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Cross-invariant manifolds for the Gross–Pitaevskii–Poisson equation

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Abstract

This study investigates the two-dimensional (2D) Gross–Pitaevskii–Poisson equations that model dipolar Bose–Einstein condensation. The new element introduced to these equations in this study is the construction of the modified cross-constrained minimization problem \(\widetilde{d_{\mathcal {M}}}\) that is associated with the nonlocal kernel \(\partial _{\mathbf {n}_{\perp } \mathbf {n}_{\perp }}-n_{3}^{2}\Delta \), resulting in the loss of \(L^2\)-norm scale invariance. We find modified cross-constrained manifolds and develop sharp criteria of blow-up and global existence as well as strong instability of standing waves.

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References

  1. Anderson, M.H., Ensher, J.R., Matthewa, M.R., Wieman, C.E., Cornell, E.A.: Observation of Bose–Einstein condensation in a dilute atomic vapor. Science 269, 198–201 (1995)

    Article  Google Scholar 

  2. Antonelli, P., Sparber, C.: Existence of solitary waves in dipolar quantum gases. Phys. D 240, 426–431 (2011)

    Article  MathSciNet  Google Scholar 

  3. Bao, W., Cai, Y.: Mathematical theory and numerical methods for Bose–Einstein condensation. Kinet. Rel. Models 6, 1–135 (2013)

    Article  MathSciNet  Google Scholar 

  4. Bao, W., Abdallah, N.B., Cai, Y.: Gross–Pitaevskii–Poisson equations for dipolar Bose–Einstein condensate with anisotropic confinement. SIAM J. Math. Anal. 44, 1713–1741 (2012)

    Article  MathSciNet  Google Scholar 

  5. Bao, W., Jaksch, D., Markowich, P.A.: Numerical solution of the Gross–Pitaevskii equation for Bose–Einstein condensation. J. Comput. Phys. 187, 318–342 (2003)

    Article  MathSciNet  Google Scholar 

  6. Bellazzini, J., Jeanjean, L.: On dipolar quantum gases in the unstable regime. SIAM J. Math. Anal. 48, 2028–2058 (2016)

    Article  MathSciNet  Google Scholar 

  7. Bellazzini, J., Forcella, L.: Asymptotic dynamic for dipolar quantum gases below the ground state energy threshold. J. Funct. Anal. 277, 1958–1998 (2019)

    Article  MathSciNet  Google Scholar 

  8. Bradley, C.C., Sackett, C.A., Tollett, J.J., Hulet, R.G.: Evidence of Bose–Einstein condensation in an atomic gas with attractive interaction. Phys. Rev. Lett. 75, 1687–1690 (1995)

    Article  Google Scholar 

  9. Bradley, C.C., Sackett, C.A., Hulet, R.G.: Bose–Einstein condensation of Lithium: observation of limited condensate number. Phys. Rev. Lett. 78, 985–989 (1997)

    Article  Google Scholar 

  10. Cazenave, T.: Semilinear Schrödinger Equations, Courant Lect. Notes Math. 10, Courant In stitute of Mathematical Sciences, New York. AMS, Providence, RI (2003)

  11. Cai, Y., Rosenkranz, M., Lei, Z., Bao, W.: Mean-field regime of trapped dipolar Bose–Einstein condensates in one and two dimensions. Phys. Rev. A 82, 043623 (2010)

    Article  Google Scholar 

  12. Carles, R., Markowich, P.A., Sparber, C.: On the Gross–Pitaevskii equation for trapped dipolar quantum gases. Nonlinearity 21, 2569–2590 (2008)

    Article  MathSciNet  Google Scholar 

  13. Davis, K.B., Mewes, M.O., Andrews, M.R., van Druten, N.J., Durfee, D.S., Kurn, D.M., Ketterle, W.: Bose–Einstein condensation in a gas of sodium atoms. Phys. Rev. Lett. 75, 3969–3973 (1995)

