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

Benedetta  Noris; Hugo  Tavares; Gianmaria Verzini

doi:10.3934/dcds.2015.35.6085

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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

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

References:

[1]	A. Aftalion, B. Noris and C. Sourdis, Thomas-Fermi approximation for coexisting two component Bose-Einstein condensates and nonexistence of vortices for small rotation,, Communications in Mathematical Physics, 336* (2015), 509. doi: 10.1007/s00220-014-2281-9. Google Scholar*
[2]	S. Alama, L. Bronsard and P. Mironescu, On the structure of fractional degree vortices in a spinor Ginzburg-Landau model,, J. Funct. Anal., 256 (2009), 1118. doi: 10.1016/j.jfa.2008.10.021. Google Scholar
[3]	A. Ambrosetti and G. Prodi, On the inversion of some differentiable mappings with singularities between Banach spaces,, Ann. Mat. Pura Appl. (4), 93 (1972), 231. doi: 10.1007/BF02412022. Google Scholar
[4]	A. Ambrosetti and E. Colorado, Standing waves of some coupled nonlinear Schrödinger equations,, J. Lond. Math. Soc. (2), 75 (2007), 67. doi: 10.1112/jlms/jdl020. Google Scholar
[5]	A. Ambrosetti and G. Prodi, A Primer of Nonlinear Analysis,, Cambridge Studies in Advanced Mathematics, (1993). Google Scholar
[6]	T. Bartsch, N. Dancer and Z.-Q. Wang, A Liouville theorem, a-priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic system,, Calc. Var. Partial Differential Equations, 37 (2010), 345. doi: 10.1007/s00526-009-0265-y. Google Scholar
[7]	T. Bartsch, Z.-Q. Wang and J. Wei, Bound states for a coupled Schrödinger system,, J. Fixed Point Theory Appl., 2 (2007), 353. doi: 10.1007/s11784-007-0033-6. Google Scholar
[8]	H. Brezis, Semilinear equations in $R^N$ without condition at infinity,, Appl. Math. Optim., 12 (1984), 271. doi: 10.1007/BF01449045. Google Scholar
[9]	T. Cazenave, Semilinear Schrödinger Equations,, Courant Lecture Notes in Mathematics, (2003). Google Scholar
[10]	S.-M. Chang, C.-S. Lin, T.-C. Lin and W.-W. Lin, Segregated nodal domains of two-dimensional multispecies Bose-Einstein condensates,, Phys. D, 196 (2004), 341. doi: 10.1016/j.physd.2004.06.002. Google Scholar
[11]	Z. Chen, C.-S. Lin and W. Zou, Multiple sign-changing and semi-nodal solutions for coupled Schrödinger equations,, J. Differential Equations, 255 (2013), 4289. doi: 10.1016/j.jde.2013.08.009. Google Scholar
[12]	E. N. Dancer, J. Wei and T. Weth, A priori bounds versus multiple existence of positive solutions for a nonlinear Schrödinger system,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 953. doi: 10.1016/j.anihpc.2010.01.009. Google Scholar
[13]	, DispersiveWiki project,, URL , (). Google Scholar
[14]	M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry. I,, J. Funct. Anal., 74 (1987), 160. doi: 10.1016/0022-1236(87)90044-9. Google Scholar
[15]	M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry. II,, J. Funct. Anal., 94 (1990), 308. doi: 10.1016/0022-1236(90)90016-E. Google Scholar
[16]	N. Ikoma, Compactness of minimizing sequences in nonlinear Schrödinger systems under multiconstraint conditions,, Adv. Nonlinear Stud., 14 (2014), 115. Google Scholar
[17]	O. Kavian and F. B. Weissler, Self-similar solutions of the pseudo-conformally invariant nonlinear Schrödinger equation,, Michigan Math. J., 41 (1994), 151. doi: 10.1307/mmj/1029004922. Google Scholar
