arXiv subjects

Giorgio Gubbiotti

Publications and source records attributed to Giorgio Gubbiotti.

At least 19 recordsLinked to original sources

arXiv2026

Classification of pairs of second-order Hamiltonian operators and hydrodynamic type systems in six components

Following a recent characterisation of hydrodynamic systems with second-order Hamiltonian structure in terms of alternating three-forms on the projective space, we present a complete classification of systems in 6 components. We use non-trivial group action techniques to obtain 5 canonical forms. A general result for every dimension is also obtained for one of these cases, whose geometric structure is strictly related to the symplectic canonical form.

math-ph↗

arXiv2026

Discrete-time maximally superintegrable systems and deformed symmetry algebras: the Calogero-Moser case

We determine the complete structure of the symmetry algebras associated with the N-body Calogero-Moser system and its maximally superintegrable discretization. We prove that the discretization naturally leads to a nontrivial deformation of the continuous symmetry algebra, with the discretization parameter playing the rôle of a deformation parameter. This phenomenon illustrates how discrete superintegrable systems can be viewed as natural sources of deformed polynomial algebraic structures. As a byproduct of these results, we also reveal a connection between the above-mentioned symmetry algebras and the Bell polynomials, as a consequence of the trace properties.

math-ph↗

arXiv2025

A novel chain of Lie algebras and its coalgebra symmetry

We study a novel $n(n+1)/2$-dimensional non-semisimple Lie algebra $\mathfrak{g}_n$, a generalisation of both $\mathfrak{sl}_2(\mathbb{K})$ and the two-photon Lie algebra $\mathfrak{h}_6$. We investigate its properties, including its structure, representations, and its Casimir elements. In particular, we prove that there exists only one non-trivial Casimir polynomial of degree $n$ given by the determinant of an $n\times n$ symmetric matrix. We then associate this Lie algebra to a hierarchy of Hamiltonian systems with integrability properties depending on $n$, and describe their first integrals as sums of squares of linear combinations of the components of the angular momentum. In particular, we obtain that these systems are integrable for $n=2$, quasi-integrable for $n=3$, and of Poincaré-Lyapunov-Nekhoroshev type for $n\geq4$.

math-ph↗

arXiv2025

Explicit isomorphisms for the symmetry algebras of continuous and discrete isotropic oscillators

We present a detailed study of a parametric Lie algebra encompassing the symmetry algebras of various models, both continuous and discrete. This algebraic structure characterizes the isotropic oscillator (with positive, purely imaginary, and zero frequency) and one of its possible nonlinear deformations. We demonstrate a novel occurrence of this Lie algebra in the framework of maximally superintegrable discretizations of the isotropic harmonic oscillator. In particular, we also show that the continuous model and one of its discretizations admit a Nambu-Hamiltonian structure. Through an in-depth analysis of the properties characterizing the Lie algebra in the abstract setting, for different values of the parameter, we find explicit expressions of the Killing forms and construct explicit isomorphism maps to $\mathfrak{u}_N$, $\mathfrak{gl}_N(\mathbb{R})$, and a semidirect sum of $\mathfrak{so}_N(\mathbb{R})$ with $\mathbb{R}^{N(N+1)/2}$. Notably, due to the above isomorphisms, our formulas hold true for $\mathfrak{su}_N$ and $\mathfrak{sl}_N(\mathbb{R})$ and are valid for arbitrary $N$.

math-ph↗

arXiv2025

Lie algebras with compatible scalar products for non-homogeneous Hamiltonian operators

We study from an algebraic and geometric viewpoint Hamiltonian operators which are sum of a non-degenerate first-order homogeneous operator and a Poisson tensor. In flat coordinates, also known as Darboux coordinates, these operators are uniquely determined by a triple composed by a Lie algebra, its most general non-degenerate quadratic Casimir and a 2-cocycle. We present some classes of operators associated to Lie algebras with non-degenerate quadratic Casimirs and we give a description of such operators in low dimensions. Finally, motivated by the example of the KdV equation we discuss the conditions of bi-Hamiltonianity of such operators.

