Journal of Nonlinear Science

, Volume 25, Issue 3, pp 709–739 | Cite as

On the Gross–Pitaevskii Equation with Pumping and Decay: Stationary States and Their Stability

  • Jesús Sierra
  • Aslan KasimovEmail author
  • Peter Markowich
  • Rada-Maria Weishäupl


We investigate the behavior of solutions of the complex Gross–Pitaevskii equation, a model that describes the dynamics of pumped decaying Bose–Einstein condensates. The stationary radially symmetric solutions of the equation are studied, and their linear stability with respect to two-dimensional perturbations is analyzed. Using numerical continuation, we calculate not only the ground state of the system, but also a number of excited states. Accurate numerical integration is employed to study the general nonlinear evolution of the system from the unstable stationary solutions to the formation of stable vortex patterns.


Complex Gross–Pitaevskii equation Numerical continuation Collocation method Bose–Einstein condensate 



J. S., A. K., and P. M. gratefully acknowledge research support by King Abdullah University of Science and Technology (KAUST). The first author acknowledges the assistance and comments from W. Bao, D. Ketcheson, P. Antonelli, N. Berloff, F. Pinsker, B. Sandstede, B. Oldeman, and the Research Computing Group from KAUST. The work of the last author has been supported by the Hertha-Firnberg Program of the FWF, Grant T402-N13.

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Jesús Sierra
    • 1
  • Aslan Kasimov
    • 1
    Email author
  • Peter Markowich
    • 1
  • Rada-Maria Weishäupl
    • 2
  1. 1.King Abdullah University of Science and TechnologyThuwalSaudi Arabia
  2. 2.Faculty of MathematicsVienna UniversityWienAustria

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