On a system of nonlinear Schrödinger equations with quadratic interaction
Annales de l'I.H.P. Analyse non linéaire, Volume 30 (2013) no. 4, p. 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 ${〈x〉}^{-1}{L}^{2}=ℱ\left({H}^{1}\right)$ under mass resonance condition, where $〈x〉={\left(1+{|x|}^{2}\right)}^{1/2}$ 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.

@article{AIHPC_2013__30_4_661_0,
author = {Hayashi, Nakao and Ozawa, Tohru and Tanaka, Kazunaga},
title = {On a system of nonlinear Schr\"odinger equations with quadratic interaction},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
publisher = {Elsevier},
volume = {30},
number = {4},
year = {2013},
pages = {661-690},
doi = {10.1016/j.anihpc.2012.10.007},
zbl = {1291.35347},
mrnumber = {3082479},
language = {en},
url = {http://www.numdam.org/item/AIHPC_2013__30_4_661_0}
}

Hayashi, Nakao; Ozawa, Tohru; Tanaka, Kazunaga. On a system of nonlinear Schrödinger equations with quadratic interaction. Annales de l'I.H.P. Analyse non linéaire, Volume 30 (2013) no. 4, pp. 661-690. doi : 10.1016/j.anihpc.2012.10.007. http://www.numdam.org/item/AIHPC_2013__30_4_661_0/

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