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Statistical mechanics of the 2-dimensional focusing nonlinear Schrödinger equation

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Abstract

We study a natural construction of an invariant measure for the 2-dimensional periodic focusing nonlinear Schrödinger equation, with the critical cubic nonlinearity. We find that a phase transition occurs as the coupling constant defining the strength of the nonlinearity is increased, but that the natural construction, successful for the 1-dimensional case and for the 2-dimensional defocusing case, cannot produce an invariant measure. Our methods rely on an analysis of a statistical mechanical model closely related to the spherical model of Berlin and Kac.

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Authors and Affiliations

  1. Department of Mathematics, University of Virginia, 22903-3199, Chalottesville, VA, USA

    David C. Brydges

  2. Department of Mathematics and Statistics, McMaster University, L8S 4K1, Hamilton, ON, Canada

    Gordon Slade

Authors
  1. David C. Brydges
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  2. Gordon Slade
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Communicated by A. Jaffe

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Brydges, D.C., Slade, G. Statistical mechanics of the 2-dimensional focusing nonlinear Schrödinger equation. Commun.Math. Phys. 182, 485–504 (1996). https://doi.org/10.1007/BF02517899

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  • DOI: https://doi.org/10.1007/BF02517899

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