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. The Laplacian growth problem is one of the central problems in the theory of pattern formation. It has many different faces and a lot of important applications. In general words, this is about dynamics of moving front (interface) between two different phases. In many cases the dynamics is governed by a scalar field that obeys the Laplace equation; that is why this class of growth problems is called Laplacian. Here we shall confine ourselves to the two-dimensional (2D) case only. To be definite, we shall speak about two incompressible fluids with different viscosities on the plane. In practice, the 2D geometry is realized in the narrow gap between two plates. In this version, this is known as the Saffman–Taylor problem or viscous fingering in the Hele–Shaw cell. For a review, see [1]. We shall mostly concentrate on the external radial problem for it turns out to be the simplest case in the frame of the suggested approach. Let the exterior of a simply connected domain on the plane be occupied by a viscous fluid (oil) while the interior be occupied by a fluid with small viscosity (water). The oil/water interface is assumed to be a simple analytic curve. Other versions such as internal radial problem, wedge or channel geometry are briefly discussed at the end of the paper. Basically, they allow for the same approach. Let p(x, y) be the pressure, then p is constant in the water domain. We set it equal to zero. In the case of zero surface tension p is a continuous function across the interface, so p = 0 on the interface. In the oil domain the gradient of p is proportional to local velocity V = (Vx, Vy) of the fluid (Darcy’s law): V = −κ grad p, (1) where κ is called the filtration coefficient1. In particular, this law holds on the interface thus governing its dynamics: