Searcharxiv⌕ Search

arXiv subjects

Avy Soffer

Publications and source records attributed to Avy Soffer.

At least 91 records · Page 5Linked to original sources
arXiv2015

Dynamics of a Classical Particle in a Quasi Periodic Potential

We study the dynamics of a one-dimensional classical particle in a space and time dependent potential with randomly chosen parameters. The focus of this work is a quasi-periodic potential, which only includes a finite number of Fourier components. The momentum is calculated analytically for short time within a self-consistent approximation, under certain conditions. We find that the dynamics can be described by a model of a random walk between the Chirikov resonances, which are resonances between the particle momentum and the Fourier components of the potential. We use numerical methods to test these results and to evaluate the important properties, such as the characteristic hopping time between the resonances. This work sheds light on the short time dynamics induced by potentials which are relevant for optics and atom optics.

cond-mat.stat-mech↗
arXiv2014

Dynamics of Noncommutative Solitons II: Spectral Theory, Dispersive Estimates and Stability

We consider the Schrödinger equation with a (matrix) Hamiltonian given by a second order difference operator with nonconstant growing coefficients, on the half one dimensional lattice. This operator appeared first naturally in the construction and dynamics of noncommutative solitons in the context of noncommutative field theory. We completely determine the spectrum of the Hamiltonian linearized around a ground state soliton and prove the optimal decay rate of $t^{-1}\log^{-2}t$ for the associated time decay estimate. We use a novel technique involving generating functions of orthogonal polynomials to achieve this estimate.

math-ph↗
arXiv2014

Scattering for the Klein-Gordon equation with quadratic and variable coefficient cubic nonlinearities

We study the 1D Klein-Gordon equation with variable coefficient cubic nonlinearity. This problem exhibits a striking resonant interaction between the spatial frequencies of the nonlinear coefficients and the temporal oscillations of the solutions. In the case where the worst of this resonant behavior is absent, we prove L-Infinity scattering as well as a certain kind of strong smoothness for the solution at time-like infinity with the help of several new normal-forms transformations. Some explicit examples are also given which suggest qualitatively different behavior in the case where the strongest cubic resonances are present.

math.AP↗
arXiv2013

Multiscale time averaging, Reloaded

We develop a rigorously controlled multi-time scale averaging technique; the averaging is done on a finite time interval, properly chosen, and then, via iterations and normal form transformations, the time intervals are scaled to arbitrary order. Here, we consider as an example the problem of finite dimensional conservative dynamical system, which is quasiperiodic and dominated by slow frequencies, leading to small divisor problems in perturbative schemes.

math-ph↗
arXiv2011

Friction in a Model of Hamiltonian Dynamics

We study the motion of a heavy tracer particle weakly coupled to a dense ideal Bose gas exhibiting Bose-Einstein condensation. In the so-called mean-field limit, the dynamics of this system approaches one determined by nonlinear Hamiltonian evolution equations describing a process of emission of Cerenkov radiation of sound waves into the Bose-Einstein condensate along the particle's trajectory. The emission of Cerenkov radiation results in a friction force with memory acting on the tracer particle and causing it to decelerate until it comes to rest.

math-ph↗
arXiv2011

Monotonic Local Decay Estimates

For the Hamiltonian operator H = -Δ+V(x) of the Schrödinger Equation with a repulsive potential, the problem of local decay is considered. It is analyzed by a direct method, based on a new, L^2 bounded, propagation observable. The resulting decay estimate, is in certain cases monotonic in time, with no "Quantum Corrections". This method is then applied to some examples in one and higher dimensions. In particular the case of the Wave Equation on a Schwarzschild manifold is redone: Local decay, stronger than the known ones are proved (minimal loss of angular derivatives and lower order of radial derivatives of initial data). The method developed here can be an alternative in some cases to the Morawetz type estimates, with L^2-multipliers replacing the first order operators. It provides an alternative to Mourre's method, by including thresholds and high energies.

math.AP↗
arXiv2011

On pointwise decay of linear waves on a Schwarzschild black hole background

We prove sharp pointwise $t^{-3}$ decay for scalar linear perturbations of a Schwarzschild black hole without symmetry assumptions on the data. We also consider electromagnetic and gravitational perturbations for which we obtain decay rates $t^{-4}$, and $t^{-6}$, respectively. We proceed by decomposition into angular momentum $\ell$ and summation of the decay estimates on the Regge-Wheeler equation for fixed $\ell$. We encounter a dichotomy: the decay law in time is entirely determined by the asymptotic behavior of the Regge-Wheeler potential in the far field, whereas the growth of the constants in $\ell$ is dictated by the behavior of the Regge-Wheeler potential in a small neighborhood around its maximum. In other words, the tails are controlled by small energies, whereas the number of angular derivatives needed on the data is determined by energies close to the top of the Regge-Wheeler potential. This dichotomy corresponds to the well-known principle that for initial times the decay reflects the presence of complex resonances generated by the potential maximum, whereas for later times the tails are determined by the far field. However, we do not invoke complex resonances at all, but rely instead on semiclassical Sigal-Soffer type propagation estimates based on a Mourre bound near the top energy.