    Article  Google Scholar 

  14. Gao, Y., Wang, Z.: Blow-up for trapped dipolar quantum gases with large energy. J. Math. Phys. 60, 121501 (2019)

    Article  MathSciNet  Google Scholar 

  15. Dalfovo, F., Giorgini, S., Pitaevskii, L.P., Stringari, S.: Theory of Bose–Einstein condensation in trapped gases. Rev. Mod. Phys. 71(1999), 463–512 (1999)

    Article  Google Scholar 

  16. Eychenne, A., Rougerie, N.: On the stability of 2D dipolar Bose–Einstein condensates. SIAM J. Math. Anal. 51, 1371–1386 (2019)

    Article  MathSciNet  Google Scholar 

  17. Griesmaier, A., Werner, J., Hensler, S., Stuhler, J., Pfau, T.: Bose–Einstein condensation of chromium. Phys. Rev. Lett. 94, 160401 (2005)

    Article  Google Scholar 

  18. Huang, J., Zhang, J.: Exact value of cross-constrain problem and strong instability of standing waves in trapped dipolar quantum gases. Appl. Math. Lett. 70, 32–38 (2017)

    Article  MathSciNet  Google Scholar 

  19. Kenig, C.E., Merle, F.: Global well-posedness, scattering, and blow-up for the energy-critical focusing nonlinear Schrödinger equation in the radial case. Invent. Math. 166, 645–675 (2006)

    Article  MathSciNet  Google Scholar 

  20. Lu, M., Burdick, N.Q., Youn, S.H., Lev, B.L.: A strongly dipolar Bose–Einstein condensate of dysprosium. Phys. Rev. Lett. 107, 190401 (2011)

    Article  Google Scholar 

  21. Ma, L., Cao, P.: The threshold for the focusing Gross–Pitaevskii equation with trapped dipolar quantum gases. J. Math. Anal. Appl. 381, 240–246 (2011)

    Article  MathSciNet  Google Scholar 

  22. Ma, L., Wang, J.: Sharp threshold of the Gross–Pitaevskii equation with trapped dipolar quantum gases. Canad. Math. Bull. 56, 378–387 (2013)

    Article  MathSciNet  Google Scholar 

  23. Merle, F., Raphaël, P.: On a sharp lower bound on the blow-up rate for the \(L^2\)-critical nonlinear Schrödinger equation. J. Am. Math. Soc. 19, 37–90 (2006)

    Article  Google Scholar 

  24. Yi, S., You, L.: Trapped atomic condensates with anisotropic interactions. Phys. Rev. A 61, 041604 (2000)

    Article  Google Scholar 

  25. Zhang, J.: Sharp conditions of global existence for nonlinear Schrodinger and Klein-Gordon equations. Nonlinear Anal. 48, 191–207 (2002)

    Article  MathSciNet  Google Scholar 

  26. Zhang, J.: Sharp threshold for blowup and global existence in nonlinear Schroödinger equations under a harmonic potential. Commun. Partial Differ. Equ. 30, 1429–1443 (2005)

    Article  Google Scholar 

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Acknowledgements

This study was supported by the NSFC of China (Grant Nos. 12071323 and 11771314) and the Sichuan Sciences and Technology Program (Grant Nos. 2020YJ0146 and 2020YJ0357).

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Authors and Affiliations

  1. School of Mathematical Sciences and V.C. & V.R. Key Lab, Sichuan Normal University, Chengdu, 610066, People’s Republic of China

    Xiaoyu Zhou & Shihui Zhu

  2. College of Mathematics and Physics, Chengdu University of Technology, Chengdu, 610059, People’s Republic of China

    Shihui Zhu

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  1. Xiaoyu Zhou
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  2. Shihui Zhu
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Correspondence to Shihui Zhu.

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Zhou, X., Zhu, S. Cross-invariant manifolds for the Gross–Pitaevskii–Poisson equation. Z. Angew. Math. Phys. 73, 160 (2022). https://doi.org/10.1007/s00033-022-01806-9

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