4,554 research outputs found
Lattice symmetry breaking perturbations for spiral waves
Spiral waves in two-dimensional excitable media have been observed
experimentally and studied extensively. It is now well-known that the symmetry
properties of the medium of propagation drives many of the dynamics and
bifurcations which are experimentally observed for these waves. Also,
symmetry-breaking induced by boundaries, inhomogeneities and anisotropy have
all been shown to lead to different dynamical regimes as to that which is
predicted for mathematical models which assume infinite homogeneous and
isotropic planar geometry. Recent mathematical analyses incorporating the
concept of forced symmetry-breaking from the Euclidean group of all planar
translations and rotations have given model-independent descriptions of the
effects of media imperfections on spiral wave dynamics. In this paper, we
continue this program by considering rotating waves in dynamical systems which
are small perturbations of a Euclidean-equivariant dynamical system, but for
which the perturbation preserves only the symmetry of a regular square lattice
Nonlinear waves in Newton's cradle and the discrete p-Schroedinger equation
We study nonlinear waves in Newton's cradle, a classical mechanical system
consisting of a chain of beads attached to linear pendula and interacting
nonlinearly via Hertz's contact forces. We formally derive a spatially discrete
modulation equation, for small amplitude nonlinear waves consisting of slow
modulations of time-periodic linear oscillations. The fully-nonlinear and
unilateral interactions between beads yield a nonstandard modulation equation
that we call the discrete p-Schroedinger (DpS) equation. It consists of a
spatial discretization of a generalized Schroedinger equation with p-Laplacian,
with fractional p>2 depending on the exponent of Hertz's contact force. We show
that the DpS equation admits explicit periodic travelling wave solutions, and
numerically find a plethora of standing wave solutions given by the orbits of a
discrete map, in particular spatially localized breather solutions. Using a
modified Lyapunov-Schmidt technique, we prove the existence of exact periodic
travelling waves in the chain of beads, close to the small amplitude modulated
waves given by the DpS equation. Using numerical simulations, we show that the
DpS equation captures several other important features of the dynamics in the
weakly nonlinear regime, namely modulational instabilities, the existence of
static and travelling breathers, and repulsive or attractive interactions of
these localized structures
Travelling wave solutions in a negative nonlinear diffusion-reaction model
We use a geometric approach to prove the existence of smooth travelling wave
solutions of a nonlinear diffusion-reaction equation with logistic kinetics and
a convex nonlinear diffusivity function which changes sign twice in our domain
of interest. We determine the minimum wave speed, c*, and investigate its
relation to the spectral stability of the travelling wave solutions.Comment: 23 pages, 10 figure
Normal form for travelling kinks in discrete Klein-Gordon lattices
We study travelling kinks in the spatial discretizations of the nonlinear
Klein--Gordon equation, which include the discrete lattice and the
discrete sine--Gordon lattice. The differential advance-delay equation for
travelling kinks is reduced to the normal form, a scalar fourth-order
differential equation, near the quadruple zero eigenvalue. We show numerically
non-existence of monotonic kinks (heteroclinic orbits between adjacent
equilibrium points) in the fourth-order equation. Making generic assumptions on
the reduced fourth-order equation, we prove the persistence of bounded
solutions (heteroclinic connections between periodic solutions near adjacent
equilibrium points) in the full differential advanced-delay equation with the
technique of center manifold reduction. Existence and persistence of multiple
kinks in the discrete sine--Gordon equation are discussed in connection to
recent numerical results of \cite{ACR03} and results of our normal form
analysis
Kinks Dynamics in One-Dimensional Coupled Map Lattices
We examine the problem of the dynamics of interfaces in a one-dimensional
space-time discrete dynamical system. Two different regimes are studied : the
non-propagating and the propagating one. In the first case, after proving the
existence of such solutions, we show how they can be described using Taylor
expansions. The second situation deals with the assumption of a travelling wave
to follow the kink propagation. Then a comparison with the corresponding
continuous model is proposed. We find that these methods are useful in simple
dynamical situations but their application to complex dynamical behaviour is
not yet understood.Comment: 17pages, LaTex,3 fig available on cpt.univ-mrs.fr directory
pub/preprints/94/dynamical-systems/94-P.307
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