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Bifurcation of Nonlinear Bloch Waves from the Spectrum in the Gross–Pitaevskii Equation

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Abstract

We rigorously analyze the bifurcation of stationary so-called nonlinear Bloch waves (NLBs) from the spectrum in the Gross–Pitaevskii (GP) equation with a periodic potential, in arbitrary space dimensions. These are solutions which can be expressed as finite sums of quasiperiodic functions and which in a formal asymptotic expansion are obtained from solutions of the so-called algebraic coupled mode equations. Here we justify this expansion by proving the existence of NLBs and estimating the error of the formal asymptotics. The analysis is illustrated by numerical bifurcation diagrams, mostly in 2D. In addition, we illustrate some relations of NLBs to other classes of solutions of the GP equation, in particular to so-called out-of-gap solitons and truncated NLBs, and present some numerical experiments concerning the stability of these solutions.

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Notes

  1. Strictly speaking, the problem should first be rewritten in real variables to define a Jacobian, see the discussion above Lemma 10, but for brevity we use this compact symbolic notation here.

  2. A symbolic notation for the Jacobian used again.

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Acknowledgments

The authors thank Michael I. Weinstein for fruitful discussions, in particular for inquiring about the possibility to generalize the bifurcation assumptions to multiple Bloch eigenvalues, as formulated in (H3). The research of T.D. is partly supported by the German Research Foundation, DFG Grant No. DO1467/3-1.

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Authors and Affiliations

  1. Department of Mathematics, Technical University Dortmund, 44221, Dortmund, Germany

    Tomáš Dohnal

  2. Institute of Mathematics, Carl von Ossietzky University Oldenburg, 26111, Oldenburg, Germany

    Hannes Uecker

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  1. Tomáš Dohnal
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  2. Hannes Uecker
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Correspondence to Tomáš Dohnal.

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Communicated by Michael I. Weinstein.

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Dohnal, T., Uecker, H. Bifurcation of Nonlinear Bloch Waves from the Spectrum in the Gross–Pitaevskii Equation. J Nonlinear Sci 26, 581–618 (2016). https://doi.org/10.1007/s00332-015-9281-6

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  • DOI: https://doi.org/10.1007/s00332-015-9281-6

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