Validity of nonlinear geometric optics with times growing logarithmically (Q2701585)

The theory of nonlinear geometrical optics yields a family of approximate solutions to the hyperbolic system of partial differential equations; the percentage error in the approximation tends to zero uniformly on bounded time intervals as the wavelength \(\varepsilon\to 0\). It is shown in this paper that under certain conditions when the amplitude is uniformly bounded in space and time, the percentage error tends to zero uniformly on time intervals, which grow logarithmically. An illustration is given at the end that includes all the mathematical essentials needed in the more complicated models; it is shown that on time intervals growing logarithmically, the approximate solution differs from the exact solution by \(o(\sqrt\varepsilon)\).