2,385 research outputs found
Diffraction of Bloch Wave Packets for Maxwell's Equations
We study, for times of order 1/h, solutions of Maxwell's equations in an
O(h^2) modulation of an h-periodic medium. The solutions are of slowly varying
amplitude type built on Bloch plane waves with wavelength of order h. We
construct accurate approximate solutions of three scale WKB type. The leading
profile is both transported at the group velocity and dispersed by a
Schr\"odinger equation given by the quadratic approximation of the Bloch
dispersion relation. A weak ray average hypothesis guarantees stability.
Compared to earlier work on scalar wave equations, the generator is no longer
elliptic. Coercivity holds only on the complement of an infinite dimensional
kernel. The system structure requires many innovations
Nonlinear stability of periodic traveling wave solutions of systems of viscous conservation laws in the generic case
Extending previous results of Oh--Zumbrun and Johnson--Zumbrun, we show that
spectral stability implies linearized and nonlinear stability of spatially
periodic traveling-wave solutions of viscous systems of conservation laws for
systems of generic type, removing a restrictive assumption that wave speed be
constant to first order along the manifold of nearby periodic solutions.Comment: Fixed minor typo
Whitham Averaged Equations and Modulational Stability of Periodic Traveling Waves of a Hyperbolic-Parabolic Balance Law
In this note, we report on recent findings concerning the spectral and
nonlinear stability of periodic traveling wave solutions of
hyperbolic-parabolic systems of balance laws, as applied to the St. Venant
equations of shallow water flow down an incline. We begin by introducing a
natural set of spectral stability assumptions, motivated by considerations from
the Whitham averaged equations, and outline the recent proof yielding nonlinear
stability under these conditions. We then turn to an analytical and numerical
investigation of the verification of these spectral stability assumptions.
While spectral instability is shown analytically to hold in both the Hopf and
homoclinic limits, our numerical studies indicates spectrally stable periodic
solutions of intermediate period. A mechanism for this moderate-amplitude
stabilization is proposed in terms of numerically observed "metastability" of
the the limiting homoclinic orbits.Comment: 27 pages, 5 figures. Minor changes throughou
Dispersive homogenized models and coefficient formulas for waves in general periodic media
We analyze a homogenization limit for the linear wave equation of second
order. The spatial operator is assumed to be of divergence form with an
oscillatory coefficient matrix that is periodic with
characteristic length scale ; no spatial symmetry properties are
imposed. Classical homogenization theory allows to describe solutions
well by a non-dispersive wave equation on fixed time intervals
. Instead, when larger time intervals are considered, dispersive effects
are observed. In this contribution we present a well-posed weakly dispersive
equation with homogeneous coefficients such that its solutions
describe well on time intervals . More
precisely, we provide a norm and uniform error estimates of the form for . They are accompanied by computable formulas for all
coefficients in the effective models. We additionally provide an
-independent equation of third order that describes dispersion
along rays and we present numerical examples.Comment: 28 pages, 7 figure
Linear Asymptotic Stability and Modulation Behavior near Periodic Waves of the Korteweg-de Vries Equation
We provide a detailed study of the dynamics obtained by linearizing the
Korteweg-de Vries equation about one of its periodic traveling waves, a cnoidal
wave. In a suitable sense, linearly analogous to space-modulated stability, we
prove global-in-time bounded stability in any Sobolev space, and asymptotic
stability of dispersive type. Furthermore, we provide both a leading-order
description of the dynamics in terms of slow modulation of local parameters and
asymptotic modulation systems and effective initial data for the evolution of
those parameters. This requires a global-in-time study of the dynamics
generated by a non normal operator with non constant coefficients. On the road
we also prove estimates on oscillatory integrals particularly suitable to
derive large-time asymptotic systems that could be of some general interest
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