Singular Levy processes and dispersive effects of generalized Schrödinger equations

Abstract

We introduce new models for Schrödinger-type equations, which generalize standard NLS and for which different dispersion occurs depending on the directions. Our purpose is to understand dispersive properties depending on the directions of propagation, in the spirit of waveguide manifolds, but where the diffusion is of different types. We mainly consider the standard Euclidean space and the waveguide case but our arguments extend easily to other types of manifolds (like product spaces). Our approach unifies in a natural way several previous results. Those models are also generalizations of some appearing in seminal works in mathematical physics, such as relativistic strings. In particular, we prove the large data scattering on waveguide manifolds ${\mathbb{R}}^d\times \mathbb{T}$, $d\ge 3$. This result can be regarded as the analogue of \cite{TV2, YYZ2} in our setting and the waveguide analogue investigated in \cite{GSWZ}. A key ingredient of the proof is a Morawetz-type estimate for the setting of this model.

22 pages. Comments are welcome! This is an expanded version with the complete theory developed which incorporates an earlier research note of Y. Sire and Z. Zhao. A few typos are corrected

References in corpus (3)

• Global well-posedness and long-time behavior of the fractional NLS
• Large data global well-posedness and scattering for the focusing cubic nonlinear Schrödinger equation on ${\mathbb{R}}^2\times \mathbb{T}$
• Global well-posedness and scattering of the two dimensional cubic focusing nonlinear Schrödinger system