* Stephan Fauve (ENS Paris)*

This course aims to provide simple methods to handle so-called complex
phenomena described by nonlinear partial differential
equations. Systems governed by nonlinear equations display multiple
solutions with different symmetries. We study the bifurcations i.e.
the transitions between these solutions when a parameter of the system
is varied. We show that in the vicinity of these bifurcations, the
system is governed by universal equations, normal forms, that mostly
depend on the broken symmetries at the transition. We emphasize the
analogy with phase transitions, but also point out differences such
that limit cycles or chaotic behaviors that do not occur at
equilibrium. The following problems are considered:

1. Nonlinear
waves in dispersive media: a universal equation that describes the
motion of a wave packet, the nonlinear Schrödinger
equation. Instability of a quasi-monochromatic wave, generation of
solitons. Solitons as particles.

2. Pattern-forming instabilities in
hydrodynamics. Amplitude equations from symmetry arguments.

3. Analogy with the mean field description of phase transitions:
superfluidity, superconductivity, magnetic domains,
commensurable-incommensurable transitions.

4. Broken symmetries and
neutral modes. Secondary instabilities described by phase
dynamics. Topological defects.

5. Subcritical bifurcations and
metastable states. Localized structures. Analogy with the liquid-vapor
transition. Nucleation and Maxwell construction. Non potential
effects.

** Prerequisite:**

Overall, a basic Master 1 level in
applied maths, physics or mechanics is required

**Bibliography:**

1. G. B. Witham, Linear and nonlinear waves, Wiley (New York, 1974)

2. A. C. Newell, Solitons in mathematics and physics, SIAM (1985)

3. Hydrodynamics and nonlinear instabilities, edited by C. Godrèche and P. Manneville, Cambridge University Press (1998)

**Timing:**
The Course is offered in the second part (december-february) of the M2 year.

It consists of 8 Lectures on Wednesdays from 1.45pm to 5.45pm at Sorbonne University, **room to be announced**

First lecture on **Wednesday 4th December**

**ECTS Credits:**3

**Hours:** 30 hours.