Introduction to Dispersive Equations

Jacek Jendrej

Email: jendrej AT imj-prg.fr

Schedule

Sorbonne University, Jussieu Campus, room 15/16 1.01

January 5 – February 13, 2026

Wednesday 8:50–10:50

Thursday 8:50–10:50 (lecture) and 16:00–18:00 (tutorial)

Exam

Thursday, February 19, 2026, 9:00–12:00

Room 15/16 1.01

Notes are allowed

Electronic devices are prohibited

Course Description

The aim of this course is to introduce fundamental notions in the theory of so-called dispersive partial differential equations. These equations model dispersive waves, that is, waves whose propagation speed depends on the wave number. Dispersion plays a crucial role in the description of many physical phenomena; see for instance Dispersion (optics).

In the first part of the course, we present a general theory of linear dispersive equations. We then turn to nonlinear dispersive waves, focusing in particular on the nonlinear Klein–Gordon equation.

Content

• Linear dispersive equations and their resolution via the Fourier transform. Dispersion relation. Stationary phase lemma. Group velocity. Asymptotic description for large times. Dispersive estimates.
• Nonlinear Klein–Gordon equation. Well-posedness (Cauchy problem). Asymptotic description of solutions for large times in different asymptotic regimes. Nonlinear scattering.
• If time permits: modified scattering in dimension 1. Normal forms.

Prerequisites

Differential calculus; ordinary differential equations; Fourier transform; basic notions of functional analysis and partial differential equations.

It is recommended to have attended the M2 courses “HFE” and “EDP”.

Bibliography

• Muscalu, Camil & Schlag, Wilhelm, Classical and Multilinear Harmonic Analysis, Cambridge University Press, 2013.
• Sulem, Catherine & Sulem, Pierre-Louis, The Nonlinear Schrödinger Equation. Self-Focusing and Wave Collapse, Springer, 1999.
• Tao, Terence, Nonlinear Dispersive Equations: Local and Global Analysis, AMS, 2006.
• Whitham, Gerald B., Linear and Nonlinear Waves, John Wiley & Sons, 1974.

Lecture Notes

Lecture notes (PDF) (updated regularly, usually before each lecture)