Toda Lattice and Tomimatsu-Sato Solutions

We discuss an analytic proof of a conjecture (Nakamura) that solutions of Toda molecule equation give those of Ernst equation giving Tomimatsu-Sato solutions of Einstein equation. Using Pfaffian identities it is shown for Weyl solutions completely and for generic cases partially.Comment: LaTeX 8 page

Two-dimensional soliton cellular automaton of deautonomized Toda-type

A deautonomized version of the two-dimensional Toda lattice equation is presented. Its ultra-discrete analogue and soliton solutions are also discussed.Comment: 11 pages, LaTeX fil

Continuous vacua in bilinear soliton equations

We discuss the freedom in the background field (vacuum) on top of which the solitons are built. If the Hirota bilinear form of a soliton equation is given by A(D_{\vec x})\bd GF=0,\, B(D_{\vec x})(\bd FF - \bd GG)=0 where both $A$ and $B$ are even polynomials in their variables, then there can be a continuum of vacua, parametrized by a vacuum angle $\phi$. The ramifications of this freedom on the construction of one- and two-soliton solutions are discussed. We find, e.g., that once the angle $\phi$ is fixed and we choose $u=\arctan G/F$ as the physical quantity, then there are four different solitons (or kinks) connecting the vacuum angles $\pm\phi$, $\pm\phi\pm\Pi2$ (defined modulo $\pi$). The most interesting result is the existence of a ``ghost'' soliton; it goes over to the vacuum in isolation, but interacts with ``normal'' solitons by giving them a finite phase shift.Comment: 9 pages in Latex + 3 figures (not included

Solutions of a discretized Toda field equation for DrD_{r} from Analytic Bethe Ansatz

Commuting transfer matrices of $U_{q}(X_{r}^{(1)})$ vertex models obey the functional relations which can be viewed as an $X_{r}$ type Toda field equation on discrete space time. Based on analytic Bethe ansatz we present, for $X_{r}=D_{r}$, a new expression of its solution in terms of determinants and Pfaffians.Comment: Latex, 14 pages, ioplppt.sty and iopl12.sty assume

Pfaffian and Determinant Solutions to A Discretized Toda Equation for Br,CrB_r, C_r and DrD_r

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 $X_r$. For $X_r = B_r, C_r$ and $D_r$, 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

An integrable generalization of the Toda law to the square lattice

We generalize the Toda lattice (or Toda chain) equation to the square lattice; i.e., we construct an integrable nonlinear equation, for a scalar field taking values on the square lattice and depending on a continuous (time) variable, characterized by an exponential law of interaction in both discrete directions of the square lattice. We construct the Darboux-Backlund transformations for such lattice, and the corresponding formulas describing their superposition. We finally use these Darboux-Backlund transformations to generate examples of explicit solutions of exponential and rational type. The exponential solutions describe the evolution of one and two smooth two-dimensional shock waves on the square lattice.Comment: 14 pages, 4 figures, to appear in Phys. Rev. E http://pre.aps.org

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

Handbook for MAP, volume 32. Part 1: MAP summary. Part 2: MAPSC minutes, reading, August 1989. MAP summaries from nations. Part 3: MAP data catalogue

Extended abstracts from the fourth workshop on the technical and scientific aspects of mesosphere stratosphere troposphere (MST) radar are presented. Individual sessions addressed the following topics: meteorological applications of MST and ST radars, networks, and campaigns; the dynamics of the equatorial middle atmosphere; interpretation of radar returns from clear air; techniques for studying gravity waves and turbulence, intercomparison and calibration of wind and wave measurements at various frequencies; progress in existing and planned MST and ST radars; hardware design for MST and ST radars and boundary layer/lower troposphere profilers; signal processing; and data management

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 4$\times$4 R-matrix with certain vectors in its "quantum" space. Components of the vectors are identified with $\tau$-functions of the model. This construction generalizes the known representation of M-operators in continuous time models in terms of Lax operators and classical $r$-matrix.Comment: 10 pages, LaTeX (misprints are corrected

Multilinear Operators: The Natural Extension Of Hirota's Bilinear Formalism

We introduce multilinear operators, that generalize Hirota's bilinear $D$ operator, based on the principle of gauge invariance of the $\tau$ functions. We show that these operators can be constructed systematically using the bilinear $D$'s as building blocks. We concentrate in particular on the trilinear case and study the possible integrability of equations with one dependent variable. The 5th order equation of the Lax-hierarchy as well as Satsuma's lowest-order gauge invariant equation are shown to have simple trilinear expressions. The formalism can be extended to an arbitrary degree of multilinearity.Comment: 9 pages in plain Te