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Localization & Partial Differential Equations
Org: T. Minzoni (IIMAS-UNAM) and M. Ward (UBC)
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- LUIS CISNEROS, Universidad Nacional Autónoma de México
A numerical study on the two-dimensional discrete sine-Gordon
equation
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It is known that the continuous two-dimensional sine-Gordon equation
does not support radial symmetric solutions. Radial initial
conditions collapse in finite time. We study numerically the discrete
version of this problem, and we obtain radial symmetric solutions that
do not collapse. We show using the modulation theory that the
collapse is prevented by the Peierls-Nabarro potential generated by
the discreteness of the problem. It is also shown that there is a
threshold of radial velocities above which the Peierls-Nabarro
potential is not enough to stop the collapse. The modulation theory
is shown to compare favorably with the numerical solution.
- JORGE FUJIOKA, Universidad Nacional Autónoma de México
Systems with Embedded Solitons
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At the end of the nineties a brand-new type of solitons were
discovered: the embedded solitons. Initially they were found
in optical systems, and afterwards they were also found in
hydrodynamical models, liquid crystal theory and discrete systems.
These peculiar solitary waves are interesting because they exist under
conditions in which, until recently, it was considered that the
propagation of solitons was impossible. In the beginning it was
considered that these nonlinear waves were necessarily isolated and
unstable, but later on it was found that they can be stable and may
exist in families. In the present communication it is explained what
these embedded solitons are, in which models they have been
found, and what variants exist (stable, unstable, continuous,
discrete, etc.).
- MARI CARMEN JORGE, Universidad Nacional Autónoma de México
A note on interface and bubble type asymptotic solutions to
the two-dimensional Cahn-Hilliard equation
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In this work we construct asymptotic solutions to the two-dimensional
steady-state Cahn-Hilliard equation. Using variational methods we
construct explicit approximate solutions of interface type. We also
study the determination of the bubble type solutions inside a
container with rigid walls. In this case the center of the bubble is
completely determined.
- THEODORE KOLOKOLNIKOV, Dalhousie University
Spot patterns in the BZ reaction
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We study spot patterns in the BZ reaction. We show that these
patterns can undergo two different types of self-replication. The
first type of self-replication is due to fold point corresponding to a
dissapearence of the steady state. The second type is due to an
instability of the steady state. In the former case, the spot splits
into a ring. In the latter case, the spot replicates into two spots,
or forms finger-like patterns.
- RACHEL KUSKE, University of British Columbia
Noise-sensitivity in bursting: new approaches for
quantitative analysis
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The phenomenon of bursting, composed of alternating periods of active
spiking and near quiescence, is observed in a variety of biological
applications including neural and cellular dynamics. It has a complex
sensitivity to noise which exhibits dynamical features from both the
underlying deterministic behavior and the stochastic elements. We use
a combined approach of approximating both the time dependent
probability density and the stochastic multi-scale dynamics in order
to understand contributions from both the deterministic and stochastic
features. This approach leads to simplified approximate models which
can be analyzed or simulated efficiently, providing quantitative
measures of the noise sensitivity. To illustrate the new approaches,
we focus on a model of bursting in dendritic spines. Generalizations
to other applications with similar dynamics will also be discussed.
- PANAYOTIS PANAYOTAROS, IIMAS-UNAM
Localized invariant tori in the discrete NLS with diffraction
management
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We present results on the existence of localized invariant tori
solutions in a discrete NLS equation with periodic parametric
forcing. The equation models a system of coupled waveguide arrays
with a special geometry that reduces diffraction effects. The
solutions are obtained by continuing breather periodic solutions of an
approximate autonomous system. We review some results on localized
and multipeak solutions of this system and sketch a continuation
argument that is based on general ideas on the continuation of
invariant tori in Hamiltonian systems with symmetries.
- ARTURO VARGAS, Universidad Nacional Autónoma de México, IIMAS
Evolution of Benjamin-Ono Solitons in the Presence of Weak
Z-K Lateral Dispersion
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The talk is about the effect of weak lateral dispersion of Z-K
type on a Benjamin-Ono solitary wave. The asymptotic solution is
based on an approximate variational solution for the solitary wave,
which is then modulated in time through the use of conservation
equations. The effect of the dispersive radiation shed as the
solitary wave evolves is also included in the modulation equations.
It is found that the weak lateral dispersion produces a strongly
anisotropic, stable solitary wave which decays algebraically in the
direction of propagation, as for the Benjamin-Ono solitary wave, and
exponentially in the transverse direction. Also, it is found that the
initial conditions with amplitude above a threshold evolve into
solitary waves, while those with amplitude below the threshold evolve
as lumps for a short time, then merge into radiation.
- MICHAEL WARD, Univeristy of British Columbia, Vancouver, BC
Eigenvalue Optimization, Spikes, and the Neumann Green's
Function
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An optimization problem for the fundamental eigenvalue l0 of
the Laplacian in a planar simply-connected domain that contains N
small identically-shaped holes, each of a small radius e << 1, is
considered. A Neumann boundary condition is imposed on the outer
boundary of the domain and a Dirichlet condition is imposed on the
boundary of each of the holes. For small hole radii e, we derive
an asymptotic expansion for l0 in terms of certain properties of
the Neumann Green's function for the Laplacian. This expansion
depends on the locations xi, for i=1,...,N, of the small
holes. For the unit disk, ring-type configurations of holes are
constructed to optimize the eigenvalue with respect to the hole
locations. This eigenvalue optimization problem is shown to be
closely related to the problem of determining equilibrium vortex
configurations in the Ginzburg-Landau theory of superconductivity,
and is also relevant for constructing localized spike-type solutions
to certain singularly perturbed reaction-diffusion systems. For these
spike solutions, some stability results are also given.
- RALF WITTENBERG, Department of Mathematics, Simon Fraser University, Burnaby,
BC, Canada
Attractors in one-dimensional spatiotemporal chaos
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We discuss the dynamics on the attractor of a family of
one-dimensional PDEs displaying spatiotemporally chaotic solutions,
including the Kuramoto-Sivashinsky (KS) equation. We obtain bounds
and estimates on the L2 norm and attractor dimension. A
sixth-order analogue of the KS equation, the Nikolaevskii model for
short-wave pattern formation with Galilean invariance, displays a
novel multiple-scale attractor. We show that existing modulation
equation descriptions coupling the amplitudes for the patterned mode
and mean flow, while asymptotically consistent, are incomplete. The
attractor features spatiotemporal chaos with strong scale separation,
coexistence of scaling regimes, anomalous exponents and Burgers-like
viscous shocks. These are captured by higher-order corrections to the
amplitude equations.
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