[18]	T.-C. Lin and J. Wei, Ground state of $N$ coupled nonlinear Schrödinger equations in $R^n$, $n\leq 3$,, Comm. Math. Phys., 255 (2005), 629. doi: 10.1007/s00220-005-1313-x. Google Scholar
[19]	Z. Liu and Z.-Q. Wang, Multiple bound states of nonlinear Schrödinger systems,, Comm. Math. Phys., 282 (2008), 721. doi: 10.1007/s00220-008-0546-x. Google Scholar
[20]	L. A. Maia, E. Montefusco and B. Pellacci, Positive solutions for a weakly coupled nonlinear Schrödinger system,, J. Differential Equations, 229 (2006), 743. doi: 10.1016/j.jde.2006.07.002. Google Scholar
[21]	L. A. Maia, E. Montefusco and B. Pellacci, Orbital stability property for coupled nonlinear Schrödinger equations,, Adv. Nonlinear Stud., 10 (2010), 681. Google Scholar
[22]	N. V. Nguyen, On the orbital stability of solitary waves for the 2-coupled nonlinear Schrödinger system,, Commun. Math. Sci., 9 (2011), 997. doi: 10.4310/CMS.2011.v9.n4.a3. Google Scholar
[23]	N. V. Nguyen and Z.-Q. Wang, Orbital stability of solitary waves for a nonlinear Schrödinger system,, Adv. Differential Equations, 16 (2011), 977. Google Scholar
[24]	B. Noris and M. Ramos, Existence and bounds of positive solutions for a nonlinear Schrödinger system,, Proc. Amer. Math. Soc., 138 (2010), 1681. doi: 10.1090/S0002-9939-10-10231-7. Google Scholar
[25]	B. Noris, H. Tavares, S. Terracini and G. Verzini, Convergence of minimax structures and continuation of critical points for singularly perturbed systems,, J. Eur. Math. Soc. (JEMS), 14 (2012), 1245. 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]	B. Noris and G. Verzini, A remark on natural constraints in variational methods and an application to superlinear Schrödinger systems,, J. Differential Equations, 254 (2013), 1529. doi: 10.1016/j.jde.2012.11.003. Google Scholar
[28]	M. Ohta, Stability of solitary waves for coupled nonlinear Schrödinger equations,, Nonlinear Anal., 26 (1996), 933. doi: 10.1016/0362-546X(94)00340-8. Google Scholar
[29]	J. Royo-Letelier, Segregation and symmetry breaking of strongly coupled two-component Bose-Einstein condensates in a harmonic trap,, Calc. Var. Partial Differential Equations, 49 (2014), 103. doi: 10.1007/s00526-012-0571-7. Google Scholar
[30]	J. Shatah, Stable standing waves of nonlinear Klein-Gordon equations,, Commun. Math. Phys., 91 (1983), 313. doi: 10.1007/BF01208779. Google Scholar
[31]	B. Sirakov, Least energy solitary waves for a system of nonlinear Schrödinger equations in $\mathbbR^n$,, Comm. Math. Phys., 271 (2007), 199. 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]	H. Tavares and T. Weth, Existence and symmetry results for competing variational systems,, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 715. doi: 10.1007/s00030-012-0176-z. Google Scholar
[34]	H. Tavares and S. Terracini, Sign-changing solutions of competition-diffusion elliptic systems and optimal partition problems,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 29 (2012), 279. doi: 10.1016/j.anihpc.2011.10.006. Google Scholar
[35]	S. Terracini and G. Verzini, Multipulse phases in $k$-mixtures of Bose-Einstein condensates,, Arch. Ration. Mech. Anal., 194 (2009), 717. doi: 10.1007/s00205-008-0172-y. Google Scholar
[36]	R. Tian and Z.-Q. Wang, Multiple solitary wave solutions of nonlinear Schrödinger systems,, Topol. Methods Nonlinear Anal., 37 (2011), 203. Google Scholar
[37]	W. P. Ziemer, Weakly Differentiable Functions,, Graduate Texts in Mathematics, (1989). doi: 10.1007/978-1-4612-1015-3. Google Scholar

show all references

References:

