# American Institute of Mathematical Sciences

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

 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.
Citation: Benedetta Noris, Hugo Tavares, Gianmaria Verzini. Stable solitary waves with prescribed $L^2$-mass for the cubic Schrödinger system with trapping potentials. Discrete and Continuous Dynamical Systems, 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), 509-579. doi: 10.1007/s00220-014-2281-9. [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), 1118-1136. doi: 10.1016/j.jfa.2008.10.021. [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), 231-246. doi: 10.1007/BF02412022. [4] A. Ambrosetti and E. Colorado, Standing waves of some coupled nonlinear Schrödinger equations, J. Lond. Math. Soc. (2), 75 (2007), 67-82. doi: 10.1112/jlms/jdl020. [5] A. Ambrosetti and G. Prodi, A Primer of Nonlinear Analysis, Cambridge Studies in Advanced Mathematics, 34, Cambridge University Press, Cambridge, 1993. [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), 345-361. doi: 10.1007/s00526-009-0265-y. [7] T. Bartsch, Z.-Q. Wang and J. Wei, Bound states for a coupled Schrödinger system, J. Fixed Point Theory Appl., 2 (2007), 353-367. doi: 10.1007/s11784-007-0033-6. [8] H. Brezis, Semilinear equations in $R^N$ without condition at infinity, Appl. Math. Optim., 12 (1984), 271-282. doi: 10.1007/BF01449045. [9] T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10, New York University Courant Institute of Mathematical Sciences, New York, 2003. [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), 341-361. doi: 10.1016/j.physd.2004.06.002. [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), 4289-4311. doi: 10.1016/j.jde.2013.08.009. [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), 953-969. doi: 10.1016/j.anihpc.2010.01.009. [13] , DispersiveWiki project,, URL , (). [14] M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry. I, J. Funct. Anal., 74 (1987), 160-197. doi: 10.1016/0022-1236(87)90044-9. [15] M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry. II, J. Funct. Anal., 94 (1990), 308-348. doi: 10.1016/0022-1236(90)90016-E. [16] N. Ikoma, Compactness of minimizing sequences in nonlinear Schrödinger systems under multiconstraint conditions, Adv. Nonlinear Stud., 14 (2014), 115-136. [17] O. Kavian and F. B. Weissler, Self-similar solutions of the pseudo-conformally invariant nonlinear Schrödinger equation, Michigan Math. J., 41 (1994), 151-173. doi: 10.1307/mmj/1029004922. [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), 629-653. doi: 10.1007/s00220-005-1313-x. [19] Z. Liu and Z.-Q. Wang, Multiple bound states of nonlinear Schrödinger systems, Comm. Math. Phys., 282 (2008), 721-731. doi: 10.1007/s00220-008-0546-x. [20] L. A. Maia, E. Montefusco and B. Pellacci, Positive solutions for a weakly coupled nonlinear Schrödinger system, J. Differential Equations, 229 (2006), 743-767. doi: 10.1016/j.jde.2006.07.002. [21] L. A. Maia, E. Montefusco and B. Pellacci, Orbital stability property for coupled nonlinear Schrödinger equations, Adv. Nonlinear Stud., 10 (2010), 681-705. [22] N. V. Nguyen, On the orbital stability of solitary waves for the 2-coupled nonlinear Schrödinger system, Commun. Math. Sci., 9 (2011), 997-1012. doi: 10.4310/CMS.2011.v9.n4.a3. [23] N. V. Nguyen and Z.-Q. Wang, Orbital stability of solitary waves for a nonlinear Schrödinger system, Adv. Differential Equations, 16 (2011), 977-1000. [24] B. Noris and M. Ramos, Existence and bounds of positive solutions for a nonlinear Schrödinger system, Proc. Amer. Math. Soc., 138 (2010), 1681-1692. doi: 10.1090/S0002-9939-10-10231-7. [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), 1245-1273. doi: 10.4171/JEMS/332. [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,, , (). [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), 1529-1547. doi: 10.1016/j.jde.2012.11.003. [28] M. Ohta, Stability of solitary waves for coupled nonlinear Schrödinger equations, Nonlinear Anal., 26 (1996), 933-939. doi: 10.1016/0362-546X(94)00340-8. [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), 103-124. doi: 10.1007/s00526-012-0571-7. [30] J. Shatah, Stable standing waves of nonlinear Klein-Gordon equations, Commun. Math. Phys., 91 (1983), 313-327. doi: 10.1007/BF01208779. [31] B. Sirakov, Least energy solitary waves for a system of nonlinear Schrödinger equations in $\mathbbR^n$, Comm. Math. Phys., 271 (2007), 199-221. doi: 10.1007/s00220-006-0179-x. [32] N. Soave, On existence and phase separation of positive solutions to nonlinear elliptic systems modelling simultaneous cooperation and competition,, , (). [33] H. Tavares and T. Weth, Existence and symmetry results for competing variational systems, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 715-740. doi: 10.1007/s00030-012-0176-z. [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), 279-300. doi: 10.1016/j.anihpc.2011.10.006. [35] S. Terracini and G. Verzini, Multipulse phases in $k$-mixtures of Bose-Einstein condensates, Arch. Ration. Mech. Anal., 194 (2009), 717-741. doi: 10.1007/s00205-008-0172-y. [36] R. Tian and Z.-Q. Wang, Multiple solitary wave solutions of nonlinear Schrödinger systems, Topol. Methods Nonlinear Anal., 37 (2011), 203-223. [37] W. P. Ziemer, Weakly Differentiable Functions, Graduate Texts in Mathematics, Vol. 120, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4612-1015-3.

