In terms of the methods of asymptotic analysis, so far we have studied integral asymptotics associated with Fourier integrals that represent solutions of linear PDEs. Now, suppose we want to study physical problems like the propagation of waves in the ocean, or the propagation of light in optical fibers; the general equations obtained from first principles in these cases are the Euler or Navier–Stokes equations governing fluid motion on a free surface and Maxwell's electromagnetic (optical wave) equations with nonlinear induced polarization terms. These equations are too difficult to handle using linear methods or, in most situations, by direct numerical simulation. Loosely speaking, these physical equations describe “too much”.

Mathematical complications often arise when one has widely separated scales in the problem, e.g., the wavelength of a typical ocean wave is small compared to the ocean's depth and the wavelength of light in a fiber is much smaller than the fiber's length or transmission distance. For example, the typical wavelength of light in an optical fiber is of the order of 10-6 m, whereas the length (distance) of an undersea telecommunications fiber is of the order of 10,000 km or 107 m, i.e., 13 orders of magnitude larger than the wavelength! Therefore, if we were to try solving the original equations numerically – and resolve both the smallest scales as well as keep the largest ones – we would require vast amounts of computer time and memory.