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

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

Temperature independent diffuse scattering and elastic lattice deformations in relaxor PbMg1/3Nb2/3O3

The results of diffuse neutron scattering experiment on PbMg1/3Nb2/3O3 single crystal above the Burns temperature are reported. It is shown that the high temperature elastic diffuse component is highly anisotropic in low-symmetry Brillouin zones and this anisotropy can be described using Huang scattering formalism assuming that the scattering originates from mesoscopic lattice deformations due to elastic defects. The qualitative agreement between this model and the experimental data is achieved with simple isotropic defects. It is demonstrated that weak satellite maxima near the Bragg reflections can be interpreted as the finite resolution effect.Comment: 7 pages, 7 figure

Finite-dimensional reductions of the discrete Toda chain

The problem of construction of integrable boundary conditions for the discrete Toda chain is considered. The restricted chains for properly chosen closure conditions are reduced to the well known discrete Painlev\'e equations $dP_{III}$, $dP_{V}$, $dP_{VI}$. Lax representations for these discrete Painlev\'e equations are found.Comment: Submitted to Jornal of Physics A: Math. Gen., 14 page

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

Relation between crystal and magnetic structures of the layered manganites La2-2xSr1+2xMn2O7 (0.30 =< x =< 0.50)

Comprehensive neutron-powder diffraction and Rietveld analyses were carried out to clarify the relation between the crystal and magnetic structures of La2-2xSr1+2xMn2O7 (0.30 =< x =< 0.50). The Jahn-Teller (JT) distortion of Mn-O6 octahedra, i.e., the ratio of the averaged apical Mn-O bond length to the equatorial Mn-O bond length, is Delta_JT=1.042(5) at x=0.30, where the magnetic easy-axis at low temperature is parallel to the c axis. As the JT distortion becomes suppressed with increasing x, a planar ferromagnetic structure appears at x =< 0.32, which is followed by a canted antiferromagnetic (AFM) structure at x =< 0.39. The canting angle between neighboring planes continuously increases from 0 deg (planar ferromagnet: 0.32 =< x < 0.39) to 180 deg (A-type AFM: x=0.48 where Delta_JT=1.013(5)). Dominance of the A-type AF structure with decrease of JT distortion can be ascribed to the change in the eg orbital state from d3z^2-r^2 to dx^2-y^2

Doping dependence of the exchange energies in bilayer manganites: Role of orbital degrees of freedom

Recently, an intriguing doping dependence of the exchange energies in the bilayer manganites $La_{2-2x}Sr_{1+2x}Mn_2O_7$ has been observed in the neutron scattering experiments. The intra-layer exchange only weakly changed with doping while the inter-layer one drastically decreased. Here we propose a theory which accounts for these experimental findings. We argue, that the observed striking doping dependence of the exchange energies can be attributed to the evaluation of the orbital level splitting with doping. The latter is handled by the interplay between Jahn-Teller effect (supporting an axial orbital) and the orbital anisotropy of the electronic band in the bilayer structure (promoting an in-plane orbital), which is monitored by the Coulomb repulsion. The presented theory, while being a mean-field type, describes well the experimental data and also gives the estimates of the several interesting energy scales involved in the problem.Comment: Added references, corrected typos. To appear in Phys. Rev.

Dual Resonance Model Solves the Yang-Baxter Equation

The duality of dual resonance models is shown to imply that the four point string correlation function solves the Yang-Baxter equation. A reduction of transfer matrices to $A_l$ symmetry is described by a restriction of the KP $\tau$ function to Toda molecules.Comment: 10 pages, LaTe