References are to notes by Palais
unless otherwise indicated. Relevant Homework problems are listed.
January 14: Examples of equations Problem 1.1; Problem 1.4; Problem 1.5
January 16: Fundamental solutions; Solitons Problem 1.2, 1.3,1.6
January 21: No lecture
January 23: Lecture by M. Vishik
January 28: The equation u t = F(u) u x and the characteristics for this equation.
January 30: Rarefaction waves and shock formation (1.5 and chapters from )
See problem 1.7 and the exercises from 1.5 of notes.
February 4: Split-stepping and some unsolved problems (1.5)
Feb ruary 6: Limits of Lattice Models (Appendix B) Problem 2.1
February 11: Multi-soliton formulas; constants of the motion.
Feb ruary 13: More on constants of the motion; Hamiltonian formalism
February 18: Poisson brackets and concerved quantities
Febrary 20: Important examples from field theory
February 25: Vector fields, commuting flows and Lie brackets (Lecture by Dan Freed)
February 27: Abstract symplectic structures and the most important formulas (Lecture by Dan Freed)
March 4: KdV, Sine Gordon and non-linear Schroedinger as Hamiltonian systems
March 6: Finite-dimensional
examples of flows preserving
March 26: The behavior of the flows on the scattering data (the material in the Palais' notes is too bried; there is a better explanation in Solitons: an introduction" by P.G. Drazin and R.S. Johnson, Cambridge texts in applied mathematics)
March 28 No Lecture
April 1 An introdcution to inverse scattering theory (there is not much in the notes; this lecture was taken from section 3.3 and 3.4 of "Soliton: and introduction", reference given above)
April 3: The inverse scattering
theory for pure solitons (