On a system of nonlinear Schrödinger equations with quadratic interaction

Hayashi, Nakao; Ozawa, Tohru; Tanaka, Kazunaga

Annales de l'I.H.P. Analyse non linéaire, Tome 30 (2013) no. 4, pp. 661-690.

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 ${\langle x\rangle }^{-1}{L}^2=ℱ\left(H^1\right)$ under mass resonance condition, where $\langle x\rangle ={(1+{\left|x\right|}^2)}^{1/2}$ 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.

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