Noriko Saitoh

Gauge and Dual Symmetries and Linearization of Hirota's Bilinear Equations

In Hirota’s [Hiroshima University Technical Report Nos. A 6, A 9, 1981; J. Phys. Soc. Jpn. 50, 3785 (1981)] bilinear difference equation which is satisfied by solutions to the Kadomtsev–Petviashvili (KP) hierarchy, gauge and dual symmetries are found, which enable one to reduce the problem of solving the nonlinear equation to solving a single linear equation.

A Transformation Connecting the Toda Lattice and the K-dV Equation

A generalized equation is derived from the Toda lattice by a transformation of variables including a characteristic parameter h . The equation reduces to the original Toda lattice for the value h =1, while to the K-dV equation at the limiting value h =0. Soliton solutions of the equation are obtained by the inverse scattering method, reproducing the corresponding solution of the K-dV equation in every step of the calculation.

The Classical Specific Heat of the Exponential Lattice

The classical specific heat of the exponential lattice at constant length is derived and its asymptotic behaviours are studied.

Experiment on Lattice Soliton by Nonlinear LC Circuit –Observation of a Dark Soliton

We examine the propagation of a dark soliton in a nonlinear LC circuit which is equivalent to the Toda lattice. At first, we derive the nonlinear Schrodinger equation to describe a nonlinear wave of strong dispersion. It is shown theoretically that the wave of envelope is modulationally stable and propagates as a dark soliton, if the circuit is equivalent to the Toda lattice. In the experiment, it is confirmed that the dark soliton is generated from an initial wave and propagates stably in the circuit. The width of the soliton agrees with the theory.

A Characterization of Discrete Time Soliton Equations

We propose a method to characterize discrete time evolution equations, which generalize discrete time soliton equations, including the q -difference Painleve IV equations discussed recently [K. Kaj...

Linearization of Multi-Dimensional Toda Lattice and Bäcklund Transformation

An infinite number of linear differential equations whose solutions satisfy multi-dimensional Toda lattice (MDTL) equation are found. The coefficients of any of these linear equations are given by a solution of MDTL equation, thus the equation provides a kind of linear Backlund transformation of the NDTL equation.

Invariant Varieties of Periodic Points for Some Higher Dimensional Integrable Maps

By studying various rational integrable maps on \(\hat{\mathbf{C}}^{d}\) with p invariants, we show that periodic points form an invariant variety of dimension ≥ p for each period, in contrast to t...

On recurrence equations associated with invariant varieties of periodic points

A recurrence equation is a discrete integrable equation whose solutions are all periodic and the period is fixed. We show that infinitely many recurrence equations can be derived from the information about invariant varieties of periodic points of higher dimensional integrable maps.

Solutions of two- and three-dimensional Toda lattice equations in terms of various forms of Bessel functions

New solutions of two- and three-(space)-dimensional generalized Toda lattice equations are given. The solutions are written by various forms of Bessel functions, and their properties are discussed in relation to cylindrical and spherical nonlinear modes in the systems. A discussion is also given to another type of nonlinear difference-differential equations, having the form of a generalized version of higher dimensional Toda lattice equation and exhibiting spherical nonlinear modes.

Complex analysis of a piece of Toda lattice

We study a small piece of two-dimensional Toda lattice as a complex dynamical system. In particular, it is shown analytically how the Julia set, which appears when the piece is deformed, disappears as the system approaches the integrable limit.