The global small-data scattering conjecture for one-dimensional cubic defocusing dispersive flows

One-dimensional dispersive flows with cubic defocusing nonlinearities and small initial data are considered. Global small-data scattering conjecture. Such flows should have global-in-time scattering solutions. This conjecture concerns the existence and scattering of small-data solutions for a broad class of one-dimensional cubic dispersive equations; the paper proves it for a much larger class of semilinear problems with general dispersion relations, while the full general formulation is not established here.