Nonlinear evolution equations, rescalings, model PDES and their integrability: I

The authors study the problem of wave modulation for a large and quite general class of nonlinear evolution equations. They demonstrate that only a very limited number of 'universal' model equations, on relevant time and space scales, describe the phenomena of interest under all circumstances. Classical among the model equations is of course the nonlinear Schrodinger equation (NLS); however, under certain conditions, modulations occur on shorter time and space scales than those relevant for the NLS. On the other hand, if the NLS becomes linear by cancellation of terms, then appropriate model equations exist on longer time and space scales. The limited number of model equations suggests that they should have wide applicability. The method of analysis consists basically of Fourier decomposition followed by rescalings and appropriate limits. If one adheres to the principle that integrability properties are inherited through such limit procedures, then the model equations should be integrable (in some sense) under a wide range of conditions. Through their investigation of integrability properties, they find this expectation largely confirmed.