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Dynamic analysis on an almost periodic predator-prey system with impulsive effects and time delays
On 3d dipolar Bose-Einstein condensates involving quantum fluctuations and three-body interactions
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 |
$ 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, $ |
$ K $ |
$ \lambda_3>0 $ |
$ p\in(4,6] $ |
$ p = 6 $ |
$ p = 5 $ |
$ p\neq 6 $ |
References:
[1] |
H. Abels, Pseudodifferential and Singular Integral Operators, An introduction with applications. De Gruyter Graduate Lectures. De Gruyter, Berlin, 2012. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[33] |
W. A. Strauss,
Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162.
doi: 10.1007/BF01626517. |
[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. |
[35] |
A. Triay,
Derivation of the dipolar Gross-Pitaevskii energy, SIAM J. Math. Anal., 50 (2018), 33-63.
doi: 10.1137/17M112378X. |
[36] |
M. I. Weinstein,
Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys., 87 (1982/83), 567-576.
doi: 10.1007/BF01208265. |
[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. |
show all references
References:
[1] |
H. Abels, Pseudodifferential and Singular Integral Operators, An introduction with applications. De Gruyter Graduate Lectures. De Gruyter, Berlin, 2012. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[33] |
W. A. Strauss,
Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162.
doi: 10.1007/BF01626517. |
[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. |
[35] |
A. Triay,
Derivation of the dipolar Gross-Pitaevskii energy, SIAM J. Math. Anal., 50 (2018), 33-63.
doi: 10.1137/17M112378X. |
[36] |
M. I. Weinstein,
Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys., 87 (1982/83), 567-576.
doi: 10.1007/BF01208265. |
[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. |
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