In this paper, we consider the quadratic nonlinear Schrödinger system in three space dimensions. Our aim is to obtain sharp scattering criteria. Because of the mass-subcritical nature, it is difficult to do so in terms of conserved quantities. The corresponding single equation is studied by the second author and a sharp scattering criterion is established by introducing a distance from a trivial scattering solution, the zero solution. By the structure of the nonlinearity we are dealing with, the system admits a scattering solution which is a pair of the zero function and a linear Schrödinger flow. Taking this fact into account, we introduce a new optimizing quantity and give a sharp scattering criterion in terms of it.
Citation: |
[1] |
T. Cazenave and F. B. Weissler, Rapidly decaying solutions of the nonlinear Schrödinger equation, Comm. Math. Phys., 147 (1992), 75-100.
doi: 10.1007/BF02099529.![]() ![]() ![]() |
[2] |
M. Christ, J. Colliander and T. Tao, Ill-posedness for nonlinear Schrödinger and wave equations, preprint, arXiv: math/0311048.
![]() |
[3] |
M. Colin, Th. Colin and M. Ohta, Stability of solitary waves for a system of nonlinear Schrödinger equations with three wave interaction, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 2211-2226.
doi: 10.1016/j.anihpc.2009.01.011.![]() ![]() ![]() |
[4] |
V. D. Dinh, Existence, stability of standing waves and the characterization of finite time blow-up solutions for a system NLS with quadratic interaction, Nonlinear Anal., 190 (2020), 111589, 39 pp.
doi: 10.1016/j.na.2019.111589.![]() ![]() ![]() |
[5] |
B. Dodson, Global well-posedness and scattering for the mass critical nonlinear Schrödinger equation with mass below the mass of the ground state, Adv. Math., 285 (2015), 1589-1618.
doi: 10.1016/j.aim.2015.04.030.![]() ![]() ![]() |
[6] |
J. Ginibre, T. Ozawa and G. Velo, On the existence of the wave operators for a class of nonlinear Schrödinger equations, Ann. Inst. H. Poincar Phys. Théor., 60 (1994), 211-239.
![]() ![]() |
[7] |
J. Ginibre and G. Velo, Smoothing properties and retarded estimates for some dispersive evolution equations, Comm. Math. Phys., 144 (1992), 163-188.
doi: 10.1007/BF02099195.![]() ![]() ![]() |
[8] |
M. Hamano, Global dynamics below the ground state for the quadratic Schrödinger system in 5d, preprint, arXiv: 1805.12245.
![]() |
[9] |
N. Hayashi, C. Li and T. Ozawa, Small data scattering for a system of nonlinear Schrödinger equations, Differ. Equ. Appl., 3 (2011), 415-426.
doi: 10.7153/dea-03-26.![]() ![]() ![]() |
[10] |
N. Hayashi, T. Ozawa and K. Tanaka, On a system of nonlinear Schrödinger equations with quadratic interaction, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 661-690.
doi: 10.1016/j.anihpc.2012.10.007.![]() ![]() ![]() |
[11] |
R. A. Hunt, On $L(p, q)$ spaces, Enseignement Math. (2), 12 (1966), 249–276.
![]() ![]() |
[12] |
T. Inui, N. Kishimoto and K. Nishimura, Scattering for a mass critical NLS system below the ground state with and without mass-resonance condition, Discrete Contin. Dyn. Syst., 39 (2019), 6299-6353.
doi: 10.3934/dcds.2019275.![]() ![]() ![]() |
[13] |
T. Kato, An $L^{q, r}$-theory for nonlinear Schrödinger equations, Spectral and scattering theory and applications, Adv. Stud. Pure Math., 23, Math. Soc. Japan, Tokyo, (1994), 223–238.
doi: 10.2969/aspm/02310223.![]() ![]() ![]() |
[14] |
M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980.
doi: 10.1353/ajm.1998.0039.![]() ![]() ![]() |
[15] |
C. E. Kenig and F. Merle, Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case, Invent. Math., 166 (2006), 645-675.
doi: 10.1007/s00222-006-0011-4.![]() ![]() ![]() |
[16] |
R. Killip, S. Masaki, J. Murphy and M. Visan, Large data mass-subcritical NLS: critical weighted bounds imply scattering, NoDEA Nonlinear Differential Equations Appl., 24 (2017), Art. 38, 33 pp.
doi: 10.1007/s00030-017-0463-9.![]() ![]() ![]() |
[17] |
R. Killip, S. Masaki, J. Murphy and M. Visan, The radial mass-subcritical NLS in negative order Sobolev spaces, Discrete Contin. Dyn. Syst., 39 (2019), 553-583.
doi: 10.3934/dcds.2019023.![]() ![]() ![]() |
[18] |
R. Killip, T. Tao and M. Visan, The cubic nonlinear Schrödinger equation in two dimensions with radial data, J. Eur. Math. Soc. (JEMS), 11 (2009), 1203-1258.
doi: 10.4171/JEMS/180.![]() ![]() ![]() |
[19] |
R. Killip, M. Visan and X. Zhang, The mass-critical nonlinear Schrödinger equation with radial data in dimensions three and higher, Anal. PDE, 1 (2008), 229-266.
doi: 10.2140/apde.2008.1.229.![]() ![]() ![]() |
[20] |
S. Masaki, A sharp scattering condition for focusing mass-subcritical nonlinear Schrödinger equation, Commun. Pure Appl. Anal., 14 (2015), 1481-1531.
doi: 10.3934/cpaa.2015.14.1481.![]() ![]() ![]() |
[21] |
S. Masaki, On minimal nonscattering solution for focusing mass-subcritical nonlinear Schrödinger equation, Comm. Partial Differential Equations, 42 (2017), 626-653.
doi: 10.1080/03605302.2017.1286672.![]() ![]() ![]() |
[22] |
S. Masaki, Two minimization problems on non-scattering solutions to mass-subcritical nonlinear Schrödinger equation, preprint, arXiv: 1605.09234.
![]() |
[23] |
S. Masaki and J.-I. Segata, Existence of a minimal non-scattering solution to the mass-subcritical generalized Korteweg-de Vries equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 35 (2018), 283-326.
doi: 10.1016/j.anihpc.2017.04.003.![]() ![]() ![]() |
[24] |
K. Nakanishi and T. Ozawa, Remarks on scattering for nonlinear Schrodinger equations, NoDEA Nonlinear Differential Equations Appl., 9 (2002), 45-68.
doi: 10.1007/s00030-002-8118-9.![]() ![]() ![]() |
[25] |
R. O'Neil, Convolution operators and L(p, q) spaces, Duke Math. J., 30 (1963), 129-142.
doi: 10.1215/S0012-7094-63-03015-1.![]() ![]() ![]() |
[26] |
R. S. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J., 44 (1977), 705-714.
doi: 10.1215/S0012-7094-77-04430-1.![]() ![]() ![]() |