Homoclinic snaking in discrete systems

Hadi Susanto, University of Essex. Part of the applied mathematics seminar series.

‘Snaking’ bifurcations, describing localised solutions existing within a small region of parameter space, are widely observed in in numerous physical applications. In this talk, I will present some of our recent works on the snaking of localised patterns in discrete systems. Particularly I am going to consider three different equations: the discrete Swift-Hohenberg equation (i.e., obtained from discretizing the spatial derivatives of the Swift-Hohenberg equation using central finite differences), coupled discrete nonlinear Schrodinger equations with parity-time symmetric potential, and two-dimensional Allen-Cahn equations. The coupled Schrodinger equations are proposed as a classical model of the parity-time symmetric quantum physics. Our study shows that discrete systems can yield bifurcation diagrams that are different from their continuum counterparts. In particular, we provide analytical methods for the width of the snaking region in the weak and strong coupling region.