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June  2021, 26(6): 3455-3477. doi: 10.3934/dcdsb.2020239

On 3d dipolar Bose-Einstein condensates involving quantum fluctuations and three-body interactions

Yongming Luo 1,, and  Athanasios Stylianou 2,

1. 

Institut für Wissenschaftliches Rechnen, Technische Universität Dresden, Willers-Bau, Zellescher Weg 12-14, 01069 Dresden, Germany
2. 

Institut für Mathematik, Universität Kassel, Heinrich-Plett-Straße 40, 34132 Kassel, Germany

* Corresponding author: Yongming Luo

Received  May 2020 Revised  June 2020 Published  August 2020

We study the following nonlocal mixed order Gross-Pitaevskii equation
$ i\,\partial_t \psi = -\frac{1}{2}\,\Delta \psi+V_{ext}\,\psi+\lambda_1\,|\psi|^2\,\psi+\lambda_2\,(K*|\psi|^2)\,\psi+\lambda_3\,|\psi|^{p-2}\,\psi, $
where
$ K $
 is the classical dipole-dipole interaction kernel,
$ \lambda_3>0 $
 and
$ p\in(4,6] $
; the case
$ p = 6 $
 being energy critical. For
$ p = 5 $
 the equation is considered currently as the state-of-the-art model for describing the dynamics of dipolar Bose-Einstein condensates (Lee-Huang-Yang corrected dipolar GPE). We prove existence and nonexistence of standing waves in different parameter regimes; for
$ p\neq 6 $
 we prove global well-posedness and small data scattering.
Keywords: Dipolar Gross-Pitaevskii equation, Lee-Huang-Young correction, competing nonlinearities, ground states, well-posedness.
Mathematics Subject Classification: Primary: 35Q55, 49J35; Secondary: 35B09.
Citation: Yongming Luo, Athanasios Stylianou. On 3d dipolar Bose-Einstein condensates involving quantum fluctuations and three-body interactions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3455-3477. doi: 10.3934/dcdsb.2020239
References:
[1]

H. Abels, Pseudodifferential and Singular Integral Operators, An introduction with applications. De Gruyter Graduate Lectures. De Gruyter, Berlin, 2012.  Google Scholar

[2]

P. Antonelli and C. Sparber, Existence of solitary waves in dipolar quantum gases, Physica D: Nonlinear Phenomena, 240 (2011), 426 – 431. doi: 10.1016/j.physd.2010.10.004.  Google Scholar

[3]

D. Baillie, R. M. Wilson, R. N. Bisset and P. B. Blakie, Self-bound dipolar droplet: A localized matter wave in free space, Phys. Rev. A, 94 (2016), 021602(R). doi: 10.1103/PhysRevA.94.021602.  Google Scholar

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W. Bao, N. Ben Abdallah and Y. Cai, Gross-Pitaevskii-Poisson equations for dipolar Bose-Einstein condensate with anisotropic confinement, SIAM J. Math. Anal., 44 (2012), 1713-1741.  doi: 10.1137/110850451.  Google Scholar

[5]

W. Bao, L. Le Treust and F. Méhats, Dimension reduction for dipolar Bose-Einstein condensates in the strong interaction regime, Kinet. Relat. Models, 10 (2017), 553-571.  doi: 10.3934/krm.2017022.  Google Scholar

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J. Bellazzini, N. Boussaïd, L. Jeanjean and N. Visciglia, Existence and stability of standing waves for supercritical NLS with a partial confinement, Comm. Math. Phys., 353 (2017), 229-251.  doi: 10.1007/s00220-017-2866-1.  Google Scholar

[7]

J. Bellazzini and L. Jeanjean, On dipolar quantum gases in the unstable regime, SIAM J. Math. Anal., 48 (2016), 2028-2058.  doi: 10.1137/15M1015959.  Google Scholar

[8]

H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. Ⅱ. Existence of infinitely many solutions, Arch. Rational Mech. Anal., 82 (1983), 347-375.  doi: 10.1007/BF00250556.  Google Scholar

[9]

