44,379 research outputs found
Nonlinear dynamical boundary-value problem of hydrogen thermal desorption
The nonlinear boundary-value problem for the diffusion equation,
which models gas interaction with solids, is considered. The
model includes diffusion and the sorption/desorption processes on
the surface, which leads to dynamical nonlinear boundary
conditions. The boundary-value problem is reduced to an
integro-differential equation of a special kind; existence and
uniqueness of the classical (differentiable) solution theorems
are proved. The results of numerical experiments are presented
Stochastic FitzHugh-Nagumo equations on networks with impulsive noise
We consider a system of nonlinear partial differential equations with
stochastic dynamical boundary conditions that arises in models of
neurophysiology for the diffusion of electrical potentials through a finite
network of neurons. Motivated by the discussion in the biological literature,
we impose a general diffusion equation on each edge through a generalized
version of the FitzHugh-Nagumo model, while the noise acting on the boundary is
described by a generalized stochastic Kirchhoff law on the nodes. In the
abstract framework of matrix operators theory, we rewrite this stochastic
boundary value problem as a stochastic evolution equation in infinite
dimensions with a power-type nonlinearity, driven by an additive L\'evy noise.
We prove global well-posedness in the mild sense for such stochastic partial
differential equation by monotonicity methods.Comment: 18 pages. Minor revisio
Nonlinear mobility continuity equations and generalized displacement convexity
We consider the geometry of the space of Borel measures endowed with a
distance that is defined by generalizing the dynamical formulation of the
Wasserstein distance to concave, nonlinear mobilities. We investigate the
energy landscape of internal, potential, and interaction energies. For the
internal energy, we give an explicit sufficient condition for geodesic
convexity which generalizes the condition of McCann. We take an eulerian
approach that does not require global information on the geodesics. As
by-product, we obtain existence, stability, and contraction results for the
semigroup obtained by solving the homogeneous Neumann boundary value problem
for a nonlinear diffusion equation in a convex bounded domain. For the
potential energy and the interaction energy, we present a non-rigorous argument
indicating that they are not displacement semiconvex.Comment: 33 pages, 1 figur
Quantum Trajectories, State Diffusion and Time Asymmetric Eventum Mechanics
We show that the quantum stochastic unitary dynamics Langevin model for
continuous in time measurements provides an exact formulation of the Heisenberg
uncertainty error-disturbance principle. Moreover, as it was shown in the 80's,
this Markov model induces all stochastic linear and non-linear equations of the
phenomenological "quantum trajectories" such as quantum state diffusion and
spontaneous localization by a simple quantum filtering method. Here we prove
that the quantum Langevin equation is equivalent to a Dirac type boundary-value
problem for the second-quantized input "offer waves from future" in one extra
dimension, and to a reduction of the algebra of the consistent histories of
past events to an Abelian subalgebra for the "trajectories of the output
particles". This result supports the wave-particle duality in the form of the
thesis of Eventum Mechanics that everything in the future is constituted by
quantized waves, everything in the past by trajectories of the recorded
particles. We demonstrate how this time arrow can be derived from the principle
of quantum causality for nondemolition continuous in time measurements.Comment: 21 pages. See also relevant publications at
http://www.maths.nott.ac.uk/personal/vpb/publications.htm
The Fermi-Pasta-Ulam problem: 50 years of progress
A brief review of the Fermi-Pasta-Ulam (FPU) paradox is given, together with
its suggested resolutions and its relation to other physical problems. We focus
on the ideas and concepts that have become the core of modern nonlinear
mechanics, in their historical perspective. Starting from the first numerical
results of FPU, both theoretical and numerical findings are discussed in close
connection with the problems of ergodicity, integrability, chaos and stability
of motion. New directions related to the Bose-Einstein condensation and quantum
systems of interacting Bose-particles are also considered.Comment: 48 pages, no figures, corrected and accepted for publicatio
Pattern forming pulled fronts: bounds and universal convergence
We analyze the dynamics of pattern forming fronts which propagate into an
unstable state, and whose dynamics is of the pulled type, so that their
asymptotic speed is equal to the linear spreading speed v^*. We discuss a
method that allows to derive bounds on the front velocity, and which hence can
be used to prove for, among others, the Swift-Hohenberg equation, the Extended
Fisher-Kolmogorov equation and the cubic Complex Ginzburg-Landau equation, that
the dynamically relevant fronts are of the pulled type. In addition, we
generalize the derivation of the universal power law convergence of the
dynamics of uniformly translating pulled fronts to both coherent and incoherent
pattern forming fronts. The analysis is based on a matching analysis of the
dynamics in the leading edge of the front, to the behavior imposed by the
nonlinear region behind it. Numerical simulations of fronts in the
Swift-Hohenberg equation are in full accord with our analytical predictions.Comment: 27 pages, 9 figure
Strongly nonlinear dynamics of electrolytes in large ac voltages
We study the response of a model micro-electrochemical cell to a large ac
voltage of frequency comparable to the inverse cell relaxation time. To bring
out the basic physics, we consider the simplest possible model of a symmetric
binary electrolyte confined between parallel-plate blocking electrodes,
ignoring any transverse instability or fluid flow. We analyze the resulting
one-dimensional problem by matched asymptotic expansions in the limit of thin
double layers and extend previous work into the strongly nonlinear regime,
which is characterized by two novel features - significant salt depletion in
the electrolyte near the electrodes and, at very large voltage, the breakdown
of the quasi-equilibrium structure of the double layers. The former leads to
the prediction of "ac capacitive desalination", since there is a time-averaged
transfer of salt from the bulk to the double layers, via oscillating diffusion
layers. The latter is associated with transient diffusion limitation, which
drives the formation and collapse of space-charge layers, even in the absence
of any net Faradaic current through the cell. We also predict that steric
effects of finite ion sizes (going beyond dilute solution theory) act to
suppress the strongly nonlinear regime in the limit of concentrated
electrolytes, ionic liquids and molten salts. Beyond the model problem, our
reduced equations for thin double layers, based on uniformly valid matched
asymptotic expansions, provide a useful mathematical framework to describe
additional nonlinear responses to large ac voltages, such as Faradaic
reactions, electro-osmotic instabilities, and induced-charge electrokinetic
phenomena.Comment: 30 pages, 17 eps-figures, RevTe
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