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250076 VO Nonlinear Schrödinger and Wave Equations (2018S)

7.00 ECTS (4.00 SWS), SPL 25 - Mathematik

Details

Sprache: Englisch

Prüfungstermine

Lehrende

Termine

Tuesday 12h15 - 13h45

Wednesday 12h15 - 13h45

Preliminary meeting: Thursday 1. March 13h15

Room: Seminarraum 8.135 (WPI Seminarroom 8th floor)


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”, i.e. = “Modeling + Analysis + 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, …
Numerical schemes: Finite Difference schemes, spectral methods, time splitting, Absorbing Boundary Layers ("optical potential")

Art der Leistungskontrolle und erlaubte Hilfsmittel

oral exam (on the blackboard)

Mindestanforderungen und Beurteilungsmaßstab

The presentation is self-contained based on material distributed to the students.
Basic knowledge of functional analysis, PDEs and numerical mathematics 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: Di 03.08.2021 00:23