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August  2017, 37(7): 3749-3786. doi: 10.3934/dcds.2017159

Blow-up solutions for two coupled Gross-Pitaevskii equations with attractive interactions

1. 

Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, P.O. Box 71010, Wuhan 430071, China

2. 

Department of Mathematics, School of Science, Wuhan University of Technology, Wuhan 430070, China

* Corresponding author

Received  August 2016 Revised  March 2017 Published  April 2017

Fund Project: This work was supported by NSFC (grants 11322104 and 11671394 for YJGuo, grant 11501555 for XYZeng and grant 11471331 for HSZhou).

The paper is concerned with a system of two coupled time-independent Gross-Pitaevskii equations in $\mathbb{R}^2$, which is used to model two-component Bose-Einstein condensates with both attractive intraspecies and attractive interspecies interactions. This system is essentially an eigenvalue problem of a stationary nonlinear Schrödinger system in $\mathbb{R}^2$, and solutions of the problem are obtained by seeking minimizers of the associated variational functional with constrained mass (i.e. $L^2-$norm constaints). Under a certain type of trapping potentials $V_i(x)$ ($i=1, 2$), the existence, non-existence and uniqueness of this kind of solutions are studied. Moreover, by establishing some delicate energy estimates, we show that each component of the solutions blows up at the same point (i.e., one of the global minima of $V_i(x)$) when the total interaction strength of intraspecies and interspecies goes to a critical value. An optimal blowing up rate for the solutions of the system is also given.

Citation: Yujin Guo, Xiaoyu Zeng, Huan-Song Zhou. Blow-up solutions for two coupled Gross-Pitaevskii equations with attractive interactions. Discrete and Continuous Dynamical Systems, 2017, 37 (7) : 3749-3786. doi: 10.3934/dcds.2017159
References:
[1]

M. H. AndersonJ. R. EnsherM. R. MatthewsC. E. Wieman and E. A. Cornell, Observation of Bose-Einstein condensation in a dilute atomic vapor, Collected Papers of Carl Wieman, (2008), 453-456.  doi: 10.1142/9789812813787_0062.

[2]

W. H. AschbacherJ. FröhlichG. M. GrafK. Schnee and M. Troyer, Symmetry breaking regime in the nonlinear Hartree equation, J. Math. Phys., 43 (2002), 3879-3891.  doi: 10.1063/1.1488673.

[3]

W. Z. Bao and Y. Y. Cai, Ground states of two-component Bose-Einstein condensates with an internal atomic Josephson junction, East Asian J. Appl. Math., 1 (2011), 49-81.  doi: 10.4208/eajam.190310.170510a.

[4]

W. Z. Bao and Y. Y. Cai, Mathematical theory and numerical methods for Bose-Einstein condensation, Kinetic and Related Models, 6 (2013), 1-135.  doi: 10.3934/krm.2013.6.1.

[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.

[6]

T. Bartsch and Z.-Q. Wang, Existence and multiplicity results for some superlinear elliptic problems on $\mathbb{R}^N$, Comm. Partial Differential Equations, 20 (1995), 1725-1741.  doi: 10.1080/03605309508821149.

[7]

M. Caliari and M. Squassina, Location and phase segregation of ground and excited states for 2D Gross-Pitaevskii systems, Dynamics of PDE, 5 (2008), 117-137.  doi: 10.4310/DPDE.2008.v5.n2.a2.

[8]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics 10, Courant Institute of Mathematical Science/AMS, New York, 2003. doi: 10.1090/cln/010.

[9]

Z. J. Chen and W. M. Zou, Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent, Arch. Ration. Mech Anal., 205 (2012), 515-551.  doi: 10.1007/s00205-012-0513-8.

[10]

Z. J. Chen and W. M. Zou, Standing waves for coupled Schrödinger equations with decaying potentials, J. Math. Phys., 54 (2013), 111505, 21pp.  doi: 10.1063/1.4833795.

[11]

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 Differential Equations, 48 (2013), 695-711.  doi: 10.1007/s00526-012-0568-2.