math.AP↗
arXiv2011

The Nonlinear Schroedinger Equation with a random potential: Results and Puzzles

The Nonlinear Schroedinger Equation (NLSE) with a random potential is motivated by experiments in optics and in atom optics and is a paradigm for the competition between the randomness and nonlinearity. The analysis of the NLSE with a random (Anderson like) potential has been done at various levels of control: numerical, analytical and rigorous. Yet, this model equation presents us with a highly inconclusive and often contradictory picture. We will describe the main recent results obtained in this field and propose a list of specific problems to focus on, that we hope will enable to resolve these outstanding questions.

math-ph↗
arXiv2011

The Maximal Velocity of a Photon

The probability of a photon to move faster than light, under the Einstein Quantum dynamics is shown to vanish for large times. The proof is based on using propagation estimates on multiscaled energy intervals, to control the singular symbol of the Hamiltonian, at zero.

math-ph↗
arXiv2011

The Schroedinger Equation with Potential in Rough Motion

This paper proves endpoint Strichartz estimates for the linear Schroedinger equation in $R^3$, with a time-dependent potential that keeps a constant profile and is subject to a rough motion, which need not be differentiable and may be large in norm. The potential is also subjected to a time-dependent rescaling, with a non-differentiable dilation parameter. We use the Strichartz estimates to prove the non-dispersion of bound states, when the path is small in norm, as well as boundedness of energy. We also include a sample nonlinear application of the linear results.

math.AP↗
arXiv2011

Eigenvalue repulsion estimates and some applications for the one-dimensional Anderson model

We show that the spacing between eigenvalues of the discrete 1D Hamiltonian with arbitrary potentials which are bounded, and with Dirichlet or Neumann Boundary Conditions is bounded away from zero. We prove an explicit lower bound, given by $Ce^{-bN}$, where $N$ is the lattice size, and $C$ and $b$ are some finite constants. In particular, the spectra of such Hamiltonians have no degenerate eigenvalues. As applications we show that to leading order in the coupling, the solution of a nonlinearly perturbed Anderson model in one-dimension (on the lattice) remains exponentially localized, in probability and average sense for initial conditions given by a unique eigenfunction of the linear problem. We also bound the derivative of the eigenfunctions of the linear Anderson model with respect to a potential change.

cond-mat.dis-nn↗
arXiv2010

Some Hamiltonian Models of Friction

Mathematical results on some models describing the motion of a tracer particle through a Bose-Einstein condensate are described. In the limit of a very dense, very weakly interacting Bose gas and for a very large particle mass, the dynamics of the coupled system is determined by classical non-linear Hamiltonian equations of motion. The particle's motion exhibits deceleration corresponding to friction (with memory) caused by the emission of Cerenkov radiation of gapless modes into the gas. Precise results are stated and outlines of proofs are presented. Some technical details are deferred to forthcoming papers.

math.AP↗
arXiv2010

A proof of Price's Law on Schwarzschild black hole manifolds for all angular momenta

Price's Law states that linear perturbations of a Schwarzschild black hole fall off as $t^{-2\ell-3}$ for $t \to \infty$ provided the initial data decay sufficiently fast at spatial infinity. Moreover, if the perturbations are initially static (i.e., their time derivative is zero), then the decay is predicted to be $t^{-2\ell-4}$. We give a proof of $t^{-2\ell-2}$ decay for general data in the form of weighted $L^1$ to $L^\infty$ bounds for solutions of the Regge--Wheeler equation. For initially static perturbations we obtain $t^{-2\ell-3}$. The proof is based on an integral representation of the solution which follows from self--adjoint spectral theory. We apply two different perturbative arguments in order to construct the corresponding spectral measure and the decay bounds are obtained by appropriate oscillatory integral estimates.

gr-qc↗
arXiv2009

A numerical and symbolical approximation of the Nonlinear Anderson Model

A modified perturbation theory in the strength of the nonlinear term is used to solve the Nonlinear Schroedinger Equation with a random potential. It is demonstrated that in some cases it is more efficient than other methods. Moreover we obtain error estimates. This approach can be useful for the solution of other nonlinear differential equations of physical relevance.

cond-mat.mes-hall↗
arXiv2009

Perturbation theory for the Nonlinear Schroedinger Equation with a random potential

A perturbation theory for the Nonlinear Schroedinger Equation (NLSE) in 1D on a lattice was developed. The small parameter is the strength of the nonlinearity. For this purpose secular terms were removed and a probabilistic bound on small denominators was developed. It was shown that the number of terms grows exponentially with the order. The results of the perturbation theory are compared with numerical calculations. An estimate on the remainder is obtained and it is demonstrated that the series is asymptotic.

cond-mat.dis-nn↗
arXiv2008

Improved Error Bounds for Dirichlet-to-Neumann Absorbing Boundaries

It has long been known how to construct radiation boundary conditions for the time dependent wave equation. Although arguments suggesting that they are accurate have been given, it is only recently that rigorous error bounds have been proved. Previous estimates show that the error caused by these methods behaves like epsilon C exp(gamma t) for any gamma > 0. We improve these results and show that the error behaves like epsilon t^2.

math.NA↗
arXiv2008

On the distribution of linear combinations of eigenvalues of the Anderson model

Probabilistic estimates on linear combinations of eigenvalues of the one dimensional Anderson model are derived. So far only estimates on the density of eigenvalues and of pairs were found by Wegner and by Minami. Our work was motivated by perturbative explorations of the Nonlinear Schroedinger Equation, where linear combinations of eigenvalues are the denominators and evaluation of their smallness is crucial.

math-ph↗