4,913 research outputs found
On Bounded Positive Stationary Solutions for a Nonlocal Fisher-KPP Equation
We study the existence of stationary solutions for a nonlocal version of the
Fisher-Kolmogorov-Petrovskii-Piscounov (FKPP) equation. The main motivation is
a recent study by Berestycki et {al.} [Nonlinearity 22 (2009),
{pp.}~2813--2844] where the nonlocal FKPP equation has been studied and it was
shown for the spatial domain andsufficiently small nonlocality
that there are only two bounded non-negative stationary solutions. Here we
provide a similar result for using a completely different
approach. In particular, an abstract perturbation argument is used in suitable
weighted Sobolev spaces. One aim of the alternative strategy is that it can
eventually be generalized to obtain persistence results for hyperbolic
invariant sets for other nonlocal evolution equations on unbounded domains with
small nonlocality, {i.e.}, to improve our understanding in applications when a
small nonlocal influence alters the dynamics and when it does not.Comment: 24 pages, 1 figure; revised versio
On the existence of oscillating solutions in non-monotone Mean-Field Games
For non-monotone single and two-populations time-dependent Mean-Field Game
systems we obtain the existence of an infinite number of branches of
non-trivial solutions. These non-trivial solutions are in particular shown to
exhibit an oscillatory behaviour when they are close to the trivial (constant)
one. The existence of such branches is derived using local and global
bifurcation methods, that rely on the analysis of eigenfunction expansions of
solutions to the associated linearized problem. Numerical analysis is performed
on two different models to observe the oscillatory behaviour of solutions
predicted by bifurcation theory, and to study further properties of branches
far away from bifurcation points.Comment: 24 pages, 10 figure
String Instabilities in Black Hole Spacetimes
We study the emergence of string instabilities in - dimensional black
hole spacetimes (Schwarzschild and Reissner - Nordstr\o m), and De Sitter space
(in static coordinates to allow a better comparison with the black hole case).
We solve the first order string fluctuations around the center of mass motion
at spatial infinity, near the horizon and at the spacetime singularity. We find
that the time components are always well behaved in the three regions and in
the three backgrounds. The radial components are {\it unstable}: imaginary
frequencies develop in the oscillatory modes near the horizon, and the
evolution is like , , near the spacetime
singularity, , where the world - sheet time , and the
proper string length grows infinitely. In the Schwarzschild black hole, the
angular components are always well - behaved, while in the Reissner - Nordstr\o
m case they develop instabilities inside the horizon, near where the
repulsive effects of the charge dominate over those of the mass. In general,
whenever large enough repulsive effects in the gravitational background are
present, string instabilities develop. In De Sitter space, all the spatial
components exhibit instability. The infalling of the string to the black hole
singularity is like the motion of a particle in a potential
where depends on the spacetime
dimensions and string angular momentum, with for Schwarzschild and
for Reissner - Nordstr\o m black holes. For the
string ends trapped by the black hole singularity.Comment: 26pages, Plain Te
Nonlinear Eigenvalues and Bifurcation Problems for Pucci's Operator
In this paper we extend existing results concerning generalized eigenvalues
of Pucci's extremal operators. In the radial case, we also give a complete
description of their spectrum, together with an equivalent of Rabinowitz's
Global Bifurcation Theorem. This allows us to solve equations involving Pucci's
operators
Coupled complex Ginzburg-Landau systems with saturable nonlinearity and asymmetric cross-phase modulation
We formulate and study dynamics from a complex Ginzburg-Landau system with
saturable nonlinearity, including asymmetric cross-phase modulation (XPM)
parameters. Such equations can model phenomena described by complex
Ginzburg-Landau systems under the added assumption of saturable media. When the
saturation parameter is set to zero, we recover a general complex cubic
Ginzburg-Landau system with XPM. We first derive conditions for the existence
of bounded dynamics, approximating the absorbing set for solutions. We use this
to then determine conditions for amplitude death of a single wavefunction. We
also construct exact plane wave solutions, and determine conditions for their
modulational instability. In a degenerate limit where dispersion and
nonlinearity balance, we reduce our system to a saturable nonlinear
Schr\"odinger system with XPM parameters, and we demonstrate the existence and
behavior of spatially heterogeneous stationary solutions in this limit. Using
numerical simulations we verify the aforementioned analytical results, while
also demonstrating other interesting emergent features of the dynamics, such as
spatiotemporal chaos in the presence of modulational instability. In other
regimes, coherent patterns including uniform states or banded structures arise,
corresponding to certain stable stationary states. For sufficiently large yet
equal XPM parameters, we observe a segregation of wavefunctions into different
regions of the spatial domain, while when XPM parameters are large and take
different values, one wavefunction may decay to zero in finite time over the
spatial domain (in agreement with the amplitude death predicted analytically).
While saturation will often regularize the dynamics, such transient dynamics
can still be observed - and in some cases even prolonged - as the saturability
of the media is increased, as the saturation may act to slow the timescale.Comment: 36 page
Asymptotic solutions of forced nonlinear second order differential equations and their extensions
Using a modified version of Schauder's fixed point theorem, measures of
non-compactness and classical techniques, we provide new general results on the
asymptotic behavior and the non-oscillation of second order scalar nonlinear
differential equations on a half-axis. In addition, we extend the methods and
present new similar results for integral equations and Volterra-Stieltjes
integral equations, a framework whose benefits include the unification of
second order difference and differential equations. In so doing, we enlarge the
class of nonlinearities and in some cases remove the distinction between
superlinear, sublinear, and linear differential equations that is normally
found in the literature. An update of papers, past and present, in the theory
of Volterra-Stieltjes integral equations is also presented
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