Nonlinear systems related to an arbitrary space–time dependence of the spectral transform

A general algebraic analytic scheme for the spectral transform of solutions of nonlinear evolution equations is proposed. This allows one to give the general nonlinear evolution corresponding to an arbitrary time and space dependence of the spectral transform (in general nonlinear and with nonanalytic dispersion relations). The main theorem is that the compatibility conditions always give a true nonlinear evolution because it can always be written as an identity between polynomials in the spectral variable k. This general result is then used to obtain first a method to generate a new class of solutions to the nonlinear Schrödinger equation, and second to construct the spectral transform theory for solving initial-boundary value problems for resonant wave-coupling processes (like self-induced transparency in two-level media, or stimulated Brillouin scattering of plasma waves, or else stimulated Raman scattering in nonlinear optics, etc.).