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), p. 661-690 / Harvested from Numdam

Résumé

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

Détail

Publié le : 2013-01-01

MR 3082479 | Zbl 1291.35347

DOI : https://doi.org/10.1016/j.anihpc.2012.10.007

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