250086 VO Nonlinear Schrödinger and Wave Equations (2024W)

6.00 ECTS (4.00 SWS), SPL 25 - Mathematik

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Details

Sprache: Englisch

Prüfungstermine

Lehrende

Termine

Place: MMM - WPI Seminarrraum im 8. Stock Fak.Math, OMP1, 8.135
Time: Wednesday 13.00-14.00
Tuesday 12.30- 14.00

! First lecture = organizational meeting, on wednesday 2nd Oct 13h !

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 NLS and NLW in one course (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 aspects of "Applied Mathematics”, i.e. Modeling, 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.

Methods:
Functional analysis, Semigroup theory, Sobolev embeddings, Strichartz estimates, energy estimates, linear PDE theory,...,

3) Numerics:
Finite Element Methods for NLS,
Time Splitting,
Spectral methods,
Boundary conditions

Art der Leistungskontrolle und erlaubte Hilfsmittel

Oral exam to prove the understanding of important concepts.
Students can put more weight on 2 of the 3 aspects (application, analyis , numerics)

Student's version of the lecture notes should be brought and be used during exam

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/should 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 Schrödinger equations'', Hokkaido Univ. Technical Report, Series in Math. 43 (1996), pp. 80-128.

Zuordnung im Vorlesungsverzeichnis

MAMV; MANV

Letzte Änderung: Mi 19.02.2025 17:06