The 2001 Chern Lectures will be delivered by Peter Lax on February 9-March 12, 2001.
Department of Mathematics, University of California, Berkeley, presents
The 2001 Chern Lectures
Professor of Mathematics
Courant Institute, NYU
Review of Dispersive Equations
Numerical evidence of oscillations and weak convergence. The culprit: too many conservation laws.
Introductory remarks about finite dimensional Hamiltonian systems. The KdV equation as a Hamiltonian system; two Hamiltonian structures. The conservation laws. Solitary waves. The periodic case.
Scattering and Inverse Scattering
Transmission and reflection coefficients. Eigenvalues and norming constants. Uniqueness theorem. The Gelfand-Levitan-Marchenko equation. Reflectionless potentials. Trace formula.
Solution of KdV via Inverse Scattering
Linear equations for the reflection coefficients and the norming constants. The Fredholm determinant in the reflectionless case.
The Zero Dispersion Limit
Asymptotic analysis of direct scattering problem. The minimum problem. The integral equation. Solution of the integral equation in the smooth regime. Solution of the integral equation in the presence of shocks.