467 citations · 2k across the 86 of their papers we have counts for
23 papers · 1 filter
Direct linearization, Cauchy matrix and Sato Grassmannian
Kanehisa Takasaki
Fu and Nijhoff's direct linearization scheme for the KP hierarchy and its reductions employs a system of evolution equations of an infinite matrix with quadratic nonlinearity.…
Generalized ILW hierarchy: Solutions and limit to extended lattice GD hierarchy
Kanehisa Takasaki
The intermediate long wave (ILW) hierarchy and its generalization, labelled by a positive integer , can be formulated as reductions of the lattice KP hierarchy. The integrabilit…
Extended lattice Gelfand-Dickey hierarchy
Kanehisa Takasaki
The lattice Gelfand-Dickey hierarchy is a lattice version of the Gelfand-Dickey hierarchy. A special case is the lattice KdV hierarchy. Inspired by recent work of Buryak and Rossi,…
Pfaffian structures and certain solutions to BKP hierarchies II. Multiple integrals
A. Orlov, T. Shiota, K. Takasaki
We introduce a useful and rather simple classes of BKP tau functions which which we shall shall call "easy tau functions". We consider the "large BKP hiearchy" related to $O(2\inft…
Two extensions of 1D Toda hierarchy
Kanehisa Takasaki
The extended Toda hierarchy of Carlet, Dubrovin and Zhang is reconsidered in the light of a 2+1D extension of the 1D Toda hierarchy constructed by Ogawa. These two extensions of th…
Auxiliary Linear Problem, Difference Fay Identities and Dispersionless Limit of Pfaff-Toda Hierarchy
Kanehisa Takasaki
Recently the study of Fay-type identities revealed some new features of the DKP hierarchy (also known as "the coupled KP hierarchy" and "the Pfaff lattice"). Those results are now…
Loewner equations, Hirota equations and reductions of universal Whitham hierarchy
Kanehisa Takasaki, Takashi Takebe
This paper reconsiders finite variable reductions of the universal Whitham hierarchy of genus zero in the perspective of dispersionless Hirota equations. In the case of one-variabl…
Differential Fay identities and auxiliary linear problem of integrable hiearchies
Kanehisa Takasaki
We review the notion of differential Fay identities and demonstrate, through case studies, its new role in integrable hierarchies of the KP type. These identities are known to be a…
Hamiltonian Structure of PI Hierarchy
Kanehisa Takasaki
The string equation of type may be thought of as a higher order analogue of the first Painlevé equation that corresponds to the case of . For , this equati…
Universal Whitham hierarchy, dispersionless Hirota equations and multi-component KP hierarchy
Kanehisa Takasaki, Takashi Takebe
The goal of this paper is to identify the universal Whitham hierarchy of genus zero with a dispersionless limit of the multi-component KP hierarchy. To this end, the multi-componen…
Dispersionless Hirota equations of two-component BKP hierarchy
Kanehisa Takasaki
The BKP hierarchy has a two-component analogue (the 2-BKP hierarchy). Dispersionless limit of this multi-component hierarchy is considered on the level of the -function. The so…
Radial Loewner equation and dispersionless cmKP hierarchy
Kanehisa Takasaki, Takashi Takebe
It has been shown that the dispersionless KP hierarchy (or the Benney hierarchy) is reduced to the chordal Löwner equation. We show that the radial Löwner equation also gives reduc…
Tyurin parameters of commuting pairs and infinite dimensional Grassmann manifold
Kanehisa Takasaki
Commuting pairs of ordinary differential operators are classified by a set of algebro-geometric data called ``algebraic spectral data''. These data consist of an algebraic curve (`…
-analogue of modified KP hierarchy and its quasi-classical limit
Kanehisa Takasaki
A -analogue of the tau function of the modified KP hierarchy is defined by a change of independent variables. This tau function satisfies a system of bilinear -difference equ…
Landau-Lifshitz hierarchy and infinite dimensional Grassmann variety
Kanehisa Takasaki
The Landau-Lifshitz equation is an example of soliton equations with a zero-curvature representation defined on an elliptic curve. This equation can be embedded into an integrable…
Elliptic spectral parameter and infinite dimensional Grassmann variety
Kanehisa Takasaki
Recent results on the Grassmannian perspective of soliton equations with an elliptic spectral parameter are presented along with a detailed review of the classical case with a rati…
Tyurin parameters and elliptic analogue of nonlinear Schrödinger hierarchy
Kanehisa Takasaki
Two "elliptic analogues'' of the nonlinear Schrödinger hiererchy are constructed, and their status in the Grassmannian perspective of soliton equations is elucidated. In addition t…
Integrable systems whose spectral curve is the graph of a function
Kanehisa Takasaki
For some integrable systems, such as the open Toda molecule, the spectral curve of the Lax representation becomes the graph of a function . Thos…
Spectral curve, Darboux coordinates and Hamiltonian structure of periodic dressing chains
Kanehisa Takasaki
A chain of one-dimensional Schrödinger operators connected by successive Darboux transformations is called the ``Darboux chain'' or ``dressing chain''. The periodic dressing chain…
An integrable system on the moduli space of rational functions and its variants
Kanehisa Takasaki, Takashi Takebe
We study several integrable Hamiltonian systems on the moduli spaces of meromorphic functions on Riemann surfaces (the Riemann sphere, a cylinder and a torus). The action-angle var…
Spectral curve and Hamiltonian structure of isomonodromic SU(2) Calogero-Gaudin system
Kanehisa Takasaki
This paper presents a new approach to the Hamiltonian structure of isomonodromic deformations of a matrix system of ODE's on a torus. An isomonodromic analogue of the $\rmSU(2)$ Ca…
Hierarchy of (2+1)-dimensional nonlinear Schroedinger equation, self-dual Yang-Mills equation, and toroidal Lie algebras
Saburo Kakei, Takeshi Ikeda, Kanehisa Takasaki
The hierarchy structure associated with a (2+1)-dimensional Nonlinear Schroedinger equation is discussed as an extension of the theory of the KP hierarchy. Several methods to const…
Toroidal Lie algebras and Bogoyavlensky's 2+1-dimensional equation
T. Ikeda, K. Takasaki
We introduce an extension of the \ell-reduced KP hierarchy, which we call the \ell-Bogoyavlensky hierarchy. Bogoyavlensky's 2+1-dimensional extension of the KdV equation is the low…