# American Institute of Mathematical Sciences

May  2020, 40(5): 2739-2766. doi: 10.3934/dcds.2020148

## Stable standing waves for cubic nonlinear Schrödinger systems with partial confinement

 1 School of Mathematics and Statistics, Central China Normal University, Wuhan, 430079, China 2 School of Mathematics and Information Science, Guangzhou University, Guangzhou, 510405, China 3 School of Mathematics, Hefei University of Technology, Hefei, 230009, China

* Corresponding author: Gongbao Li

Received  June 2019 Published  March 2020

Fund Project: The project is supported by National Natural Science Foundation of China(Grant No.11771166), Hubei Key Laboratory of Mathematical Sciences and Program for Changjiang Scholars and Innovative Research Team in University #IRT_17R46. H.F Jia is also supported by National Postdoctoral Science Foundation of China(Grant No.2019M662833). X. Luo is also supported by National Natural Science Foundation of China(Grant No.11901147)

In this article, we deal with the existence, qualitative and symmetry properties of normalized solutions to the following nonlinear Schrödinger system
 $\begin{equation*} \begin{cases} -\Delta u+(x_{1}^{2}+x_{2}^{2})u = \lambda_{1}u+\mu_{1}u^{3}+\beta uv^{2}, &\quad x\in \mathbb{R}^3,\\ -\Delta v+(x_{1}^{2}+x_{2}^{2})v = \lambda_{2}v+\mu_{2}v^{3}+\beta u^{2}v, &\quad x\in \mathbb{R}^3, \end{cases} \end{equation*}$
where
 $\mu_{i}>0$
(
 $i$
= 1, 2),
 $\beta>0$
, and the frequencies
 $\lambda_{1}$
,
 $\lambda_{2}$
are unknown and appear as Lagrange multipliers. In addition, we study the stability of the corresponding standing waves for the related time-dependent Schrödinger systems. We mainly extend the results in J. Bellazzini et al. (Commun. Math. Phys. 2017), which dealt with mass-supercritical nonlinear Schrödinger equation with partial confinement, to cubic nonlinear Schrödinger systems with partial confinement.
Citation: Huifang Jia, Gongbao Li, Xiao Luo. Stable standing waves for cubic nonlinear Schrödinger systems with partial confinement. Discrete and Continuous Dynamical Systems, 2020, 40 (5) : 2739-2766. doi: 10.3934/dcds.2020148
##### References:
 [1] N. Akhmediev and A. Ankiewicz, Partially coherent solitons on a finite background, Phys. Rev. Lett., 82 (1999), 2661. doi: 10.1103/PhysRevLett.82.2661. [2] P. Antonelli, R. Carles and J. Drumond Silva, Scattering for nonlinear Schrödinger equation under partial harmonic confinement, Commun. Math. Phys., 334 (2015), 367-396.  doi: 10.1007/s00220-014-2166-y. [3] 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 Differ. Equ., 37 (2010), 345-361.  doi: 10.1007/s00526-009-0265-y. [4] T. Bartsch and L. Jeanjean, Normalized solutions for nonlinear Schrödinger systems, Proc. Roy. Soc. Edinburgh Sect. A, 148 (2018), 225-242.  doi: 10.1017/S0308210517000087. [5] T. Bartsch, L. Jeanjean and N. Soave, Normalized solutions for a system of coupled cubic Schrödinger equations on $\mathbb{R}^{3}$, J. Math. Pures Appl., 106 (2016), 583-614.  doi: 10.1016/j.matpur.2016.03.004. [6] T. Bartsch and N. Soave, A natural constraint approach to normalized solutions of nonlinear Schrödinger equations and systems, Journal of Functional Analysis, 272 (2017), 4998-5037.  