R. N. Bisset, R. M. Wilson, D. Baillie and P. B. Blakie, Ground-state phase diagram of a dipolar condensate with quantum fluctuations, Phys. Rev. A, 94 (2016), 033619. doi: 10.1103/PhysRevA.94.033619.  Google Scholar

[10]

P. B. Blakie, Properties of a dipolar condensate with three-body interactions, Phys. Rev. A, 93 (2016), 033644. doi: 10.1103/PhysRevA.93.033644.  Google Scholar

[11]

R. Carles, Sharp weights in the Cauchy problem for nonlinear Schrödinger equations with potential, Z. Angew. Math. Phys., 66 (2015), 2087-2094.  doi: 10.1007/s00033-015-0501-6.  Google Scholar

[12]

R. Carles and H. Hajaiej, Complementary study of the standing wave solutions of the Gross-Pitaevskii equation in dipolar quantum gases, Bull. Lond. Math. Soc., 47 (2015), 509-518.  doi: 10.1112/blms/bdv024.  Google Scholar

[13]

R. Carles, P. A. Markowich and C. Sparber, On the Gross-Pitaevskii equation for trapped dipolar quantum gases, Nonlinearity, 21 (2008), 2569-2590.  doi: 10.1088/0951-7715/21/11/006.  Google Scholar

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T. Cazenave, Semilinear Schrödinger Equations, volume 10 of Courant Lecture Notes in Mathematics, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. doi: 10.1090/cln/010.  Google Scholar

[15]

S. Giovanazzi, A. Görlitz and T. Pfau, Tuning the dipolar interaction in quantum gases, Phys. Rev. Lett., 89 (2002), 130401. doi: 10.1103/PhysRevLett.89.130401.  Google Scholar

[16]

M. D. Groves, D. J. B. Lloyd and A. Stylianou, Pattern formation on the free surface of a ferrofluid: Spatial dynamics and homoclinic bifurcation, Physica D: Nonlinear Phenomena, 350 (2017), 1-12.  doi: 10.1016/j.physd.2017.03.004.  Google Scholar

[17]

Y. He and X. Luo, Concentrating standing waves for the Gross–Pitaevskii equation in trapped dipolar quantum gases, J. Differential Equations, 266 (2019), 600-629.  doi: 10.1016/j.jde.2018.07.047.  Google Scholar

[18]

H. Kadau, M. Schmitt, M. Wenzel, C. Wink, T. Maier, I. Ferrier-Barbut and T. Pfau, Observing the Rosensweig instability of a quantum ferrofluid, Nature, 530 (2016), 194-197.  doi: 10.1038/nature16485.  Google Scholar

[19]

P. G. Kevrekidis, D. J. Frantzeskakis and Ricardo Carretero-González, The Defocusing Nonlinear Schrödinger Equation, From dark solitons to vortices and vortex rings., Society for Industrial and Applied Mathematics, Philadelphia, PA, 2015. doi: 10.1137/1.9781611973945.  Google Scholar

[20]

T. Lahaye, C. Menotti, L. Santos, M. Lewenstein and T. Pfau, The physics of dipolar bosonic quantum gases, Reports on Progress in Physics, 72 (2009), 126401. doi: 10.1088/0034-4885/72/12/126401.  Google Scholar

[21]

P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. Ⅰ, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109–145, . doi: 10.1016/S0294-1449(16)30428-0.  Google Scholar

[22]

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

[23]

B. Liu, L. Ma and J. Wang, Blow up threshold for the Gross-Pitaevskii system with trapped dipolar quantum gases, ZAMM Z. Angew. Math. Mech., 96 (2016), 344-360.  doi: 10.1002/zamm.201400189.  Google Scholar

[24]

O. Lopes and M. Mariş, Symmetry of minimizers for some nonlocal variational problems, J. Funct. Anal., 254 (2008), 535-592.  doi: 10.1016/j.jfa.2007.10.004.  Google Scholar

[25]

L. Ma and P. Cao, The threshold for the focusing Gross-Pitaevskii equation with trapped dipolar quantum gases, J. Math. Anal. Appl., 381 (2011), 240-246.  doi: 10.1016/j.jmaa.2011.02.031.  Google Scholar

