A Super-Integrable Discretization of the Calogero Model

A time-discretization that preserves the super-integrability of the Calogero model is obtained by application of the integrable time-discretization of the harmonic oscillator to the projection method for the Calogero model with continuous time. In particular, the difference equations of motion, which provide an explicit scheme for time-integration, are explicitly presented for the two-body case. Numerical results exhibit that the scheme conserves all the$(=3)$ conserved quantities of the (two-body) Calogero model with a precision of the machine epsilon times the number of iterations.Comment: 22 pages, 5 figures. Added references. Corrected typo

Construction of Integrals of Higher-Order Mappings

We find that certain higher-order mappings arise as reductions of the integrable discrete A-type KP (AKP) and B-type KP (BKP) equations. We find conservation laws for the AKP and BKP equations, then we use these conservation laws to derive integrals of the associated reduced maps.Comment: appear to Journal of the Physical Society of Japa

Supersymmetric Modified Korteweg-de Vries Equation: Bilinear Approach

A proper bilinear form is proposed for the N=1 supersymmetric modified Korteweg-de Vries equation. The bilinear B\"{a}cklund transformation of this system is constructed. As applications, some solutions are presented for it.Comment: 8 pages, LaTeX using packages amsmath and amssymb, some corrections mad

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 the Julia set, which appears when the piece is deformed, is shown analytically how it disappears as the system approaches to the integrable limit.Comment: 17 pages, LaTe

Hypothesis of two-dimensional stripe arrangement and its implications for the superconductivity in high-Tc cuprates

The hypothesis that holes doped into high-Tc cuprate superconductors organize themselves in two-dimensional (2D) array of diagonal stripes is discussed, and, on the basis of this hypothesis, a new microscopic model of superconductivity is proposed and solved. The model describes two kinds of hole states localized either inside the stripes or in the antiferromagnetic domains between the stripes. The characteristic energy difference between these two kinds of states is identified with the pseudogap. The superconducting (SC) order parameter predicted by the model has two components, whose phases exhibit a complex dependence on the the center-of-mass coordinate. The model predictions for the tunneling characteristics and for the dependence of the critical temperature on the superfluid density show good quantitative agreement with a number of experiments. The model, in particular, predicts that the SC peaks in the tunneling spectra are asymmetric, only when the ratio of the SC gap to the critical temperature is greater than 4. It is also proposed that, at least in some high-Tc cuprates, there exist two different superconducting states corresponding to the same doping concentration and the same critical temperature. Finally, the checkerboard pattern in the local density of states observed by scanning tunneling microscopy in Bi-2212 is interpreted as coming from the states localized around the centers of stripe elements forming the 2D superstructure.Comment: Text close to the published version. This version is 10 per cent shorter than the previous one. All revisions are mino

On a family of solutions of the KP equation which also satisfy the Toda lattice hierarchy

We describe the interaction pattern in the $x$-$y$ plane for a family of soliton solutions of the Kadomtsev-Petviashvili (KP) equation, $(-4u_{t}+u_{xxx}+6uu_x)_{x}+3u_{yy}=0$. Those solutions also satisfy the finite Toda lattice hierarchy. We determine completely their asymptotic patterns for $y\to \pm\infty$, and we show that all the solutions (except the one-soliton solution) are of {\it resonant} type, consisting of arbitrary numbers of line solitons in both aymptotics; that is, arbitrary $N_-$ incoming solitons for $y\to -\infty$ interact to form arbitrary $N_+$ outgoing solitons for $y\to\infty$. We also discuss the interaction process of those solitons, and show that the resonant interaction creates a {\it web-like} structure having $(N_--1)(N_+-1)$ holes.Comment: 18 pages, 16 figures, submitted to JPA; Math. Ge

Invariant varieties of periodic points for some higher dimensional integrable maps

By studying various rational integrable maps on $\mathbf{\hat C}^d$ with $p$ invariants, we show that periodic points form an invariant variety of dimension $\ge p$ for each period, in contrast to the case of nonintegrable maps in which they are isolated. We prove the theorem: {\it `If there is an invariant variety of periodic points of some period, there is no set of isolated periodic points of other period in the map.'}Comment: 24 page

A refined invariant subspace method and applications to evolution equations

The invariant subspace method is refined to present more unity and more diversity of exact solutions to evolution equations. The key idea is to take subspaces of solutions to linear ordinary differential equations as invariant subspaces that evolution equations admit. A two-component nonlinear system of dissipative equations was analyzed to shed light on the resulting theory, and two concrete examples are given to find invariant subspaces associated with 2nd-order and 3rd-order linear ordinary differential equations and their corresponding exact solutions with generalized separated variables.Comment: 16 page

An integrable multicomponent quad equation and its Lagrangian formulation

We present a hierarchy of discrete systems whose first members are the lattice modified Korteweg-de Vries equation, and the lattice modified Boussinesq equation. The N-th member in the hierarchy is an N-component system defined on an elementary plaquette in the 2-dimensional lattice. The system is multidimensionally consistent and a Lagrangian which respects this feature, i.e., which has the desirable closure property, is obtained.Comment: 10 page

On a discrete Davey-Stewartson system

We propose a differential difference equation in ${\mathcal R}^1\times {\mathcal Z}^2$ and study it by Hirota's bilinear method. This equation has a singular continuum limit into a system which admits the reduction to the Davey-Stewartson equation. The solutions of this discrete DS system are characterized by Casorati and Grammian determinants. Based on the bilinear form of this discrete DS system, we construct the bilinear B\"{a}cklund transformation which enables us to obtain its Lax pair.Comment: 12 pages, 2 figure