Universität Wien FIND

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250073 VO Nonlinear Schrödinger and Wave equations (2020W)

7.00 ECTS (4.00 SWS), SPL 25 - Mathematik

Registration/Deregistration

Details

Language: English

Examination dates

Lecturers

Classes (iCal) - next class is marked with N

This course on "Schrödinger and wave equations" for master and PhD students (in MINT - mathematics and physics) is planned as "hybrid" = "half presence teaching in the classroom" plus "half distance teaching with zoom + whiteboard".

Depending on the situation with corona and on the number of students we might also switch to classroom teaching only or distance teaching only.
The classroom lectures will be available on moodle as video, too.

Tuesday 06.10. 13:15 - 14:45 Hörsaal 3 Oskar-Morgenstern-Platz 1 Erdgeschoß
Wednesday 07.10. 11:30 - 13:00 Hörsaal 12 Oskar-Morgenstern-Platz 1 2.Stock
Tuesday 13.10. 13:15 - 14:45 Hörsaal 3 Oskar-Morgenstern-Platz 1 Erdgeschoß
Wednesday 14.10. 11:30 - 13:00 Hörsaal 12 Oskar-Morgenstern-Platz 1 2.Stock
Tuesday 20.10. 13:15 - 14:45 Hörsaal 3 Oskar-Morgenstern-Platz 1 Erdgeschoß
Wednesday 21.10. 11:30 - 13:00 Hörsaal 12 Oskar-Morgenstern-Platz 1 2.Stock
Tuesday 27.10. 13:15 - 14:45 Hörsaal 3 Oskar-Morgenstern-Platz 1 Erdgeschoß
Wednesday 28.10. 11:30 - 13:00 Hörsaal 12 Oskar-Morgenstern-Platz 1 2.Stock
Tuesday 03.11. 13:15 - 14:45 Hörsaal 3 Oskar-Morgenstern-Platz 1 Erdgeschoß
Wednesday 04.11. 11:30 - 13:00 Hörsaal 12 Oskar-Morgenstern-Platz 1 2.Stock
Tuesday 10.11. 13:15 - 14:45 Hörsaal 3 Oskar-Morgenstern-Platz 1 Erdgeschoß
Wednesday 11.11. 11:30 - 13:00 Hörsaal 12 Oskar-Morgenstern-Platz 1 2.Stock
Tuesday 17.11. 13:15 - 14:45 Hörsaal 3 Oskar-Morgenstern-Platz 1 Erdgeschoß
Wednesday 18.11. 11:30 - 13:00 Hörsaal 12 Oskar-Morgenstern-Platz 1 2.Stock
Tuesday 24.11. 13:15 - 14:45 Hörsaal 3 Oskar-Morgenstern-Platz 1 Erdgeschoß
Wednesday 25.11. 11:30 - 13:00 Hörsaal 12 Oskar-Morgenstern-Platz 1 2.Stock
Tuesday 01.12. 13:15 - 14:45 Hörsaal 3 Oskar-Morgenstern-Platz 1 Erdgeschoß
Wednesday 02.12. 11:30 - 13:00 Hörsaal 12 Oskar-Morgenstern-Platz 1 2.Stock
Wednesday 09.12. 11:30 - 13:00 Hörsaal 12 Oskar-Morgenstern-Platz 1 2.Stock
Tuesday 15.12. 13:15 - 14:45 Hörsaal 3 Oskar-Morgenstern-Platz 1 Erdgeschoß
Wednesday 16.12. 11:30 - 13:00 Hörsaal 12 Oskar-Morgenstern-Platz 1 2.Stock
Tuesday 12.01. 13:15 - 14:45 Hörsaal 3 Oskar-Morgenstern-Platz 1 Erdgeschoß
Wednesday 13.01. 11:30 - 13:00 Hörsaal 12 Oskar-Morgenstern-Platz 1 2.Stock
Tuesday 19.01. 13:15 - 14:45 Hörsaal 3 Oskar-Morgenstern-Platz 1 Erdgeschoß
Wednesday 20.01. 11:30 - 13:00 Hörsaal 12 Oskar-Morgenstern-Platz 1 2.Stock
Tuesday 26.01. 13:15 - 14:45 Hörsaal 3 Oskar-Morgenstern-Platz 1 Erdgeschoß
Wednesday 27.01. 11:30 - 13:00 Hörsaal 12 Oskar-Morgenstern-Platz 1 2.Stock

Information

Aims, contents and method of the course

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, Optimal Control theory...
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")
Optimal Control theory

Assessment and permitted materials

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

Minimum requirements and assessment criteria

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

Examination topics

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

Reading list

.) 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.

Association in the course directory

MAMV, MANV

Last modified: Tu 02.02.2021 18:09