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This article is devoted to a general class of one dimensional NLS problemswith a cubic nonlinearity. The question of obtaining scattering, global in timesolutions for such problems has attracted a lot of attention in recent years,and many global well-posedness results have been proved for a number of modelsunder the assumption that the initial data is both \emph{small} and\emph{localized}. However, except for the completely integrable case, no suchresults have been known for small but non-localized initial data.  In this article we introduce a new, nonperturbative method, to prove globalwell-posedness and scattering for $L^2$ initial data which is \emph{small} but\emph{non-localized}. Our main structural assumption is that our nonlinearityis \emph{defocusing}. However, we do not assume that our problem has any exactconservation laws. Our method is based on a robust reinterpretation of the ideaof interaction Morawetz estimates, developed almost 20 years ago by the I-team.  In terms of scattering, we prove that our global solutions satisfy bothglobal $L^6$ Strichartz estimates and bilinear $L^2$ bounds. This is a Galileaninvariant result, which is new even for the classical defocusing cubic NLS.There, by scaling our result also admits a large data counterpart.

Global solutions for 1D cubic defocusing dispersive equations: Part I