A sharp scattering threshold level for mass-subcritical nonlinear Schrödinger system

Masaru Hamano; Satoshi Masaki

doi:10.3934/dcds.2020323

Article Contents

2021, Volume 41, Issue 3: 1415-1447. Doi: 10.3934/dcds.2020323 open access

This issue Previous Article Global large solutions and optimal time-decay estimates to the Korteweg system Next Article Mean-square random invariant manifolds for stochastic differential equations

A sharp scattering threshold level for mass-subcritical nonlinear Schrödinger system

Masaru Hamano^1, , and
Satoshi Masaki^2,

1.
Department of Mathematics, Graduate School of Science and Engineering, Saitama University, 255 Shimo-Okubo, Sakura-ku, Saitama-shi, Saitama, 338-8570, Japan
2.
Department of Systems Innovation, Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka, 560-8531, Japan

^* Corresponding author: Masaru Hamano
^* Corresponding author: Masaru Hamano

The first author is supported by JSPS KAKENHI Grant Number JP19J13300

Received: November 30, 2019

Revised: May 31, 2020

Early access: September 2020

Published: March 2021

The second author is supported by JSPS KAKENHI Grant Numbers JP17K14219, JP17H02854, JP17H02851, and JP18KK0386

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Abstract

In this paper, we consider the quadratic nonlinear Schrödinger 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 criterion 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 the zero function and a linear Schrödinger flow. Taking this fact into account, we introduce a new optimizing quantity and give a sharp scattering criterion in terms of it.

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Mathematics Subject Classification: Primary: 35Q55.

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