[12]

F. DalfovoS. GiorginiL. P. Pitaevskii and S. Stringari, Theory of Bose-Einstein condensation in trapped gases, Rev. Modern Phys., 71 (1999), 463-512. 

[13] B. GidasW. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\mathbb{R}^N$, in Mathematical analysis and applications Part A, Adv. in Math. Suppl. Stud., Academic Press, New York, 1981. 
[14]

Y. J. Guo and R. Seiringer, On the mass concentration for Bose-Einstein condensates with attractive interactions, Lett. Math. Phys., 104 (2014), 141-156.  doi: 10.1007/s11005-013-0667-9.

[15]

Y. J. Guo, Z. -Q. Wang, X. Y. Zeng and H. -S. Zhou, Properties of ground states of attractive Gross-Pitaevskii equations with multi-well potentials, preprint, arXiv: 1502.01839.

[16]

Y. J. GuoX.Y. Zeng and H.-S. Zhou, Energy estimates and symmetry breaking in attractive Bose-Einstein condensates with ring-shaped potentials, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 809-828.  doi: 10.1016/j.anihpc.2015.01.005.

[17]

Y. J. Guo, X. Y. Zeng and H. -S. Zhou, Blow-up solutions for a nonlinear Schrödinger system with attractive intraspecies and repulsive interspecies interactions, preprint.

[18]

D. S. HallM. R. MatthewsJ. R. EnsherC. E. Wieman and E. A. Cornell, Dynamics of component separation in a binary mixture of Bose-Einstein condensates, Collected Papers of Carl Wieman, (2008), 515-518.  doi: 10.1142/9789812813787_0071.

[19]

N. Ikoma and K. Tanaka, A local mountain pass type result for a system of nonlinear Schrödinger equations, Calc. Var. Partial Differential Equations, 40 (2011), 449-480.  doi: 10.1007/s00526-010-0347-x.

[20]

R. K. Jackson and M. I. Weinstein, Geometric analysis of bifurcation and symmetry breaking in a Gross-Pitaevskii equation, J. Stat. Phys., 116 (2004), 881-905.  doi: 10.1023/B:JOSS.0000037238.94034.75.

[21]

O. Kavian and F. B. Weissler, Self-similar solutions of the pseudo-conformally invariant nonlinear Schrödinger equation, Michigan Math. J., 41 (1994), 151-173.  doi: 10.1307/mmj/1029004922.

[22]

E. W. KirrP. G. Kevrekidis and D. E. Pelinovsky, Symmetry-breaking bifurcation in the nonlinear Schrödinger equation with symmetric potentials, Comm. Math. Phys., 308 (2011), 795-844.  doi: 10.1007/s00220-011-1361-3.

[23]

E. W. KirrP. G. KevrekidisE. Shlizerman and M. I. Weinstein, Symmetry-breaking bifurcation in nonlinear Schrödinger/Gross-Pitaevskii equations, SIAM J. Math. Anal., 40 (2008), 566-604.  doi: 10.1137/060678427.

[24]

Y. C. KuoW. W. Lin and S. F. Shieh, Bifurcation analysis of a two-component Bose-Einstein condensate, Phys. D, 211 (2005), 311-346.  doi: 10.1016/j.physd.2005.09.003.

[25]

T. C. Lin and J. C. Wei, Spikes in two-component systems of nonlinear Schrödinger equations with trapping potentials, J. Differential Equations, 229 (2006), 538-569.  doi: 10.1016/j.jde.2005.12.011.

[26]

P. L. Lions, The concentration-compactness principle in the Caclulus of Variations. The locally compact case, Part Ⅰ: Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109-145. Part Ⅱ: Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 223-283. doi: 10. 1016/S0294-1449(16)30422-X.

[27]

C. Y. LiuN.V. Nguyen and Z.-Q. Wang, Orbital stability of spatially synchronized solitary waves of an m-coupled nonlinear Schrödinger system, J. Math. Phys., 57 (2016), 101501, 20pp.  doi: 10.1063/1.4964255.

