Growth processes related to the dispersionless Lax equations

Abstract

This paper is a short review of the connection between certain types of growth processes and the integrable systems theory, written from the viewpoint of the latter. Starting from the dispersionless Lax equations for the 2D Toda hierarchy, we interpret them as evolution equations for conformal maps in the plane. This provides a unified approach to evolution of smooth domains (such as Laplacian growth) and growth of slits. We show that the Löwner differential equation for a parametric family of conformal maps of slit domains arises as a consistency condition for reductions of the dispersionless Toda hierarchy. It is also demonstrated how the both types of growth processes can be simulated by the large $N$ limit of the Dyson gas picture for the model of normal random matrices.

16 pages, 2 figures, typos corrected, references added

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• Conformal Mappings and Dispersionless Toda hierarchy
• Symmetric solutions of the dispersionless Toda hierarchy and associated conformal dynamics
• Dispersionless version of the constrained Toda hierarchy and symmetric radial Löwner equation
• Loewner equations and reductions of dispersionless hierarchies
• Kernel Formula Approach to the Universal Whitham Hierarchy