dorsal/arxiv
View SchemaA New Class of Nonsingular Exact Solutions for Laplacian Pattern Formation
| Authors | Mark B. Mineev-Weinstein, Silvina Ponce Dawson |
|---|---|
| Categories | |
| ArXiv ID | patt-sol/9305010 |
| URL | https://arxiv.org/abs/patt-sol/9305010 |
| DOI | 10.1103/PhysRevE.50.R24 |
Abstract
We present a new class of exact solutions for the so-called {\it Laplacian Growth Equation} describing the zero-surface-tension limit of a variety of 2D pattern formation problems. Contrary to common belief, we prove that these solutions are free of finite-time singularities (cusps) for quite general initial conditions and may well describe real fingering instabilities. At long times the interface consists of N separated moving Saffman-Taylor fingers, with ``stagnation points'' in between, in agreement with numerous observations. This evolution resembles the N-soliton solution of classical integrable PDE's.
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"abstract": "We present a new class of exact solutions for the so-called {\\it Laplacian\nGrowth Equation} describing the zero-surface-tension limit of a variety of 2D\npattern formation problems. Contrary to common belief, we prove that these\nsolutions are free of finite-time singularities (cusps) for quite general\ninitial conditions and may well describe real fingering instabilities. At long\ntimes the interface consists of N separated moving Saffman-Taylor fingers, with\n``stagnation points\u0027\u0027 in between, in agreement with numerous observations. This\nevolution resembles the N-soliton solution of classical integrable PDE\u0027s.",
"arxiv_id": "patt-sol/9305010",
"authors": [
"Mark B. Mineev-Weinstein",
"Silvina Ponce Dawson"
],
"categories": [
"patt-sol",
"nlin.PS"
],
"doi": "10.1103/PhysRevE.50.R24",
"title": "A New Class of Nonsingular Exact Solutions for Laplacian Pattern Formation",
"url": "https://arxiv.org/abs/patt-sol/9305010"
},
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