Graph Ginzburg–Landau: discrete dynamics, continuum limits, and applications. An overview

Abstract

In [BF12, BF16] the graph Ginzburg–Landau functional was introduced. Here u is a real-valued function on the node set V of a simple1, undirected graph (with ui its value at node i), ωij ≥ 0 are edge weights which are assumed to be positive on all edges in the graph and zero between non-neighbouring nodes i and j, ε is a positive parameter, and W is a double well potential with wells of equal depth. A typical choice is the quartic polynomial W(x) = x2(x − 1)2 which has wells of depth 0 at x = 0 and x = 1, but we will encounter some situations where other choices are useful or even necessary.