In this article we begin a systematic investigation via multiscale expansions of nonlinear evolution PDEs (partial differential equations). In this first article we restrict consideration to a single, autonomous, but otherwise generic, PDE in 1+1 variables (space+time), of first order in time, whose 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 Schrödinger type. A systematic analysis of such equations is presented, corresponding to various assumptions on the “resonances” occurring for the first few harmonics.

A technique to perform a convenient Change of (independent) variables in a PDE is reported, and it is used to generate C-integrable nonlinear PDEs, i.e., nonlinear PDEs solvable by an appropriate Change of variables. Several examples of such PDEs are exhibited.

A technique to generate C-integrable nonlinear partial differentiation equations (PDEs) (i.e., nonlinear PDEs linearizable by an appropriate Change of variables) is reported, and several examples of such PDEs are exhibited.