First and Second Order Approximations for a Nonlinear Wave Equation

In this paper we consider the following nonlinear half-wave equation:

i∂tV−|D|V=|V|2V,i∂tV−|D|V=|V|2V,                                                        (NLW)

where D=−i∂xD=−i∂x, both on RR and TT. On RR, we prove that, if the initial condition is of order O(ε)O(ε)and supported on positive frequencies only, then the corresponding solution can be approximated by the solution of the Szegő equation. The Szegő equation i∂tu=Π+(|u|2u)i∂tu=Π+(|u|2u), where Π+Π+ is the Szegő projector onto non-negative frequencies, is a completely integrable system that gives an accurate description of solutions of (NLW). The approximation holds for a long time 0≤t≤Cε−2[log(1/εδ)]1−2α0≤t≤Cε−2[log⁡(1/εδ)]1−2α, 0≤α≤1/20≤α≤1/2. The proof is based on the renormalization group method. As a corollary, we give an example of a solution of (NLW) on RR whose high Sobolev norms grow over time, relative to the norm of the initial condition. An analogous result of approximation was proved by Gérard and Grellier (Anal PDEs, arXiv:1110.5719v1) on TT using Birkhoff normal forms. We improve their result by finding a second order approximation with the help of an averaging method. We show, in particular, that the effective dynamics is no longer given by the Szegő equation.