[28]

Z.L. Liu and Z.-Q. Wang, Ground states and bound states of a nonlinear Schrödinger system, Advanced Nonlinear Studies, 10 (2010), 175-193.  doi: 10.1515/ans-2010-0109.

[29]

M. Maeda, On the symmetry of the ground states of nonlinear Schrödinger equation with potential, Adv. Nonlinear Stud., 10 (2010), 895-925.  doi: 10.1515/ans-2010-0409.

[30]

E. MontefuscoB. Pellacci and M. Squassina, Semiclassical states for weakly coupled nonlinear Schrödinger systems, J. Eur. Math. Soc., 10 (2008), 47-71.  doi: 10.4171/JEMS/103.

[31]

N.V. Nguyen and Z.-Q. Wang, Orbital stability of solitary waves for a nonlinear Schrödinger system, Adv. Differ. Equations, 16 (2011), 977-1000. 

[32]

W.-M. Ni and I. Takagi, On the shape of least-energy solutions to a semilinear Neumann problem, Comm. Pure Appl. Math., 44 (1991), 819-851.  doi: 10.1002/cpa.3160440705.

[33]

A. Pomponio, Coupled nonlinear Schrödinger systems with potentials, J. Differ. Equations, 227 (2006), 258-281.  doi: 10.1016/j.jde.2005.09.002.

[34] M. Reed and B. Simon, Methods of Modern Mathematical Physics. ô. Analysis of Operators, Academic Press,, New York-London, 1978. 
[35]

J. Royo-Letelier, Segregation and symmetry breaking of strongly coupled two-component Bose-Einstein condensates in a harmonic trap, Calc. Var. Partial Differential Equations, 49 (2014), 103-124.  doi: 10.1007/s00526-012-0571-7.

[36]

W. A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162.  doi: 10.1007/BF01626517.

[37]

M. Struwe, Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Ergebnisse Math. 34, Springer, 2008.

[38]

E. Timmermans, Phase separation of Bose-Einstein condensates, Phys. Rev. Lett., 81 (1998), 5718-5721.  doi: 10.1103/PhysRevLett.81.5718.

[39]

X. F. Wang, On concentration of positive bound states of nonlinear Schrödinger equations, Comm. Math. Phys., 153 (1993), 229-244.  doi: 10.1007/BF02096642.

[40]

J. Wei and T. Weth, Radial solutions and phase separation in a system of two coupled schrödinger equations, Arch. Rational. Mech. Anal., 190 (2008), 83-106.  doi: 10.1007/s00205-008-0121-9.

[41]

M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolations estimates, Comm. Math. Phys., 87 (1983), 567-576. 

[42]

X. Y. ZengY.M. Zhang and H.-S. Zhou, Existence and stability of standing waves for a coupled nonlinear schrödinger system, Acta Mathematica Scientia, 35 (2015), 45-70.  doi: 10.1016/S0252-9602(14)60138-7.

show all references

References:
[1]

M. H. AndersonJ. R. EnsherM. R. MatthewsC. E. Wieman and E. A. Cornell, Observation of Bose-Einstein condensation in a dilute atomic vapor, Collected Papers of Carl Wieman, (2008), 453-456.  doi: 10.1142/9789812813787_0062.

[2]

W. H. AschbacherJ. FröhlichG. M. GrafK. Schnee and M. Troyer, Symmetry breaking regime in the nonlinear Hartree equation, J. Math. Phys., 43 (2002), 3879-3891.  doi: 10.1063/1.1488673.

[3]

W. Z. Bao and Y. Y. Cai, Ground states of two-component Bose-Einstein condensates with an internal atomic Josephson junction, East Asian J. Appl. Math., 1 (2011), 49-81.  doi: 10.4208/eajam.190310.170510a.

[4]

W. Z. Bao and Y. Y. Cai, Mathematical theory and numerical methods for Bose-Einstein condensation, Kinetic and Related Models, 6 (2013), 1-135.  doi: 10.3934/krm.2013.6.1.

[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.

