The Role of Symmetry and Separation in Surface Evolution and Curve Shortening

  title={The Role of Symmetry and Separation in Surface Evolution and Curve Shortening},
  author={Philip Broadbridge and Peter J. Vassiliou},
  journal={Symmetry Integrability and Geometry-methods and Applications},
With few exceptions, known explicit solutions of the curve shortening flow (CSE) of a plane curve, can be constructed by classical Lie point symmetry reductions or by functional separation of variables. One of the functionally separated solutions is the exact curve shortening flow of a closed, convex \oval"-shaped curve and another is the smoothing of an initial periodic curve that is close to a square wave. The types of anisotropic evaporation coefficient are found for which the evaporation… Expand

Figures from this paper

Asymptotically self-similar solutions to curvature flow equations with prescribed contact angle and their applications to groove profiles due to evaporation-condensation
We study the asymptotic behavior of solutions to fully nonlinear second order parabolic equations including a generalized curvature flow equation which was introduced by Mullins in 1957 as a model ofExpand
The fundamental solutions of the curve shortening problem via the Schwarz function
Curve shortening in the z-plane in which, at a given point on the curve, the normal velocity of the curve is equal to the curvature, is shown to satisfy StSz = Szz , where S(z, t) is the SchwarzExpand
Enhanced group classification of nonlinear diffusion–reaction equations with gradient-dependent diffusivity
Abstract We carry out the enhanced group classification of a class of (1+1)-dimensional nonlinear diffusion–reaction equations with gradient-dependent diffusivity using the two-step version of theExpand
Applications of Integrable Nonlinear Diffusion Equations in Industrial Modelling
There are useful integrable nonlinear diffusion equations that can be transformed directly to linear partial differential equations. The possibility of linearisation allows us to incorporate a muchExpand
On Some Simple Methods to Derive the Hairclip and Paperclip Solutions of the Curve Shortening Flow
We use two simple methods to derive four important explicit graphical solutions of the curve shortening flow in the plane. They are well-known as the circle, hairclip, paperclip, and grim reaperExpand
Self-similar solutions to the mean curvature flow in the Minkowski plane $\mathbf R^{1,1}$
We introduce the mean curvature flow of curves in the Minkowski plane $\mathbf R^{1,1}$ and give a classification of all the self-similar solutions. In addition, we describe five other exactExpand
On an asymptotically log-periodic solution to the graphical curve shortening flow equation
With the help of heat equation, we first construct an example of a graphical solution to the curve shortening flow. This solution $ y\left(x, t\right) \ $has the interesting property that itExpand
The Impact of Applications on Mathematics
Biological cells require active fluxes of matter to maintain their internal organization and perform multiple tasks to live. In particular they rely on cytoskeletal transport driven by motorExpand


The normalized curve shortening flow and homothetic solutions
The curve shortening problem, by now widely known, is to understand the evolution of regular closed curves γ: R/Z -> M moving according to the curvature normal vector: dy/dt = kN = -"the ZΛgradientExpand
Integrable nonlinear evolution equations applied to solidification and surface redistribution
Members of a hierarchy of integrable nonlinear evolution equations are applied to a problem in solidification and various problems in surface redistribution of crystalline materials. The members ofExpand
An integrable fourth-order nonlinear evolution equation applied to surface redistribution due to capillarity
Members of an hierarchy of integrable nonlinear evolution equations, related to the well-known linearizable diffusion equation which has the diffusivity form as the reciprocal of the square of theExpand
Grain boundary grooving by surface diffusion: an analytic nonlinear model for a symmetric groove
  • P. Tritscher, P. Broadbridge
  • Mathematics
  • Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences
  • 1995
The fourth-order nonlinear boundary-value problem for the evolution of a single symmetric grain-boundary groove by surface diffusion is modelled analytically. A solution is achieved by partitioningExpand
Overview no. 113 surface motion by surface diffusion
Abstract Geometry growth laws for morphological change are developed and examined for the class of dynamic problems where surface diffusion is the only transport mechanism and hence volume isExpand
Temperature-dependent surface diffusion near a grain boundary
Metal surface evolution is described by a nonlinear fourth-order partial differential equation for curvature-driven flow. The standard boundary conditions for grain-boundary grooving, at aExpand
Exact solvability of the Mullins nonlinear diffusion model of groove development
The Mullins equation for the development of a surface groove by evaporation–condensation is yt=yxx/1+y2x. It is pointed out that this is the equation of the potential for the field variable ΘExpand
Separation of variables for the 1-dimensional non-linear diffusion equation
Abstract The class of separable solutions of a 1-dimensional sourceless diffusion equation is stabilized by the action of the generic symmetry group. It includes all solutions invariant under aExpand
Invariant Geometric Evolutions of Surfaces and Volumetric Smoothing
This paper presents the simplest affine invariant flow for (convex) surfaces in three-dimensional space, which, like the affine-invariant curve shortening flow, will be of fundamental importance in the processing of three- dimensional images. Expand
An integrable fourth-order nonlinear evolution equation applied to thermal grooving of metal surfaces
The fourth-order nonlinear partial differential equation for surface diffusion is approximated by a new integrable nonlinear evolution equation. Exact solutions are obtained for thermal grooving,Expand