×

Stable standing waves for cubic nonlinear Schrödinger systems with partial confinement. (English) Zbl 1437.35189

This paper deals with the existence, qualitative and symmetry properties of normalized solutions to a coupled nonlinear Schrödinger system with partial confinement. The stability of the corresponding standing waves for the related time-dependent Schrödinger systems are also studied. These results generalize those in [J. Bellazzini et al., Commun. Math. Phys. 353, No. 1, 229–251 (2017; Zbl 1367.35150)] where the mass-supercritical nonlinear Schrödinger equation with partial confinement was studied.

MSC:

35J10 Schrödinger operator, Schrödinger equation
35J47 Second-order elliptic systems
35J91 Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian

Citations:

Zbl 1367.35150
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] N. Akhmediev and A. Ankiewicz, Partially coherent solitons on a finite background, Phys. Rev. Lett., 82 (1999), 2661.
[2] P. Antonelli; R. Carles; J. Drumond Silva, Scattering for nonlinear Schrödinger equation under partial harmonic confinement, Commun. Math. Phys., 334, 367-396 (2015) · Zbl 1309.35124 · doi:10.1007/s00220-014-2166-y
[3] T. Bartsch; N. Dancer; 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, 345-361 (2010) · Zbl 1189.35074 · doi:10.1007/s00526-009-0265-y
[4] T. Bartsch; L. Jeanjean, Normalized solutions for nonlinear Schrödinger systems, Proc. Roy. Soc. Edinburgh Sect. A, 148, 225-242 (2018) · Zbl 1393.35035 · doi:10.1017/S0308210517000087
[5] T. Bartsch; L. Jeanjean; N. Soave, Normalized solutions for a system of coupled cubic Schrödinger equations on \(\mathbb{R}^3 \), J. Math. Pures Appl., 106, 583-614 (2016) · Zbl 1347.35107 · doi:10.1016/j.matpur.2016.03.004
[6] T. Bartsch; N. Soave, A natural constraint approach to normalized solutions of nonlinear Schrödinger equations and systems, Journal of Functional Analysis, 272, 4998-5037 (2017) · Zbl 1485.35173 · 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. · Zbl 1409.35076
[8] T. Bartsch; Z.-Q. Wang; J. C. Wei, Bound states for a coupled Schrödinger system, J. Fixed Point Theory Appl., 2, 353-367 (2007) · Zbl 1153.35390 · doi:10.1007/s11784-007-0033-6
[9] J. Bellazzini; N. Boussaïd; L. Jeanjean; N. Visciglia, Existence and stability of standing waves for supercritical NLS with a partial confinement, Commun. Math. Phys., 353, 229-251 (2017) · Zbl 1367.35150 · doi:10.1007/s00220-017-2866-1
[10] H. Brézis; E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88, 486-490 (1983) · Zbl 0526.46037 · doi:10.1090/S0002-9939-1983-0699419-3
[11] T. Cazenave; P.-L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Commun. Math. Phys., 85, 549-561 (1982) · Zbl 0513.35007 · doi:10.1007/BF01403504
[12] Z. J. Chen; W. M. Zou, An optimal constant for the existence of least energy solutions of a coupled Schrödinger system, Calc. Var. Partial Differ. Equ., 48, 695-711 (2013) · Zbl 1286.35104 · doi:10.1007/s00526-012-0568-2
[13] B. D. Esry; C. H. Greene; J. P. Burke Jr.; J. L. Bohn, Hartree-Fock theory for double condensates, Phys. Rev. Lett., 78, 3594-3597 (1997) · doi:10.1103/PhysRevLett.78.3594
[14] L. Fanelli; E. Montefusco, On the blow-up threshold for weakly coupled nonlinear Schrödinger equations, J. Phys. A: Math. and Theor., 40, 14139-14150 (2007) · Zbl 1134.35099 · 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, 132-145 (2016) · Zbl 1342.35330 · 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. · Zbl 1192.82033
[17] R. Fukuizumi, Stability and instability of standing waves for the nonlinear Schrödinger equation with harmonic potential, Discrete Contin. Dyn. Syst., 7, 525-544 (2001) · Zbl 0992.35094 · 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. · Zbl 1436.35288
