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THE NONLINEAR HEAT EQUATION ON DENSE GRAPHS AND GRAPH LIMITS
Journal article   Open access   Peer reviewed

THE NONLINEAR HEAT EQUATION ON DENSE GRAPHS AND GRAPH LIMITS

Georgi S. Medvedev
SIAM journal on mathematical analysis, v 46(4), pp 2743-2766
01 Jan 2014
DOI: https://doi.org/10.1137/130943741
url
http://arxiv.org/abs/1302.5804View
SubmittedOpen Access (License Unspecified),  Open

Abstract

Mathematics Mathematics, Applied Physical Sciences Science & Technology
The continuum limit of coupled dynamical systems is an approximate procedure, by which the dynamical problem on a sequence of large graphs is replaced by an evolution integral equation on a continuous spatial domain. While this method has been widely used in the analysis of pattern formation in nonlocally coupled networks, its mathematical basis remained little understood. In this paper, we use the combination of ideas and results from the theory of graph limits and nonlinear evolution equations to provide a rigorous mathematical justification for taking the continuum limit and to extend this method to cover many complex networks, for which it has not been applied before. Specifically, for dynamical networks on convergent sequences of simple and weighted graphs, we prove convergence of solutions of the initial-value problems for discrete models to those of the limiting continuous equations. In addition, for sequences of simple graphs converging to {0, 1}-valued graphons, it is shown that the convergence rate depends on the fractal dimension of the boundary of the support of the graph limit. These results are then used to study the coexistence of coherence and incoherence in chimera states and the attractors of the nonlocal Kuramoto equation on certain multipartite graphs. Furthermore, the analytical tools developed in this work are used in the rigorous justification of the continuum limit for networks on random graphs that we undertake in a companion paper [Arch. Ration. Mech. Anal., 21 (2014), pp. 781-803]. As a by-product of the analysis of the continuum limit on deterministic and random graphs, we identify the link between this problem and the convergence analysis of several classical numerical schemes: the collocation, Galerkin, and Monte Carlo methods. Therefore, our results can be used to characterize convergence of these approximate methods of solving initial-value problems for nonlinear evolution equations with nonlocal interactions.

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