Gibbs measure dynamics for the fractional NLS

Abstract

We construct global solutions on a full measure set with respect to the Gibbs measure for the one dimensional cubic fractional nonlinear Schrödinger equation (FNLS) 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 to go beyond the available deterministic theory of the equation. When $1<\alpha \le \frac{6}{5}$, we are not able so far to get the recurrence properties but we succeeded to use a method of Bourgain-Bulut to prove the convergence of the solutions of the FNLS equation with regularized both data and nonlinearity. Finally, if $\frac{7}{8}<\alpha \le 1$ we can construct global solutions in a much weaker sense by a classical compactness argument.

Correction of minor errors in the previous version

References in corpus (7)

• Almost sure well-posedness of the cubic nonlinear Schrödinger equation below L^2(T)
• Almost sure global well posedness for the radial nonlinear Schrodinger equation on the unit ball I: the 2D case
• Invariant Gibbs Measures and a.s. Global Well-Posedness for Coupled KdV Systems
• Invariant Gibbs measures and global strong solutions for nonlinear Schrödinger equations in dimension two
• Nonlinear fractional Schrödinger equations in one dimension
• On the transport of Gaussian measures under the one-dimensional fractional nonlinear Schrödinger equations
• 2D-Defocusing Nonlinear Schrödinger Equation with Random Data on Irrational Tori