1,509 research outputs found
Rational solutions of the discrete time Toda lattice and the alternate discrete Painleve II equation
The Yablonskii-Vorob'ev polynomials , which are defined by a second
order bilinear differential-difference equation, provide rational solutions of
the Toda lattice. They are also polynomial tau-functions for the rational
solutions of the second Painlev\'{e} equation (). Here we define
two-variable polynomials on a lattice with spacing , by
considering rational solutions of the discrete time Toda lattice as introduced
by Suris. These polynomials are shown to have many properties that are
analogous to those of the Yablonskii-Vorob'ev polynomials, to which they reduce
when . They also provide rational solutions for a particular
discretisation of , namely the so called {\it alternate discrete}
, and this connection leads to an expression in terms of the Umemura
polynomials for the third Painlev\'{e} equation (). It is shown that
B\"{a}cklund transformation for the alternate discrete Painlev\'{e} equation is
a symplectic map, and the shift in time is also symplectic. Finally we present
a Lax pair for the alternate discrete , which recovers Jimbo and Miwa's
Lax pair for in the continuum limit .Comment: 23 pages, IOP style. Title changed, and connection with Umemura
polynomials adde
Bilinear Discrete Painleve-II and its Particular Solutions
By analogy to the continuous Painlev\'e II equation, we present particular
solutions of the discrete Painlev\'e II (d-P) equation. These
solutions are of rational and special function (Airy) type. Our analysis is
based on the bilinear formalism that allows us to obtain the function
for d-P. Two different forms of bilinear d-P are obtained
and we show that they can be related by a simple gauge transformation.Comment: 9 pages in plain Te
On a q-difference Painlev\'e III equation: I. Derivation, symmetry and Riccati type solutions
A q-difference analogue of the Painlev\'e III equation is considered. Its
derivations, affine Weyl group symmetry, and two kinds of special function type
solutions are discussed.Comment: arxiv version is already officia
Nambu-Hamiltonian flows associated with discrete maps
For a differentiable map that has
an inverse, we show that there exists a Nambu-Hamiltonian flow in which one of
the initial value, say , of the map plays the role of time variable while
the others remain fixed. We present various examples which exhibit the map-flow
correspondence.Comment: 19 page
Spin Seebeck insulator
Thermoelectric generation is an essential function of future energy-saving
technologies. However, this generation has been an exclusive feature of
electric conductors, a situation which inflicts a heavy toll on its
application; a conduction electron often becomes a nuisance in thermal design
of devices. Here we report electric-voltage generation from heat flowing in an
insulator. We reveal that, despite the absence of conduction electrons, a
magnetic insulator LaY2Fe5O12 converts a heat flow into spin voltage. Attached
Pt films transform this spin voltage into electric voltage by the inverse spin
Hall effect. The experimental results require us to introduce thermally
activated interface spin exchange between LaY2Fe5O12 and Pt. Our findings
extend the range of potential materials for thermoelectric applications and
provide a crucial piece of information for understanding the physics of the
spin Seebeck effect.Comment: 19 pages, 5 figures (including supplementary information
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