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December  2015, 35(12): 6085-6112. doi: 10.3934/dcds.2015.35.6085

Stable solitary waves with prescribed $L^2$-mass for the cubic Schrödinger system with trapping potentials

Benedetta Noris 1, , Hugo Tavares 2, and  Gianmaria Verzini 3,

1. 

INdAM-COFUND Marie Curie Fellow, Laboratoire de Mathématiques de Versailles, Université de Versailles Saint-Quentin, 45 avenue des Etats-Unis, 78035 Versailles Cédex, France
2. 

Center for Mathematical Analysis, Geometry and Dynamical Systems, Mathematics Department, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal
3. 

Dipartimento di Matematica "Francesco Brioschi", Politecnico di Milano, p.za Leonardo da Vinci 32, 20133 Milano

Received  May 2014 Published  May 2015

For the cubic Schrödinger system with trapping potentials in $\mathbb{R}^N$, $N\leq3$, or in bounded domains, we investigate the existence and the orbital stability of standing waves having components with prescribed $L^2$-mass. We provide a variational characterization of such solutions, which gives information on the stability through a condition of Grillakis-Shatah-Strauss type. As an application, we show existence of conditionally orbitally stable solitary waves when: a) the masses are small, for almost every scattering lengths, and b) in the defocusing, weakly interacting case, for any masses.
Keywords: Gross-Pitaevskii systems, constrained critical points, Cooperative and competitive elliptic systems, orbital stability., Ambrosetti-Prodi type problem.
Mathematics Subject Classification: Primary: 35C08, 35Q55, 35J50; Secondary: 35B4.
Citation: Benedetta Noris, Hugo Tavares, Gianmaria Verzini. Stable solitary waves with prescribed $L^2$-mass for the cubic Schrödinger system with trapping potentials. Discrete & Continuous Dynamical Systems, 2015, 35 (12) : 6085-6112. doi: 10.3934/dcds.2015.35.6085
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show all references

References:
[1]

Communications in Mathematical Physics, 336 (2015), 509-579. doi: 10.1007/s00220-014-2281-9.  Google Scholar

[2]

J. Funct. Anal., 256 (2009), 1118-1136. doi: 10.1016/j.jfa.2008.10.021.  Google Scholar

[3]

Ann. Mat. Pura Appl. (4), 93 (1972), 231-246. doi: 10.1007/BF02412022.  Google Scholar

[4]

J. Lond. Math. Soc. (2), 75 (2007), 67-82. doi: 10.1112/jlms/jdl020.  Google Scholar

[5]

Cambridge Studies in Advanced Mathematics, 34, Cambridge University Press, Cambridge, 1993.  Google Scholar

[6]

Calc. Var. Partial Differential Equations, 37 (2010), 345-361. doi: 10.1007/s00526-009-0265-y.  Google Scholar

[7]

J. Fixed Point Theory Appl., 2 (2007), 353-367. doi: 10.1007/s11784-007-0033-6.  Google Scholar

[8]

Appl. Math. Optim., 12 (1984), 271-282. doi: 10.1007/BF01449045.  Google Scholar

[9]

Courant Lecture Notes in Mathematics, 10, New York University Courant Institute of Mathematical Sciences, New York, 2003.  Google Scholar

[10]

Phys. D, 196 (2004), 341-361. doi: 10.1016/j.physd.2004.06.002.  Google Scholar

[11]

J. Differential Equations, 255 (2013), 4289-4311. doi: 10.1016/j.jde.2013.08.009.  Google Scholar

[12]

Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 953-969. doi: 10.1016/j.anihpc.2010.01.009.  Google Scholar

[13]

, DispersiveWiki project,, URL , ().   Google Scholar

[14]

J. Funct. Anal., 74 (1987), 160-197. doi: 10.1016/0022-1236(87)90044-9.  Google Scholar

[15]

J. Funct. Anal., 94 (1990), 308-348. doi: 10.1016/0022-1236(90)90016-E.  Google Scholar

[16]

Adv. Nonlinear Stud., 14 (2014), 115-136.  Google Scholar

[17]

Michigan Math. J., 41 (1994), 151-173. doi: 10.1307/mmj/1029004922.  Google Scholar

[18]

Comm. Math. Phys., 255 (2005), 629-653. doi: 10.1007/s00220-005-1313-x.  Google Scholar

[19]

Comm. Math. Phys., 282 (2008), 721-731. doi: 10.1007/s00220-008-0546-x.  Google Scholar

[20]

J. Differential Equations, 229 (2006), 743-767. doi: 10.1016/j.jde.2006.07.002.  Google Scholar

[21]

Adv. Nonlinear Stud., 10 (2010), 681-705.  Google Scholar

[22]

Commun. Math. Sci., 9 (2011), 997-1012. doi: 10.4310/CMS.2011.v9.n4.a3.  Google Scholar

[23]

Adv. Differential Equations, 16 (2011), 977-1000.  Google Scholar

[24]

Proc. Amer. Math. Soc., 138 (2010), 1681-1692. doi: 10.1090/S0002-9939-10-10231-7.  Google Scholar

[25]

J. Eur. Math. Soc. (JEMS), 14 (2012), 1245-1273. doi: 10.4171/JEMS/332.  Google Scholar

[26]

B. Noris, H. Tavares and G. Verzini, Existence and orbital stability of the ground states with prescribed mass for the $L^2$-critical and supercritical NLS on bounded domains,, , ().   Google Scholar

[27]

J. Differential Equations, 254 (2013), 1529-1547. doi: 10.1016/j.jde.2012.11.003.  Google Scholar

[28]

Nonlinear Anal., 26 (1996), 933-939. doi: 10.1016/0362-546X(94)00340-8.  Google Scholar

[29]

Calc. Var. Partial Differential Equations, 49 (2014), 103-124. doi: 10.1007/s00526-012-0571-7.  Google Scholar

[30]

Commun. Math. Phys., 91 (1983), 313-327. doi: 10.1007/BF01208779.  Google Scholar

[31]

Comm. Math. Phys., 271 (2007), 199-221. doi: 10.1007/s00220-006-0179-x.  Google Scholar

[32]

N. Soave, On existence and phase separation of positive solutions to nonlinear elliptic systems modelling simultaneous cooperation and competition,, , ().   Google Scholar

[33]

NoDEA Nonlinear Differential Equations Appl., 20 (2013), 715-740. doi: 10.1007/s00030-012-0176-z.  Google Scholar

[34]

Ann. Inst. H. Poincaré Anal. Non Linéaire, 29 (2012), 279-300. doi: 10.1016/j.anihpc.2011.10.006.  Google Scholar

[35]

Arch. Ration. Mech. Anal., 194 (2009), 717-741. doi: 10.1007/s00205-008-0172-y.  Google Scholar

[36]

Topol. Methods Nonlinear Anal., 37 (2011), 203-223.  Google Scholar

[37]

Graduate Texts in Mathematics, Vol. 120, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4612-1015-3.  Google Scholar

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