[26]

L. Ma and J. Wang, Sharp threshold of the Gross-Pitaevskii equation with trapped dipolar quantum gases, Canad. Math. Bull., 56 (2013), 378-387.  doi: 10.4153/CMB-2011-181-2.  Google Scholar

[27]

B. A. Malomed, Suppression of quantum-mechanical collapse in bosonic gases with intrinsic repulsion: A brief review, 3 2018. arXiv: 1804.06607. To appear in Condensed Matter (Special Issue "Proceedings of the conference SuperFluctuations 2017"). doi: 10.3390/condmat3020015.  Google Scholar

[28]

M. Mariş, On some minimization problems in RN, In New Trends in Differential Equations, Control Theory and Optimization, pages 215–230. World Sci. Publ., Hackensack, NJ, 2016.  Google Scholar

[29]

J. Metz, T. Lahaye, B. Fröhlich, A. Griesmaier, T. Pfau, H. Saito, Y. Kawaguchi and M. Ueda, Coherent collapses of dipolar bose-einstein condensates for different trap geometries, New Journal of Physics, 11 (2009), 055032. doi: 10.1088/1367-2630/11/5/055032.  Google Scholar

[30]

E. Parini and A. Stylianou, A free boundary approach to the Rosensweig instability of ferrofluids, Z. Angew. Math. Phys., 69 (2018), no. 2, Paper No. 32, 18 pp. doi: 10.1007/s00033-018-0924-y.  Google Scholar

[31]

R. Richter and A. Lange, Surface instabilities of ferrofluids, In S. Odenbach, editor, Colloidal Magnetic Fluids, volume 763 of Lecture Notes in Physics, pages 1–91. Springer Berlin Heidelberg, 2009. doi: 10.1007/978-3-540-85387-9_3.  Google Scholar

[32]

M. Schmitt, M. Wenzel, F. Böttcher, I. Ferrier-Barbut and T. Pfau, Self-bound droplets of a dilute magnetic quantum liquid, Nature, 539 (2016), 259-262.  doi: 10.1038/nature20126.  Google Scholar

[33]

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

[34]

T. Tao, M. Visan and X. Zhang, The nonlinear Schrödinger equation with combined power-type nonlinearities, Comm. Partial Differential Equations, 32 (2007), 1281-1343.  doi: 10.1080/03605300701588805.  Google Scholar

[35]

A. Triay, Derivation of the dipolar Gross-Pitaevskii energy, SIAM J. Math. Anal., 50 (2018), 33-63.  doi: 10.1137/17M112378X.  Google Scholar

[36]

M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys., 87 (1982/83), 567-576.  doi: 10.1007/BF01208265.  Google Scholar

[37]

J. Zhang, Stability of standing waves for nonlinear Schrödinger equations with unbounded potentials, Z. Angew. Math. Phys., 51 (2000), 498-503.  doi: 10.1007/PL00001512.  Google Scholar

show all references

References:
[1]

H. Abels, Pseudodifferential and Singular Integral Operators, An introduction with applications. De Gruyter Graduate Lectures. De Gruyter, Berlin, 2012.  Google Scholar

[2]

P. Antonelli and C. Sparber, Existence of solitary waves in dipolar quantum gases, Physica D: Nonlinear Phenomena, 240 (2011), 426 – 431. doi: 10.1016/j.physd.2010.10.004.  Google Scholar

[3]

D. Baillie, R. M. Wilson, R. N. Bisset and P. B. Blakie, Self-bound dipolar droplet: A localized matter wave in free space, Phys. Rev. A, 94 (2016), 021602(R). doi: 10.1103/PhysRevA.94.021602.  Google Scholar

[4]

W. Bao, N. Ben Abdallah and Y. Cai, Gross-Pitaevskii-Poisson equations for dipolar Bose-Einstein condensate with anisotropic confinement, SIAM J. Math. Anal., 44 (2012), 1713-1741.  doi: 10.1137/110850451.  Google Scholar