doi: 10.1016/j.jfa.2017.01.025. [7] T. Bartsch and N. Soave, Multiple normalized solutions for a competing system of Schrödinger equations, Calc. Var. Partial Differ. Equ., 58 (2019), Art. 22, 24 pp. doi: 10.1007/s00526-018-1476-x. [8] T. Bartsch, Z.-Q. Wang and J. C. Wei, Bound states for a coupled Schrödinger system, J. Fixed Point Theory Appl., 2 (2007), 353-367.  doi: 10.1007/s11784-007-0033-6. [9] J. Bellazzini, N. Boussaïd, L. Jeanjean and N. Visciglia, Existence and stability of standing waves for supercritical NLS with a partial confinement, Commun. Math. Phys., 353 (2017), 229-251.  doi: 10.1007/s00220-017-2866-1. [10] H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490.  doi: 10.1090/S0002-9939-1983-0699419-3. [11] T. Cazenave and P.-L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Commun. Math. Phys., 85 (1982), 549-561.  doi: 10.1007/BF01403504. [12] Z. J. Chen and W. M. Zou, An optimal constant for the existence of least energy solutions of a coupled Schrödinger system, Calc. Var. Partial Differ. Equ., 48 (2013), 695-711.  doi: 10.1007/s00526-012-0568-2. [13] B. D. Esry, C. H. Greene, J. P. Burke Jr. and J. L. Bohn, Hartree-Fock theory for double condensates, Phys. Rev. Lett., 78 (1997), 3594-3597.  doi: 10.1103/PhysRevLett.78.3594. [14] L. Fanelli and E. Montefusco, On the blow-up threshold for weakly coupled nonlinear Schrödinger equations, J. Phys. A: Math. and Theor., 40 (2007), 14139-14150.  doi: 10.1088/1751-8113/40/47/007. [15] B. H. Feng, Sharp threshold of global existence and instability of standing wave for the Schrödinger-Hartree equation with a harmonic potential, Nonlinear Analysis Real World Applications, 31 (2016), 132-145.  doi: 10.1016/j.nonrwa.2016.01.012. [16] D. J. Frantzeskakis, Dark solitons in atomic Bose Einstein condensates: From theory to experiments, J. Phys. A: Math. Theor., 43 (2010), 213001, 68 pp. doi: 10.1088/1751-8113/43/21/213001. [17] R. Fukuizumi, Stability and instability of standing waves for the nonlinear Schrödinger equation with harmonic potential, Discrete Contin. Dyn. Syst., 7 (2001), 525-544.  doi: 10.3934/dcds.2001.7.525. [18] T. X. Gou, Existence and orbital stability of standing waves to nonlinear Schrödinger system with partial confinement, J. Math. Phys., 59 (2018), 071508, 12 pp. doi: 10.1063/1.5028208. [19] T. X. Gou and L. Jeanjean, Existence and orbital stability of standing waves for nonlinear Schrödinger systems, Nonlinear Anal., 144 (2016), 10-22.  doi: 10.1016/j.na.2016.05.016. [20] T. X. Gou and L. Jeanjean, Multiple positive normalized solutions for nonlinear Schrödinger systems, Nonlinearity, 31 (2018), 2319-2345.  doi: 10.1088/1361-6544/aab0bf. [21] Y. J. Guo, S. Li, J. C. Wei and X. Y. Zeng, Ground states of two-component attractive Bose-Einstein condensates Ⅰ: Existence and uniqueness, Journal of Functional Analysis, 276 (2019), 183-230.  doi: 10.1016/j.jfa.2018.09.015. [22] Y. J. Guo, S. Li, J. C. Wei and X. Y. Zeng, Ground states of two-component attractive Bose-Einstein condensates Ⅱ: Semi-trivial limit behavior, Trans. Amer. Math. Soc., 371 (2019), 6903-6948.  doi: 10.1090/tran/7540. [23] Y. J. Guo, X. Y. Zeng and H.-S. Zhou, Blow-up solutions for two coupled Gross-Pitaevskii equations with attractive interactions, Discrete Contin. Dyn. Syst., 37 (2017), 3749-3786.  doi: 10.3934/dcds.2017159. [24] Y. J. Guo, X. Y. Zeng and H.