Threshold solutions for the intercritical inhomogeneous NLS

TL;DR

This paper classifies the behavior of solutions to the focusing inhomogeneous nonlinear Schrödinger equation at the critical mass-energy threshold, identifying special solutions and their dynamics in relation to blowup or scattering.

Contribution

It introduces solutions at the threshold that approach standing waves and classifies all possible behaviors at this critical level.

Findings

Existence of solutions $Q^\\pm$ at the threshold that approach standing waves.

Classification of all threshold solutions into blowup, scattering, or standing wave behaviors.

Extension of the blowup/scattering dichotomy to the critical threshold case.

Abstract

We consider the focusing inhomogeneous nonlinear Schr\"odinger equation in $H^1(\mathbb{R}^3)$, \begin{equation} i\partial_t u + \Delta u + |x|^{-b}|u|^{2}u=0,{equation} where $0 < b <\tfrac{1}{2}$. Previous works have established a blowup/scattering dichotomy below a mass-energy threshold determined by the ground state solution $Q$. In this work, we study solutions exactly at this mass-energy threshold. In addition to the ground state solution, we prove the existence of solutions $Q^\pm$, which approach the standing wave in the positive time direction, but either blow up or scatter in the negative time direction. Using these particular solutions, we classify all possible behaviors for threshold solutions. In particular, the solution either behaves as in the sub-threshold case, or it agrees with $e^{it}Q$, $Q^+$, or $Q^-$ up to the symmetries of the equation.