[5]

W. Bao, L. Le Treust and F. Méhats, Dimension reduction for dipolar Bose-Einstein condensates in the strong interaction regime, Kinet. Relat. Models, 10 (2017), 553-571.  doi: 10.3934/krm.2017022.  Google Scholar

[6]

J. Bellazzini, N. Boussaïd, L. Jeanjean and N. Visciglia, Existence and stability of standing waves for supercritical NLS with a partial confinement, Comm. Math. Phys., 353 (2017), 229-251.  doi: 10.1007/s00220-017-2866-1.  Google Scholar

[7]

J. Bellazzini and L. Jeanjean, On dipolar quantum gases in the unstable regime, SIAM J. Math. Anal., 48 (2016), 2028-2058.  doi: 10.1137/15M1015959.  Google Scholar

[8]

H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. Ⅱ. Existence of infinitely many solutions, Arch. Rational Mech. Anal., 82 (1983), 347-375.  doi: 10.1007/BF00250556.  Google Scholar

[9]

R. N. Bisset, R. M. Wilson, D. Baillie and P. B. Blakie, Ground-state phase diagram of a dipolar condensate with quantum fluctuations, Phys. Rev. A, 94 (2016), 033619. doi: 10.1103/PhysRevA.94.033619.  Google Scholar

[10]

P. B. Blakie, Properties of a dipolar condensate with three-body interactions, Phys. Rev. A, 93 (2016), 033644. doi: 10.1103/PhysRevA.93.033644.  Google Scholar

[11]

R. Carles, Sharp weights in the Cauchy problem for nonlinear Schrödinger equations with potential, Z. Angew. Math. Phys., 66 (2015), 2087-2094.  doi: 10.1007/s00033-015-0501-6.  Google Scholar

[12]

R. Carles and H. Hajaiej, Complementary study of the standing wave solutions of the Gross-Pitaevskii equation in dipolar quantum gases, Bull. Lond. Math. Soc., 47 (2015), 509-518.  doi: 10.1112/blms/bdv024.  Google Scholar

[13]

R. Carles, P. A. Markowich and C. Sparber, On the Gross-Pitaevskii equation for trapped dipolar quantum gases, Nonlinearity, 21 (2008), 2569-2590.  doi: 10.1088/0951-7715/21/11/006.  Google Scholar

[14]

T. Cazenave, Semilinear Schrödinger Equations, volume 10 of Courant Lecture Notes in Mathematics, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. doi: 10.1090/cln/010.  Google Scholar

[15]

S. Giovanazzi, A. Görlitz and T. Pfau, Tuning the dipolar interaction in quantum gases, Phys. Rev. Lett., 89 (2002), 130401. doi: 10.1103/PhysRevLett.89.130401.  Google Scholar

[16]

M. D. Groves, D. J. B. Lloyd and A. Stylianou, Pattern formation on the free surface of a ferrofluid: Spatial dynamics and homoclinic bifurcation, Physica D: Nonlinear Phenomena, 350 (2017), 1-12.  doi: 10.1016/j.physd.2017.03.004.  Google Scholar

[17]

Y. He and X. Luo, Concentrating standing waves for the Gross–Pitaevskii equation in trapped dipolar quantum gases, J. Differential Equations, 266 (2019), 600-629.  doi: 10.1016/j.jde.2018.07.047.  Google Scholar

[18]

H. Kadau, M. Schmitt, M. Wenzel, C. Wink, T. Maier, I. Ferrier-Barbut and T. Pfau, Observing the Rosensweig instability of a quantum ferrofluid, Nature, 530 (2016), 194-197.  doi: 10.1038/nature16485.  Google Scholar

[19]

P. G. Kevrekidis, D. J. Frantzeskakis and Ricardo Carretero-González, The Defocusing Nonlinear Schrödinger Equation, From dark solitons to vortices and vortex rings., Society for Industrial and Applied Mathematics, Philadelphia, PA, 2015. doi: 10.1137/1.9781611973945.  Google Scholar

[20]

