Blowup and Scattering problems for the Nonlinear Schr\"odinger equations

TL;DR

This paper studies the long-term behavior of solutions to certain focusing nonlinear Schrödinger equations, identifying conditions under which solutions either scatter or blow up, and revealing the ground state's unstable directions.

Contribution

It introduces a subset of initial data with a division into two classes, analyzing their asymptotic behaviors and uncovering the ground state's two unstable directions.

Findings

Solutions in $PW_{+}$ scatter and behave asymptotically freely.

Solutions in $PW_{-}$ blow up or grow unbounded.

The ground state has two unstable directions.

Abstract

We consider $L^{2}$-supercritical and $H^{1}$-subcritical focusing nonlinear Schr\"odinger equations. We introduce a subset $PW$ of $H^{1}(\mathbb{R}^{d})$ for $d\ge 1$, and investigate behavior of the solutions with initial data in this set. For this end, we divide $PW$ into two disjoint components $PW_{+}$ and $PW_{-}$. Then, it turns out that any solution starting from a datum in $PW_{+}$ behaves asymptotically free, and solution starting from a datum in $PW_{-}$ blows up or grows up, from which we find that the ground state has two unstable directions. We also investigate some properties of generic global and blowup solutions.