Yoshimasa Nakamura

Completely integrable gradient systems on the manifolds of Gaussian and multinomial distributions

Gradient systems on the manifolds of Gaussian and multinomial distributions are shown to be completely integrable Hamiltonian systems. The corresponding flows converge exponentially to equilibrium. A Lax representation of the gradient systems is found.

Transformation group acting on a self‐dual Yang–Mills hierarchy

A GL(n,C) self‐dual Yang–Mills hierarchy is introduced; it is an infinite system of self‐dual Yang–Mills equations having an infinite number of independent variables. Cauchy problems for the hierarchy are formally solved by using Lie transforms of a wave matrix. A relationship between the Kadomtsev–Petviashvili hierarchy and the self‐dual Yang–Mills hierarchy is discussed. Furthermore, it is shown that an infinite‐dimensional transformation group acts on a solution space to the (n≥2) self‐dual Yang–Mills hierarchy. A parametric solution to the hierarchy is also given as a representation of the transformation group.

A new nonlinear dynamical system that leads to eigenvalues

This paper presents a new dynamical system of Lax type which solves the skew-Hermitian eigenvalue problem. The solution of the system is found to converge to a diagonal matrix which is a permutation of the eigenvalues of the initial value matrix.

Moduli space ofSU(2) monopoles and complex cyclic-Toda hierarchy

We study the problem of complete parametrization of the moduli space ofSU(2) Yang-Mills-Higgs monopoles in terms of a nonlinear integrable system. It is shown that the moduli space is homeomorphic to the solution space of a new generalization of finite nonperiodic Toda equation called the complex cyclic-Toda hierarchy.

On Heisenberg models with values in Hermitian symmetric spaces

Heisenberg models with values in Hermitian symmetric spaces are introduced. A matrix form of the corresponding nonlinear Schrödinger equations is also presented. It is shown that a hidden symmetry of the generalized Heisenberg models generates a one-parameter family of solutions.

Fractional transformation group induced by QR factorization and linear prediction problems

Abstract A new fractional transformation group is found which acts transitively on the space of linear predictors for nonstationary processes by using the QR factorization of nonsingular matrices.

Self‐dual Yang–Mills hierarchy: A nonlinear integrable dynamical system in higher dimensions

A self‐dual Yang–Mills (SDYM) hierarchy is presented that has some properties similar to those of the celebrated Kadomtsev–Petviashvili (KP) hierarchy having one spacial variable. The SDYM hierarchy is an infinite system of GL(∞)‐invariant compatible SDYM equations depending on an infinite number of spacial variables, on which an exponential operator acts as a time evolution operator. A nontrivial GL(∞) symmetry of the SDYM hierarchy called the Bruhat transformation is found. The existence of the GL(∞) SDYM hierarchy supports the complete integrability of the single GL(∞)SDYM equation with a reduction condition, however, our observation suggests that the SDYM equation without the condition may not be completely integrable in sharp contrast with the KP equation.

Remarks on the moduli space of SU(2) monopoles and Toda flow

Rational functions of degreek which parametrize the moduli space of SU(2)k-monopoles, can be regarded as transfer functions for certain linear dynamical systems. It is shown that a time shift for linear systems induces a finite nonperiodic complexk×k Toda flow on the parameter space of generic SU(2)k-monopoles. Thus, there exists an integrable flow of the Toda type over the moduli space of SU(2) monopoles.

Nonlinear integrable flow on the framed moduli space of instantons

A nonlinear integrable flow of the Lax-type on the framed moduli space of SU(n) Yang-Mills instantons is found.

On self-dual yang-Mills hierarchy☆

Abstract In this note, motivated by the Kadomtsev-Petviashvili (KP) hierarchy of integrable nonlinear evolution equations, a GL(n, C ) self-dual Yang-Mills (SDYM) hierarchy is presented; it is an infinite system of SDYM equations having an infinite number of independent variables and being outside of the KP hierarchy. A relationship between the KP hierarchy and the SDYM hierarchy is discussed. It is also shown that GL(∞) SDYM equations introduced in this note are reduced to the GL(n, C ) SDYM hierarchy by imposing an algebraic constraiint.