Long time dynamics of non-radial solutions to inhomogeneous nonlinear Schr\"odinger equations

TL;DR

This paper investigates the long-term behavior of non-radial solutions to the focusing inhomogeneous nonlinear Schrödinger equation, establishing scattering and blow-up criteria, and analyzing solutions relative to the ground state threshold.

Contribution

It introduces new scattering and blow-up criteria for non-radial solutions, and demonstrates the existence of finite-time blow-up solutions with cylindrical symmetry.

Findings

Established a scattering criterion for non-radial solutions.

Proved a blow-up criterion using localized virial estimates.

Showed existence of finite-time blow-up solutions with cylindrical symmetry.

Abstract

We study long time dynamics of non-radial solutions to the focusing inhomogeneous nonlinear Schr\"odinger equation. By using the concentration/compactness and rigidity method, we establish a scattering criterion for non-radial solutions to the equation. We also prove a non-radial blow-up criterion for the equation whose proof makes use of localized virial estimates. As a byproduct of these criteria, we study long time dynamics of non-radial solutions to the equation with data lying below, at, and above the ground state threshold. In addition, we provide a new argument showing the existence of finite time blow-up solution to the equation with cylindrically symmetric data. The ideas developed in this paper are robust and can be applicable to other types of nonlinear Schr\"odinger equations.