Lorentz Center - Coherent structures in evolutionary equations from 12 Jul 2010 through 16 Jul 2010
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    Coherent structures in evolutionary equations
    from 12 Jul 2010 through 16 Jul 2010

 

Abstracts

 

 

 

Joris Bierkens

“Invariant measures for stochastic differential equations in infinite dimensions”

 

When an evolution described by a PDE is perturbed by random noise, the resulting stochastic evolution may be described by a stochastic partial differential equation (SPDE). This is an example of an infinite dimensional stochastic differential equation (SDE). Similarly, when a differential equation is perturbed by effects from the past (delay) and random noise, we obtain an infinite dimensional SDE. Stationary behaviour of such evolutions may be characterized by means of the invariant measure. We will discuss some general ideas on the existence and uniqueness of such invariant measures for infinite dimensional SDEs.

 

 

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Dirk Blömker

“Evolution of dominant pattern under degenerate forcing”

 

We discuss stochastic partial differential equations (SPDEs). Near a change of stability the evolution of dominant pattern on a slow time-scale is described by amplitude equations. If we force the SPDE with degenerate noise, then in some cases the noise appears as a stabilizing deterministic  effect in the amplitude equation. As an example we discuss the Swift-Hohenberg equation on bounded or unbounded domains and present rigorous results as well as numerical simulations.

 

 

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Lee DeVille

“Coherent Structures in Cascading Systems”

 

Many complex systems exhibit “cascades'' of activity. Speaking loosely, complex systems are dynamical systems made up of many interacting subunits, each of which has potentially complicated nonlinear dynamics; cascades are bursts of activity in which a significant fraction of the subunits in an entire system performs a dynamical change in a time much shorter than the typical timescale of the system.  Numerous well-known examples of such systems exist in neuroscience, material science, economics, and statistical physics.  A particularly interesting class of cascading systems are those in

which the cascades are noise-driven, and which are noisy at the unit level, yet are coherent at the system level.  We will show how a class of such systems can be understood, and present examples of limit theorems for the dynamics of such systems.  The techniques presented will combine elements of nonlinear dynamical systems, large-deviation theory, and random graph theory.  We will also present applications in

neuroscience, including a study of synchrony-breaking in stochastic neuronal networks.

 

 

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Patrick Dondl

“Pinning of interfaces in random media”

 

We consider the evolution of an interface, modeled by a parabolic equation, in a random environment. The randomness is given by a distribution of smooth obstacles of random strength. To provide a barrier for the moving interface, we construct a positive, steady state supersolution. This construction depends on the existence, after rescaling, of a Lipschitz hypersurface separating the domain into a top and a bottom part, consisting of boxes that contain at least one obstacle of sufficient strength. We prove this percolation result.

Joint work with N. Dirr (Bath University) and M. Scheutzow (TU Berlin).

 

 

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Julia Ehrt

“Global attractors for viscous and in-viscous balance laws on the circle ^”

 

The talk investigates the relation between the global attractors of viscous balance laws and hyperbolic balance laws on the circle for vanishing viscosity given by: \begin{equation} \label{eq:P} \tag{P} u_t + [f(u)]_x = \varepsilon u_{xx} + g(u) \end{equation} and \begin{equation}\label{eq:H} \tag{H} u_t + [f(u)]_x = g(u) \end{equation} The talk focuses on the question of convergence of heteroclinic orbits of the parabolic equation (\ref{eq:P}) to solutions of the limit equation (\ref{eq:H}) when $\varepsilon\rightarrow0$. Despite a result of Fan and Hale from 1995 that proves persistence of heteroclinics I will present a surprisingly easy necessary condition for their persistence which proves the Fan Hale result to be wrong. In case of non-convergence to a singular heteroclinic the talk will show convergence to a cascade of heteroclinic orbits of the limiting equation . The talk closes with some remarks on the consequences for the relation of the global attractors of (\ref{eq:P}) and (\ref{eq:H}) for vanishing $\varepsilon$.

 

 

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Carlos Escudero

“Non-equilibrium growth of radial clusters: weak convergence to the macroscopic shape and implications for morphogenesis”

 

The building blocks of mathematical morphogenesis were put several decades ago in the seminal works of Turing and Eden. Their goal was understanding how a macroscopic structure, in particular one breaking the initial homogeneity, could arise out of a multiplicity of simple interactions. While the approach of Turing implied the use of reaction-diffusion equations, Eden concentrated on a probabilistic abstraction of a developing cell colony. In particular, he studied the architecture of a lattice cell colony to which new cells were added following certain probabilistic rules. The objective was studying the asymptotic colony profile. The original Eden problem can be greatly generalized by means of the use of stochastic partial differential equations. They allow a systematic study of the properties of the colony periphery, particularly of the interface fluctuations. In this work we will summarize our recent progress in this field, concentrating on the properties of the realizations of the stochastic growth process. Our goal is unveiling under which conditions the developing radial cluster asymptotically weakly converges to the concentrically propagating spherically symmetric

profile.

