2001/11/23 by D. G. Aronson, Aronson, D. G., S. I. Betelu +3
Physics and Astronomy · #Adaptation and Self-Organizing Systems (nlin.AO) #FOS: Physical sciences #Pattern Formation and Solitons (nlin.PS) #nlin.AO #nlin.PS
paper · pdf · doi:10.48550/arxiv.nlin/0111055
14 pages, 2 figure
arxiv created 2001/12/06 · arxiv updated 2009/11/30
We present a systematic computational approach to the study of self-similar dynamics. The approach, through the use of what can be thought of as a ``dynamic pinning condition" factors out self-similarity, and yields a transformed, non-local evolution equation. The approach, which is capable of treating both first and second kind self-similar solutions, yields as a byproduct the self-similarity exponents of the original dynamics. We illustrate the approach through the porous medium equation, showing how both the Barenblatt (first kind) and the Graveleau (second kind) self-similar solutions arise in this framework. We also discuss certain implications of the dynamics of the transformed equation (which we will name "MN-dynamics"); in particular we discuss the discrete-time implementation of the approach, and connections with time-stepper based methods for the "coarse" integration/bifurcation analysis of microscopic simulators.