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 n6. The Cauchy problem is studied in L 2 , in H 1 , and in the weighted L 2 space x -1 L 2 =(H 1 ) under mass resonance condition, where x=(1+|x| 2 ) 1/2 and 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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