3,149 research outputs found
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 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 - plane for a family of
soliton solutions of the Kadomtsev-Petviashvili (KP) equation,
. Those solutions also satisfy the
finite Toda lattice hierarchy. We determine completely their asymptotic
patterns for , 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 incoming
solitons for interact to form arbitrary outgoing solitons
for . We also discuss the interaction process of those solitons,
and show that the resonant interaction creates a {\it web-like} structure
having 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 with
invariants, we show that periodic points form an invariant variety of dimension
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 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
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