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 & Continuous Dynamical Systems - A, 2020, 40 (5) : 2739-2766. doi: 10.3934/dcds.2020148
References:
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N. Akhmediev and A. Ankiewicz, Partially coherent solitons on a finite background, Phys. Rev. Lett., 82 (1999), 2661. doi: 10.1103/PhysRevLett.82.2661.  Google Scholar

[2]

P. AntonelliR. 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.  Google Scholar

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T. BartschN. 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.  Google Scholar

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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.  Google Scholar

[5]

T. BartschL. 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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[8]

T. BartschZ.-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.  Google Scholar

[9]

J. BellazziniN. BoussaïdL. 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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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[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.  Google Scholar

[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.  Google Scholar

[16]

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[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

Y. J. GuoS. LiJ. 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.  Google Scholar

[22]

Y. J. GuoS. LiJ. 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.  Google Scholar

[23]

Y. J. GuoX. 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.  Google Scholar

[24]

Y. J. GuoX. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

B. Malomed, Multi-component Bose-Einstein condensates: Theory, Emergent Nonlinear Phenomena in Bose-Einstein Condensation, Springer-Verlag, Berlin, (2008), 287–305. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[34]

B. NorisH. TavaresS. 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.  Google Scholar

[35]

B. NorisH. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

P. AntonelliR. 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.  Google Scholar

[3]

T. BartschN. 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.  Google Scholar

[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.  Google Scholar

[5]

T. BartschL. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

T. BartschZ.-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.  Google Scholar

[9]

J. BellazziniN. BoussaïdL. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

B. D. EsryC. H. GreeneJ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

Y. J. GuoS. LiJ. 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.  Google Scholar

[22]

Y. J. GuoS. LiJ. 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.  Google Scholar

[23]

Y. J. GuoX. 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.  Google Scholar

[24]

Y. J. GuoX. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

B. Malomed, Multi-component Bose-Einstein condensates: Theory, Emergent Nonlinear Phenomena in Bose-Einstein Condensation, Springer-Verlag, Berlin, (2008), 287–305. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[34]

B. NorisH. TavaresS. 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.  Google Scholar

[35]

B. NorisH. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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