Global dynamics below the ground state for the quadratic Schödinger system in 5d

2018/05/30 by Masaru Hamano, Hamano, Masaru · 2 citations
Engineering · Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.1805.12245

openalex publication_date 2018/05/30 · openalex created_date 2018/06/13 · openalex updated_date 2026/07/28

Abstract

In this paper we consider the nonlinear Schrödinger system (NLS) with quadratic interaction in five dimensions. We determine the global behavior of the solutions to the system with data below the ground state. Our proof of the scattering result is based on an argument by Kenig Merle [16]. In particular, the new part of this paper is to deal with asymmetric interaction. A blowing up or growing up result is proved by combining the argument by Du Wu Zhang in [6] and a variational characterization of minimizers. Moreover, we show a blowing-up result if the data has finite variance or is radial.

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• A sharp scattering threshold level for mass-subcritical nonlinear Schrödinger system
• Blow-up of the radially symmetric solutions for the quadratic nonlinear Schrödinger system without mass-resonance

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