LONG-TIME BEHAVIOR OF THE N-FINGER SOLUTION OF THE LAPLACIAN GROWTH EQUATION

The non-singular N-finger solutions of the Laplacian Growth Equation, Im [(partial derivative f (x, t)/partial derivative t)BAR (partial derivative f (x, t)/partial derivative 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 Schrodinger, and sine-Gordon equations, is discussed. Using the asymptotic properties of the N-finger solution, canonical variables of ''action-angle''-type are introduced.