Global solutions for 1D cubic defocusing dispersive equations, Part IV: general dispersion relations

Abstract

A broad conjecture, formulated by the authors in earlier work, reads as follows: "Cubic defocusing dispersive one dimensional flows with small initial data have global dispersive solutions". Notably, here smallness is only assumed in $H^s$ Sobolev spaces, without any localization assumption. The conjecture was initially proved by the authors first for a class of semilinear Schr\"odinger type models, and then for quasilinear Schr\"odinger flows. In this work we take the next natural step, and prove the above conjecture for a much larger class of one dimensional semilinear dispersive problems with a cubic nonlinearity, where the dispersion relation is no longer of Schr\"odinger type. This result is the first of its kind, for any 1D cubic problem not of Schr\"odinger type. Furthermore, it only requires initial data smallness at critical regularity, a threshold that has never been reached before for any 1D cubic dispersive flow. In terms of dispersive decay, we prove that our global in time solutions satisfy both global $L^6_{t,x}$ Strichartz estimates and bilinear $L^2_{t,x}$ bounds.