[1]	A. Aftalion, B. Noris and C. Sourdis, Thomas-Fermi approximation for coexisting two component Bose-Einstein condensates and nonexistence of vortices for small rotation,, Communications in Mathematical Physics, 336* (2015), 509. doi: 10.1007/s00220-014-2281-9. Google Scholar*
[2]	S. Alama, L. Bronsard and P. Mironescu, On the structure of fractional degree vortices in a spinor Ginzburg-Landau model,, J. Funct. Anal., 256 (2009), 1118. doi: 10.1016/j.jfa.2008.10.021. Google Scholar
[3]	A. Ambrosetti and G. Prodi, On the inversion of some differentiable mappings with singularities between Banach spaces,, Ann. Mat. Pura Appl. (4), 93 (1972), 231. doi: 10.1007/BF02412022. Google Scholar
[4]	A. Ambrosetti and E. Colorado, Standing waves of some coupled nonlinear Schrödinger equations,, J. Lond. Math. Soc. (2), 75 (2007), 67. doi: 10.1112/jlms/jdl020. Google Scholar
[5]	A. Ambrosetti and G. Prodi, A Primer of Nonlinear Analysis,, Cambridge Studies in Advanced Mathematics, (1993). Google Scholar
[6]	T. Bartsch, N. Dancer and Z.-Q. Wang, A Liouville theorem, a-priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic system,, Calc. Var. Partial Differential Equations, 37 (2010), 345. doi: 10.1007/s00526-009-0265-y. Google Scholar
[7]	T. Bartsch, Z.-Q. Wang and J. Wei, Bound states for a coupled Schrödinger system,, J. Fixed Point Theory Appl., 2 (2007), 353. doi: 10.1007/s11784-007-0033-6. Google Scholar
[8]	H. Brezis, Semilinear equations in $R^N$ without condition at infinity,, Appl. Math. Optim., 12 (1984), 271. doi: 10.1007/BF01449045. Google Scholar
[9]	T. Cazenave, Semilinear Schrödinger Equations,, Courant Lecture Notes in Mathematics, (2003). Google Scholar
[10]	S.-M. Chang, C.-S. Lin, T.-C. Lin and W.-W. Lin, Segregated nodal domains of two-dimensional multispecies Bose-Einstein condensates,, Phys. D, 196 (2004), 341. doi: 10.1016/j.physd.2004.06.002. Google Scholar
[11]	Z. Chen, C.-S. Lin and W. Zou, Multiple sign-changing and semi-nodal solutions for coupled Schrödinger equations,, J. Differential Equations, 255 (2013), 4289. doi: 10.1016/j.jde.2013.08.009. Google Scholar
[12]	E. N. Dancer, J. Wei and T. Weth, A priori bounds versus multiple existence of positive solutions for a nonlinear Schrödinger system,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 953. doi: 10.1016/j.anihpc.2010.01.009. Google Scholar
[13]	, DispersiveWiki project,, URL , (). Google Scholar
[14]	M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry. I,, J. Funct. Anal., 74 (1987), 160. doi: 10.1016/0022-1236(87)90044-9. Google Scholar
[15]	M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry. II,, J. Funct. Anal., 94 (1990), 308. doi: 10.1016/0022-1236(90)90016-E. Google Scholar
[16]	N. Ikoma, Compactness of minimizing sequences in nonlinear Schrödinger systems under multiconstraint conditions,, Adv. Nonlinear Stud., 14 (2014), 115. Google Scholar
[17]	O. Kavian and F. B. Weissler, Self-similar solutions of the pseudo-conformally invariant nonlinear Schrödinger equation,, Michigan Math. J., 41 (1994), 151. doi: 10.1307/mmj/1029004922. Google Scholar
[18]	T.-C. Lin and J. Wei, Ground state of $N$ coupled nonlinear Schrödinger equations in $R^n$, $n\leq 3$,, Comm. Math. Phys., 255 (2005), 629. doi: 10.1007/s00220-005-1313-x. Google Scholar
[19]	Z. Liu and Z.-Q. Wang, Multiple bound states of nonlinear Schrödinger systems,, Comm. Math. Phys., 282 (2008), 721. doi: 10.1007/s00220-008-0546-x. Google Scholar
