Mathematics 748: Topics in Mathematical Physics
2012-2013: Hamiltonian partial differential equations
Instructor: W. Craig
Meeting times: Mon & Thurs 9:00 - 11:00 in HH 410
Office hours: Thurs 11:00 - 12:30 or by appointment, HH418
Syllabus:
1) Introduction: evolution equations and Hamiltonian systems
2) Hamiltonian systems and transformation theory
i) definition and many examples
ii) symplectic forms and symplectic transformations
iii) generating functions and flows
iv) Lagrangian systems and the Legendre transform
v) Hamilton - Jacobi theory and local symplectic invariants
vi) Poisson brackets
3) Existence theory for evolution equations
i) symmetric hyperbolic systems
ii) Cauchy - Kowalevsky and Nirenberg - Nishida theorems
iii) application to the problem of water waves
iv) global existence of NLS in energy space
v) solitons and finite gap solutions of KdV
4) Water waves in Hamiltonian formulation
i) Zakharov's Hamiltonian
ii) Dirichlet - Neumann operator
iii) derivation from first principles of mechanics
iv) Existence theory for the initial value problem
5) Birkhoff normal forms
i) NLS on the circle and the quantum harmonic oscillator
ii) water waves
iii) NLS on Euclidian space
iv) scattering for the defocusing NLS
6) Stability and Nekhoroshev stability
i) finite dimensional results
ii) NLS: neighborhoods of finite dimensional invariant sets
7) KAM theory (optional)
8) Lower bounds on growth of Sobolev norms (optional)