Validity of nonlinear geometric optics with times growing logarithmically

The profiles (a.k.a. amplitudes) which enter in the approximate solutions o f nonlinear geometric optics satisfy equations, sometimes called the slowly varying amplitude equations, which are simpler than the original hyperboli c systems. When the underlying problem is conservative one often finds that the amplitudes are defined for all time and are uniformly bounded. The app roximations of nonlinear geometric optics typically have percentage error w hich tends to zero uniformly on bounded time intervals as the wavelength ep silon tends to zero. Under suitable hypotheses when the amplitude is unifor mly bounded in space and time we show that the percentage error tends to ze ro uniformly on time intervals [0, C\ ln epsilon \] which grow logarithmica lly. The proof relies in an essential way on the fact that one has a correc tor to the leading term of geometric optics.