On the collapse arresting effects of discreteness

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.