Nonlinear Schrödinger equations
MATH-514 / 5 credits
Teacher:
Language: English
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
In the programs
- Semester: Spring
- Exam form: Oral (summer session)
- Subject examined: Nonlinear Schrödinger equations
- Lecture: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Type: optional
- Semester: Spring
- Exam form: Oral (summer session)
- Subject examined: Nonlinear Schrödinger equations
- Lecture: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Type: optional
- Semester: Spring
- Exam form: Oral (summer session)
- Subject examined: Nonlinear Schrödinger equations
- Lecture: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Type: optional
Reference week
Mo | Tu | We | Th | Fr | |
8-9 | |||||
9-10 | |||||
10-11 | |||||
11-12 | |||||
12-13 | |||||
13-14 | |||||
14-15 | |||||
15-16 | |||||
16-17 | |||||
17-18 | |||||
18-19 | |||||
19-20 | |||||
20-21 | |||||
21-22 |
Légendes:
Lecture
Exercise, TP
Project, labs, other