T. Lahaye, C. Menotti, L. Santos, M. Lewenstein and T. Pfau, The physics of dipolar bosonic quantum gases, Reports on Progress in Physics, 72 (2009), 126401. doi: 10.1088/0034-4885/72/12/126401.  Google Scholar

[21]

P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. Ⅰ, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109–145, . doi: 10.1016/S0294-1449(16)30428-0.  Google Scholar

[22]

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

[23]

B. Liu, L. Ma and J. Wang, Blow up threshold for the Gross-Pitaevskii system with trapped dipolar quantum gases, ZAMM Z. Angew. Math. Mech., 96 (2016), 344-360.  doi: 10.1002/zamm.201400189.  Google Scholar

[24]

O. Lopes and M. Mariş, Symmetry of minimizers for some nonlocal variational problems, J. Funct. Anal., 254 (2008), 535-592.  doi: 10.1016/j.jfa.2007.10.004.  Google Scholar

[25]

L. Ma and P. Cao, The threshold for the focusing Gross-Pitaevskii equation with trapped dipolar quantum gases, J. Math. Anal. Appl., 381 (2011), 240-246.  doi: 10.1016/j.jmaa.2011.02.031.  Google Scholar

[26]

L. Ma and J. Wang, Sharp threshold of the Gross-Pitaevskii equation with trapped dipolar quantum gases, Canad. Math. Bull., 56 (2013), 378-387.  doi: 10.4153/CMB-2011-181-2.  Google Scholar

[27]

B. A. Malomed, Suppression of quantum-mechanical collapse in bosonic gases with intrinsic repulsion: A brief review, 3 2018. arXiv: 1804.06607. To appear in Condensed Matter (Special Issue "Proceedings of the conference SuperFluctuations 2017"). doi: 10.3390/condmat3020015.  Google Scholar

[28]

M. Mariş, On some minimization problems in RN, In New Trends in Differential Equations, Control Theory and Optimization, pages 215–230. World Sci. Publ., Hackensack, NJ, 2016.  Google Scholar

[29]

J. Metz, T. Lahaye, B. Fröhlich, A. Griesmaier, T. Pfau, H. Saito, Y. Kawaguchi and M. Ueda, Coherent collapses of dipolar bose-einstein condensates for different trap geometries, New Journal of Physics, 11 (2009), 055032. doi: 10.1088/1367-2630/11/5/055032.  Google Scholar

[30]

E. Parini and A. Stylianou, A free boundary approach to the Rosensweig instability of ferrofluids, Z. Angew. Math. Phys., 69 (2018), no. 2, Paper No. 32, 18 pp. doi: 10.1007/s00033-018-0924-y.  Google Scholar

[31]

R. Richter and A. Lange, Surface instabilities of ferrofluids, In S. Odenbach, editor, Colloidal Magnetic Fluids, volume 763 of Lecture Notes in Physics, pages 1–91. Springer Berlin Heidelberg, 2009. doi: 10.1007/978-3-540-85387-9_3.  Google Scholar

[32]

M. Schmitt, M. Wenzel, F. Böttcher, I. Ferrier-Barbut and T. Pfau, Self-bound droplets of a dilute magnetic quantum liquid, Nature, 539 (2016), 259-262.  doi: 10.1038/nature20126.  Google Scholar

[33]

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

[34]

T. Tao, M. Visan and X. Zhang, The nonlinear Schrödinger equation with combined power-type nonlinearities, Comm. Partial Differential Equations, 32 (2007), 1281-1343.  doi: 10.1080/03605300701588805.  Google Scholar

[35]

A. Triay, Derivation of the dipolar Gross-Pitaevskii energy, SIAM J. Math. Anal., 50 (2018), 33-63.  doi: 10.1137/17M112378X.  Google Scholar

[36]

M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys., 87 (1982/83), 567-576.  doi: 10.1007/BF01208265.  Google Scholar

[37]

J. Zhang, Stability of standing waves for nonlinear Schrödinger equations with unbounded potentials, Z. Angew. Math. Phys., 51 (2000), 498-503.  doi: 10.1007/PL00001512.  Google Scholar

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