for fluid turbulence models, and the authors discuss some unexplored directions in the area of low-dimensional behaviour in the complex Ginzburg-Landau equation.

Weak and strong solutions of the complex Ginzburg-Landau equation

Abstract The generalized complex Ginzburg-Landau equation, ∂ t A=RA+(1+iv) δA-(1+iμ)¦A¦ 2σ A , has been proposed and studied as a model for “turbulent” dynamics in nonlinear partial differential equations. It is a particularly interesting model in this respect because it is a dissipative version of the Hamiltonian nonlinear Schrodinger equation possessing solutions that form localized singularities in finite time. In this paper we investigate existence and regularity of solutions to this equation subject to periodic boundary conditions in various spatial dimensions. Appropriately defined weak solutions are established globally in time, and unique strong solutions are found locally. A new collection of a priori estimates are presented, and we discuss the relationship of our results for the complex Ginzburg-Landau equation to analogous issues for fluid turbulence described by the incompressible Navier-Stokes equations.

On the possibility of soft and hard turbulence in the complex Ginzburg-Landau equation

Abstract We present analytical methods which predict the occurrence of both soft (weak) and hard (strong) turbulence in the complex Ginzburg-Landau (CGL) equation: A t =RA+(1+iν)δA−(1+iμ)A¦A¦ 2 on a periodic domain [0,1] D in D spatial dimensions. Hard turbulence is characterised by large fluctuations away from spatial and temporal averages with a cascade of energy to small scales. This form of hard turbulence appears to occur not in 1D but only in 2D and 3D in parameter regions which are bounded by hyperbolic curves in the second and fourth quadrants of the μ-ν planes where the system is modulationally unstable (ϵ=1+μν 2n: F n =∫(¦∇ n−1 A¦ 2 +α n ¦A¦ 2n )dx , for αn > 0. For large times and large R, upper bounds exist for the infinite set of Fns, constructed from the hierarchy of differential inequalities Fn≤(2nR+cn‖A‖2∞)Fn−bnF2n/Fn−1, for cn, bn > 0 (F0≡1). Estimates for the “bottom rung” F2 give upper bounds for the whole ladder. Long time upper bounds on F2 and ‖A‖2∞ (and hence all Fn) are well controlled in the soft region but become much larger in the hard region, whereas spati al and temporal averages remain comparatively small. When the nonlinearity is A¦A¦2q, the critical case qD=2 gives parallel results.

Statics and dynamics of a diffusion-limited reaction: Anomalous kinetics, nonequilibrium self-ordering, and a dynamic transition

We solve exactly the one-dimensional diffusion-limited single-species coagulation process (A+A→A) with back reactions (A→A+A) and/or a steady input of particles (B→A). The exact solution yields not only the steady-state concentration of particles, but also the exact time-dependent concentration as well as the time-dependent probability distribution for the distance between neighboring particles, i. e., the interparticle distribution function (IPDF). The concentration for this diffusion-limited reaction process does not obey the classical “mean-field” rate equation. Rather, the kinetics is described by a finite set of ordinary differential equations only in particular cases, with no such description holding in general. The reaction kinetics is linked to the spatial distribution of particles as reflected in the IPDFs. The spatial distribution of particles is totally random, i. e., the maximum entropy distribution, only in the steady state of the strictly reversible process A+A↔A, a true equilibrium state with detailed balance. Away from this equilibrium state the particles display a static or dynamic self-organization imposed by the nonequilibrium reactions. The strictly reversible process also exhibits a sharp transition in its relaxation dynamics when switching between equilibria of different values of the system parameters. When the system parameters are suddenly changed so that the new equilibrium concentration is greater than exactly twice the old equilibrium concentration, the exponential relaxation time depends on the initial concentration.

Exponential decay rate of the power spectrum for solutions of the Navier–Stokes equations

Using a method developed by Foias and Temam [J. Funct. Anal. 87, 359 (1989)], exponential decay of the spatial Fourier power spectrum for solutions of the incompressible Navier–Stokes equations is established and explicit rigorous lower bounds on a small length scale defined by the exponential decay rate are obtained.

NUMERICAL AND ANALYTICAL STUDIES OF NONEQUILIBRIUM FLUCTUATION-INDUCED TRANSPORT PROCESSES

We present a numerical simulation algorithm that is well suited for the study of noise-induced transport processes. The algorithm has two advantages over standard techniques: (1) it preserves the property of detailed balance for systems in equilibrium and (2) it provides an efficient method for calculating nonequilibrium currents. Numerical results are compared with exact solutions from two different types of correlation ratchets, and are used to verify the results of perturbation calculations done on a three-state ratchet.

Mean exit times over fluctuating barriers

Abstract We investigate the problem of thermal activation over a fluctuating barrier. Three regimes are considered: the fluctuations slow compared to the mean crossing time τA of the average barrier height, fluctuations on roughly the same timescale as τA, and fluctuations extremely fast compared to τA. In the latter two cases, the mean barrier crossing time is reduced. The relevance of these results to a variety of problems in complex systems is discussed.

Heat transfer in convective turbulence

We investigate the bulk scaling of the convective heat transport in the Boussinesq equations. An a priori bound on the scaling exponent is obtained without making any assumptions on the turbulent fluctuations. If horizontal gradients of temperature fluctuations are relatively small near the boundary then different scaling regimes are obtained. PACS classification scheme numbers: 4727T, 0340G, 4727C, 4727A

On the shape and dimension of the Lorenz attractor

It is shown how the global attractor of the Lorenz equations is contained in a volume bounded by a sphere, a cylinder, the volume between two parabolic sheets, an ellipsoid and a cone. The first four are absorbing volumes while the interior of the cone is expelling. By a numerical search over these volumes, it is found that the origin is the most unstable point on the attractor and that an upper bound for the attractors Lyapunov dimension is 2.401 when b = 8/3, r = 28 and a = 10.

Fluctuations and correlations in a diffusion-reaction system: Exact hydrodynamics

We present an exact closed formulation of the reversible diffusion-limited coagulation-growth reactions 2A ↔ A with irreversible input B → A in one spatial dimension. The treatment here accommodates spatial as well as temporal variations in the particle density with a complete account of microscopic fluctuations and correlations. Moreover, spatial and/or temporal variations in the transport and reaction coefficients can be included in the model. A general solution to the reversible process is presented, and we explore the phenomenon of wavefront propagation.