250073 VO Nonlinear Schrödinger and Wave equations (2019W)
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An/Abmeldung
Hinweis: Ihr Anmeldezeitpunkt innerhalb der Frist hat keine Auswirkungen auf die Platzvergabe (kein "first come, first served").
Details
Sprache: Englisch
Prüfungstermine
Lehrende
Termine
Wednesday and Thursday, 03.10.2019 - 30.01.2020 12:30-14:00
WPI SeminarRaum 8.135 (Oskar-Morgenstern-Platz 1)
Beginning time may be shifted by request of the students .
Information
Ziele, Inhalte und Methode der Lehrveranstaltung
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.
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.
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
"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
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")Preliminary meeting: Thursday 3. Oct 12h30
Zeiten können noch auf Wunsch der Studierenden verschoben werden !