Non-Laplacian growth, algebraic domains and finite reflection groups

TL;DR

This paper investigates the algebraic nature of moving boundary domains driven by elliptic PDEs, revealing a connection with Calogero-Moser systems and finite reflection groups, thus linking geometric growth patterns with integrable systems.

Contribution

It establishes a novel link between non-Laplacian growth processes, algebraic domains, and the theory of finite reflection groups and Calogero-Moser systems.

Findings

Domains are algebraic for specific elliptic PDEs with singularities

Connection between growth dynamics and integrable Calogero-Moser systems

Identification of conditions for algebraic domains in non-Laplacian growth

Abstract

Dynamics of planar domains with moving boundaries driven by the gradient of a scalar field that satisfies an elliptic PDE is studied. We consider the question: For which kind of PDEs the domains are algebraic, provided the field has singularities at a fixed point inside the domain? The construction reveals a direct connection with the theory of the Calogero-Moser systems related to finite reflection groups and their integrable deformations.