Whitham-Toda Hierarchy in the Laplacian Growth Problem

The Laplacian growth problem in the limit of zero surface tension is proved to be equivalent to finding a particular solution to the dispersionless Toda lattice hierarchy. The hierarchical times are harmonic moments of the growing domain. The Laplacian growth equation itself is the quasiclassical version of the string equation that selects the solution to the hierarchy.Comment: 7 pages, no figures, Talk given at the Workshop NEEDS 99 (Crete, Greece, June 1999

Velocity selection (without surface tension) in multi-connected Laplacian growth

We predict a novel selection phenomenon in nonlinear interface dynamics out of equilibrium. Using a recently developed formalism based on the Schottky-Klein prime functions, we extended the existing integrable theory from a single interface to multiple moving interfaces. After applying this extended theory to the two-dimensional Laplacian growth, we derive a new rich class of exact (non-singular) solutions for the unsteady dynamics of an arbitrary assembly of air bubbles within a layer of a viscous fluid in a Hele-Shaw cell. These solutions demonstrate that all bubbles reach an asymptotic velocity, $U$, which is {\it precisely twice} greater than the velocity, $V$, of the uniform background flow, i.e., $U=2V$. The result does not depend on the number of bubbles. It is worth to mention that contrary to common belief, the predicted velocity selection does not require surface tension.Comment: 5 pages, 1 figure. Updated versio

Planar elliptic growth

The planar elliptic extension of the Laplacian growth is, after a proper parametrization, given in a form of a solution to the equation for area-preserving diffeomorphisms. The infinite set of conservation laws associated with such elliptic growth is interpreted in terms of potential theory, and the relations between two major forms of the elliptic growth are analyzed. The constants of integration for closed form solutions are identified as the singularities of the Schwarz function, which are located both inside and outside the moving contour. Well-posedness of the recovery of the elliptic operator governing the process from the continuum of interfaces parametrized by time is addressed and two examples of exact solutions of elliptic growth are presented.Comment: 27 page

Bubble dynamics in a Hele-Shaw cell: Exact solutions and velocity selection

Unsteady evolution of an inviscid bubble, surrounded by a viscous fluid moving with velocity $V$ far from the bubble, in a Hele-Shaw channel is described by a general class of exact solutions obtained without surface tension. Several numerical examples are provided. These solutions select the bubble velocity $U = 2V$ in a long-time asymptotics from a continuum of all possible values of $U$, because the selected value is shown to represent the only attractor of this infinitely-dimensional nonlinear dynamical system. This work is an expanded version of results announced earlier by two of us (Phys.~Rev.~E {\bf 89}, 061003(R), 2014).Comment: 33 pages, 13 figure

Multidimensional Pattern Formation Has an Infinite Number of Constants of Motion

Extending our previous work on 2D growth for the Laplace equation we study here {\it multidimensional} growth for {\it arbitrary elliptic} equations, describing inhomogeneous and anisotropic pattern formations processes. We find that these nonlinear processes are governed by an infinite number of conservation laws. Moreover, in many cases {\it all dynamics of the interface can be reduced to the linear time--dependence of only one ``moment" $M_0$} which corresponds to the changing volume while {\it all higher moments, $M_l$, are constant in time. These moments have a purely geometrical nature}, and thus carry information about the moving shape. These conserved quantities (eqs.~(7) and (8) of this article) are interpreted as coefficients of the multipole expansion of the Newtonian potential created by the mass uniformly occupying the domain enclosing the moving interface. Thus the question of how to recover the moving shape using these conserved quantities is reduced to the classical inverse potential problem of reconstructing the shape of a body from its exterior gravitational potential. Our results also suggest the possibility of controlling a moving interface by appropriate varying the location and strength of sources and sinks.Comment: CYCLER Paper 93feb00