Equations aux dérivées partielles non-linéaires

Enseignant : Stephan Fauve
Chargé de TD : Guillaume Michel
ECTS : 3
Langue d’enseignement : English


1. Partial differential equations (PDEs) in physics

  • Examples
  • Why do we get PDEs of similar forms in different domains of physics ?
  • What are the mechanisms that generate nonlinearities ?

2. The nonlinear advection equation

  • Kinematic waves
  • The method of characteristics
  • Wave breaking
  • Shock waves
  • Examples from traffic flow, compressible gas dynamics, shallow water waves

3. Self-similar solutions of partial differential equations

  • Dimensional analysis and relation with group theory
  • Nonlinear diffusion and anomalous exponents
  • Advection-diffusion and some problems related to mixing of a passive scalar in a turbulent flow

4. The nonlinear Schrödinger equation (NLS)

  • Weakly nonlinear waves in a dispersive medium
  • The method of multiple scales
  • Side-band instability of a monochromatic wave
  • Solitary waves ; continuous family of solutions from dilatation and Galilean invariances
  • From NLS to the Korteweg-de Vries equation (KdV)

5. Some exact solutions of nonlinear PDEs

  • Burgers equation and the Hopf-Cole transformation
  • Hirota’s method to solve KdV equation
  • Collisions of solitons
  • Other methods to find exact solutions in integrable systems

6. Ginzburg-Landau equations

  • Examples from magnetism, supraconductivity and commensurate-incommensurate transitions
  • Long wavelength instabilities and defects
  • Dynamics of topological defects ; describing a system as a field or particles.