2,829 research outputs found
Strong asymptotic stability of a compactly coupled system of wave equations
AbstractWe prove the well-posedness and study the strong asymptotic stability of a compactly coupled system of wave equations with a nonlinear feedback acting on one end only
Gaussian solitary waves and compactons in Fermi-Pasta-Ulam lattices with Hertzian potentials
We consider a class of fully-nonlinear Fermi-Pasta-Ulam (FPU) lattices,
consisting of a chain of particles coupled by fractional power nonlinearities
of order . This class of systems incorporates a classical Hertzian
model describing acoustic wave propagation in chains of touching beads in the
absence of precompression. We analyze the propagation of localized waves when
is close to unity. Solutions varying slowly in space and time are
searched with an appropriate scaling, and two asymptotic models of the chain of
particles are derived consistently. The first one is a logarithmic KdV
equation, and possesses linearly orbitally stable Gaussian solitary wave
solutions. The second model consists of a generalized KdV equation with
H\"older-continuous fractional power nonlinearity and admits compacton
solutions, i.e. solitary waves with compact support. When , we numerically establish the asymptotically Gaussian shape of exact FPU
solitary waves with near-sonic speed, and analytically check the pointwise
convergence of compactons towards the limiting Gaussian profile
Asymptotic stability of the multidimensional wave equation coupled with classes of positive-real impedance boundary conditions
This paper proves the asymptotic stability of the multidimensional wave
equation posed on a bounded open Lipschitz set, coupled with various classes of
positive-real impedance boundary conditions, chosen for their physical
relevance: time-delayed, standard diffusive (which includes the
Riemann-Liouville fractional integral) and extended diffusive (which includes
the Caputo fractional derivative). The method of proof consists in formulating
an abstract Cauchy problem on an extended state space using a dissipative
realization of the impedance operator, be it finite or infinite-dimensional.
The asymptotic stability of the corresponding strongly continuous semigroup is
then obtained by verifying the sufficient spectral conditions derived by Arendt
and Batty (Trans. Amer. Math. Soc., 306 (1988)) as well as Lyubich and V\~u
(Studia Math., 88 (1988))
A proof of anomalous invasion speeds in a system of coupled Fisher-KPP equations
This article is concerned with the rigorous validation of anomalous spreading
speeds in a system of coupled Fisher-KPP equations of cooperative type.
Anomalous spreading refers to a scenario wherein the coupling of two equations
leads to faster spreading speeds in one of the components. The existence of
these spreading speeds can be predicted from the linearization about the
unstable state. We prove that initial data consisting of compactly supported
perturbations of Heaviside step functions spreads asymptotically with the
anomalous speed. The proof makes use of a comparison principle and the explicit
construction of sub and super solutions
Non-stationary Spectra of Local Wave Turbulence
The evolution of the Kolmogorov-Zakharov (K-Z) spectrum of weak turbulence is
studied in the limit of strongly local interactions where the usual kinetic
equation, describing the time evolution of the spectral wave-action density,
can be approximated by a PDE. If the wave action is initially compactly
supported in frequency space, it is then redistributed by resonant interactions
producing the usual direct and inverse cascades, leading to the formation of
the K-Z spectra. The emphasis here is on the direct cascade. The evolution
proceeds by the formation of a self-similar front which propagates to the right
leaving a quasi-stationary state in its wake. This front is sharp in the sense
that the solution remains compactly supported until it reaches infinity. If the
energy spectrum has infinite capacity, the front takes infinite time to reach
infinite frequency and leaves the K-Z spectrum in its wake. On the other hand,
if the energy spectrum has finite capacity, the front reaches infinity within a
finite time, t*, and the wake is steeper than the K-Z spectrum. For this case,
the K-Z spectrum is set up from the right after the front reaches infinity. The
slope of the solution in the wake can be related to the speed of propagation of
the front. It is shown that the anomalous slope in the finite capacity case
corresponds to the unique front speed which ensures that the front tip contains
a finite amount of energy as the connection to infinity is made. We also
introduce, for the first time, the notion of entropy production in wave
turbulence and show how it evolves as the system approaches the stationary K-Z
spectrum.Comment: revtex4, 19 pages, 10 figure
- …