57 research outputs found
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, ,
which is {\it precisely twice} greater than the velocity, , of the uniform
background flow, i.e., . 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
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" }
which corresponds to the changing volume while {\it all higher moments, ,
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
Integrable Structure of Interface Dynamics
We establish the equivalence of a 2D contour dynamics to the dispersionless
limit of the integrable Toda hierarchy constrained by a string equation.
Remarkably, the same hierarchy underlies 2D quantum gravity.Comment: 5 pages, no figures, submitted to Phys. Rev. Lett, typos correcte
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