Scattering Theory of Linear and Nonlinear Waves: A Unified New Paradigm

Avy Soffer (Rutgers U)

Jun 18. 2024, 10:30 — 11:15

Abstract:

I will present a new approach to Mathematical Scattering of multichannel Dispersive and Hyperbolic Equations.

In this approach we identify the targe time behavior of such equations, both linear and non-linear, for general (large) data, and interactions terms which can be space-time dependent.

In particular, for the NLS equations with spherically symmetric data and Interaction terms, we prove that all global solutions in H^1 converge to a smooth and localized function plus a free wave, in 5 or more dimensions.

Similar result holds for  3,4 dimensions, though the argument proving localization is different.

We also show similar results in any dimension for localized type of interactions, provided they decay fast enough.

We show breakdown of the standard Asymptotic Completeness conjecture if the interaction is time dependent and decays like r^{-2} at infinity.

Many of these results extend to the non-radial case, for NLS, NLKG and Bi-harmonic NLS in three or more dimensions.

Furthermore, we prove Local-Decay Estimates for Time dependent potentials in 5 or more dimensions.

Finally, we apply this approach to N-body scattering, and prove AC for three quasi-particle scattering.

This is based on joint works with Baoping Liu and Xiaoxu Wu.

Further Information
Venue:
ESI Boltzmann Lecture Hall
Associated Event:
Nonlinear Waves and Relativity (Thematic Programme)
Organizer(s):
Roland Donninger (U of Vienna)
David Fajman (U of Vienna)
Birgit Schörkhuber (U of Innsbruck)