Symmetric solutions of the dispersionless Toda hierarchy and associated conformal dynamics

Under certain reality conditions, a general solution to the dispersionless Toda lattice hierarchy describes deformations of simply-connected plane domains with a smooth boundary. The solution depends on an arbitrary (real positive) function of two variables which plays the role of a density or a conformal metric in the plane. We consider in detail the important class of symmetric solutions characterized by the density functions that depend only on the distance from the origin and that are positive and regular in an annulus $r_0< |z|<r_1$. We construct the dispersionless tau-function which gives formal local solution to the inverse potential problem and to the Riemann mapping problem and discuss the associated conformal dynamics related to viscous flows in the Hele-Shaw cell.Comment: 28 pages, 3 figures. arXiv admin note: text overlap with arXiv:1302.728

Quantum Gaudin model and classical KP hierarchy

This short note is a review of the intriguing connection between the quantum Gaudin model and the classical KP hierarchy recently established in [1]. We construct the generating function of integrals of motion for the quantum Gaudin model with twisted boundary conditions (the master T-operator) and show that it satisfies the bilinear identity and Hirota equations for the classical KP hierarchy. This implies that zeros of eigenvalues of the master $T$-operator in the spectral parameter have the same dynamics as the Calogero-Moser system of particles.Comment: 12 pages, written for proceedings of the International conference "Physics and Mathematics of Nonlinear Phenomena", Gallipoli, 22-29 June 201