The Role of Symmetry and Separation in Surface Evolution and Curve Shortening
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, conve...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2011 |
| Main Authors: | , |
| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2011
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/147169 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | The Role of Symmetry and Separation in Surface Evolution and Curve Shortening / P. Broadbridge, P. Vassiliou // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 28 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862561834310041600 |
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| author | Broadbridge, P. Vassiliou, P. |
| author_facet | Broadbridge, P. Vassiliou, P. |
| citation_txt | The Role of Symmetry and Separation in Surface Evolution and Curve Shortening / P. Broadbridge, P. Vassiliou // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 28 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | 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.
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| first_indexed | 2025-11-25T23:28:42Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-147169 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-11-25T23:28:42Z |
| publishDate | 2011 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Broadbridge, P. Vassiliou, P. 2019-02-13T18:06:47Z 2019-02-13T18:06:47Z 2011 The Role of Symmetry and Separation in Surface Evolution and Curve Shortening / P. Broadbridge, P. Vassiliou // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 28 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 35A30; 35K55; 58J70; 74E10 DOI:10.3842/SIGMA.2011.052 https://nasplib.isofts.kiev.ua/handle/123456789/147169 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. This paper is a contribution to the Special Issue “Symmetry, Separation, Super-integrability and SpecialFunctions (S⁴)”. The full collection is available at http://www.emis.de/journals/SIGMA/S4.html.
 This paper is submitted in appreciation of the valuable on-going contributions of Professor
 Willard Miller Jr. The first author gratefully acknowledges support by the Australian Research Council under project DP1095044. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications The Role of Symmetry and Separation in Surface Evolution and Curve Shortening Article published earlier |
| spellingShingle | The Role of Symmetry and Separation in Surface Evolution and Curve Shortening Broadbridge, P. Vassiliou, P. |
| title | The Role of Symmetry and Separation in Surface Evolution and Curve Shortening |
| title_full | The Role of Symmetry and Separation in Surface Evolution and Curve Shortening |
| title_fullStr | The Role of Symmetry and Separation in Surface Evolution and Curve Shortening |
| title_full_unstemmed | The Role of Symmetry and Separation in Surface Evolution and Curve Shortening |
| title_short | The Role of Symmetry and Separation in Surface Evolution and Curve Shortening |
| title_sort | role of symmetry and separation in surface evolution and curve shortening |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/147169 |
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