4,309 research outputs found
Solutions of a discretized Toda field equation for from Analytic Bethe Ansatz
Commuting transfer matrices of vertex models obey the
functional relations which can be viewed as an type Toda field equation
on discrete space time. Based on analytic Bethe ansatz we present, for
, a new expression of its solution in terms of determinants and
Pfaffians.Comment: Latex, 14 pages, ioplppt.sty and iopl12.sty assume
Zero curvature representation for classical lattice sine-Gordon equation via quantum R-matrix
Local M-operators for the classical sine-Gordon model in discrete space-time
are constructed by convolution of the quantum trigonometric 44 R-matrix
with certain vectors in its "quantum" space. Components of the vectors are
identified with -functions of the model. This construction generalizes
the known representation of M-operators in continuous time models in terms of
Lax operators and classical -matrix.Comment: 10 pages, LaTeX (misprints are corrected
Pfaffian and Determinant Solutions to A Discretized Toda Equation for and
We consider a class of 2 dimensional Toda equations on discrete space-time.
It has arisen as functional relations in commuting family of transfer matrices
in solvable lattice models associated with any classical simple Lie algebra
. For and , we present the solution in terms of
Pfaffians and determinants. They may be viewed as Yangian analogues of the
classical Jacobi-Trudi formula on Schur functions.Comment: Plain Tex, 9 page
A survey of Hirota's difference equations
A review of selected topics in Hirota's bilinear difference equation (HBDE)
is given. This famous 3-dimensional difference equation is known to provide a
canonical integrable discretization for most important types of soliton
equations. Similarly to the continuous theory, HBDE is a member of an infinite
hierarchy. The central point of our exposition is a discrete version of the
zero curvature condition explicitly written in the form of discrete
Zakharov-Shabat equations for M-operators realized as difference or
pseudo-difference operators. A unified approach to various types of M-operators
and zero curvature representations is suggested. Different reductions of HBDE
to 2-dimensional equations are considered. Among them discrete counterparts of
the KdV, sine-Gordon, Toda chain, relativistic Toda chain and other typical
examples are discussed in detail.Comment: LaTeX, 43 pages, LaTeX figures (with emlines2.sty
Multiple addition theorem for discrete and continuous nonlinear problems
The addition relation for the Riemann theta functions and for its limits,
which lead to the appearance of exponential functions in soliton type equations
is discussed. The presented form of addition property resolves itself to the
factorization of N-tuple product of the shifted functions and it seems to be
useful for analysis of soliton type continuous and discrete processes in the
N+1 space-time. A close relation with the natural generalization of bi- and
tri-linear operators into multiple linear operators concludes the paper.Comment: 9 page
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
Analytical three-dimensional bright solitons and soliton-pairs in Bose-Einstein condensates with time-space modulation
We provide analytical three-dimensional bright multi-soliton solutions to the
(3+1)-dimensional Gross-Pitaevskii (GP) equation with time and space-dependent
potential, time-dependent nonlinearity, and gain/loss. The zigzag propagation
trace and the breathing behavior of solitons are observed. Different shapes of
bright solitons and fascinating interactions between two solitons can be
achieved with different parameters. The obtained results may raise the
possibility of relative experiments and potential applications.Comment: 5 pages, 4 figure
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
Integrable dynamics of Toda-type on the square and triangular lattices
In a recent paper we constructed an integrable generalization of the Toda law
on the square lattice. In this paper we construct other examples of integrable
dynamics of Toda-type on the square lattice, as well as on the triangular
lattice, as nonlinear symmetries of the discrete Laplace equations on the
square and triangular lattices. We also construct the - function
formulations and the Darboux-B\"acklund transformations of these novel
dynamics.Comment: 22 pages, 4 figure
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