Nonlinear Schrodinger-type equations from multiscale reduction of PDEs. I.Systematic derivation

In this article we begin a systematic investigation via multiscale expansio ns of nonlinear evolution PDEs (partial differential equations). In this fi rst article we restrict consideration to a single, autonomous, but otherwis e generic, PDE in 1 + 1 variables (space+time), of first order in time, who se linear part is dispersive, and to solutions dominated by a single plane wave satisfying the linear part of the PDE. The expansion parameter is an, assumedly small, coefficient multiplying this plane wave. The main (indeed, asymptotically exact) effect of the (weak) nonlinearity is then to cause a modulation of the amplitude of the plane wave and of its harmonics, which is generally described, in (appropriately defined) coarse-grained time and space variables, by evolution equations of nonlinear Schrodinger type. A sy stematic analysis of such equations is presented, corresponding to various assumptions on the "resonances" occurring for the first few harmonics. (C) 2000 American Institute of Physics. [S0022-2488(00)03209-6].