论文标题
1D立方分散方程的长期解决方案，第二部分：聚焦案例
Long time solutions for 1D cubic dispersive equations, Part II: the focusing case
论文作者
论文摘要
本文涉及一维分散流，并在真实线上具有立方非线性。在最近的一项工作中，作者对这种流动提出了广泛的猜想，并断言在散落的情况下，少量初始数据产生了全球，散射解决方案。然后，在Schrödinger分散关系的情况下证明了这一猜想。在散射方面，我们的全球解决方案被证明满足了全球$ l^6 $ strichartz估计和双线性$ l^2 $界限。值得注意的是，在初始数据上没有定位假设。 在本文中，我们考虑了聚焦方案。可能有一个小孤子，所以人们总体上不希望拥有全球散射解决方案。取而代之的是，我们寻求长期的解决方案，并询问解决方案存在的时间尺度是什么并满足良好的分散估计。我们的主要结果也适用于Schrödinger分散关系，他断言，对于$ε$的初始数据，该解决方案存在于时间尺度$ε^{ - 8} $上，并且满足所需的$ l^6 $ strichartz估计值和Birinear $ l^2 $ lirinear $ l^2 $ bounds time time-time-scale $ scale $ε据我们所知，这是达到这样一个门槛的第一个结果。
This article is concerned with one dimensional dispersive flows with cubic nonlinearities on the real line. In a very recent work, the authors have introduced a broad conjecture for such flows, asserting that in the defocusing case, small initial data yields global, scattering solutions. Then this conjecture was proved in the case of a Schrödinger dispersion relation. In terms of scattering, our global solutions were proved to satisfy both global $L^6$ Strichartz estimates and bilinear $L^2$ bounds. Notably, no localization assumption is made on the initial data. In this article we consider the focusing scenario. There potentially one may have small solitons, so one cannot hope to have global scattering solutions in general. Instead, we look for long time solutions, and ask what is the time-scale on which the solutions exist and satisfy good dispersive estimates. Our main result, which also applies in the case of the Schrödinger dispersion relation, asserts that for initial data of size $ε$, the solutions exist on the time-scale $ε^{-8}$, and satisfy the desired $L^6$ Strichartz estimates and bilinear $L^2$ bounds on the time-scale $ε^{-6}$. To the best of our knowledge, this is the first result to reach such a threshold.
