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

@article{Broadbridge2011TheRO,
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},
year={2011},
volume={7},
pages={052}
}
• Published 1 June 2011
• Physics
• 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…
10 Citations

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## References

SHOWING 1-10 OF 30 REFERENCES

### The normalized curve shortening flow and homothetic solutions

• Mathematics
• 1986
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 ZΛgradient

### 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 of

### An integrable fourth-order nonlinear evolution equation applied to surface redistribution due to capillarity

• P. Tritscher
• Mathematics
The Journal of the Australian Mathematical Society. Series B. Applied Mathematics
• 1997
Abstract 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

### Grain boundary grooving by surface diffusion: an analytic nonlinear model for a symmetric groove

• 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 partitioning

### Temperature-dependent surface diffusion near a grain boundary

• Mathematics
• 2010
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 a

### 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 Θ

### Invariant Geometric Evolutions of Surfaces and Volumetric Smoothing

• Mathematics
SIAM J. Appl. Math.
• 1997
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.

### An integrable fourth-order nonlinear evolution equation applied to thermal grooving of metal surfaces

• Mathematics
• 1994
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,