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.
Dispersionless version of multi-component Pfaff-Toda hierarchy
Skolkovo Institute of Science and Technology（斯科尔科沃科学技术研究所）；National Research University Higher School of Economics（国立研究高等经济学院）；NRC “Kurchatov institute”（库尔恰托夫国家研究中心）
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 …
We explain how dispersionless integrable hierarchy in 2d topological field theory arises from the Kodaira-Spencer gravity (BCOV theory). The infinitely many commuting Hamiltonians are given by the current observables associated to the infinite abelian symmetries of the Kodaira-Spencer gravity. We describe a BV framework of effective field theories that leads to the B-model interpretation of dispersionless integrable hierarchy.
The spectral problem for the quantum dispersionless Korteweg-de Vries (KdV) hierarchy, aka the quantum Hopf hierarchy, is solved by Dubrovin. In this article, following Dubrovin, we study Buryak-Rossi's quantum KdV hierarchy. In particular, we prove a symmetry property and a non-degeneracy property for the quantum KdV Hamiltonians. On the basis of this we construct a complete set of common eigenvectors. The analysis underlying this spectral problem implies certain vanishing identities for combinations of characters of the symmetric group. We also comment on the geometry of the spectral curves …
We continue the study of the B-Toda hierarchy (the Toda lattice with the constraint of type B) which can be regarded as a discretization of the BKP hierarchy. We introduce the tau-function of the B-Toda hierarchy and obtain the bilinear equations for it. Examples of soliton tau-functions are presented in the explicit form.
We construct a certain reduction of the 2D Toda hierarchy and obtain a tau-symmetric Hamiltonian integrable hierarchy. This reduced integrable hierarchy controls the linear Hodge integrals in the way that one part of its flows yields the intermediate long wave hierarchy, and the remaining flows coincide with a certain limit of the flows of the fractional Volterra hierarchy which controls the special cubic Hodge integrals.
We consider solutions of the 2D Toda lattice hierarchy which are trigonometric functions of the ``zeroth'' time $t_0=x$. It is known that their poles move as particles of the trigonometric Ruijsenaars-Schneider model. We extend this correspondence to the level of hierarchies: the dynamics of poles with respect to the $m$-th hierarchical time $t_m$ (respectively, $\bar t_m$) of the 2D Toda lattice hierarchy is shown to be governed by the Hamiltonian which is proportional to the $m$-th Hamiltonian $\mbox{tr}\, L^m$ (respectively, $\mbox{tr}\, L^{-m}$) of the Ruijsenaars-Schneider model, where $L…
We introduce a new integrable hierarchy of nonlinear differential-difference equations which we call constrained Toda hierarchy (C-Toda). It can be regarded as a certain subhierarchy of the 2D Toda lattice obtained by imposing the constraint $\bar {\cal L}={\cal L}^†$ on the two Lax operators (in the symmetric gauge). We prove the existence of the tau-function of the C-Toda hierarchy and show that it is the square root of the 2D Toda lattice tau-function. In this and some other respects the C-Toda is a Toda analogue of the CKP hierarchy. It is also shown that zeros of the tau-function of ellip…
We study elliptic families of solutions to the recently introduced constrained Toda hierarchy, i.e., solutions which are elliptic functions of some linear combination of the hierarchical times. Equations of motion for poles of such solutions are obtained.