-S. Zhou, Blow-up behavior of ground states for a nonlinear Schrödinger system with attractive and repulsive interactions, J. Differential Equations, 264 (2018), 1411-1441.  doi: 10.1016/j.jde.2017.09.039. [25] H. Hajaiej, Orbital stability of standing waves of two-component Bose-Einstein condensates with internal atomic Josephson junction, J. Math. Anal. Appl., 420 (2014), 195-206.  doi: 10.1016/j.jmaa.2014.04.072. [26] C. E. Kenig, Carleman estimates, uniform Sobolev inequalities for second-order differential operators, and unique continuation theorems, Proceedings of the International Congress of Mathematicians, Amer. Math. Soc., Providence, RI, 1, 2 (1987), 948–960. doi: 10.1007/s10444-011-9194-3. [27] E. H. Lieb and M. Loss, Analysis, Second edition, Graduate Studies in Mathematics, 14. American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/014. [28] P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. Ⅱ, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 223-283.  doi: 10.1016/S0294-1449(16)30422-X. [29] B. Malomed, Multi-component Bose-Einstein condensates: Theory, Emergent Nonlinear Phenomena in Bose-Einstein Condensation, Springer-Verlag, Berlin, (2008), 287–305. [30] N. V. Nguyen, R. S. Tian, B. Deconinck and N. Sheils, Global existence of a coupled system of Schrödinger equations with power-type nonlinearities, J. Math. Phys., 54 (2013), 011503, 19 pp. doi: 10.1063/1.4774149. [31] N. V. Nguyen and Z.-Q. Wang, Existence and stability of a two-parameter family of solitary waves for a 2-coupled nonlinear Schrödinger system, Discrete Contin. Dyn. Syst., 36 (2016), 1005-1021.  doi: 10.3934/dcds.2016.36.1005. [32] N. V. Nguyen and Z.-Q. Wang, Orbital stability of solitary waves for a nonlinear Schrödinger system, Adv. Differ. Equ., 16 (2011), 977-1000. [33] N. V. Nguyen and Z.-Q. Wang, Orbital stability of solitary waves of a 3-coupled nonlinear Schrödinger system, Nonlinear Anal., 90 (2013), 1-26.  doi: 10.1016/j.na.2013.05.027. [34] 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., 14 (2012), 1245-1273.  doi: 10.4171/JEMS/332. [35] B. Noris, H. Tavares and G. Verzini, Stable solitary waves with prescribed $L^{2}$-mass for the cubic Schrödinger system with trapping potentials, Discrete Contin. Dyn. Syst., 35 (2015), 6085-6112.  doi: 10.3934/dcds.2015.35.6085. [36] S. J. Peng and Z.-Q. Wang, Segregated and synchronized vector solutions for nonlinear Schrödinger systems, Arch. Ration. Mech. Anal., 208 (2013), 305-339.  doi: 10.1007/s00205-012-0598-0. [37] J. Royo-Letelier, Segregation and symmetry breaking of strongly coupled two-component Bose-Einstein condensates in a harmonic trap, Calc. Var. Partial Differ. Equ., 49 (2014), 103-124.  doi: 10.1007/s00526-012-0571-7. [38] M. Shibata, A new rearrangement inequality and its application for $L^{2}$-constraint minimizing problems, Math. Z., 287 (2017), 341-359.  doi: 10.1007/s00209-016-1828-1. [39] H. Tavares and S. Terracini, Sign-changing solutions of competition-diffusion elliptic systems and optimal partition problems, Ann. Inst. Henri Poincaré Anal. Non Linéaire, 29 (2012), 279-300.  doi: 10.1016/j.anihpc.2011.10.006. [40] J. Zhang, Sharp threshold for blow up and global existence in nonlinear Schrödinger equations under a harmonic potential, Comm. Partial Differential Equations, 30 (2005), 1429-1443.  doi: 10.1080/03605300500299539.

