Dispersionless version of the constrained Toda hierarchy and symmetric radial Löwner equation

We study the dispersionless version of the recently introduced constrained Toda hierarchy. Like the Toda lattice itself, it admits three equivalent formulations: the formulation in terms of Lax equations, the formulation of the Zakharov–Shabat type and the formulation through the generating equation...

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Bibliographic Details
Published in:Letters in mathematical physics 2022-10, Vol.112 (5), Article 105
Main Authors: Takebe, Takashi, Zabrodin, Anton
Format: Article
Language:English
Subjects:
Complex Systems
Conformal mapping
Differential equations
Dispersion
Geometry
Group Theory and Generalizations
Mathematical and Computational Physics
Physics
Physics and Astronomy
Slits
Theoretical
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Summary:We study the dispersionless version of the recently introduced constrained Toda hierarchy. Like the Toda lattice itself, it admits three equivalent formulations: the formulation in terms of Lax equations, the formulation of the Zakharov–Shabat type and the formulation through the generating equation for the dispersionless limit of logarithm of the tau-function. We show that the dispersionless constrained Toda hierarchy describes conformal maps of reflection-symmetric planar domains to the exterior of the unit disc. We also find finite-dimensional reductions of the hierarchy and show that they are characterized by a differential equation of the Löwner type which we call the symmetric radial Löwner equation. It is also shown that solutions to the symmetric radial Löwner equation are conformal maps of the exterior of the unit circle with two symmetric slits to the exterior of the unit circle.
ISSN:0377-9017
1573-0530
DOI:10.1007/s11005-022-01599-y