Effective dynamics of a nonlinear wave equation
Effective dynamics of a nonlinear wave equation

Oana Pocovnicu, Imperial College, London
Fine Hall 314
We consider the nonlinear wave equation on the real line $i u_tDu=u^2u$. Its resonant dynamics is given by the Szego equation, which is a completely integrable nondispersive nonlinear equation. We show that the solution of the wave equation can be approximated by that of the resonant dynamics for a long time. The proof uses the renormalization group method introduced by Chen, Goldenfeld, and Oono in the context of theoretical physics. As a consequence, we obtain growth of high Sobolev norms of certain solutions of the nonlinear wave equation, since this phenomenon was already exhibited for the Szego equation.