Long-Time Behavior of the N-finger Solution of the Laplacian Growth Equation

SILVINA PONCE DAWSON; MARK B. MINEEV-WEINSTEIN

PHYSICA D - NONLINEAR PHENOMENA

Año: 1994 vol. 73 p. 373 - 387

0167-2789

The non-singular N-finger solutions of the Laplacian Growth Equation, Im [ovbar∂f(x,t)∂t) (∂f(x,t)/∂ x)] = 1, describing the motion of the interface in numerous non-equilibrium processes, such as dendritic growth, flows through porous media, electrodeposition, etc., is analyzed. The motion of the interface is described by N + 1 moving singularities (simple poles) in the upper-half of an auxiliar “mathematical plane”. In the long-time limit these singularities tend to the real axis, following an exponential law. Meanwhile, the physical interface develops at most N separated fingers. In the case of enough separation, each of the gaps between fingers corresponds to one singularity while each finger is locally similar to the Saffman-Taylor one. The analogy with the N-soliton solutions of exactly integrable PDE's, such as Korteweg-de Vries, Nonlinear Schrödinger, and sine-Gordon equations, is discussed. Using the asymptotic properties of the N-finger solution, canonical variables of “action-angle”-type are introduced.