14,302 research outputs found
Dynamics of localized waves in 1D random potentials: statistical theory of the coherent forward scattering peak
As recently discovered [PRL 190601(2012)], Anderson localization
in a bulk disordered system triggers the emergence of a coherent forward
scattering (CFS) peak in momentum space, which twins the well-known coherent
backscattering (CBS) peak observed in weak localization experiments. Going
beyond the perturbative regime, we address here the long-time dynamics of the
CFS peak in a 1D random system and we relate this novel interference effect to
the statistical properties of the eigenfunctions and eigenspectrum of the
corresponding random Hamiltonian. Our numerical results show that the dynamics
of the CFS peak is governed by the logarithmic level repulsion between
localized states, with a time scale that is, with good accuracy, twice the
Heisenberg time. This is in perfect agreement with recent findings based on the
nonlinear -model. In the stationary regime, the width of the CFS peak
in momentum space is inversely proportional to the localization length,
reflecting the exponential decay of the eigenfunctions in real space, while its
height is exactly twice the background, reflecting the Poisson statistical
properties of the eigenfunctions. Our results should be easily extended to
higher dimensional systems and other symmetry classes.Comment: See the published article for the updated versio
Computation of Spiral Spectra
A computational linear stability analysis of spiral waves in a
reaction-diffusion equation is performed on large disks. As the disk radius R
increases, eigenvalue spectra converge to the absolute spectrum predicted by
Sandstede and Scheel. The convergence rate is consistent with 1/R, except
possibly near the edge of the spectrum. Eigenfunctions computed on large disks
are compared with predicted exponential forms. Away from the edge of the
absolute spectrum the agreement is excellent, while near the edge computed
eigenfunctions deviate from predictions, probably due to finite-size effects.
In addition to eigenvalues associated with the absolute spectrum, computations
reveal point eigenvalues. The point eigenvalues and associated eigenfunctions
responsible for both core and far-field breakup of spiral waves are shown.Comment: 20 pages, 13 figures, submitted to SIAD
A minimization principle for the description of time-dependent modes associated with transient instabilities
We introduce a minimization formulation for the determination of a
finite-dimensional, time-dependent, orthonormal basis that captures directions
of the phase space associated with transient instabilities. While these
instabilities have finite lifetime they can play a crucial role by either
altering the system dynamics through the activation of other instabilities, or
by creating sudden nonlinear energy transfers that lead to extreme responses.
However, their essentially transient character makes their description a
particularly challenging task. We develop a minimization framework that focuses
on the optimal approximation of the system dynamics in the neighborhood of the
system state. This minimization formulation results in differential equations
that evolve a time-dependent basis so that it optimally approximates the most
unstable directions. We demonstrate the capability of the method for two
families of problems: i) linear systems including the advection-diffusion
operator in a strongly non-normal regime as well as the Orr-Sommerfeld/Squire
operator, and ii) nonlinear problems including a low-dimensional system with
transient instabilities and the vertical jet in crossflow. We demonstrate that
the time-dependent subspace captures the strongly transient non-normal energy
growth (in the short time regime), while for longer times the modes capture the
expected asymptotic behavior
Classical nonlinear response of a chaotic system: Langevin dynamics and spectral decomposition
We consider the classical response of a strongly chaotic Hamiltonian system.
The spectrum of such a system consists of discrete complex Ruelle-Pollicott
(RP) resonances which manifest themselves in the behavior of the correlation
and response functions. We interpret the RP resonances as the eigenstates and
eigenvalues of the Fokker-Planck operator obtained by adding an infinitesimal
noise term to the first-order Liouville operator. We demonstrate how the
deterministic expression for the linear response is reproduced in the limit of
vanishing noise. For the second-order response we establish an equivalence of
the spectral decomposition with infinitesimal noise and the long-time
asymptotic expansion for the deterministic case.Comment: 16 pages, 1 figur
Very Singular Similarity Solutions and Hermitian Spectral Theory for Semilinear Odd-Order PDEs
Very singular self-similar solutions of semilinear odd-order PDEs are studied
on the basis of a Hermitian-type spectral theory for linear rescaled odd-order
operators.Comment: 49 pages, 12 Figure
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