Existence, stability of standing waves and the characterization of finite time blow-up solutions for a system NLS with quadratic interaction

Abstract

We study the existence and stability of standing waves for a system of nonlinear Schrödinger equations with quadratic interaction in dimensions $d\le 3$. We also study the characterization of finite time blow-up solutions with minimal mass to the system under mass resonance condition in dimension $d=4$. Finite time blow-up solutions with minimal mass are showed to be (up to symmetries) pseudo-conformal transformations of a ground state standing wave.

arXiv comment39 pages