15,345 research outputs found
Nonlinear stability of source defects in the complex Ginzburg-Landau equation
In an appropriate moving coordinate frame, source defects are time-periodic
solutions to reaction-diffusion equations that are spatially asymptotic to
spatially periodic wave trains whose group velocities point away from the core
of the defect. In this paper, we rigorously establish nonlinear stability of
spectrally stable source defects in the complex Ginzburg-Landau equation. Due
to the outward transport at the far field, localized perturbations may lead to
a highly non-localized response even on the linear level. To overcome this, we
first investigate in detail the dynamics of the solution to the linearized
equation. This allows us to determine an approximate solution that satisfies
the full equation up to and including quadratic terms in the nonlinearity. This
approximation utilizes the fact that the non-localized phase response,
resulting from the embedded zero eigenvalues, can be captured, to leading
order, by the nonlinear Burgers equation. The analysis is completed by
obtaining detailed estimates for the resolvent kernel and pointwise estimates
for the Green's function, which allow one to close a nonlinear iteration
scheme.Comment: 53 pages, 5 figure
Propagation and Structure of Planar Streamer Fronts
Streamers often constitute the first stage of dielectric breakdown in strong
electric fields: a nonlinear ionization wave transforms a non-ionized medium
into a weakly ionized nonequilibrium plasma. New understanding of this old
phenomenon can be gained through modern concepts of (interfacial) pattern
formation. As a first step towards an effective interface description, we
determine the front width, solve the selection problem for planar fronts and
calculate their properties. Our results are in good agreement with many
features of recent three-dimensional numerical simulations.
In the present long paper, you find the physics of the model and the
interfacial approach further explained. As a first ingredient of this approach,
we here analyze planar fronts, their profile and velocity. We encounter a
selection problem, recall some knowledge about such problems and apply it to
planar streamer fronts. We make analytical predictions on the selected front
profile and velocity and confirm them numerically.
(abbreviated abstract)Comment: 23 pages, revtex, 14 ps file
A topological approximation of the nonlinear Anderson model
We study the phenomena of Anderson localization in the presence of nonlinear
interaction on a lattice. A class of nonlinear Schrodinger models with
arbitrary power nonlinearity is analyzed. We conceive the various regimes of
behavior, depending on the topology of resonance-overlap in phase space,
ranging from a fully developed chaos involving Levy flights to pseudochaotic
dynamics at the onset of delocalization. It is demonstrated that quadratic
nonlinearity plays a dynamically very distinguished role in that it is the only
type of power nonlinearity permitting an abrupt localization-delocalization
transition with unlimited spreading already at the delocalization border. We
describe this localization-delocalization transition as a percolation
transition on a Cayley tree. It is found in vicinity of the criticality that
the spreading of the wave field is subdiffusive in the limit
t\rightarrow+\infty. The second moment grows with time as a powerlaw t^\alpha,
with \alpha = 1/3. Also we find for superquadratic nonlinearity that the analog
pseudochaotic regime at the edge of chaos is self-controlling in that it has
feedback on the topology of the structure on which the transport processes
concentrate. Then the system automatically (without tuning of parameters)
develops its percolation point. We classify this type of behavior in terms of
self-organized criticality dynamics in Hilbert space. For subquadratic
nonlinearities, the behavior is shown to be sensitive to details of definition
of the nonlinear term. A transport model is proposed based on modified
nonlinearity, using the idea of stripes propagating the wave process to large
distances. Theoretical investigations, presented here, are the basis for
consistency analysis of the different localization-delocalization patterns in
systems with many coupled degrees of freedom in association with the asymptotic
properties of the transport.Comment: 20 pages, 2 figures; improved text with revisions; accepted for
publication in Physical Review
Nonlocalized modulation of periodic reaction diffusion waves: The Whitham equation
In a companion paper, we established nonlinear stability with detailed
diffusive rates of decay of spectrally stable periodic traveling-wave solutions
of reaction diffusion systems under small perturbations consisting of a
nonlocalized modulation plus a localized perturbation. Here, we determine
time-asymptotic behavior under such perturbations, showing that solutions
consist to leading order of a modulation whose parameter evolution is governed
by an associated Whitham averaged equation
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
Space-modulated Stability and Averaged Dynamics
In this brief note we give a brief overview of the comprehensive theory,
recently obtained by the author jointly with Johnson, Noble and Zumbrun, that
describes the nonlinear dynamics about spectrally stable periodic waves of
parabolic systems and announce parallel results for the linearized dynamics
near cnoidal waves of the Korteweg-de Vries equation. The latter are expected
to contribute to the development of a dispersive theory, still to come.Comment: Proceedings of the "Journ\'ees \'Equations aux d\'eriv\'ees
partielles", Roscoff 201
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