Teruhisa Tsuda

Integrable mappings via rational elliptic surfaces

We present a geometric description of the QRT map (which is an integrable mapping introduced by Quispel, Roberts and Thompson) in terms of the addition formula of a rational elliptic surface. By this formulation, we classify all the cases when the QRT map is periodic; and show that its period is 2, 3, 4, 5 or 6. A generalization of the QRT map which acts birationally on a pencil of K3 surfaces, or Calabi–Yau manifolds, is also presented.

FROM KP/UC HIERARCHIES TO PAINLEVÉ EQUATIONS

We study the underlying relationship between Painleve equations and infinite-dimensional integrable systems, such as the KP and UC hierarchies. We show that a certain reduction of these hierarchies by requiring homogeneity and periodicity yields Painleve equations, including their higher order generalization. This result allows us to clearly understand various aspects of the equations, e.g., Lax formalism, Hirota bilinear relations for tau-functions, Weyl group symmetry, and algebraic solutions in terms of the character polynomials, i.e., the Schur function and the universal character.

On an Integrable System of q-Difference Equations Satisfied by the Universal Characters: Its Lax Formalism and an Application to q-Painlevé Equations

The universal character is a generalization of the Schur function attached to a pair of partitions. We study an integrable system of q-difference equations satisfied by the universal characters, which is an extension of the q-KP hierarchy and is called the lattice q-UC hierarchy. We describe the lattice q-UC hierarchy as a compatibility condition of its associated linear system (Lax formalism) and explore an application to the q-Painlevé equations via similarity reduction. In particular a higher order analogue of the q-Painlevé VI equation is presented.

Ultradiscretization of solvable one-dimensional chaotic maps

We consider the ultradiscretization of a solvable one-dimensional chaotic map which arises from the duplication formula of the elliptic functions. It is shown that the ultradiscrete limit of the map and its solution yield the tent map and its solution simultaneously. A geometric interpretation of the dynamics of the tent map is given in terms of the tropical Jacobian of a certain tropical curve. Generalization to the maps corresponding to the mth multiplication formula of the elliptic functions is also discussed.

Hermite–Padé approximation, isomonodromic deformation and hypergeometric integral

We develop an underlying relationship between the theory of rational approximations and that of isomonodromic deformations. We show that a certain duality in Hermite’s two approximation problems for functions leads to the Schlesinger transformations, i.e. transformations of a linear differential equation shifting its characteristic exponents by integers while keeping its monodromy invariant. Since approximants and remainders are described by block-Toeplitz determinants, one can clearly understand the determinantal structure in isomonodromic deformations. We demonstrate our method in a certain family of Hamiltonian systems of isomonodromy type including the sixth Painlevé equation and Garnier systems; particularly, we present their solutions written in terms of iterated hypergeometric integrals. An algorithm for constructing the Schlesinger transformations is also discussed through vector continued fractions.

Determinant Structure for \({\tau}\)-Function of Holonomic Deformation of Linear Differential Equations

In our previous works, a relationship between Hermite’s two approximation problems and Schlesinger transformations of linear differential equations has been clarified. In this paper, we study

UC hierarchy and monodromy preserving deformation

{\tau}

Toda equation and special polynomials associated with the Garnier system

τ-functions associated with holonomic deformations of linear differential equations by using Hermite’s two approximation problems. As a result, we present a determinant formula for the ratio of