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 (