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

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Abstract

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.

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Acknowledgments

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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Authors and Affiliations

  1. King Abdullah University of Science and Technology, Box 4700, Thuwal, 23955-6900, Saudi Arabia

    Jesús Sierra, Aslan Kasimov & Peter Markowich

  2. Faculty of Mathematics, Vienna University, Oskar-Morgenstern-Platz 1, 1090, Wien, Austria

    Rada-Maria Weishäupl

Authors
  1. Jesús Sierra
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  2. Aslan Kasimov
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  3. Peter Markowich
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  4. Rada-Maria Weishäupl
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Correspondence to Aslan Kasimov.

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Communicated by Edriss S. Titi.

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Sierra, J., Kasimov, A., Markowich, P. et al. On the Gross–Pitaevskii Equation with Pumping and Decay: Stationary States and Their Stability. J Nonlinear Sci 25, 709–739 (2015). https://doi.org/10.1007/s00332-015-9239-8

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