# Analysis Seminar

#### Effective Dynamics of a Non-Linear Wave Equation

**Speaker:**
Oana Pocovnicu, Imperial College, London

**Location:**
Warren Weaver Hall 1302

**Date:**
Thursday, December 1, 2011, 11 a.m.

**Synopsis:**

We consider the non-linear wave equation on the real line \(iu_t-|D|u=|u|^2u\). Its resonant dynamics is given by the Szego equation, which is a completely integrable non-dispersive non-linear 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 non-linear wave equation, since this phenomenon was already exhibited for the Szego equation.