show all references

##### References:
 [1] N. Akhmediev and A. Ankiewicz, Partially coherent solitons on a finite background, Phys. Rev. Lett., 82 (1999), 2661. doi: 10.1103/PhysRevLett.82.2661. [2] P. Antonelli, R. Carles and J. Drumond Silva, Scattering for nonlinear Schrödinger equation under partial harmonic confinement, Commun. Math. Phys., 334 (2015), 367-396.  doi: 10.1007/s00220-014-2166-y. [3] 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 Differ. Equ., 37 (2010), 345-361.  doi: 10.1007/s00526-009-0265-y. [4] T. Bartsch and L. Jeanjean, Normalized solutions for nonlinear Schrödinger systems, Proc. Roy. Soc. Edinburgh Sect. A, 148 (2018), 225-242.  doi: 10.1017/S0308210517000087. [5] T. Bartsch, L. Jeanjean and N. Soave, Normalized solutions for a system of coupled cubic Schrödinger equations on $\mathbb{R}^{3}$, J. Math. Pures Appl., 106 (2016), 583-614.  doi: 10.1016/j.matpur.2016.03.004. [6] T. Bartsch and N. Soave, A natural constraint approach to normalized solutions of nonlinear Schrödinger equations and systems, Journal of Functional Analysis, 272 (2017), 4998-5037.  doi: 10.1016/j.jfa.2017.01.025. [7] T. Bartsch and N. Soave, Multiple normalized solutions for a competing system of Schrödinger equations, Calc. Var. Partial Differ. Equ., 58 (2019), Art. 22, 24 pp. doi: 10.1007/s00526-018-1476-x. [8] T. Bartsch, Z.-Q. Wang and J. C. Wei, Bound states for a coupled Schrödinger system, J. Fixed Point Theory Appl., 2 (2007), 353-367.  doi: 10.1007/s11784-007-0033-6. [9] J. Bellazzini, N. Boussaïd, L. Jeanjean and N. Visciglia, Existence and stability of standing waves for supercritical NLS with a partial confinement, Commun. Math. Phys., 353 (2017), 229-251.  doi: 10.1007/s00220-017-2866-1. [10] H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490.  doi: 10.1090/S0002-9939-1983-0699419-3. [11] T. Cazenave and P.-L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Commun. Math. Phys., 85 (1982), 549-561.  doi: 10.1007/BF01403504. [12] Z. J. Chen and W. M. Zou, An optimal constant for the existence of least energy solutions of a coupled Schrödinger system, Calc. Var. Partial Differ. Equ., 48 (2013), 695-711.  doi: 10.1007/s00526-012-0568-2. [13] B. D. Esry, C. H. Greene, J. P. Burke Jr. and J. L. Bohn, Hartree-Fock theory for double condensates, Phys. Rev. Lett., 78 (1997), 3594-3597.  doi: 10.1103/PhysRevLett.78.3594. [14] L. Fanelli and E. Montefusco, On the blow-up threshold for weakly coupled nonlinear Schrödinger equations, J. Phys. A: Math. and Theor., 40 (2007), 14139-14150.  doi: 10.1088/1751-8113/40/47/007. [15] B. H. Feng, Sharp threshold of global existence and instability of standing wave for the Schrödinger-Hartree equation with a harmonic potential, Nonlinear Analysis Real World Applications, 31 (2016), 132-145.  doi: 10.1016/j.nonrwa.2016.01.012. [16] D. J. Frantzeskakis, Dark solitons in atomic Bose Einstein condensates: From theory to experiments, J. Phys. A: Math. Theor., 43 (2010), 213001, 68 pp. doi: 10.1088/1751-8113/43/21/213001. [17] R. Fukuizumi, Stability and instability of standing waves for the nonlinear Schrödinger equation with harmonic potential, Discrete Contin. Dyn. Syst., 7 (2001), 525-544.  doi: 10.3934/dcds.2001.7.525. [18] T. X. Gou, Existence and orbital stability of standing waves to nonlinear Schrödinger system with partial confinement, J. Math. Phys., 59 (2018), 071508, 12 pp. doi: 10.1063/1.5028208. [19] T. X. Gou and L. Jeanjean, Existence and orbital stability of standing waves for nonlinear Schrödinger systems, Nonlinear Anal., 144 (2016), 10-22.  doi: 10.1016/j.na.2016.05.016. [20] T. X. Gou and L. Jeanjean, Multiple positive normalized solutions for nonlinear Schrödinger systems, Nonlinearity, 31 (2018), 2319-2345.  doi: 10.1088/1361-6544/aab0bf. [21] Y. J. Guo, S. Li, J. C. Wei and X. Y. Zeng, Ground states of two-component attractive Bose-Einstein condensates Ⅰ: Existence and uniqueness, Journal of Functional Analysis, 276 (2019), 183-230.  