Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2509.03155

On vector Schwarz-KdV equation

M. Balakhnev·V. Sokolov
View original source ↗Read source document ↗Export metadata ↓

Abstract

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.

Explore related subjects

nlin.SI

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

M. Balakhnev, V. Sokolov. 2026-09-18. On vector Schwarz-KdV equation. https://arxiv.org/abs/2509.03155

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

KEEP EXPLORING

Related papers

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↗
arXiv2026

Maxwell's relations as Hamilton's equations: a symplectic and variational framework

We develop a geometric framework in which classical thermodynamics is reformulated as a Hamiltonian dynamical system. An explicit canonical mapping $(q,p,t,H)\leftrightarrow(V,-P,S,-T)$ identifies Maxwell's relations as the characteristic equations of the thermodynamic Poincaré--Cartan one-form, in direct parallel with Hamilton's equations of motion. Treating thermodynamic potentials as action functionals, a variational principle recovers both the Maxwell relations and the thermodynamic constraints (adiabaticity, isothermality) as conserved first integrals. For adiabatic processes in an ideal gas, this yields explicit second-order ordinary differential equations for $V(T)$ and $P(T)$, each governed by a temperature-dependent Lagrangian. Assembling the component Lagrangians into a multi-parameter Lagrangian 1-form $\mathcal{L}$, we prove that the closure condition $d\mathcal{L}=0$ holds on the solution manifold, establishing that path-independence of thermodynamic state functions is a geometric consequence of multi-time integrability rather than an independent postulate.

nlin.SI↗