For initial value problems associated to linear partial differential equations, we know very well that the Fourier transform provides an avenue towards a complete description of solutions. Long time behavior, regularization of singularities in initial data, and other singular limits, are reduced to analysis of integrals. For initial value problems associated to pdes that are not linear, much less detail is afforded by Fourier based methods, and frequently we are left with results which, in essence, explain that solutions often behave like solutions to linear pdes. The integrable nonlinear Schroedinger equation (a nonlinear partial differential equation) is an example which may be integrated via scattering and inverse scattering theory. I will describe some asymptotic analyses of this equation which exhibit how solutions differ from solutions to linear equations.