[20]	L. A. Maia, E. Montefusco and B. Pellacci, Positive solutions for a weakly coupled nonlinear Schrödinger system,, J. Differential Equations, 229 (2006), 743. doi: 10.1016/j.jde.2006.07.002. Google Scholar
[21]	L. A. Maia, E. Montefusco and B. Pellacci, Orbital stability property for coupled nonlinear Schrödinger equations,, Adv. Nonlinear Stud., 10 (2010), 681. Google Scholar
[22]	N. V. Nguyen, On the orbital stability of solitary waves for the 2-coupled nonlinear Schrödinger system,, Commun. Math. Sci., 9 (2011), 997. doi: 10.4310/CMS.2011.v9.n4.a3. Google Scholar
[23]	N. V. Nguyen and Z.-Q. Wang, Orbital stability of solitary waves for a nonlinear Schrödinger system,, Adv. Differential Equations, 16 (2011), 977. Google Scholar
[24]	B. Noris and M. Ramos, Existence and bounds of positive solutions for a nonlinear Schrödinger system,, Proc. Amer. Math. Soc., 138 (2010), 1681. doi: 10.1090/S0002-9939-10-10231-7. Google Scholar
[25]	B. Noris, H. Tavares, S. Terracini and G. Verzini, Convergence of minimax structures and continuation of critical points for singularly perturbed systems,, J. Eur. Math. Soc. (JEMS), 14 (2012), 1245. 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]	B. Noris and G. Verzini, A remark on natural constraints in variational methods and an application to superlinear Schrödinger systems,, J. Differential Equations, 254 (2013), 1529. doi: 10.1016/j.jde.2012.11.003. Google Scholar
[28]	M. Ohta, Stability of solitary waves for coupled nonlinear Schrödinger equations,, Nonlinear Anal., 26 (1996), 933. doi: 10.1016/0362-546X(94)00340-8. Google Scholar
[29]	J. Royo-Letelier, Segregation and symmetry breaking of strongly coupled two-component Bose-Einstein condensates in a harmonic trap,, Calc. Var. Partial Differential Equations, 49 (2014), 103. doi: 10.1007/s00526-012-0571-7. Google Scholar
[30]	J. Shatah, Stable standing waves of nonlinear Klein-Gordon equations,, Commun. Math. Phys., 91 (1983), 313. doi: 10.1007/BF01208779. Google Scholar
[31]	B. Sirakov, Least energy solitary waves for a system of nonlinear Schrödinger equations in $\mathbbR^n$,, Comm. Math. Phys., 271 (2007), 199. 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]	H. Tavares and T. Weth, Existence and symmetry results for competing variational systems,, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 715. doi: 10.1007/s00030-012-0176-z. Google Scholar
[34]	H. Tavares and S. Terracini, Sign-changing solutions of competition-diffusion elliptic systems and optimal partition problems,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 29 (2012), 279. doi: 10.1016/j.anihpc.2011.10.006. Google Scholar
[35]	S. Terracini and G. Verzini, Multipulse phases in $k$-mixtures of Bose-Einstein condensates,, Arch. Ration. Mech. Anal., 194 (2009), 717. doi: 10.1007/s00205-008-0172-y. Google Scholar
[36]	R. Tian and Z.-Q. Wang, Multiple solitary wave solutions of nonlinear Schrödinger systems,, Topol. Methods Nonlinear Anal., 37 (2011), 203. Google Scholar
[37]	W. P. Ziemer, Weakly Differentiable Functions,, Graduate Texts in Mathematics, (1989). doi: 10.1007/978-1-4612-1015-3. Google Scholar

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

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

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