TY - JOUR
T1 - On the collapse arresting effects of discreteness
AU - Tzirakis, N.
AU - Kevrekidis, P. G.
N1 - Funding Information:
This work was supported by a University of Massachusetts Faculty Research Grant, the Eppley Foundation for Research and NSF-DMS-0204585 and NSF-CAREER. NT gratefully acknowledges discussions with A.R. Nahmod. PGK is grateful to P. Rosenau for discussions regarding the regularization methods and for relevant references, and also particularly to I.G. Kevrekidis for numerous illuminating discussions regarding blowup problems.
PY - 2005/8/5
Y1 - 2005/8/5
N2 - 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.
AB - 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.
KW - Collapse arrest
KW - DNLS equation
KW - Discreteness
KW - Well-posedness
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U2 - 10.1016/j.matcom.2005.03.013
DO - 10.1016/j.matcom.2005.03.013
M3 - Conference article
AN - SCOPUS:21244432305
SN - 0378-4754
VL - 69
SP - 553
EP - 566
JO - Mathematics and Computers in Simulation
JF - Mathematics and Computers in Simulation
IS - 5-6
T2 - Nonlinear Waves: Computation and Theory IV
Y2 - 7 April 2003 through 10 April 2003
ER -