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250075 VO Nonlinear Schrödinger and Wave equations (2022W)

5.00 ECTS (3.00 SWS), SPL 25 - Mathematik

VOR-ORT

Moodle

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Details

Sprache: Englisch

Prüfungstermine

Lehrende

Termine

Place: MMM - WPI Seminarrraum im 8. Stock Fak.Math, OMP1
Time: Tuesday 12.00 - 13.30,
Wednesday 12.00-13.00

times have been adapted to student wishes

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 all 3 aspects of "Applied Mathematics”, Modeling and Analysis and Numerics, 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:
Spectral methods, finite difference and relaxation schemes, Absorbing Boundary Conditions,

Methods:
functional analysis, semigroup theory, Sobolev embeddings, Strichartz estimates, energy estimates, linear PDE theory, …
Finite Difference schemes, spectral methods, time splitting, Boundary Conditions

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 and PDEs required,
Basic knowledge of numerics 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: Do 15.06.2023 09:07