doi: 10.1016/j.jfa.2018.09.015. [22] Y. J. Guo, S. Li, J. C. Wei and X. Y. Zeng, Ground states of two-component attractive Bose-Einstein condensates Ⅱ: Semi-trivial limit behavior, Trans. Amer. Math. Soc., 371 (2019), 6903-6948.  doi: 10.1090/tran/7540. [23] Y. J. Guo, X. Y. Zeng and H.-S. Zhou, Blow-up solutions for two coupled Gross-Pitaevskii equations with attractive interactions, Discrete Contin. Dyn. Syst., 37 (2017), 3749-3786.  doi: 10.3934/dcds.2017159. [24] Y. J. Guo, X. Y. Zeng and H.-S. Zhou, Blow-up behavior of ground states for a nonlinear Schrödinger system with attractive and repulsive interactions, J. Differential Equations, 264 (2018), 1411-1441.  doi: 10.1016/j.jde.2017.09.039. [25] H. Hajaiej, Orbital stability of standing waves of two-component Bose-Einstein condensates with internal atomic Josephson junction, J. Math. Anal. Appl., 420 (2014), 195-206.  doi: 10.1016/j.jmaa.2014.04.072. [26] C. E. Kenig, Carleman estimates, uniform Sobolev inequalities for second-order differential operators, and unique continuation theorems, Proceedings of the International Congress of Mathematicians, Amer. Math. Soc., Providence, RI, 1, 2 (1987), 948–960. doi: 10.1007/s10444-011-9194-3. [27] E. H. Lieb and M. Loss, Analysis, Second edition, Graduate Studies in Mathematics, 14. American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/014. [28] P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. Ⅱ, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 223-283.  doi: 10.1016/S0294-1449(16)30422-X. [29] B. Malomed, Multi-component Bose-Einstein condensates: Theory, Emergent Nonlinear Phenomena in Bose-Einstein Condensation, Springer-Verlag, Berlin, (2008), 287–305. [30] N. V. Nguyen, R. S. Tian, B. Deconinck and N. Sheils, Global existence of a coupled system of Schrödinger equations with power-type nonlinearities, J. Math. Phys., 54 (2013), 011503, 19 pp. doi: 10.1063/1.4774149. [31] N. V. Nguyen and Z.-Q. Wang, Existence and stability of a two-parameter family of solitary waves for a 2-coupled nonlinear Schrödinger system, Discrete Contin. Dyn. Syst., 36 (2016), 1005-1021.  doi: 10.3934/dcds.2016.36.1005. [32] N. V. Nguyen and Z.-Q. Wang, Orbital stability of solitary waves for a nonlinear Schrödinger system, Adv. Differ. Equ., 16 (2011), 977-1000. [33] N. V. Nguyen and Z.-Q. Wang, Orbital stability of solitary waves of a 3-coupled nonlinear Schrödinger system, Nonlinear Anal., 90 (2013), 1-26.  doi: 10.1016/j.na.2013.05.027. [34] 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., 14 (2012), 1245-1273.  doi: 10.4171/JEMS/332. [35] B. Noris, H. Tavares and G. Verzini, Stable solitary waves with prescribed $L^{2}$-mass for the cubic Schrödinger system with trapping potentials, Discrete Contin. Dyn. Syst., 35 (2015), 6085-6112.  doi: 10.3934/dcds.2015.35.6085. [36] S. J. Peng and Z.-Q. Wang, Segregated and synchronized vector solutions for nonlinear Schrödinger systems, Arch. Ration. Mech. Anal., 208 (2013), 305-339.  doi: 10.1007/s00205-012-0598-0. [37] J. Royo-Letelier, Segregation and symmetry breaking of strongly coupled two-component Bose-Einstein condensates in a harmonic trap, Calc. Var. Partial Differ. Equ., 49 (2014), 103-124.  doi: 10.1007/s00526-012-0571-7. [38] M. Shibata, A new rearrangement inequality and its application for $L^{2}$-constraint minimizing problems, Math. Z., 287 (2017), 341-359.  doi: 10.1007/s00209-016-1828-1. [39] H. Tavares and S. Terracini, Sign-changing solutions of competition-diffusion elliptic systems and optimal partition problems, Ann. Inst. Henri Poincaré Anal. Non Linéaire, 29 (2012), 279-300.  doi: 10.1016/j.anihpc.2011.10.006. [40] J. Zhang, Sharp threshold for blow up and global existence in nonlinear Schrödinger equations under a harmonic potential, Comm. Partial Differential Equations, 30 (2005), 1429-1443.  doi: 10.1080/03605300500299539.
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