Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · solv-int/9901006

The hunting for the discrete Painlevé VI is over

B. Grammaticos·A. Ramani
View original source ↗Read source document ↗Export metadata ↓

Abstract

We present the discrete, q-, form of the Painlevé VI equation written as a three-point mapping and analyse the structure of its singularities. This discrete equation goes over to P_{VI} at the continuous limit and degenerates towards the discrete q-P_{V} through coalescence. It possesses special solutions in terms of the q-hypergeometric function. It can bilinearised and, under the appropriate assumptions, ultradiscretised. A new discrete form for P_{V} is also obtained which is of difference type, in contrast with the `standard' form of the discrete P_{V}. Finally, we present the `asymmetric' form of q-P_{VI}$ as a system of two first-order mappings involving seven arbitrary parameters.

Explore related subjects

solv-intnlin.SI

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

B. Grammaticos, A. Ramani. 1999-01-19. The hunting for the discrete Painlevé VI is over. https://arxiv.org/abs/solv-int/9901006

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2017

The averaging of non-local Hamiltonian structures in Whitham's method

We consider the $m$-phase Whitham's averaging method and propose a procedure of "averaging" of non-local Hamiltonian structures. The procedure is based on the existence of a sufficient number of local commuting integrals of a system and gives a Poisson bracket of Ferapontov type for the Whitham's system. The method can be considered as a generalization of the Dubrovin-Novikov procedure for the local field-theoretical brackets.

solv-int↗
arXiv2013

On two aspects of the Painleve analysis

We use the Calogero equation to illustrate the following two aspects of the Painleve analysis of nonlinear PDEs. First, if a nonlinear equation passes the Painleve test for integrability, the singular expansions of its solutions around characteristic hypersurfaces can be neither single-valued functions of independent variables nor single-valued functionals of data. Second, if the truncation of singular expansions of solutions is consistent, the truncation not necessarily leads to the simplest, or elementary, auto-Backlund transformation related to the Lax pair.

solv-int↗
arXiv2009

The tetrahedral analog of Veneziano amplitude

In solv-int/9812016 it was shown that the Veneziano amplitude in string theory comes naturally from one of the simplest solutions of the functional pentagon equation (FPE). More generally, FPE is intimately connected with the duality condition for scattering processes. Here I find the amplitude that comes the same way from a solution of the functional tetrahedron equation, with the duality replaced by the local Yang - Baxter equation.

solv-int↗