250075 VO Nonlinear Schrödinger and Wave equations (2022W)

5.00 ECTS (3.00 SWS), SPL 25 - Mathematik

ON-SITE

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Registration/Deregistration

Note: The time of your registration within the registration period has no effect on the allocation of places (no first come, first served).

Details

Language: English

Examination dates

Wednesday 08.02.2023 Thursday 16.02.2023 Thursday 27.04.2023 Monday 05.06.2023

Lecturers

Classes

Place: MMM - WPI Seminar room 8. floor Fak.Math, OMP1
Time: Tuesday 12.00 - 13.30,
Wednesday 12.00-13.00

times have been adapted to student wishes

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”, 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

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