Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1509.08996

Singularities of integrable systems and algebraic curves

Anton Izosimov
View original source ↗Read source document ↗Export metadata ↓

Abstract

We study the relationship between singularities of finite-dimensional integrable systems and singularities of the corresponding spectral curves. For the large class of integrable systems on matrix polynomials, which is a general framework for various multidimensional spinning tops, as well as Beauville systems, we prove that if the spectral curve is nodal, then all singularities on the corresponding fiber of the system are non-degenerate. We also show that the type of a non-degenerate singularity can be read off from the behavior of double points on the spectral curve under an appropriately defined antiholomorphic involution. Our analysis is based on linearization of integrable flows on generalized Jacobian varieties, as well as on the possibility to express the eigenvalues of an integrable vector field linearized at a singular point in terms of residues of certain meromorphic differentials.

Explore related subjects

nlin.SImath-phmath.AGmath.MP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Anton Izosimov. 2016-08-03. Singularities of integrable systems and algebraic curves. https://arxiv.org/abs/1509.08996

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↗