Lorentz Center

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## Coherent structures in evolutionary equations |

“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. ---
“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. --
“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. ---
“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). --
“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
$\varepsilon$. ---
“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. --
"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). ---
“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. --
“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. ---
“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. ---
"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? ---
“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. ---
“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. ---
"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. ---
"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. ---
“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. ---
"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. ---
"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. ---
“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. ---
“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. --- [Back] |