  We consider the nonlinear wave equation (NLW) iv_t-|D|v=|v|^2v on the real
line. We show that in a certain regime, the resonant dynamics is given by a
completely integrable nonlinear equation called the Szego equation. Moreover,
the Szego equation provides a first order approximation for NLW for a large
time. The proof is based on the renormalization group method of Chen,
Goldenfeld, and Oono. As a corollary, we give an example of solution of NLW
whose high Sobolev norms exhibit relative growth.
  An analogous result of approximation was proved by Gerard and Grellier on the
torus using the theory of Birkhoff normal forms. We improve this result by
finding the second order approximation on the torus with the help of an
averaging method introduced by Temam and Wirosoetisno. We show that the
effective dynamics will no longer be given by the Szego equation.