 

 

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Barbara Gentz

"Metastability in systems of coupled diffusions"

 

For systems of coupled Brownian particles in a potential landscape, the metastable transition times depend on coupling strength and noise intensity. For weak coupling, the system behaves like a stochastic particle system, while for strong coupling the system synchronizes and transitions between metastable states occur almost simultaneously for all particles. The transition between these extremal regimes involves a sequence of symmetry-breaking bifurcations. Away from such bifurcation points, the small-noise asymptotics of the transition times is given by the Eyring-Kramers formula. However, this formula breaks down in the presence of degenerate (non-quadratic) saddles or wells as present in our system. We will show how results by Bovier, Eckhoff, Gayrard and Klein, providing the first rigorous proof of the Eyring-Kramers formula, can be extended to such degenerate landscapes. We will conclude by discussing an analogous result for the Ginzburg-Landau partial differential equation with noise. This is joint work with Florent Barret (Palaiseau), Nils Berglund (Orléans) and Bastien Fernandez (CPT-CNRS, Marseille).

 

 

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Mark Groves

“Existence and stability of fully localised three-dimensional gravity-capillary solitary water waves"

 

A solitary wave of the type advertised in the title is a critical point of the wave energy, which is given in dimensionless coordinates by $$H(\eta,\xi) = \int_{{\mathbb R}2} \left\{\frac{1}{2}\xi G(\eta)\xi + \frac{1}{2}\eta2+ \beta \sqrt{1+\eta_x2+\eta_z2} - \beta \right\},$$ subject to the constraint that the wave impulse $$I(\eta,\xi) = \int_{{\mathbb R}2} \eta_x \xi$$ is fixed. Here $\eta(x,z)$ is the free-surface elevation, $\xi$ is the trace of the velocity potential on the free surface, $G(\eta)$ is a Dirichlet-Neumann operator and $\beta>1/3$ is the Bond number.

In this talk I show that there exists a minimiser of $H$ subject to the constraint $I=2\mu$, where $0<\mu \ll 1$. The existence of a solitary wave is thus assured, and since $H$ and $I$ are both conserved quantities its stability follows by a standard argument. `Stability' must however be understood in a qualified sense due to the lack of a global well-posedness theory for three-dimensional water waves.

 

 

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Martin Hairer

“How hot can a heat bath get?”

 

We consider one of the simplest possible models of non-equilibrium statistical mechanics: two coupled oscillators in contact with two Langevin heat baths. The twist is that one of the heat baths is at "infinite" temperature in the sense that no friction acts on the corresponding degree of freedom. We explore the question of the existence of a stationary state in this situation and its properties if it exists. In particular, we will see that the question "Is the corresponding degree of freedom at infinite temperature?" can have a surprising variety of answers.

 

 

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Michael Herrmann

“Oscillatory patterns in discrete Hamiltonian PDEs”

 

This talk is concerned with modulated oscillatory patterns in Hamiltonian lattice equations (discrete Burgers, FPU). We give an overview on typical patterns that arise from simple initial value problems and discuss some open mathematical problems.

 

 

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Panos Kevrekidis

"Dynamics of Dark Soliton and Vortex Matter Waves"

 

In this talk, we will present some recent results on the dynamics of dark soliton and multi-soliton solitary waves, as well as of vortex and multi-vortex states that arise in Bose-Einstein condensates. We will start with a brief overview of the initial attempts to identify such states in BECs and subsequently turn to recent experimental efforts and the theoretical challenges that they pose. In attempting to address these challenges, we will formulate an ODE-based particle picture for the soliton and vortex dynamics and will attempt to compare equilibrium, near-equilibrium and far from equilibrium How hot can a heat bath get?

 

 

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Grant Lythe

“Kink stochastics

 

Kinks are localised coherent structures are a striking feature of noisy, nonlinear, spatially-extended systems in one space dimension with local bistability.  At late times, a steady-state density is dynamically maintained: kinks are nucleated in pairs, diffuse and annihilate on collision. Long-term averages can be calculated using the transfer-integral method, developed in the 1970s, giving exact results that can be compared with large-scale numerical solutions of the SPDE.  In this talk, diffusion-limited reaction is the name given to a reduced model of the SPDE dynamics, in which kinks are treated as point particles. Some quantities, such as the mean number of particles per unit length, can be calculated exactly. Mathods for estimating of the width of a newly-nucleated region, and the rate of nucleation,will be discussed.

