20 research outputs found
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
Rational Solutions of the Painleve' VI Equation
In this paper, we classify all values of the parameters , ,
and of the Painlev\'e VI equation such that there are
rational solutions. We give a formula for them up to the birational canonical
transformations and the symmetries of the Painlev\'e VI equation.Comment: 13 pages, 1 Postscript figure Typos fixe
Universal Drinfeld-Sokolov Reduction and Matrices of Complex Size
We construct affinization of the algebra of ``complex size''
matrices, that contains the algebras for integral values of the
parameter. The Drinfeld--Sokolov Hamiltonian reduction of the algebra
results in the quadratic Gelfand--Dickey structure on the
Poisson--Lie group of all pseudodifferential operators of fractional order.
This construction is extended to the simultaneous deformation of orthogonal and
simplectic algebras that produces self-adjoint operators, and it has a
counterpart for the Toda lattices with fractional number of particles.Comment: 29 pages, no figure
Symplectic Structures for the Cubic Schrodinger equation in the periodic and scattering case
We develop a unified approach for construction of symplectic forms for 1D
integrable equations with the periodic and rapidly decaying initial data. As an
example we consider the cubic nonlinear Schr\"{o}dinger equation.Comment: This is expanded and corrected versio