Distinguished Professor of Mathematics
Rutgers University
Nonlinear Long Range Scattering and Normal Form Analysis
First I will describe the source and nature of long range dynamics in general.
This fundamental effect is responsible to the change in the asymptotic
behavior of the system at large times.
It is present in Coulomb and Gravitational dynamics, in theories with
massless particles (gauge theories) and in low power nonlinear
dispersive and hyperbolic equations.
Then, I will describe new results and new Normal Form techniques to deal
with the nonlinear Klein-Gordon equation in one dimension, with
quadratic and variable coefficient cubic nonlinearity. This problem
exhibits a striking resonant interaction between the spatial frequencies
of the nonlinear coefficients and the temporal oscillations of the
solutions. We prove global existence and (in L-infinity) scattering as
well as a certain kind of strong smoothness for the solution at
time-like infinity; it is based on several new classes of normal-form
transformations. The analysis also shows the limited smoothness of the
solution, in the presence of the resonances. In particular we observe
the phenomena of growth of some Invariant Sobolev norm of high order.
This seems to be generic for such nonlinear systems.
Thursday February 18th, 4:30pm, Math 507
Partially supported by a Simons Foundation Math + X Investigator Award