Nonlinear Schrödinger equations

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

Resources

Moodle Link

• https://go.epfl.ch/MATH-514