Math
743 (Craig 2007-8 semester
I)
Hamiltonian Partial Differential
Equations
Syllabus:
1. General theory
a. examples
nonlinear wave equations
nonlinear Schrödinger
equation
generalized KdV equation
Boussinesq systems
b.
linearized wave equation on the torus Td
c. the
problem of surface water waves
2. Transformation theory
a.
canonical transformations in finite dimensions
definition
examples
Hamiltonian flows are canonical transformations
generating
functions
geometrical theory
b.
transformation theory in Hilbert space
representation of a symplectic
structure
transformation of symplectic
structures
examples - formal derivation
of the
Boussinesq system and KdV equation from
the problem of water waves
3. Birkhoff normal form
a. normal forms
in finite dimensions
Lie transformations
integrability of nonresonant
normal forms Hamiltonians
resonant normal forms: case
of one resonance relation
example: Gross - Pitaevski equation
b. Birkhoff
normal forms for PDE
NLS on the torus T1
general theory of Bambusi
the problem of water waves
4. Nekhoroshev theory
a. finite
dimensional cases (Lochak's approach, convex Hamiltonians)
b. Nekhoroshev
stability for the NLS on T1
5. KAM theory
a. general exposition
b. detailed study in a particular case (to be
seleced)
c.
Lagrangian tori
6. Wave turbulence
a. normal forms
transformations with finite Hs norm
b. infrared
cuttoff
c. Wigner
transforms
d. future
directions