MATH-514 / 5 credits

Teacher:

Language: English

Remark: pas donné en 2022-23

## 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.

## Keywords

nonlinear Schrödinger equations; Hamiltonian dynamics; conservation laws; symmetries; standing waves; orbital stability; finite time blow-up

## 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
• 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
• Prove (or sketch the proof of) the main results given in the lectures

## Teaching methods

blackboard lectures + exercise sessions

oral

## In the programs

• Semester: Fall
• Exam form: Oral (winter session)
• Subject examined: Nonlinear Schrödinger equations
• Lecture: 2 Hour(s) per week x 14 weeks
• Exercises: 2 Hour(s) per week x 14 weeks
• Semester: Fall
• Exam form: Oral (winter session)
• Subject examined: Nonlinear Schrödinger equations
• Lecture: 2 Hour(s) per week x 14 weeks
• Exercises: 2 Hour(s) per week x 14 weeks
• Semester: Fall
• Exam form: Oral (winter session)
• Subject examined: Nonlinear Schrödinger equations
• Lecture: 2 Hour(s) per week x 14 weeks
• Exercises: 2 Hour(s) per week x 14 weeks
• Semester: Fall
• Exam form: Oral (winter session)
• Subject examined: Nonlinear Schrödinger equations
• Lecture: 2 Hour(s) per week x 14 weeks
• Exercises: 2 Hour(s) per week x 14 weeks

## 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