[6]

T. Bartsch and Z.-Q. Wang, Existence and multiplicity results for some superlinear elliptic problems on $\mathbb{R}^N$, Comm. Partial Differential Equations, 20 (1995), 1725-1741.  doi: 10.1080/03605309508821149.

[7]

M. Caliari and M. Squassina, Location and phase segregation of ground and excited states for 2D Gross-Pitaevskii systems, Dynamics of PDE, 5 (2008), 117-137.  doi: 10.4310/DPDE.2008.v5.n2.a2.

[8]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics 10, Courant Institute of Mathematical Science/AMS, New York, 2003. doi: 10.1090/cln/010.

[9]

Z. J. Chen and W. M. Zou, Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent, Arch. Ration. Mech Anal., 205 (2012), 515-551.  doi: 10.1007/s00205-012-0513-8.

[10]

Z. J. Chen and W. M. Zou, Standing waves for coupled Schrödinger equations with decaying potentials, J. Math. Phys., 54 (2013), 111505, 21pp.  doi: 10.1063/1.4833795.

[11]

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 Differential Equations, 48 (2013), 695-711.  doi: 10.1007/s00526-012-0568-2.

[12]

F. DalfovoS. GiorginiL. P. Pitaevskii and S. Stringari, Theory of Bose-Einstein condensation in trapped gases, Rev. Modern Phys., 71 (1999), 463-512. 

[13] B. GidasW. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\mathbb{R}^N$, in Mathematical analysis and applications Part A, Adv. in Math. Suppl. Stud., Academic Press, New York, 1981. 
[14]

Y. J. Guo and R. Seiringer, On the mass concentration for Bose-Einstein condensates with attractive interactions, Lett. Math. Phys., 104 (2014), 141-156.  doi: 10.1007/s11005-013-0667-9.

[15]

Y. J. Guo, Z. -Q. Wang, X. Y. Zeng and H. -S. Zhou, Properties of ground states of attractive Gross-Pitaevskii equations with multi-well potentials, preprint, arXiv: 1502.01839.

[16]

Y. J. GuoX.Y. Zeng and H.-S. Zhou, Energy estimates and symmetry breaking in attractive Bose-Einstein condensates with ring-shaped potentials, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 809-828.  doi: 10.1016/j.anihpc.2015.01.005.

[17]

Y. J. Guo, X. Y. Zeng and H. -S. Zhou, Blow-up solutions for a nonlinear Schrödinger system with attractive intraspecies and repulsive interspecies interactions, preprint.

[18]

D. S. HallM. R. MatthewsJ. R. EnsherC. E. Wieman and E. A. Cornell, Dynamics of component separation in a binary mixture of Bose-Einstein condensates, Collected Papers of Carl Wieman, (2008), 515-518.  doi: 10.1142/9789812813787_0071.

[19]

N. Ikoma and K. Tanaka, A local mountain pass type result for a system of nonlinear Schrödinger equations, Calc. Var. Partial Differential Equations, 40 (2011), 449-480.  doi: 10.1007/s00526-010-0347-x.

[20]

R. K. Jackson and M. I. Weinstein, Geometric analysis of bifurcation and symmetry breaking in a Gross-Pitaevskii equation, J. Stat. Phys., 116 (2004), 881-905.  doi: 10.1023/B:JOSS.0000037238.94034.75.

[21]

O. Kavian and F. B. Weissler, Self-similar solutions of the pseudo-conformally invariant nonlinear Schrödinger equation, Michigan Math. J., 41 (1994), 151-173.  doi: 10.1307/mmj/1029004922.

[22]

E. W. KirrP. G. Kevrekidis and D. E. Pelinovsky, Symmetry-breaking bifurcation in the nonlinear Schrödinger equation with symmetric potentials, Comm. Math. Phys., 308 (2011), 795-844.  doi: 10.1007/s00220-011-1361-3.

[23]

E. W. KirrP. G. KevrekidisE. Shlizerman and M. I. Weinstein, Symmetry-breaking bifurcation in nonlinear Schrödinger/Gross-Pitaevskii equations, SIAM J. Math. Anal., 40 (2008), 566-604.  doi: 10.1137/060678427.

