Planar elliptic growth

Abstract

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.

27 pages

References in corpus (5)

• Integrable Hierarchies and Dispersionless Limit
• Integrable Structure of Interface Dynamics
• Laplacian Growth and Whitham Equations of Soliton Theory
• Self-similarity in Laplacian Growth
• Multidimensional Pattern Formation Has an Infinite Number of Constants of Motion

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• On a two-phase Hele-Shaw problem with a time-dependent gap and distributions of sinks and sources
• Non-univalent solutions of the Polubarinova-Galin equation
• On Integrability and Exact Solvability in Deterministic and Stochastic Laplacian Growth
• Integrability-preserving regularizations of Laplacian Growth
• From reflections to elliptic growth
• Variation of Green Functions: Normal Derivatives
• Variational Methods and Planar Elliptic Growth