On the collapse arresting effects of discreteness

N. Tzirakis, P. G. Kevrekidis

Research output: Contribution to journalConference articlepeer-review


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.

Original languageEnglish (US)
Pages (from-to)553-566
Number of pages14
JournalMathematics and Computers in Simulation
Issue number5-6
StatePublished - Aug 5 2005
Externally publishedYes
EventNonlinear Waves: Computation and Theory IV -
Duration: Apr 7 2003Apr 10 2003


  • Collapse arrest
  • DNLS equation
  • Discreteness
  • Well-posedness

ASJC Scopus subject areas

  • Theoretical Computer Science
  • General Computer Science
  • Numerical Analysis
  • Modeling and Simulation
  • Applied Mathematics


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