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On the collapse arresting effects of discreteness

Author

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  • Tzirakis, N.
  • Kevrekidis, P.G.

Abstract

We examine the effects of discreteness on a prototypical example of a collapse exhibiting partial differential equation (PDE). As our benchmark example, we select the discrete nonlinear Schrödinger (DNLS) equation. We provide a number of physical settings where issues of the interplay of collapse and discreteness may arise and focus on the quintic, one-dimensional DNLS. We justify that collapse in the sense of continuum limit (i.e., of the L∞ norm becoming infinite) cannot occur in the discrete setting. We support our qualitative arguments both with numerical simulations as well as with an analysis of a quasi-continuum, pseudo-differential approximation to the discrete model. Global well-posedness is proved for the latter problem in Hs, for s>1/2. While the collapse arresting nature of discreteness can be immediately realized, our estimates elucidate the “approach” towards the collapse-bearing continuum limit and the mechanism through which focusing arises in the latter.

Suggested Citation

  • Tzirakis, N. & Kevrekidis, P.G., 2005. "On the collapse arresting effects of discreteness," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 69(5), pages 553-566.
  • Handle: RePEc:eee:matcom:v:69:y:2005:i:5:p:553-566
    DOI: 10.1016/j.matcom.2005.03.013
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