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

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.