A New Class of Nonsingular Exact Solutions for Laplacian Pattern Formation

Details A New Class of Nonsingular Exact Solutions for Laplacian Pattern Formation A New Class of Nonsingular Exact Solutions for Laplacian Pattern Formation

1993-05-26

arxiv.org/abs/patt-sol/9305010v1

Nonlinear Sciences

Pattern Formation and Solitons

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Scientific paper

10.1103/PhysRevE.50.R24

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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