On a system of nonlinear Schrödinger equations with quadratic interaction

Hayashi, Nakao; Ozawa, Tohru; Tanaka, Kazunaga

doi:10.1016/j.anihpc.2012.10.007

Hayashi, Nakao ; Ozawa, Tohru ; Tanaka, Kazunaga

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

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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 ${〈 x 〉}^{- 1} L^{2} = ℱ (H^{1})$ under mass resonance condition, where $〈 x 〉 = {(1 + {| x |}^{2})}^{1 / 2}$ 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.

MR Zbl

DOI : 10.1016/j.anihpc.2012.10.007

@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},
     pages = {661--690},
     publisher = {Elsevier},
     volume = {30},
     number = {4},
     year = {2013},
     doi = {10.1016/j.anihpc.2012.10.007},
     mrnumber = {3082479},
     zbl = {1291.35347},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2012.10.007/}
}

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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, Tome 30 (2013) no. 4, pp. 661-690. doi: 10.1016/j.anihpc.2012.10.007

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