2,988 research outputs found
Non-Linear Stability Analysis of Higher Order Dissipative Partial Differential Equations
We extend the invariant manifold method for analyzing the asymptotics of
dissipative partial differential equations on unbounded spatial domains to
treat equations in which the linear part has order greater than two. One
important example of this type of equation which we analyze in some detail is
the Cahn-Hilliard equation. We analyze the marginally stable solutions of this
equation in some detail. A second context in which such equations arise is in
the Ginzburg-Landau equation, or other pattern forming equations, near a
codimension-two bifurcation
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
Scattering Phases and Density of States for Exterior Domain
For a bounded open domain with connected complement and
piecewise smooth boundary, we consider the Dirichlet Laplacian -\DO on
and the S-matrix on the complement . Using the restriction
of to the boundary of , we establish that
is trace class when is negative and
give bounds on the energy dependence of this difference. This allows for
precise bounds on the total scattering phase, the definition of a
-function, and a Krein spectral formula, which improve similar results
found in the literature.Comment: 15 pages, Postscript, A
Geometric Stability Analysis for Periodic Solutions of the Swift-Hohenberg Equation
In this paper we describe invariant geometrical ~structures in the phase
space of the Swift-Hohenberg equation in a neighborhood of its periodic
stationary states. We show that in spite of the fact that these states are only
marginally stable (i.e., the linearized problem about these states has
continuous spectrum extending all the way up to zero), there exist finite
dimensional invariant manifolds in the phase space of this equation which
determine the long-time behavior of solutions near these stationary solutions.
In particular, using this point of view, we obtain a new demonstration of
Schneider's recent proof that these states are nonlinearly stable.Comment: 44 pages, plain tex, 0 figure
Mariner Venus/Mercury 1973 study
Mariner Venus/Mercury 1973 flyby mission and description of spacecraft and subsystem
Lyapunov Mode Dynamics in Hard-Disk Systems
The tangent dynamics of the Lyapunov modes and their dynamics as generated
numerically - {\it the numerical dynamics} - is considered. We present a new
phenomenological description of the numerical dynamical structure that
accurately reproduces the experimental data for the quasi-one-dimensional
hard-disk system, and shows that the Lyapunov mode numerical dynamics is linear
and separate from the rest of the tangent space. Moreover, we propose a new,
detailed structure for the Lyapunov mode tangent dynamics, which implies that
the Lyapunov modes have well-defined (in)stability in either direction of time.
We test this tangent dynamics and its derivative properties numerically with
partial success. The phenomenological description involves a time-modal linear
combination of all other Lyapunov modes on the same polarization branch and our
proposed Lyapunov mode tangent dynamics is based upon the form of the tangent
dynamics for the zero modes
Period Doubling Renormalization for Area-Preserving Maps and Mild Computer Assistance in Contraction Mapping Principle
It has been observed that the famous Feigenbaum-Coullet-Tresser period
doubling universality has a counterpart for area-preserving maps of {\fR}^2.
A renormalization approach has been used in a "hard" computer-assisted proof of
existence of an area-preserving map with orbits of all binary periods in
Eckmann et al (1984). As it is the case with all non-trivial universality
problems in non-dissipative systems in dimensions more than one, no analytic
proof of this period doubling universality exists to date.
In this paper we attempt to reduce computer assistance in the argument, and
present a mild computer aided proof of the analyticity and compactness of the
renormalization operator in a neighborhood of a renormalization fixed point:
that is a proof that does not use generalizations of interval arithmetics to
functional spaces - but rather relies on interval arithmetics on real numbers
only to estimate otherwise explicit expressions. The proof relies on several
instance of the Contraction Mapping Principle, which is, again, verified via
mild computer assistance
Memory Effects in Nonequilibrium Transport for Deterministic Hamiltonian Systems
We consider nonequilibrium transport in a simple chain of identical
mechanical cells in which particles move around. In each cell, there is a
rotating disc, with which these particles interact, and this is the only
interaction in the model. It was shown in \cite{eckmann-young} that when the
cells are weakly coupled, to a good approximation, the jump rates of particles
and the energy-exchange rates from cell to cell follow linear profiles. Here,
we refine that study by analyzing higher-order effects which are induced by the
presence of external gradients for situations in which memory effects, typical
of Hamiltonian dynamics, cannot be neglected. For the steady state we propose a
set of balance equations for the particle number and energy in terms of the
reflection probabilities of the cell and solve it phenomenologically. Using
this approximate theory we explain how these asymmetries affect various aspects
of heat and particle transport in systems of the general type described above
and obtain in the infinite volume limit the deviation from the theory in
\cite{eckmann-young} to first-order. We verify our assumptions with extensive
numerical simulations.Comment: Several change
Decay of Correlations in a Topological Glass
In this paper we continue the study of a topological glassy system. The state
space of the model is given by all triangulations of a sphere with nodes,
half of which are red and half are blue. Red nodes want to have 5 neighbors
while blue ones want 7. Energies of nodes with other numbers of neighbors are
supposed to be positive. The dynamics is that of flipping the diagonal between
two adjacent triangles, with a temperature dependent probability. We consider
the system at very low temperatures.
We concentrate on several new aspects of this model: Starting from a detailed
description of the stationary state, we conclude that pairs of defects (nodes
with the "wrong" degree) move with very high mobility along 1-dimensional
paths. As they wander around, they encounter single defects, which they then
move "sideways" with a geometrically defined probability. This induces a
diffusive motion of the single defects. If they meet, they annihilate, lowering
the energy of the system. We both estimate the decay of energy to equilibrium,
as well as the correlations. In particular, we find a decay like
- …