Non-linear partial differential equations - Département de Physique de l'Ecole Normale supérieure

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Non-linear partial differential equations

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Next student seminar :
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November 14 - 18, 2022 :
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Contact us - Student support and Graduate School office :
Tél : 01 44 32 35 60
enseignement@phys.ens.fr

Faculty: Stephan Fauve
Tutor: Guillaume Michel
ECTS credits: 3
Language of instruction: English

Description

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.

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Next student seminar :
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Here you can find information about your internships:
Experimental Internship - Undergraduate program
Master ICFP first year Internship

News : ICFP Research seminars
November 14 - 18, 2022 :
All information about the program

Contact us - Student support and Graduate School office :
Tél : 01 44 32 35 60
enseignement@phys.ens.fr