 

 

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Marc Pradas

“Viscous dispersion effects on bound states formation in falling liquid films”

 

Liquid films falling down a vertical wall are always unstable to sufficiently long waves and exhibit a rich dynamics that strongly depends on  the Reynolds number  Re. We focus here in the case of moderate  Re, when the interface of the liquid film appears  to be randomly covered by localised coherent structures, solitary pulses, which are stable and robust.  At sufficiently large distances from the inlet of the film, these localised pulses interact each other as quasi-particles through attractions and repulsions giving rise to the formation of bound states of two or more pulses travelling at the same speed, and separated by well-defined distances.In this work, we examine the effects of viscous dispersion on the interaction of the  solitary pulses. We make use of two models, first- and second-order, respectively, in the long-wavelength expansion. The second-order effects originate from streamwise viscous dissipation and play a dispersive role.  We extend a recently developed coherent-structures interaction theory for the solitary pulses of the generalized Kuramoto-Sivashinsky equation to the first- and second-order models and we are able to predict theoretically the distances at which bound states are formed, obtaining excellent agreement with numerical results of the fully nonlinear system.

 

 

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Marco Romito

"Non--uniqueness issues in SDE's and SPDE's"

 

The first part of the talk is a short (and by no means complete) exposition of issues of non--uniqueness for finite dimensional stochastic equations, as a preparation to infinite dimensional problems. In the second part we discuss some problems of non uniqueness for stochastic PDE, with emphasis on equations from fluid dynamics. In particular we introduce a general method which allows to show continuity and stability properties for a special class of solutions.

 

 

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Jens Rottmann-Matthes

"Using symmetries in long-time computations"

 

In the talk I will explain the method of freezing for the case of traveling waves. The method makes use of the geometry of the underlying space to allow for long-time computations. An asymptotic stability result that justifies the usability of the method in the important case of coupled hyperbolic-parabolic equations, which includes the FitzHugh-Nagumo system will also be presented.

 

 

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Guido Schneider

“The validity of the Ginzburg-Landau equation: An overview and new results”

 

The Ginzburg-Landau equation can be derived as an amplitude equation for the description of pattern forming systems in cylindrical domains close to the first instability. In this talk we give an overview about existing approximation results and  present some recent results about the validity in time-periodic pattern forming systems and in systems with a marginal stable long wave mode.

 

 

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Tony Shardlow

"Dissipative Particle Dynamics"

 

Dissipative Particle Dynamics is a simple particle model of multiphase fluid flows. We treat the model of a system of stochastic differential equations and discuss questions of numerical approximation and ergodicity.

 

 

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Hannes Uecker

"Coupled Mode Equations and Gap Solitons for the 2D Gross-Pitaevsky equation"

 

Gap solitons near a band edge of a spatially periodic nonlinear PDE can be formally approximated by solutions of Coupled Mode Equations (CMEs). Here we study this approximation for the case of the 2D Periodic Nonlinear Schrödinger / Gross-Pitaevsky Equation with a non-separable potential of finite contrast. The CME derivation is carried out in Bloch rather than physical  coordinates. Using the Lyapunov-Schmidt reduction we then give a rigorous  justification of the CMEs as an asymptotic model for reversible non-degenerate gap solitons and provide $H^s$ estimates for this approximation. The results are confirmed by numerical examples including some new families of CMEs and gap solitons absent for separable potentials.

 

 

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Peter van Heijster

“Radial Spots in a Planar Three-component Reaction-diffusion System”

 

It is a well-accepted belief that the increase of the number of components in a reaction-diusion equation increases the complexity of the dynamics of the equation. In this presentation, we analyze radially symmetric spot solutions of a certain two-dimensional three-component FitzHugh-Nagumo system. In particular, we report on results on the most unstable eigenvalues that decide through which type of instability the stationary structures become unstable. The underlying model was proposed in the nineties to describe, on a phenomenological level, the behavior of gas-discharge systems.  Numerical simulations suggest that the third component is necessary to stabilize traveling spot solutions as these solutions are unstable for its two-component analogue.

 

 

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Sergey Zelik

“Analytical proof of space-time chaos in Ginzburg-landau equations"

joint work with D.Turaev

 

We prove that the attractor of the 1D quintic complex Ginzburg-Landau equation with a broken phase symmetry has strictly positive space-time entropy for an open set of parameter values. The result is obtained by studying chaotic oscillations in grids of weakly interacting solitons in a class of Ginzburg-Landau type equations. We provide an analytic proof for the existence of two-soliton configurations with chaotic temporal behavior, and construct solutions which are closed to a grid of such chaotic soliton pairs, with every pair in the grid well spatially separated from the neighboring ones for all time. The temporal evolution of the well-separated multi-soliton structures is described by a weakly coupled lattice dynamical system (LDS) for the coordinates and phases of the solitons. We develop a version of normal hyperbolicity theory for the weakly coupled LDS’s with continuous time and establish for them the existence of space-time chaotic patterns similar to the Sinai-Bunimovich chaos in discrete-time LDS’s. While the LDS part of the theory may be of independent interest, the main difficulty addressed in the paper concerns with lifting the space-time chaotic solutions of the LDS back to the initial PDE. The equations we consider here are space-time autonomous, i.e. we impose no spatial or temporal modulation which could prevent the individual solitons in the grid from drifting towards each other and destroying the well-separated grid structure in a finite time. We however manage to show that the set of space-time chaotic solutions for which the random soliton drift is arrested is large enough, so the corresponding space-time entropy is strictly positive.

 

 

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