Nonlinear fractional Schr\"odinger equations in one dimension

TL;DR

This paper investigates the global behavior of small, smooth solutions to a one-dimensional fractional cubic nonlinear Schrödinger equation, revealing the necessity of a nonlinear logarithmic correction for global existence and modified scattering.

Contribution

It introduces a new analysis for a fractional NLS motivated by water waves, demonstrating the failure of linear scattering and establishing modified scattering with a nonlinear correction.

Findings

Global existence of solutions with small initial data

Identification of a nonlinear logarithmic correction

Proof of modified scattering behavior

Abstract

We consider the question of global existence of small, smooth, and localized solutions of a certain fractional semilinear cubic NLS in one dimension, $$i\partial_t u - \Lambda u = c_0{|u|}^2 u + c_1 u^3 + c_2 u \bar{u}^2 + c_3 \bar{u}^3, \qquad \Lambda = \Lambda(\partial_x) = {|\partial_x|}^(1/2)$$, where $c_0\in\mathbb{R}$ and $c_1,c_2,c_3\in\mathbb{C}$. This model is motivated by the two-dimensional water waves equations, which have a somewhat similar structure in the Eulerian formulation, in the case of irrotational flows. We show that one cannot expect linear scattering, even in this simplified model. More precisely, we identify a suitable nonlinear logarithmic correction, and prove global existence and modified scattering of solutions.