This Work Was Carried out under the Supervision Of

We consider Laplacian growth problems using a field theory approach. In particular we consider the Saffman-Taylor (ST) problem, in which a non-viscous fluid is pumped into the center of a HeleShaw cell filled with a highly viscous fluid generating a bubble of a distinctive fingering fractal pattern. The idealized settings of the problem, with vanishing surface tension between the bubble and the surrounding fluid, is singular due to the formation of cusps after a finite time (for generic initial conditions). A natural regularization of the cusp, is the addition of surface tension, but this complicates the mathematical description of the problem a great deal. Our first goal is to reveal the equivalence of the idealized Saffman-Taylor problem and the semiclassical limit of the quantum Hall system in a strong inhomogeneous magnetic field. This quantum system regularizes the cusp by introducing a short-scale cut-off associated with a Planck constant. Next we discuss the relation between the quantum Hall system and an integrable system of nonlinear equations known as the two-dimensional Toda lattice (2DTL). This relation allows us to employ methods from soliton theory for the description of the ST problem, and in particular it will allow us to to find a novel method for the regularization of the cusp. In the language of soliton theory, the semiclassical limit, is termed “the dispersionless limit”, since it may be associated with neglecting dispersion in the nonlinear equations (the 2DTL in our case). In the limit of small dispersion, one obtains a system of equations known as the Whitham equations. Dropping the dispersion from the outset, results in singular solutions for the Whitham equations. On the other hand, one may take the limit of small dispersion correctly and obtain well-behaved solutions. By analogy, the semiclassical limit must be taken correctly in the quantum Hall system in order to obtain solution which do not exhibit the singularities of the cusps. We study the implications of this regularization method in this thesis. We show that it amounts to allowing new small bubbles to form in the ST problem. These bubbles grow and eventually merge with the large bubble. The appearance of new bubbles is a mathematical extension of the 5 6 CONTENTS original ST evolutions, in which fluid can “tunnel” from the large bubble into a new location, where it may form a new bubble. During the evolution of this multi-bubble solution tunneling persists as fluid is constantly exchanged between the two bubbles (all the bubbles share the same pressure). In order to study the physical implications of this regularization procedure we will study the evolution obtained by this method. In particular we will study the shape of the bubble after the merging of two bubbles, and show that for some initial conditions one obtains tip-splitting solutions. Tip splitting is believed to be a process important for the formation of the fractal. We discuss the scaling form of these solutions and the possible implications for the fractal dimensions of the ST bubble.