This article is devoted to a general class of one-dimensional NLS problems with a cubic nonlinearity. The question of obtaining scattering, global in time solutions 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 models under the assumption that the initial data are both small and localized. However, except for the completely integrable case, no such results have been known for small but not necessarily localized initial data. In this article, we introduce a new, nonperturbative method to prove global well-posedness and scattering for $L^2$ initial data which are small and nonlocalized. Our main structural assumption is that our nonlinearity is defocusing. However, we do not assume that our problem has any exact conservation laws. Our method is based on a robust reinterpretation of the idea of Interaction Morawetz estimates, developed almost 20 years ago by the I-team. In terms of scattering, we prove that our global solutions satisfy both global $L^6$ Strichartz estimates and bilinear $L^2$ bounds. This is a Galilean invariant result, which is new even for the classical defocusing cubic NLS.1 There, by scaling, our result also admits a large data counterpart.

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一维三次散焦色散方程的全局解：第一部分

本文致力于讨论一类具有三次非线性的一维 NLS 问题。近年来，为此类问题获得分散的、全局的时间解的问题引起了广泛的关注，并且在初始数据均为假设的情况下，许多模型的全局适定性结果已得到证明。小的和本地化的。然而，除了完全可积的情况外，对于小但不一定是局部的初始数据，还没有这样的结果。在本文中，我们介绍了一种新的非微扰方法来证明全局适定性和散射 $L^2$ 初始数据是小的和非本地化。我们的主要结构假设是我们的非线性是散焦。然而，我们并不假设我们的问题有任何精确的守恒定律。我们的方法基于对 Interaction Morawetz 估计思想的强有力的重新解释，该思想是由 I 团队在大约 20 年前开发的。在散射方面，我们证明我们的全局解决方案满足全局 $L^6$ Strichartz 估计和双线性 $L^2$ 界限。这是伽利略不变结果，即使对于经典的散焦立方 NLS 来说也是新的。1在那里，通过缩放，我们的结果也承认了大数据对应物。