We consider a system of nonlinear Schrödinger equations with quadratic nonlinearities in two space dimensions. Our aim is to show time decay estimates of small solutions and nonexistence of the usual scattering states for a system. Furthermore we prove stability in time of small solutions in the neighborhood of solutions to a suitable approximate equation.
"On a system of nonlinear Schrödinger equations in 2d." Differential Integral Equations 24 (5/6) 417 - 434, May/June 2011.