Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · nlin/0104019

Discrete systems related to some equations of the Painlevé-Gambier classification

S. Lafortune·B. Grammaticos·A. Ramani·P. Winternitz
View original source ↗Read source document ↗Export metadata ↓

Abstract

We derive integrable discrete systems which are contiguity relations of two equations in the Painlevé-Gambier classification depending on some parameter. These studies extend earlier work where the contiguity relations for the six transcendental Painlevé equations were obtained. In the case of the Gambier equation we give the contiguity relations for both the continuous and the discrete system.

Explore related subjects

nlin.SI

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

S. Lafortune, B. Grammaticos, A. Ramani, P. Winternitz. 2001-04-06. Discrete systems related to some equations of the Painlevé-Gambier classification. https://doi.org/10.1016/s0375-9601(00)00259-0

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

KEEP EXPLORING

Related papers

arXiv2026

Elliptic solutions to matrix CKP equation

A class of elliptic solutions to the matrix CKP equation is studied. Equations of motion for their poles and matrix coefficients at the poles are obtained. As in the scalar case, they are of the first order, in contrast to what takes place in the KP and BKP hierarchies, where the equations of motion are of the second order.

nlin.SI↗
arXiv2026

On integrals of non-autonomous dynamical systems in finite characteristic

We use a difference Lax form to construct simultaneous integrals of motion of the fourth Painlevé equation and the difference second Painlevé equation over fields with finite characteristic $p>0$. For $p\neq 3$, we show that the integrals can be normalised to be completely invariant under the corresponding extended affine Weyl group action. We show that components of reducible fibres of integrals correspond to reductions to Riccatti equations. We further describe a method to construct non-rational algebraic solutions in a given positive characteristic. We also discuss a projective reduction of the integrals.

nlin.SI↗
arXiv2026

On equivalence between solution structures of the anti-self-dual Yang--Mills equations

We establish an exact correspondence between the quasi-Wronskian and quasi-Grammian constructions for the anti-self-dual Yang--Mills (ASDYM) equations with complex \(2\times2\) matrix-valued fields. Starting from vacuum wave functions associated with \(2M\) pairwise distinct nonzero constant spectral parameters, we restrict attention to a nonempty local domain on which the corresponding interpolation matrix is invertible. We show that both constructions are governed by the same uniquely solvable matrix-polynomial interpolation problem. In particular, the quasi-Wronskian Darboux polynomial \(T(λ)\) and the quasi-Grammian rational dressing matrix \(W(λ)\) satisfy \[ T(λ)=q_ξ(λ)W(λ), \] where \(q_ξ(λ)\) is the monic scalar polynomial whose zeros are the prescribed poles of \(W(λ)\). This identity yields explicit relations between the corresponding Yang and potential matrices and shows that the two constructions generate identical ASDYM gauge potentials in the same gauge.

nlin.SI↗