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250086 VO Nonlinear Schrödinger and Wave Equations (2024W)

6.00 ECTS (4.00 SWS), SPL 25 - Mathematik

Moodle

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

Lecturers

Classes

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

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

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 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

Assessment and permitted materials

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

Minimum requirements and assessment criteria

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

Examination topics

The exam is an opportunity to prove the understanding of basic concepts, own lecture notes etc can/should 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 Schrödinger equations'', Hokkaido Univ. Technical Report, Series in Math. 43 (1996), pp. 80-128.

Association in the course directory

MAMV; MANV

Last modified: We 13.11.2024 16:46