# American Institute of Mathematical Sciences

November  2019, 39(11): 6299-6353. doi: 10.3934/dcds.2019275

## 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

* Corresponding author: Kuranosuke Nishimura

Received  November 2018 Published  August 2019

Fund Project: The first author is partially supported by JSPS Grant-in-Aid for Early-Career Scientists JP18K13444. The second author is supported in part by Grant-in-Aid for Young Scientists (B) JP24740086 and JP16K17626.

We consider a mass-critical system of nonlinear Schrödinger equations
 $\left\{ \begin{split} i\partial_t u + \;\; \Delta u & = \bar{u}v,\\ i\partial_t v +\kappa \Delta v & = u^2, \end{split} \right. \qquad (t,x)\in \mathbb{R}\times \mathbb{R}^4,$
where
 $(u,v)$
is a
 $\mathbb{C}^2$
-valued unknown function and
 $\kappa >0$
is a constant. If
 $\kappa = 1/2$
, we say the equation satisfies mass-resonance condition. We are interested in the scattering problem of this equation under the condition
 $M(u,v) , where $ M(u,v) $denotes the mass and $ (\phi ,\psi) $is a ground state. In the mass-resonance case, we prove scattering by the argument of Dodson [5]. Scattering is also obtained without mass-resonance condition under the restriction that $ (u,v) $is radially symmetric. Citation: Takahisa Inui, Nobu Kishimoto, Kuranosuke Nishimura. Scattering for a mass critical NLS system below the ground state with and without mass-resonance condition. Discrete & Continuous Dynamical Systems - A, 2019, 39 (11) : 6299-6353. doi: 10.3934/dcds.2019275 ##### References:  [1] H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. Ⅰ. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345. doi: 10.1007/BF00250555. Google Scholar [2] M. Colin, T. Colin and M. Ohta, Stability of solitary waves for a system of nonlinear Schrödinger equations with three wave interaction, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 2211-2226. doi: 10.1016/j.anihpc.2009.01.011. Google Scholar [3] J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in$\mathbb{R}^3$, Ann. of Math.(2), 167 (2008), 767-865. doi: 10.4007/annals.2008.167.767. Google Scholar [4] B. Dodson, Global well-posedness and scattering for the defocusing,$L^2$-critical nonlinear Schrödinger equation when$d\geq3$, J. Amer. Math. Soc., 25 (2012), 429-463. doi: 10.1090/S0894-0347-2011-00727-3. Google Scholar [5] B. Dodson, Global well-posedness and scattering for the mass critical nonlinear Schrödinger equation with mass below the mass of the ground state, Adv. Math., 285 (2015), 1589-1618. doi: 10.1016/j.aim.2015.04.030. Google Scholar [6] B. Dodson, Global well-posedness and scattering for the defocusing,$L^2$critical, nonlinear Schrödinger equation when$d = 1$, Amer. J. Math., 138 (2016), 531-569. doi: 10.1353/ajm.2016.0016. Google Scholar [7] B. Dodson, Global well-posedness and scattering for the defocusing,$L^2$-critical, nonlinear Schrödinger equation when$d = 2$, Duke Math. J., 165 (2016), 3435-3516. doi: 10.1215/00127094-3673888. Google Scholar [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. Google Scholar [9] M. Hamano, Global dynamics below the ground state for the quadratic schödinger system in$5d$, preprint, arXiv: 1805.12245, 2018. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [9] M. Hamano, Global dynamics below the ground state for the quadratic schödinger system in$5d$, preprint, arXiv: 1805.12245, 2018. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. 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