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Scattering for a mass critical NLS system below the ground state with and without mass-resonance condition
1. | Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka, 560-0043, Japan |
2. | Research Institute for Mathematical Sciences, Kyoto University, Kyoto, 606-8502, Japan |
3. | Department of Mathematics, Graduate School of Science, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan |
$ \left\{ \begin{split} i\partial_t u + \;\; \Delta u & = \bar{u}v,\\ i\partial_t v +\kappa \Delta v & = u^2, \end{split} \right. \qquad (t,x)\in \mathbb{R}\times \mathbb{R}^4, $ |
$ (u,v) $ |
$ \mathbb{C}^2 $ |
$ \kappa >0 $ |
$ \kappa = 1/2 $ |
$ M(u,v)<M(\phi ,\psi) $ |
$ M(u,v) $ |
$ (\phi ,\psi) $ |
$ (u,v) $ |
References:
[1] |
H. Berestycki and P.-L. Lions,
Nonlinear scalar field equations. Ⅰ. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345.
doi: 10.1007/BF00250555. |
[2] |
M. Colin, T. 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. |
[3] |
J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao,
Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in $\mathbb{R}^3$, Ann. of Math.(2), 167 (2008), 767-865.
doi: 10.4007/annals.2008.167.767. |
[4] |
B. Dodson,
Global well-posedness and scattering for the defocusing, $L^2$-critical nonlinear Schrödinger equation when $d\geq3$, J. Amer. Math. Soc., 25 (2012), 429-463.
doi: 10.1090/S0894-0347-2011-00727-3. |
[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] |
B. Dodson,
Global well-posedness and scattering for the defocusing, $L^2$ critical, nonlinear Schrödinger equation when $d = 1$, Amer. J. Math., 138 (2016), 531-569.
doi: 10.1353/ajm.2016.0016. |
[7] |
B. Dodson,
Global well-posedness and scattering for the defocusing, $L^2$-critical, nonlinear Schrödinger equation when $d = 2$, Duke Math. J., 165 (2016), 3435-3516.
doi: 10.1215/00127094-3673888. |
[8] |
T. Duyckaerts, J. Holmer and S. Roudenko,
Scattering for the non-radial 3D cubic nonlinear Schrödinger equation, Math. Res. Lett., 15 (2008), 1233-1250.
doi: 10.4310/MRL.2008.v15.n6.a13. |
[9] |
M. Hamano, Global dynamics below the ground state for the quadratic schödinger system in $5d$, preprint, arXiv: 1805.12245, 2018. |
[10] |
N. Hayashi, C. H. 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. |
[11] |
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. |
[12] |
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. |
[13] |
R. Killip and M. Vișan,
Nonlinear schrödinger equations at critical regularity, Evolution equations, Clay Math. Proc., Amer. Math. Soc., Providence, RI, 17 (2013), 325-437.
|
[14] |
H. Koch, D. Tataru and M. Vișan, Dispersive Equations and Nonlinear Waves, Generalized Korteweg-de Vries, nonlinear Schrödinger, wave and Schrödinger maps. Oberwolfach Seminars, 45. Birkhäuser/Springer, Basel, 2014. |
[15] |
M. K. Kwong,
Uniqueness of positive solutions of $\Delta u-u+u^p = 0$ in $\mathbf{R}^n$, Arch. Rational Mech. Anal., 105 (1989), 243-266.
doi: 10.1007/BF00251502. |
[16] |
F. Merle and L. Vega,
Compactness at blow-up time for $L^2$ solutions of the critical nonlinear Schrödinger equation in 2D, Internat. Math. Res. Notices, (1998), 399-425.
|
[17] |
T. Ozawa and H. Sunagawa,
Small data blow-up for a system of nonlinear Schrödinger equations, J. Math. Anal. Appl., 399 (2013), 147-155.
doi: 10.1016/j.jmaa.2012.10.003. |
[18] |
T. Tao, M. Visan and X. Y. Zhang,
Global well-posedness and scattering for the defocusing mass-critical nonlinear Schrödinger equation for radial data in high dimensions, Duke Math. J., 140 (2007), 165-202.
doi: 10.1215/S0012-7094-07-14015-8. |
[19] |
T. Tao, M. Visan and X. Y. Zhang,
The nonlinear Schrödinger equation with combined power-type nonlinearities, Comm. Partial Differential Equations, 32 (2007), 1281-1343.