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), 509-579. doi: 10.1007/s00220-014-2281-9. [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), 1118-1136. doi: 10.1016/j.jfa.2008.10.021. [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), 231-246. doi: 10.1007/BF02412022. [4] A. Ambrosetti and E. Colorado, Standing waves of some coupled nonlinear Schrödinger equations, J. Lond. Math. Soc. (2), 75 (2007), 67-82. doi: 10.1112/jlms/jdl020. [5] A. Ambrosetti and G. Prodi, A Primer of Nonlinear Analysis, Cambridge Studies in Advanced Mathematics, 34, Cambridge University Press, Cambridge, 1993. [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), 345-361. doi: 10.1007/s00526-009-0265-y. [7] T. Bartsch, Z.-Q. Wang and J. Wei, Bound states for a coupled Schrödinger system, J. Fixed Point Theory Appl., 2 (2007), 353-367. doi: 10.1007/s11784-007-0033-6. [8] H. Brezis, Semilinear equations in $R^N$ without condition at infinity, Appl. Math. Optim., 12 (1984), 271-282. doi: 10.1007/BF01449045. [9] T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10, New York University Courant Institute of Mathematical Sciences, New York, 2003. [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), 341-361. doi: 10.1016/j.physd.2004.06.002. [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), 4289-4311. doi: 10.1016/j.jde.2013.08.009. [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), 953-969. doi: 10.1016/j.anihpc.2010.01.009. [13] , DispersiveWiki project,, URL , (). [14] M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry. I, J. Funct. Anal., 74 (1987), 160-197. doi: 10.1016/0022-1236(87)90044-9. [15] M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry. II, J. Funct. Anal., 94 (1990), 308-348. doi: 10.1016/0022-1236(90)90016-E. [16] N. Ikoma, Compactness of minimizing sequences in nonlinear Schrödinger systems under multiconstraint conditions, Adv. Nonlinear Stud., 14 (2014), 115-136. [17] O. Kavian and F. B. Weissler, Self-similar solutions of the pseudo-conformally invariant nonlinear Schrödinger equation, Michigan Math. J., 41 (1994), 151-173. doi: 10.1307/mmj/1029004922. [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), 629-653. doi: 10.1007/s00220-005-1313-x. [19] Z. Liu and Z.-Q. Wang, Multiple bound states of nonlinear Schrödinger systems, Comm. Math. Phys., 282 (2008), 721-731. doi: 10.1007/s00220-008-0546-x. [20] L. A. Maia, E. Montefusco and B. Pellacci, Positive solutions for a weakly coupled nonlinear Schrödinger system, J. Differential Equations, 229 (2006), 743-767. doi: 10.1016/j.jde.2006.07.002. [21] L. A. Maia, E. Montefusco and B. Pellacci, Orbital stability property for coupled nonlinear Schrödinger equations, Adv. Nonlinear Stud., 10 (2010), 681-705. [22] N. V. Nguyen, On the orbital stability of solitary waves for the 2-coupled nonlinear Schrödinger system, Commun. Math. Sci., 9 (2011), 997-1012. doi: 10.4310/CMS.2011.v9.n4.a3. [23] N. V. Nguyen and Z.-Q. Wang, Orbital stability of solitary waves for a nonlinear Schrödinger system, Adv. Differential Equations, 16 (2011), 977-1000. [24] B. Noris and M. Ramos, Existence and bounds of positive solutions for a nonlinear Schrödinger system, Proc. Amer. Math. Soc., 138 (2010), 1681-1692. doi: 10.1090/S0002-9939-10-10231-7. [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), 1245-1273. doi: 10.4171/JEMS/332. [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,, , (). [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), 1529-1547. doi: 10.1016/j.jde.2012.11.003. [28] M. Ohta, Stability of solitary waves for coupled nonlinear Schrödinger equations, Nonlinear Anal., 26 (1996), 933-939. doi: 10.1016/0362-546X(94)00340-8. [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), 103-124. doi: 10.1007/s00526-012-0571-7. [30] J. Shatah, Stable standing waves of nonlinear Klein-Gordon equations, Commun. Math. Phys., 91 (1983), 313-327. doi: 10.1007/BF01208779. [31] B. Sirakov, Least energy solitary waves for a system of nonlinear Schrödinger equations in $\mathbbR^n$, Comm. Math. Phys., 271 (2007), 199-221. doi: 10.1007/s00220-006-0179-x. [32] N. Soave, On existence and phase separation of positive solutions to nonlinear elliptic systems modelling simultaneous cooperation and competition,, , (). [33] H. Tavares and T. Weth, Existence and symmetry results for competing variational systems, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 715-740. doi: 10.1007/s00030-012-0176-z. [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), 279-300. doi: 10.1016/j.anihpc.2011.10.006. [35] S. Terracini and G. Verzini, Multipulse phases in $k$-mixtures of Bose-Einstein condensates, Arch. Ration. Mech. Anal., 194 (2009), 717-741. doi: 10.1007/s00205-008-0172-y. [36] R. Tian and Z.-Q. Wang, Multiple solitary wave solutions of nonlinear Schrödinger systems, Topol. Methods Nonlinear Anal., 37 (2011), 203-223. [37] W. P. Ziemer, Weakly Differentiable Functions, Graduate Texts in Mathematics, Vol. 120, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4612-1015-3.
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