On a system of nonlinear Schrödinger equations with quadratic interaction
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 H^{1} , and in the weighted L^{2} space 〈x〉^{−1}L^{2} = \mathcal{F}(H^{1}) under mass resonance condition, where 〈x〉 = (1 + |x|^{2})^{1/ 2} and \mathcal{F} is …