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 L ², 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.