Stable standing waves for cubic nonlinear Schrödinger systems with partial confinement

Huifang Jia; Gongbao Li; Xiao Luo

doi:10.3934/dcds.2020148

Previous Article
Measure theoretic pressure and dimension formula for non-ergodic measures
DCDS Home
This Issue
Next Article
Movement of time-delayed hot spots in Euclidean space for a degenerate initial state

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

Stable standing waves for cubic nonlinear Schrödinger systems with partial confinement

Huifang Jia ^1, , Gongbao Li ^2,, and Xiao Luo ^3,

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.

Keywords: Nonlinear Schrödinger system, normalized solutions, stability, standing waves.

Mathematics Subject Classification: Primary: 35J20, 35J25, 35J60.

Citation: Huifang Jia, Gongbao Li, Xiao Luo. Stable standing waves for cubic nonlinear Schrödinger systems with partial confinement. Discrete & 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. Google Scholar
[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. Google Scholar
[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. 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. 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. 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. 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. Google Scholar
[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. 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. 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. 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. 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. 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. 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. Google Scholar
[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. 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. 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. Google Scholar
[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. 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. 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. 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. 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. Google Scholar
[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. 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. 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. 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. 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. 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. 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. Google Scholar
[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. 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

[1]	Santosh Bhattarai. Stability of normalized solitary waves for three coupled nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems, 2016, 36 (4) : 1789-1811. doi: 10.3934/dcds.2016.36.1789
[2]	Reika Fukuizumi. Stability and instability of standing waves for the nonlinear Schrödinger equation with harmonic potential. Discrete & Continuous Dynamical Systems, 2001, 7 (3) : 525-544. doi: 10.3934/dcds.2001.7.525
[3]	François Genoud. Existence and stability of high frequency standing waves for a nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems, 2009, 25 (4) : 1229-1247. doi: 10.3934/dcds.2009.25.1229
[4]	Alex H. Ardila. Stability of standing waves for a nonlinear SchrÖdinger equation under an external magnetic field. Communications on Pure & Applied Analysis, 2018, 17 (1) : 163-175. doi: 10.3934/cpaa.2018010
[5]	Reika Fukuizumi, Louis Jeanjean. Stability of standing waves for a nonlinear Schrödinger equation wdelta potentialith a repulsive Dirac. Discrete & Continuous Dynamical Systems, 2008, 21 (1) : 121-136. doi: 10.3934/dcds.2008.21.121
[6]	Jun-ichi Segata. Initial value problem for the fourth order nonlinear Schrödinger type equation on torus and orbital stability of standing waves. Communications on Pure & Applied Analysis, 2015, 14 (3) : 843-859. doi: 10.3934/cpaa.2015.14.843
[7]	Fábio Natali, Ademir Pastor. Stability properties of periodic standing waves for the Klein-Gordon-Schrödinger system. Communications on Pure & Applied Analysis, 2010, 9 (2) : 413-430. doi: 10.3934/cpaa.2010.9.413
[8]	François Genoud, Charles A. Stuart. Schrödinger equations with a spatially decaying nonlinearity: Existence and stability of standing waves. Discrete & Continuous Dynamical Systems, 2008, 21 (1) : 137-186. doi: 10.3934/dcds.2008.21.137
[9]	Masahito Ohta. Strong instability of standing waves for nonlinear Schrödinger equations with a partial confinement. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1671-1680. doi: 10.3934/cpaa.2018080
[10]	Xiaoyu Zeng. Asymptotic properties of standing waves for mass subcritical nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems, 2017, 37 (3) : 1749-1762. doi: 10.3934/dcds.2017073
[11]	Yue Liu. Existence of unstable standing waves for the inhomogeneous nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2008, 7 (1) : 193-209. doi: 10.3934/cpaa.2008.7.193
[12]	Soohyun Bae, Jaeyoung Byeon. Standing waves of nonlinear Schrödinger equations with optimal conditions for potential and nonlinearity. Communications on Pure & Applied Analysis, 2013, 12 (2) : 831-850. doi: 10.3934/cpaa.2013.12.831
[13]	Zhi Chen, Xianhua Tang, Ning Zhang, Jian Zhang. Standing waves for Schrödinger-Poisson system with general nonlinearity. Discrete & Continuous Dynamical Systems, 2019, 39 (10) : 6103-6129. doi: 10.3934/dcds.2019266
[14]	Chuangye Liu, Rushun Tian. Normalized solutions for 3-coupled nonlinear Schrödinger equations. Communications on Pure & Applied Analysis, 2020, 19 (11) : 5115-5130. doi: 10.3934/cpaa.2020229
[15]	Songbai Peng, Aliang Xia. Normalized solutions of supercritical nonlinear fractional Schrödinger equation with potential. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021128
[16]	Nghiem V. Nguyen, Zhi-Qiang Wang. Existence and stability of a two-parameter family of solitary waves for a 2-coupled nonlinear Schrödinger system. Discrete & Continuous Dynamical Systems, 2016, 36 (2) : 1005-1021. doi: 10.3934/dcds.2016.36.1005
[17]	Jaeyoung Byeon, Louis Jeanjean. Multi-peak standing waves for nonlinear Schrödinger equations with a general nonlinearity. Discrete & Continuous Dynamical Systems, 2007, 19 (2) : 255-269. doi: 10.3934/dcds.2007.19.255
[18]	Nan Lu. Non-localized standing waves of the hyperbolic cubic nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems, 2015, 35 (8) : 3533-3567. doi: 10.3934/dcds.2015.35.3533
[19]	Jun-ichi Segata. Well-posedness and existence of standing waves for the fourth order nonlinear Schrödinger type equation. Discrete & Continuous Dynamical Systems, 2010, 27 (3) : 1093-1105. doi: 10.3934/dcds.2010.27.1093
[20]	Zaihui Gan, Jian Zhang. Blow-up, global existence and standing waves for the magnetic nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems, 2012, 32 (3) : 827-846. doi: 10.3934/dcds.2012.32.827

American Institute of Mathematical Sciences

Stable standing waves for cubic nonlinear Schrödinger systems with partial confinement

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

Stable standing waves for cubic nonlinear Schrödinger systems with partial confinement

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors