SIAM Journal on Mathematical Analysis, Volume 52, Issue 5, Page 4638-4704, January 2020.
We construct global solutions on a full measure set with respect to the Gibbs measure for the one-dimensional cubic fractional nonlinear Schrödinger (FNLS) equation with weak dispersion $(-\partial_x^2)^{\alpha/2}$, $\alpha<2$, by quite different methods, depending on the value of $\alpha$. We show that if $\alpha>\frac{6}{5}$, the sequence of smooth solutions for FNLS with truncated initial data converges almost surely, and the obtained limit has recurrence properties as the time goes to infinity. The analysis requires us to go beyond the available deterministic theory of the equation. When $1<\alpha\leq \frac{6}{5}$, we are not able so far to get the recurrence properties but we succeeded in using a method of Bourgain and Bulut to prove the convergence of the solutions of the FNLS equation with both regularized data and nonlinearity. Finally, if $\frac{7}{8}<\alpha\leq 1$ we can construct global solutions in a much weaker sense by a classical compactness argument.

中文翻译：

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分数NLS的Gibbs测度动力学

SIAM数学分析杂志，第52卷，第5期，第4638-4704页，2020年1月。
对于具有弱色散$（-\ partial_x ^ 2）^ {\ alpha / 2} $，$ \的一维立方分数阶非线性Schrödinger（FNLS）方程，我们针对Gibbs测度建立了一个完整测度集的全局解。 alpha <2 $，通过完全不同的方法，取决于$ \ alpha $的值。我们证明，如果$ \ alpha> \ frac {6} {5} $，则具有确定的初始数据的FNLS光滑解的序列几乎可以收敛，并且随着时间的增长，获得的极限具有递归性质。该分析要求我们超越方程式的可用确定性理论。当$ 1 <\ alpha \ leq \ frac {6} {5} $时，我们到目前为止还无法获得递归性质，但是我们成功地使用了Bourgain和Bulut方法来证明FNLS方程解的收敛性具有正则化数据和非线性。