Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-13T02:44:51.006Z Has data issue: false hasContentIssue false
  • English
  • Français

Sharp Threshold of the Gross-Pitaevskii Equation with Trapped Dipolar Quantum Gases

Published online by Cambridge University Press:  20 November 2018

Li Ma  and
Jing Wang
Li Ma
Affiliation:
Department of Mathematics, Henan Normal University, Xinxiang, 453007, China e-mail: nuslma@gmail.com
Jing Wang
Affiliation:
Department of Mathematics, Tsinghua University, Beijing, 100084, China e-mail: wangjing699@gmail.com
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper, we consider the Gross-Pitaevskii equation for the trapped dipolar quantum gases. We obtain the sharp criterion for the global existence and finite time blow-up in the unstable regime by constructing a variational problem and the so-called invariant manifold of the evolution flow.

Keywords

35Q5535A0581Q99Gross-Pitaevskii equationsharp thresholdglobal existenceblow-up
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 56 , Issue 2 , 01 June 2013 , pp. 378 - 387
Copyright
Copyright © Canadian Mathematical Society 2013

References

[1] Berestycki, H. and Cazenave, T., Instabilité des états stationnaires dans les équations de Schrodinger et de Klein-Gordon non linéaires. C. R. Acad. Sci. Paris Sér. I Math. 293(1981), 489-492.Google Scholar
[2] Carles, Remi, Markowich, Peter A., and Christ of Sparber, On the Gross-Pitaevskii equation for trapped dipolar quantam gases. Nonlinearity 21(2008), 2569–2590. http://dx.doi.org/10.1088/0951-7715/21/11/006 Google Scholar
[3] Ma, Li and Cao, Pei, The threshold for the focusing Gross–Pitaevskii equation with trapped dipolar quantum gases. J. Math. Anal. 2011, http://dx.doi.org/10.1016/j.jmaa.2011.02.031. Google Scholar
[4] Ma, Li and Zhao, Lin, Sharp thresholds of blow-up and global existence for the coupled nonlinear Schrodinger system. J. Math. Phys. 49(2008), 062103. http://dx.doi.org/10.1063/1.2939238 Google Scholar
[5] Weinstein, M. I., Nonlinear Schrodinger equations and sharp interpolation estimates. Comm. Math. Phys. 87(1983), 567–576. http://dx.doi.org/10.1007/BF01208265Google Scholar
[6] Ronen, S., Bortolotti, D. C. E., Blume, D., and Bohn, J. L., Dipolar Bose-Einstein condensates with dipole-dependent scattering length. Phys. Rev. A 74(2006), 033611.Google Scholar
[7] Stein, E. M., Singular integrals and Differentiability Properties. Princeton University Press, Princeton, 1970.Google Scholar
[8] Yi, S. and You, L., Trapped atomic condensates with anisotropic interactions. Phys. Rev. A 61(2000), 041604.Google Scholar
[9] Zhang, J., Sharp conditions of global existence for nonlinear Schrodinger and Klein–Gordon equations. Nonlinear Anal. 48(2002), 191–207. http://dx.doi.org/10.1016/S0362-546X(00)00180-2 Google Scholar