On integrals of non-autonomous dynamical systems in finite characteristic

Nalini Joshi; Pieter Roffelsen

Home/Archive/arXiv

arXiv · 2602.09298

On integrals of non-autonomous dynamical systems in finite characteristic

Nalini Joshi·Pieter Roffelsen

View original source ↗Read source document ↗Export metadata ↓

Abstract

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.

Explore related subjects

nlin.SI math-ph math.MP

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Nalini Joshi, Pieter Roffelsen. 2026-02-10. On integrals of non-autonomous dynamical systems in finite characteristic. https://arxiv.org/abs/2602.09298

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

KEEP EXPLORING

Related papers

arXiv2026

On vector Schwarz-KdV equation

A collection of miscellaneous continuous, semi-discrete, and discrete integrable systems can be associated with each integrable evolution equation of the KdV type. We give them for the Schwarz--KdV equation and generalize to the vector case. The existence of these vector generalizations is a non-trivial experimental fact for which no mathematical explanation is yet known.

nlin.SI↗

arXiv2026

An integrable $\mathbb{Z}_2^2$-graded extension of Camassa-Holm equation and its bi-Hamiltonian structure

By constructing Lax operators in the loop algebra of the $\mathbb{Z}_2^2$-graded extension of the Lie superalgebra $\mathfrak{osp}(1|2)$, we derive a $\mathbb{Z}_2^2$-graded extension of the Camassa-Holm equation. The resulting equation is an integrable nonlinear PDE for a system of four $\mathbb{Z}_2^2$-graded commutative functions, each associated with a distinct $\mathbb{Z}_2^2$-degree. We further show that the $\mathbb{Z}_2^2$-Camassa-Holm equation admits a bi-Hamiltonian structure. As a consequence, it possesses infinitely many conserved quantities, including one with non-trivial $\mathbb{Z}_2^2$-degree, which are mutually in involution with respect to the $\mathbb{Z}_2^2$-graded Poisson brackets.

nlin.SI↗

arXiv2026

Integrability of the deformed Toda systems

In 2020 M. Mucciconi and L. Petrov introduced a long-range deformation of the quantum open non-relativistic Toda system. We prove the integrability of the deformed Toda system by constructing a $2 \times 2$ Lax operator, which produces the commutative family of differential operators containing the Hamiltonian of the deformed Toda system. Moreover, we show that the same integrable deformation exists on both classical and quantum levels and can be applied to both non-relativistic and relativistic Toda systems. For the open non-relativistic deformed Toda systems we also present an $n \times n$ Lax matrix and prove that it produces the same family of Hamiltonians. We also show how to obtain the van Diejen-type deformed Toda system. Lastly, we show that on the quantum level the algebraic Bethe ansatz technique can be applied to the deformed Toda system.

nlin.SI↗