MENU

You are here

Home

Weak turbulence and wave kinetic equation

Lecturer: 
Ricardo Grande Izquierdo
Course Type: 
PhD Course
Academic Year: 
2025-2026
Period: 
April - May
Duration: 
20 h
Description: 
Description
 
The aim of this course is to present recent mathematical advances in the theory of non-equilibrium statistical physics for nonlinear waves, commonly known as wave turbulence. In particular, we will focus on a series of recent works by Deng and Hani, which provide the first rigorous derivations of the Wave Kinetic Equation from nonlinear dispersive PDEs such as the cubic Schrödinger equation, and establish a robust framework for understanding statistical behavior in large systems of interacting waves.
 
Course contents:
I. Overview of statistical mechanics for nonlinear waves. Formal derivation of Wave Kinetic Equation from dispersive PDEs.
II. Scaling laws, resonances, and quasi-resonances. 
II. Probability toolbox: introduction to Gaussian Hilbert Spaces and hypercontractivity estimates.
III. Picard iteration, remainder, and reduction to a combinatorics problem.
IV. Feynman diagrams, molecules, couples and counting estimates.
V. Oscillatory sums and convergence to the Wave Kinetic Equation.
 
References:
 
  1. Y. Deng and Z. Hani, On the Derivation of the Wave Kinetic equation for NLS, Forum of Math. Pi 9 (2021), e6, 1-37.
  2. Y. Deng and Z. Hani, Full Derivation of the Wave Kinetic equation, Invent. Math. (233):543-724 (2023).
  3. Y. Deng and Z. Hani, Propagation of chaos and higher order statistics in wave kinetic theory. J. Eur. Math. Soc. 28 (2026), no. 2, pp. 673–733.
  4. Svante Janson, Gaussian Hilbert Spaces, Cambridge Tracts in Mathematics Vol. 129, Cambridge University Press, 1997.
  5. Sergey Nazarenko, Wave Turbulence, Lecture Notes in Physics, Springer-Verlag Berlin.

Schedule

  • 28 april, 11:00-13:00, Room 134
  • 29 april, 11:00-13:00, Room 138
  • 30 april, 14:00-16:00, Room 134
  • 4 may, 14:00-16:00, Room 136
  • 6 may, 14:00-16:00, Room 136
  • 27 may, 14:00-16:00, Room 133
  • 29 may, 11:00-13:00, Room 133
  • 8 june, 11:00-13:00, Room 133
  • 10 june, 11:00-13:00, Room 133
  • 12 june, 11:00-13:00, Room 133

 

Research Group: 
Mathematical Analysis, Modelling, and Applications
Next Lectures: 
Wednesday, May 27, 2026 - 14:00 to 16:00
Friday, May 29, 2026 - 11:00 to 13:00
Monday, June 8, 2026 - 11:00 to 13:00
Wednesday, June 10, 2026 - 11:00 to 13:00
Friday, June 12, 2026 - 11:00 to 13:00

Sign in