[19] T. X. Gou; L. Jeanjean, Existence and orbital stability of standing waves for nonlinear Schrödinger systems, Nonlinear Anal., 144, 10-22 (2016) · Zbl 1457.35068 · doi:10.1016/j.na.2016.05.016
[20] T. X. Gou; L. Jeanjean, Multiple positive normalized solutions for nonlinear Schrödinger systems, Nonlinearity, 31, 2319-2345 (2018) · Zbl 1396.35009 · doi:10.1088/1361-6544/aab0bf
[21] Y. J. Guo; S. Li; J. C. Wei; X. Y. Zeng, Ground states of two-component attractive Bose-Einstein condensates Ⅰ: Existence and uniqueness, Journal of Functional Analysis, 276, 183-230 (2019) · Zbl 1405.35038 · doi:10.1016/j.jfa.2018.09.015
[22] Y. J. Guo; S. Li; J. C. Wei; X. Y. Zeng, Ground states of two-component attractive Bose-Einstein condensates Ⅱ: Semi-trivial limit behavior, Trans. Amer. Math. Soc., 371, 6903-6948 (2019) · Zbl 1421.35088 · doi:10.1090/tran/7540
[23] Y. J. Guo; X. Y. Zeng; H.-S. Zhou, Blow-up solutions for two coupled Gross-Pitaevskii equations with attractive interactions, Discrete Contin. Dyn. Syst., 37, 3749-3786 (2017) · Zbl 1372.35084 · doi:10.3934/dcds.2017159
[24] Y. J. Guo; X. Y. Zeng; H.-S. Zhou, Blow-up behavior of ground states for a nonlinear Schrödinger system with attractive and repulsive interactions, J. Differential Equations, 264, 1411-1441 (2018) · Zbl 1379.35078 · 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, 195-206 (2014) · Zbl 1295.35060 · 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. · Zbl 0692.35019
[27] E. H. Lieb and M. Loss, Analysis, Second edition, Graduate Studies in Mathematics, 14. American Mathematical Society, Providence, RI, 2001. · Zbl 0966.26002
[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, 223-283 (1984) · Zbl 0704.49004 · 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. · Zbl 1151.82369
[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. · Zbl 1286.35230
[31] N. V. Nguyen; 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, 1005-1021 (2016) · Zbl 1330.35411 · doi:10.3934/dcds.2016.36.1005
[32] N. V. Nguyen; Z.-Q. Wang, Orbital stability of solitary waves for a nonlinear Schrödinger system, Adv. Differ. Equ., 16, 977-1000 (2011) · Zbl 1252.35253
[33] N. V. Nguyen; Z.-Q. Wang, Orbital stability of solitary waves of a 3-coupled nonlinear Schrödinger system, Nonlinear Anal., 90, 1-26 (2013) · Zbl 1281.35080 · doi:10.1016/j.na.2013.05.027
[34] B. Noris; H. Tavares; S. Terracini; G. Verzini, Convergence of minimax structures and continuation of critical points for singularly perturbed systems, J. Eur. Math. Soc., 14, 1245-1273 (2012) · Zbl 1248.35197 · doi:10.4171/JEMS/332
[35] B. Noris; H. Tavares; G. Verzini, Stable solitary waves with prescribed \(L^2\)-mass for the cubic Schrödinger system with trapping potentials, Discrete Contin. Dyn. Syst., 35, 6085-6112 (2015) · Zbl 1336.35321 · doi:10.3934/dcds.2015.35.6085
[36] S. J. Peng; Z.-Q. Wang, Segregated and synchronized vector solutions for nonlinear Schrödinger systems, Arch. Ration. Mech. Anal., 208, 305-339 (2013) · Zbl 1260.35211 · 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, 103-124 (2014) · Zbl 1283.35101 · 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, 341-359 (2017) · Zbl 1382.35012 · doi:10.1007/s00209-016-1828-1
[39] H. Tavares; S. Terracini, Sign-changing solutions of competition-diffusion elliptic systems and optimal partition problems, Ann. Inst. Henri Poincaré Anal. Non Linéaire, 29, 279-300 (2012) · Zbl 1241.35046 · 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, 1429-1443 (2005) · Zbl 1081.35109 · doi:10.1080/03605300500299539
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.