### Abstract

Original language | English |
---|---|

Pages (from-to) | 1-19 |

Number of pages | 19 |

Journal | Symmetry, Integrability and Geometry: Methods and Applications |

Volume | 7 |

DOIs | |

Publication status | Published - 2011 |

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### Cite this

*Symmetry, Integrability and Geometry: Methods and Applications*,

*7*, 1-19. https://doi.org/10.3842/SIGMA.2011.052

}

*Symmetry, Integrability and Geometry: Methods and Applications*, vol. 7, pp. 1-19. https://doi.org/10.3842/SIGMA.2011.052

**The Role of Symmetry and Separation in Surface Evolution and Curve Shortening.** / Broadbridge, Philip; Vassiliou, Peter.

Research output: Contribution to journal › Article

TY - JOUR

T1 - The Role of Symmetry and Separation in Surface Evolution and Curve Shortening

AU - Broadbridge, Philip

AU - Vassiliou, Peter

PY - 2011

Y1 - 2011

N2 - 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-condensation evolution does or does not have solutions that are analogous to the basic solutions of the CSE, namely the grim reaper travelling wave, the homothetic shrinking closed curve and the homothetically expanding grain boundary groove. Using equivalence classes of anisotropic diffusion equations, it is shown that physical models of evaporation-condensation must have a diffusivity function that decreases as the inverse square of large slope. Some exact separated solutions are constructed for physically consistent anisotropic diffusion equations

AB - 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-condensation evolution does or does not have solutions that are analogous to the basic solutions of the CSE, namely the grim reaper travelling wave, the homothetic shrinking closed curve and the homothetically expanding grain boundary groove. Using equivalence classes of anisotropic diffusion equations, it is shown that physical models of evaporation-condensation must have a diffusivity function that decreases as the inverse square of large slope. Some exact separated solutions are constructed for physically consistent anisotropic diffusion equations

KW - curve shortening flow

KW - exact solutions

KW - symmetry

U2 - 10.3842/SIGMA.2011.052

DO - 10.3842/SIGMA.2011.052

M3 - Article

VL - 7

SP - 1

EP - 19

JO - Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

JF - Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SN - 1815-0659

ER -