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W. -M. Wang

Publications and source records attributed to W. -M. Wang.

26 records · Page 2Linked to original sources
arXiv2012

Approach to equilibrium for the stochastic NLS

We study the approach to equilibrium, described by a Gibbs measure, for a system on a $d$-dimensional torus evolving according to a stochastic nonlinear Schrödinger equation (SNLS) with a high frequency truncation. We prove exponential approach to the truncated Gibbs measure both for the focusing and defocusing cases when the dynamics is constrained via suitable boundary conditions to regions of the Fourier space where the Hamiltonian is convex. Our method is based on establishing a spectral gap for the non self-adjoint Fokker-Planck operator governing the time evolution of the measure, which is {\it uniform} in the frequency truncation $N$. The limit $N\to\infty$ is discussed.

math-ph↗
arXiv2010

Supercritical Nonlinear Schrödinger Equations II: Almost Global Existence

We prove almost global existence for supercritical nonlinear Schrödinger equations on the $d$-torus ($d$ arbitrary) on the good geometry selected in part I. This is seen as the Cauchy consequence of I, since the known invariant measure of smooth solutions are supported on KAM tori. In the high frequency limit, these quantitative solutions could also be relevant to Cauchy problems for compressible Euler equations.

math.AP↗
arXiv2008

Logarithmic bounds on Sobolev norms for time-dependent linear Schrödinger equations

We prove that in 1-D the growth of Sobolev norms for time-dependent linear Schrödinger equations is at most logarithmic in time for any (fixed) potential which is analytic (or Gevrey). Recently it was proven in [N] that almost surely the Sobolev norms are unbounded, which indicates that the log is almost surely necessary. In [W], the author showed that the Sobolev norms remain bounded for an explicit time periodic potential. This is in the exceptional set in the sense of [N]. The present paper together with [N, W] give a rather complete picture of time dependent linear Schrödinger equations on the circle.

math.SP↗
arXiv2008

Pure Point Spectrum of the Floquet Hamiltonian for the Quantum Harmonic Oscillator Under Time Quasi- Periodic Perturbations

We prove that the $1-d$ 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.

math.SP↗
arXiv2008

Long time Anderson localization for nonlinear random Schroedinger equation

We prove long time Anderson localization for nonlinear random Schroedinger equation in $\ell^2$ by making a Birkoff normal form type transform to creat an energy barrier where there is essentially no mode propagation. One of the new features is that this transform is in a small neighborhood enabling us to treat "rough" data, where there are no moment conditions. The formulation of the present result is inspired by the RAGE theorem.

math-ph↗