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Movement of time-delayed hot spots in Euclidean space for a degenerate initial state
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 |
$ \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*} $ |
$ \mu_{i}>0 $ |
$ i $ |
$ \beta>0 $ |
$ \lambda_{1} $ |
$ \lambda_{2} $ |
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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