Nonlinear Schrödinger equations
MATH-514 / 5 crédits
Enseignant:
Langue: Anglais
Remark: Pas donné en 2024-25
Summary
This course is an introduction to nonlinear Schrödinger equations (NLS) and, more generally, to nonlinear dispersive equations. We will discuss local and global well-posedness, conservation laws, the existence and stability of standing wave solutions, and solutions which blow up in finite time.
Content
Keywords
nonlinear Schrödinger equations; Hamiltonian dynamics; conservation laws; symmetries; standing waves; orbital stability; finite time blow-up
Learning Prerequisites
Required courses
Introduction to partial differential equations
Recommended courses
Equations aux dérivées partielles d'évolution; Analyse fonctionnelle I; Mesure et intégration; Equations différentielles ordinaires
Important concepts to start the course
résultats de base en intégration (convergence dominée, etc.); espaces de Sobolev, de Banach; convergence faible / forte; solutions faibles d'équations elliptiques; arguments de point fixe dans les espaces métriques
Learning Outcomes
By the end of the course, the student must be able to:
- Define the main objects studied in the course
- Prove properties of solutions of NLS, similar to the exercises
- Prove (or sketch the proof of) the main results given in the lectures
- Discuss qualitative properties of NLS solutions
- Compute quantitative estimates useful to study the NLS dynamics
- Apply the methods developed in the course to NLS and related equations
Teaching methods
blackboard lectures + exercise sessions
Assessment methods
oral
Dans les plans d'études
- Semestre: Printemps
- Forme de l'examen: Oral (session d'été)
- Matière examinée: Nonlinear Schrödinger equations
- Cours: 2 Heure(s) hebdo x 14 semaines
- Exercices: 2 Heure(s) hebdo x 14 semaines
- Type: optionnel
- Semestre: Printemps
- Forme de l'examen: Oral (session d'été)
- Matière examinée: Nonlinear Schrödinger equations
- Cours: 2 Heure(s) hebdo x 14 semaines
- Exercices: 2 Heure(s) hebdo x 14 semaines
- Type: optionnel
- Semestre: Printemps
- Forme de l'examen: Oral (session d'été)
- Matière examinée: Nonlinear Schrödinger equations
- Cours: 2 Heure(s) hebdo x 14 semaines
- Exercices: 2 Heure(s) hebdo x 14 semaines
- Type: optionnel
Semaine de référence
Lu | Ma | Me | Je | Ve | |
8-9 | |||||
9-10 | |||||
10-11 | |||||
11-12 | |||||
12-13 | |||||
13-14 | |||||
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21-22 |