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

TL;DR

This paper investigates the existence and stability of standing waves in a quadratic-interaction nonlinear Schrödinger system in low dimensions, and characterizes minimal mass blow-up solutions in four dimensions.

Contribution

It provides new results on the existence, stability, and blow-up behavior of solutions for a quadratic NLS system, including a characterization of minimal mass blow-up solutions.

Findings

Standing waves exist and are stable in dimensions d ≤ 3.

Minimal mass blow-up solutions in d=4 are pseudo-conformal transformations of ground states.

The study extends understanding of blow-up phenomena in quadratic NLS systems.

Abstract

We study the existence and stability of standing waves for a system of nonlinear Schr\"odinger equations with quadratic interaction in dimensions $d\leq 3$. 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.