4,581 research outputs found

### 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

### 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

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