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.
Original language | English (US) |
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Pages (from-to) | 553-566 |
Number of pages | 14 |
Journal | Mathematics and Computers in Simulation |
Volume | 69 |
Issue number | 5-6 |
DOIs | |
State | Published - Aug 5 2005 |
Externally published | Yes |
Event | Nonlinear Waves: Computation and Theory IV - Duration: Apr 7 2003 → Apr 10 2003 |
Keywords
- Collapse arrest
- DNLS equation
- Discreteness
- Well-posedness
ASJC Scopus subject areas
- Theoretical Computer Science
- General Computer Science
- Numerical Analysis
- Modeling and Simulation
- Applied Mathematics