M\"obius Symmetry of Discrete Time Soliton Equations

We have proposed, in our previous papers, a method to characterize integrable discrete soliton equations. In this paper we generalize the method further and obtain a $q$-difference Toda equation, from which we can derive various $q$-difference soliton equations by reductions.Comment: 21 pages, 4 figure, epsfig.st

A Characterization of Discrete Time Soliton Equations

We propose a method to characterize discrete time evolution equations, which generalize discrete time soliton equations, including the $q$-difference Painlev\'e IV equations discussed recently by Kajiwara, Noumi and Yamada.Comment: 13 page

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

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

A new integrable system related to the Toda lattice

A new integrable lattice system is introduced, and its integrable discretizations are obtained. A B\"acklund transformation between this new system and the Toda lattice, as well as between their discretizations, is established.Comment: LaTeX, 14 p

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

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

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