Small data blow-up for a system of nonlinear Schr\"odinger equations

TL;DR

This paper demonstrates small data blow-up phenomena in a three-component quadratic nonlinear Schrödinger system in one dimension, using the Hopf-Cole transformation to analyze growth and resonance effects.

Contribution

It introduces a novel construction of blow-up solutions for a multi-component NLS system via the Hopf-Cole transformation, highlighting the role of mass resonance.

Findings

Small data blow-up is possible in a three-component quadratic NLS system.

The Hopf-Cole transformation effectively reduces the problem to growth estimates.

Mass resonance is closely linked to amplification and blow-up behavior.

Abstract

We give examples of small data blow-up for a three-component system of quadratic nonlinear Schr\"odinger equations in one space dimension. Our construction of the blowing-up solution is based on the Hopf-Cole transformation, which allows us to reduce the problem to getting suitable growth estimates for a solution to the transformed system. Amplification in the reduced system is shown to have a close connection with the mass resonance.