Axisymmetric surface diffusion: Dynamics and stability of self-similar pinchoff

The dynamics of surface diffusion describes the motion of a surface with its normal velocity given by the surface Laplacian of its mean curvature. This flow conserves the volume enclosed inside the surface while minimizing its Surface area. We review the axisymmetric equilibria: the cylinder, sphere, and the Delaunay unduloid. The sphere is stable, while the cylinder is long-wave unstable. A subcritical bifurcation from the cylinder produces a continuous family of unduloid solutions. We present computations that suggest that the stable manifold of the unduloid forms a separatrix between states that relax to the cylinder in infinite time and those that tend toward finite-time pinchoff. We examine the structure of the pinchoff showing it has self-similar structure, using asymptotic, numerical, and analytical methods. In addition to a previously known similarity solution, we find a countable set of similarity solutions, each with a different asymptotic cone angle. We develop a stability theory in similarity variables that selects the original similarity solution as the only linearly stable one and consequently the only observable solution. We also consider similarity solutions describing the dynamics after the topological transition.