Universität Wien
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250324 VO Selected topics in Partial Differential Equations (2006S)

Selected topics in Partial Differential Equations: "Nonlinear Schrödinger Equations"

6.00 ECTS (3.00 SWS), SPL 25 - Mathematik

Mittwoch, 14:30 - 15:45, Donnerstag, 12:15 - 13:45, C 714 UZA 4.
Erstmals am Mittwoch, 15.3.2006

Details

Language: German

Lecturers

Classes

Currently no class schedule is known.

Information

Aims, contents and method of the course

Analysis: Existence and Uniqueness of NLS with local and non-local nonlinearity, Scattering, Blow-up, Asymptotic results for the semi-classical limit. Modeling: Motivation / Derivation of NLS models in Quantum dynamics
incl. Time Dependent Density Functional Theory and Bose Einstein
Condensates, NLS models in Nonlinear Optics,
Numerics: methods: Time Splitting Spectral methods and Relaxation schemes, Validation of Simulation results

Assessment and permitted materials

Minimum requirements and assessment criteria

Introduction to a very active field of PDE research and to some of the modern methods. Both masters thesis and PhD thesis in the field are supervised by the team of the Wissenschaftskolleg ¿Differential equations¿.

Requirements:

The presentation is selfcontained based on ma-terial distributed to the
students. Good know- ledge of functional analysis + basic knowledge of
PDEs and numerical mathematics is helpful.

Examination topics

Functional Analysis, Semigroup theory, Sobolev embeddings, Strichartz estimates, linear PDE theory, Numerical schemes: Finite difference schemes, Spectral methods, Time splitting etc.

Reading list

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

Cazenave, Th.:``Introduction to Nonlinear Schroedinger equations'', Textos de Metodos Matematicos 26, Rio de Janeiro, Instituto de Matematica - UFRJ, 1996.

Bourgain, J.: ``The nonlinear Schrödinger equation'', Colloqium Publications Vol. 46, AMS, Providence R.I. 1999

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

Last modified: Tu 03.08.2021 00:23