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250075 VO Nonlinear Schrödinger and Wave equations (2021W)

5.00 ECTS (3.00 SWS), SPL 25 - Mathematik
GEMISCHT

An/Abmeldung

Hinweis: Ihr Anmeldezeitpunkt innerhalb der Frist hat keine Auswirkungen auf die Platzvergabe (kein "first come, first served").

Details

max. 25 Teilnehmer*innen
Sprache: Englisch

Lehrende

Termine

Erster Termin: 7. Oktober 2021
Dienstag 11:00-12:10
Donnerstag 15:00 -16:20
Ort: WPI Seminarraum 8.136 Oskar-Morgenstern-Platz 1, 8. Stock

We hope that the covid-situation will allow "presence teaching" in the class room, according to the rules in vigour in october 2021 ("3G" and less dense packing of classroom).


Information

Ziele, Inhalte und Methode der Lehrveranstaltung

Nonlinear Schrödinger equations (NLS : "dispersive") and Nonlinear Wave equations (NLW : "hyperbolic") are fundamental classes of Partial Differential Equations (PDE), with many important applications. To deal with them jointly (in the spirit of e.g. Terry Tao’s book) reveals an interesting mutual crossover of ideas between these 2 different types of PDEs.

In this lecture we deal with two aspects of "Applied Mathematics”, Modeling and Analysis, based on lecture notes that are handed out to students.

1) Modeling: motivation / derivation of NLS :
a) quantum physics, where “one particle” NLS occur as approximate models for the linear N-body Schrödinger equation.
Quantum HydroDynamics.
b) nonlinear optics, where the paraxial approximation of the Helmholtz (wave) equation yields 2+1 dimensional cubic NLS

2) Analysis:
Existence and Uniqueness (“Local/Global WellPosedness) of NLS and NLW
with local and non-local nonlinearities, scattering, finite(-time) Blow-up; asymptotic results e.g. for the (semi-)classical limit of NLS.

3) numerics : we give a Brief Outlook to numerical methods for NLS - a follow up course "Computational Schrödinger equations" will present full Details.

Methods:
functional analysis, semigroup theory, Sobolev embeddings, Strichartz estimates, energy estimates, linear PDE theory, …

Art der Leistungskontrolle und erlaubte Hilfsmittel

Oral exam (presence on the blackboard or distance) where the presentation of exercises enters the grade.

Mindestanforderungen und Beurteilungsmaßstab

The presentation is self-contained based on material
distributed to the students.
Basic knowledge of functional analysis, PDEs and physics is helpful.

Prüfungsstoff

The exam is an opportunity to prove the understanding of basic
concepts, own lecture notes etc can be used during the exam.

Literatur

.) Mauser, N.J. and Stimming, H.P. "Nonlinear Schrödinger equations", lecture notes

.) Sulem, P.L., Sulem, C.: "The Nonlinear Schrödinger Equation, Self-Focusing and Wave Collapse", Applied Math. Sciences 139, Springer N.Y. 1999

.) Tao, Terence:
"Local And Global Analysis of Nonlinear Dispersive And Wave Equations (Cbms Regional Conference Series in Mathematics)", 373 p., American Mathematical Society, 2006

.) Ginibre, J.: ``An Introduction to Nonlinear Schroedinger equations'', Hokkaido Univ. Technical Report, Series in Math. 43 (1996), pp. 80-128.

Zuordnung im Vorlesungsverzeichnis

MAMV; MANV

Letzte Änderung: Mi 29.09.2021 12:29