doi: 10.1080/03605300701588805. |
[20] |
T. Tao, M. Visan and X. Y. Zhang,
Minimal-mass blowup solutions of the mass-critical NLS, Forum Math., 20 (2008), 881-919.
doi: 10.1515/FORUM.2008.042. |
[21] |
M. Visan,
The defocusing energy-critical nonlinear Schrödinger equation in higher dimensions, Duke Math. J., 138 (2007), 281-374.
doi: 10.1215/S0012-7094-07-13825-0. |
show all references
References:
[1] |
H. Berestycki and P.-L. Lions,
Nonlinear scalar field equations. Ⅰ. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345.
doi: 10.1007/BF00250555. |
[2] |
M. Colin, T. 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. |
[3] |
J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao,
Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in $\mathbb{R}^3$, Ann. of Math.(2), 167 (2008), 767-865.
doi: 10.4007/annals.2008.167.767. |
[4] |
B. Dodson,
Global well-posedness and scattering for the defocusing, $L^2$-critical nonlinear Schrödinger equation when $d\geq3$, J. Amer. Math. Soc., 25 (2012), 429-463.
doi: 10.1090/S0894-0347-2011-00727-3. |
[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] |
B. Dodson,
Global well-posedness and scattering for the defocusing, $L^2$ critical, nonlinear Schrödinger equation when $d = 1$, Amer. J. Math., 138 (2016), 531-569.
doi: 10.1353/ajm.2016.0016. |
[7] |
B. Dodson,
Global well-posedness and scattering for the defocusing, $L^2$-critical, nonlinear Schrödinger equation when $d = 2$, Duke Math. J., 165 (2016), 3435-3516.
doi: 10.1215/00127094-3673888. |
[8] |
T. Duyckaerts, J. Holmer and S. Roudenko,
Scattering for the non-radial 3D cubic nonlinear Schrödinger equation, Math. Res. Lett., 15 (2008), 1233-1250.
doi: 10.4310/MRL.2008.v15.n6.a13. |
[9] |
M. Hamano, Global dynamics below the ground state for the quadratic schödinger system in $5d$, preprint, arXiv: 1805.12245, 2018. |
[10] |
N. Hayashi, C. H. 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. |
[11] |
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. |
[12] |
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. |
[13] |
R. Killip and M. Vișan,
Nonlinear schrödinger equations at critical regularity, Evolution equations, Clay Math. Proc., Amer. Math. Soc., Providence, RI, 17 (2013), 325-437.
|
[14] |
H. Koch, D. Tataru and M. Vișan, Dispersive Equations and Nonlinear Waves, Generalized Korteweg-de Vries, nonlinear Schrödinger, wave and Schrödinger maps. Oberwolfach Seminars, 45. Birkhäuser/Springer, Basel, 2014. |
[15] |
M. K. Kwong,
Uniqueness of positive solutions of $\Delta u-u+u^p = 0$ in $\mathbf{R}^n$, Arch. Rational Mech. Anal., 105 (1989), 243-266.
doi: 10.1007/BF00251502. |
[16] |
F. Merle and L. Vega,
Compactness at blow-up time for $L^2$ solutions of the critical nonlinear Schrödinger equation in 2D, Internat. Math. Res. Notices, (1998), 399-425.
|
[17] |
T. Ozawa and H. Sunagawa,
Small data blow-up for a system of nonlinear Schrödinger equations, J. Math. Anal. Appl., 399 (2013), 147-155.
doi: 10.1016/j.jmaa.2012.10.003. |
[18] |
T. Tao, M. Visan and X. Y. Zhang,
Global well-posedness and scattering for the defocusing mass-critical nonlinear Schrödinger equation for radial data in high dimensions, Duke Math. J., 140 (2007), 165-202.
doi: 10.1215/S0012-7094-07-14015-8. |
[19] |
T. Tao, M. Visan and X. Y. Zhang,
The nonlinear Schrödinger equation with combined power-type nonlinearities, Comm. Partial Differential Equations, 32 (2007), 1281-1343.
doi: 10.1080/03605300701588805. |
[20] |
T. Tao, M. Visan and X. Y. Zhang,
Minimal-mass blowup solutions of the mass-critical NLS, Forum Math., 20 (2008), 881-919.
doi: 10.1515/FORUM.2008.042. |
[21] |
M. Visan,
The defocusing energy-critical nonlinear Schrödinger equation in higher dimensions, Duke Math. J., 138 (2007), 281-374.
doi: 10.1215/S0012-7094-07-13825-0. |
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