5,406 research outputs found

    M\"obius Symmetry of Discrete Time Soliton Equations

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    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 qq-difference Toda equation, from which we can derive various qq-difference soliton equations by reductions.Comment: 21 pages, 4 figure, epsfig.st

    A Characterization of Discrete Time Soliton Equations

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

    Continuous vacua in bilinear soliton equations

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    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 AA and BB are even polynomials in their variables, then there can be a continuum of vacua, parametrized by a vacuum angle ϕ\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 ϕ\phi is fixed and we choose u=arctanG/Fu=\arctan G/F as the physical quantity, then there are four different solitons (or kinks) connecting the vacuum angles ±ϕ\pm\phi, ±ϕ±Π2\pm\phi\pm\Pi2 (defined modulo π\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

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    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

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    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

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    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

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    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

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    Commuting transfer matrices of Uq(Xr(1))U_{q}(X_{r}^{(1)}) vertex models obey the functional relations which can be viewed as an XrX_{r} type Toda field equation on discrete space time. Based on analytic Bethe ansatz we present, for Xr=DrX_{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

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    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 XrX_r. For Xr=Br,CrX_r = B_r, C_r and DrD_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