[24]

Y. C. KuoW. W. Lin and S. F. Shieh, Bifurcation analysis of a two-component Bose-Einstein condensate, Phys. D, 211 (2005), 311-346.  doi: 10.1016/j.physd.2005.09.003.

[25]

T. C. Lin and J. C. Wei, Spikes in two-component systems of nonlinear Schrödinger equations with trapping potentials, J. Differential Equations, 229 (2006), 538-569.  doi: 10.1016/j.jde.2005.12.011.

[26]

P. L. Lions, The concentration-compactness principle in the Caclulus of Variations. The locally compact case, Part Ⅰ: Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109-145. Part Ⅱ: Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 223-283. doi: 10. 1016/S0294-1449(16)30422-X.

[27]

C. Y. LiuN.V. Nguyen and Z.-Q. Wang, Orbital stability of spatially synchronized solitary waves of an m-coupled nonlinear Schrödinger system, J. Math. Phys., 57 (2016), 101501, 20pp.  doi: 10.1063/1.4964255.

[28]

Z.L. Liu and Z.-Q. Wang, Ground states and bound states of a nonlinear Schrödinger system, Advanced Nonlinear Studies, 10 (2010), 175-193.  doi: 10.1515/ans-2010-0109.

[29]

M. Maeda, On the symmetry of the ground states of nonlinear Schrödinger equation with potential, Adv. Nonlinear Stud., 10 (2010), 895-925.  doi: 10.1515/ans-2010-0409.

[30]

E. MontefuscoB. Pellacci and M. Squassina, Semiclassical states for weakly coupled nonlinear Schrödinger systems, J. Eur. Math. Soc., 10 (2008), 47-71.  doi: 10.4171/JEMS/103.

[31]

N.V. Nguyen and Z.-Q. Wang, Orbital stability of solitary waves for a nonlinear Schrödinger system, Adv. Differ. Equations, 16 (2011), 977-1000. 

[32]

W.-M. Ni and I. Takagi, On the shape of least-energy solutions to a semilinear Neumann problem, Comm. Pure Appl. Math., 44 (1991), 819-851.  doi: 10.1002/cpa.3160440705.

[33]

A. Pomponio, Coupled nonlinear Schrödinger systems with potentials, J. Differ. Equations, 227 (2006), 258-281.  doi: 10.1016/j.jde.2005.09.002.

[34] M. Reed and B. Simon, Methods of Modern Mathematical Physics. ô. Analysis of Operators, Academic Press,, New York-London, 1978. 
[35]

J. Royo-Letelier, Segregation and symmetry breaking of strongly coupled two-component Bose-Einstein condensates in a harmonic trap, Calc. Var. Partial Differential Equations, 49 (2014), 103-124.  doi: 10.1007/s00526-012-0571-7.

[36]

W. A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162.  doi: 10.1007/BF01626517.

[37]

M. Struwe, Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Ergebnisse Math. 34, Springer, 2008.

[38]

E. Timmermans, Phase separation of Bose-Einstein condensates, Phys. Rev. Lett., 81 (1998), 5718-5721.  doi: 10.1103/PhysRevLett.81.5718.

[39]

X. F. Wang, On concentration of positive bound states of nonlinear Schrödinger equations, Comm. Math. Phys., 153 (1993), 229-244.  doi: 10.1007/BF02096642.

[40]

J. Wei and T. Weth, Radial solutions and phase separation in a system of two coupled schrödinger equations, Arch. Rational. Mech. Anal., 190 (2008), 83-106.  doi: 10.1007/s00205-008-0121-9.

[41]

M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolations estimates, Comm. Math. Phys., 87 (1983), 567-576. 

[42]

X. Y. ZengY.M. Zhang and H.-S. Zhou, Existence and stability of standing waves for a coupled nonlinear schrödinger system, Acta Mathematica Scientia, 35 (2015), 45-70.  doi: 10.1016/S0252-9602(14)60138-7.

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