A sharp scattering threshold level for mass-subcritical nonlinear Schr\"odinger system

TL;DR

This paper establishes a precise criterion for scattering in a mass-subcritical quadratic nonlinear Schrödinger system in three dimensions, overcoming challenges posed by conserved quantities.

Contribution

It introduces a new optimizing quantity to derive a sharp scattering criterion for the system, extending previous single-equation results to a coupled system.

Findings

Established a sharp scattering threshold for the system.

Identified a new quantity that characterizes scattering behavior.

Extended single-equation scattering criteria to a coupled system.

Abstract

In this paper, we consider the quadratic nonlinear Schr\"odinger system in three space dimensions. Our aim is to obtain sharp scattering criteria. Because of the mass-subcritical nature, it is difficult to do so in terms of conserved quantities. The corresponding single equation is studied by the second author and a sharp scattering criteria is established by introducing a distance from a trivial scattering solution, the zero solution. By the structure of the nonlinearity we are dealing with, the system admits a scattering solution which is a pair of zero solution and a linear solution. Taking this fact into account, we introduce a new optimizing quantity and give a sharp scattering criterion in terms of it.