MATH-514 / 5 credits

Teacher: Genoud François Samer

Language: English


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

or

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

 MoTuWeThFr
8-9   MAA330 
9-10    
10-11   MAA330 
11-12    
12-13     
13-14     
14-15     
15-16     
16-17     
17-18     
18-19     
19-20     
20-21     
21-22     

Thursday, 8h - 10h: Lecture MAA330

Thursday, 10h - 12h: Exercise, TP MAA330