# 250076 VO Nonlinear Schrödinger and Wave Equations (2018S)

## Labels

## Details

Language: English

### Examination dates

### Lecturers

### Classes

Tuesday 12h15 - 13h45

Wednesday 12h15 - 13h45Preliminary meeting: Thursday 1. March 13h15Room: Seminarraum 8.135 (WPI Seminarroom 8th floor)## Information

### Aims, contents and method of the course

### Assessment and permitted materials

oral exam (on the blackboard)

### 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.

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.

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

"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 03.08.2021 00:23*

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 NLS2) Analysis:

Existence and Uniqueness (“Local/global WellPosedness) of NLS and NLW with local and non-local nonlinearity, 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,…