11 research outputs found
Breathers as Metastable States for the Discrete NLS equation
We study metastable motions in weakly damped Hamiltonian systems. These are
believed to inhibit the transport of energy through Hamiltonian, or nearly
Hamiltonian, systems with many degrees of freedom. We investigate this question
in a very simple model in which the breather solutions that are thought to be
responsible for the metastable states can be computed perturbatively to an
arbitrary order. Then, using a modulation hypothesis, we derive estimates for
the rate at which the system drifts along this manifold of periodic orbits and
verify the optimality of our estimates numerically.Comment: Corrected typos. Added Acknowledgmen
Metastability and rapid convergence to quasi-stationary bar states for the 2D Navier-Stokes Equations
Quasi-stationary, or metastable, states play an important role in
two-dimensional turbulent fluid flows where they often emerge on time-scales
much shorter than the viscous time scale, and then dominate the dynamics for
very long time intervals. In this paper we propose a dynamical systems
explanation of the metastability of an explicit family of solutions, referred
to as bar states, of the two-dimensional incompressible Navier-Stokes equation
on the torus. These states are physically relevant because they are associated
with certain maximum entropy solutions of the Euler equations, and they have
been observed as one type of metastable state in numerical studies of
two-dimensional turbulence. For small viscosity (high Reynolds number), these
states are quasi-stationary in the sense that they decay on the slow, viscous
timescale. Linearization about these states leads to a time-dependent operator.
We show that if we approximate this operator by dropping a higher-order,
non-local term, it produces a decay rate much faster than the viscous decay
rate. We also provide numerical evidence that the same result holds for the
full linear operator, and that our theoretical results give the optimal decay
rate in this setting.Comment: 21 pages, 2 figures. Version 3: minor error from version 2 correcte
Kramers' law: Validity, derivations and generalisations
Kramers' law describes the mean transition time of an overdamped Brownian
particle between local minima in a potential landscape. We review different
approaches that have been followed to obtain a mathematically rigorous proof of
this formula. We also discuss some generalisations, and a case in which
Kramers' law is not valid. This review is written for both mathematicians and
theoretical physicists, and endeavours to link concepts and terminology from
both fields.Comment: 26 pages, 9 figure
Freezing similarity solutions in multi-dimensional Burgers’ Equation
The topic of this paper are similarity solutions occurring in multi-dimensional Burgers’ equation. We present a simple derivation of the symmetries appearing in a family of generalizations of Burgers’ equation in d-space dimensions. These symmetries we use to derive an equivalent partial differential algebraic equation (freezing system) that allows us to do long time simulations and obtain good approximations of similarity solutions by direct forward simulation. The method also allows us without further effort to observe meta-stable behavior near N-wave-like patterns