Existence of pearled patterns in the planar Functionalized Cahn-Hilliard equation

TL;DR

This paper demonstrates the existence of pearled patterns in the planar functionalized Cahn-Hilliard equation by reducing the problem to an 8th order ODE system and analyzing bifurcations leading to localized periodic equilibria.

Contribution

It introduces a novel reduction of the FCH equation to an 8th order ODE and characterizes the bifurcation leading to pearled equilibrium patterns.

Findings

Pearled equilibria are shown to exist in the FCH equation.

The onset of pearling coincides with a degenerate 1:1 resonant normal form bifurcation.

A two-parameter family of localized, periodic-in-plane solutions is identified.

Abstract

The functionalized Cahn-Hilliard (FCH) equation supports planar and circular bilayer interfaces as equilibria which may lose their stability through the pearling bifurcation: a periodic, high-frequency, in-plane modulation of the bilayer thickness. In two spatial dimensions we employ spatial dynamics and a center manifold reduction to reduce the FCH equation to an 8th order ODE system. A normal form analysis and a fixed-point-theorem argument show that the reduced system admits a degenerate 1:1 resonant normal form, from which we deduce that the onset of the pearling bifurcation coincides with the creation of a two-parameter family of pearled equilibria which are periodic in the in-plane direction and exponentially localized in the transverse direction.