47 research outputs found
Cauchy Problem for Incompressible Neo-Hookean materials
In this paper we consider the Cauchy problem for neo-Hookean incompressible elasticity in spatial dimension . We are here interested primarily in the low regularity case, . For , we prove existence and uniqueness for is necessary
Gauge theory of Faddeev-Skyrme functionals
We study geometric variational problems for a class of nonlinear sigma-models
in quantum field theory. Mathematically, one needs to minimize an energy
functional on homotopy classes of maps from closed 3-manifolds into compact
homogeneous spaces G/H. The minimizers are known as Hopfions and exhibit
localized knot-like structure. Our main results include proving existence of
Hopfions as finite energy Sobolev maps in each (generalized) homotopy class
when the target space is a symmetric space. For more general spaces we obtain a
weaker result on existence of minimizers in each 2-homotopy class.
Our approach is based on representing maps into G/H by equivalence classes of
flat connections. The equivalence is given by gauge symmetry on pullbacks of
G-->G/H bundles. We work out a gauge calculus for connections under this
symmetry, and use it to eliminate non-compactness from the minimization problem
by fixing the gauge.Comment: 34 pages, no figure
On asymptotic stability of the Skyrmion
We study the asymptotic behavior of spherically symmetric solutions in the
Skyrme model. We show that the relaxation to the degree-one soliton (called the
Skyrmion) has a universal form of a superposition of two effects: exponentially
damped oscillations (the quasinormal ringing) and a power law decay (the tail).
The quasinormal ringing, which dominates the dynamics for intermediate times,
is a linear resonance effect. In contrast, the polynomial tail, which becomes
uncovered at late times, is shown to be a \emph{nonlinear} phenomenon.Comment: 4 pages, 4 figures, minor changes to match the PRD versio
Linear vs. nonlinear effects for nonlinear Schrodinger equations with potential
We review some recent results on nonlinear Schrodinger equations with
potential, with emphasis on the case where the potential is a second order
polynomial, for which the interaction between the linear dynamics caused by the
potential, and the nonlinear effects, can be described quite precisely. This
includes semi-classical regimes, as well as finite time blow-up and scattering
issues. We present the tools used for these problems, as well as their
limitations, and outline the arguments of the proofs.Comment: 20 pages; survey of previous result
Synthesizing attractors of Hindmarsh-Rose neuronal systems
In this paper a periodic parameter switching scheme is applied to the
Hindmarsh-Rose neuronal system to synthesize certain attractors. Results show
numerically, via computer graphic simulations, that the obtained synthesized
attractor belongs to the class of all admissible attractors for the
Hindmarsh-Rose neuronal system and matches the averaged attractor obtained with
the control parameter replaced with the averaged switched parameter values.
This feature allows us to imagine that living beings are able to maintain vital
behavior while the control parameter switches so that their dynamical behavior
is suitable for the given environment.Comment: published in Nonlinear Dynamic
Concerning the Wave equation on Asymptotically Euclidean Manifolds
We obtain KSS, Strichartz and certain weighted Strichartz estimate for the
wave equation on , , when metric
is non-trapping and approaches the Euclidean metric like with
. Using the KSS estimate, we prove almost global existence for
quadratically semilinear wave equations with small initial data for
and . Also, we establish the Strauss conjecture when the metric is radial
with for .Comment: Final version. To appear in Journal d'Analyse Mathematiqu
Strichartz estimates on Schwarzschild black hole backgrounds
We study dispersive properties for the wave equation in the Schwarzschild
space-time. The first result we obtain is a local energy estimate. This is then
used, following the spirit of earlier work of Metcalfe-Tataru, in order to
establish global-in-time Strichartz estimates. A considerable part of the paper
is devoted to a precise analysis of solutions near the trapping region, namely
the photon sphere.Comment: 44 pages; typos fixed, minor modifications in several place
Pure Point Spectrum of the Floquet Hamiltonian for the Quantum Harmonic Oscillator Under Time Quasi- Periodic Perturbations
We prove that the quantum harmonic oscillator is stable under spatially
localized, time quasi-periodic perturbations on a set of Diophantine
frequencies of positive measure. This proves a conjecture raised by
Enss-Veselic in their 1983 paper \cite{EV} in the general quasi-periodic
setting. The motivation of the present paper also comes from construction of
quasi-periodic solutions for the corresponding nonlinear equation