Differential geometric bifurcation problems in pde2path -- algorithms and tutorial examples

TL;DR

This paper presents algorithms and tutorial examples for solving differential geometric bifurcation problems in PDEs using the MATLAB toolbox pde2path, focusing on immersed surfaces and symmetry-breaking bifurcations.

Contribution

It introduces methods for treating geometric PDE bifurcations with pde2path, including examples on minimal surfaces, constant mean curvature surfaces, and biomembrane models.

Findings

Identification of symmetry-breaking bifurcations in geometric PDEs

Benchmarking with analytically known bifurcations

Demonstration of algorithms on complex surface problems

Abstract

We describe how some differential geometric bifurcation problems can be treated with the MATLAB continuation and bifurcation toolbox pde2path. The basic setup consists in solving the PDEs for the normal displacement of an immersed surface $X\subset\mathbb{R}^3$ and subsequent update of $X$ in each continuation step, combined with bifurcation detection and localization, followed by possible branch switching. Examples treated include some minimal surfaces such as Enneper's surface and a Schwarz-P-family, some non-zero constant mean curvature surfaces such as liquid bridges and nodoids, and some 4th order biomembrane models. In all of these we find interesting symmetry breaking bifurcations. Some of these are (semi)analytically known and thus are used as benchmarks.