On a system of nonlinear Schrödinger equations with quadratic interaction

We study a system of nonlinear Schrödinger equations with quadratic interaction in space dimension n⩽6. The Cauchy problem is studied in L2, in H1, and in the weighted L2 space 〈x〉−1L2=F(H1) under mass resonance condition, where 〈x〉=(1+|x|2)1/2 and F is the Fourier transform. The existence of ground states is studied by variational methods. Blow-up solutions are presented in an explicit form in terms of ground states under mass resonance condition, which ensures the invariance of the system under pseudo-conformal transformations.