math-ph↗

arXiv2024

The Painlevé equivalence problem for a constrained 3D system

In this paper we propose a geometric approach to study Painlevé equations appearing as constrained systems of three first-order ordinary differential equations. We illustrate this approach on a system of three first-order differential equations arising in the theory of semi-classical orthogonal polynomials. We show that it can be restricted to a system of two first-order differential equations in two different ways on an invariant hypersurface. We build the space of initial conditions for each of these restricted systems and verify that they exhibit the Painlevé property from a geometric perspective. Utilising the Painlevé identification algorithm we also relate this system to the Painlevé VI equation and we build its global Hamiltonian structure. Finally, we prove that the autonomous limit of the original system is Liouville integrable, and the level curves of its first integrals are elliptic curves, which leads us to conjecture that the 3D system itself also possesses the Painlevé property without the need to restrict it to the invariant hypersurface.

nlin.SI↗

arXiv2024

Line geometry of pairs of second-order Hamiltonian operators and quasilinear systems

We demonstrate that a pair consisting of a second-order homogeneous Hamiltonian structure in $N$ components and its associated system of conservation laws is in bijective correspondence with an alternating three-form on a $N+2$-dimensional vector space. Additionally, we show that the three-form offers $N+2$ linear equations in the Plücker coordinates that define the associated line congruence. We utilize these results to characterize systems of conservation laws with second-order structure for $N\leq 4$. We finally comment how to extend this result for $N=6$.

math-ph↗

arXiv2024

Discrete-time systems in quasi-standard form and the $\mathfrak{h}_6$ coalgebra symmetry

In this paper, we characterize all discrete-time systems in quasi-standard form admitting coalgebra symmetry with respect to the Lie--Poisson algebra $\mathfrak{h}_{6}$. The outcome of this study is a family of systems depending on an arbitrary function of three variables, playing the rôle of the potential. Moreover, using a direct search approach, we classify discrete-time systems from this family that admit an additional invariant at most quadratic in the physical variables. We discuss the integrability properties of the obtained cases, their relationship with known systems, and their continuum limits.

math-ph↗

arXiv2024

Algebraic entropy for hex systems

Hex systems were recently introduced [A. P. Kels. Integrable systems on hexagonal lattices and consistency on polytopes with quadrilateral and hexagonal faces. 2022. arXiv:2205.02720 [math-ph]] as systems of equations defined on two-dimensional honeycomb lattices. We give a definition of algebraic entropy for such systems and use it to check the integrability of specific examples.

nlin.SI↗

arXiv2024

An Elementary Construction of Modified Hamiltonians and Modified Measures of 2D Kahan Maps

We show how to construct in an elementary way the invariant of the KHK discretisation of a cubic Hamiltonian system in two dimensions. That is, we show that this invariant is expressible as the product of the ratios of affine polynomials defining the prolongation of the three parallel sides of a hexagon. On the vertices of such a hexagon lie the indeterminacy points of the KHK map. This result is obtained analysing the structure of the singular fibres of the known invariant. We apply this construction to several examples, and we prove that a similar result holds true for a case outside the hypotheses of the main theorem, leading us to conjecture that further extensions are possible.

math.NA↗

arXiv2024

Algebraic entropy for systems of quad equations

In this work I discuss briefly the calculation of the algebraic entropy for systems of quad equations. In particular, I observe that since systems of multilinear equations can have algebraic solution, in some cases one might need to restrict the direction of evolution only to the pair of vertices yielding a birational evolution. Some examples from the exiting literature are presented and discussed within this framework.

math-ph↗

arXiv2023

Growth and integrability of some birational maps in dimension three

Motivated by the study of the Kahan--Hirota--Kimura discretisation of the Euler top, we characterise the growth and integrability properties of a collection of elements in the Cremona group of a complex projective 3-space using techniques from algebraic geometry. This collection consists of maps obtained by composing the standard Cremona transformation $\mathrm{c}_3\in\mathrm{Bir}(\mathbb{P}^3)$ with projectivities that permute the fixed points of $\mathrm{c}_3$ and the points over which $\mathrm{c}_3$ performs a divisorial contraction. More specifically, we show that three behaviour are possible: (A) integrable with quadratic degree growth and two invariants, (B) periodic with two-periodic degree sequences and more than two invariants, and (C) non-integrable with submaximal degree growth and one invariant.

math.AG↗

arXiv2023

Determination of the symmetry group for some QRT roots

We determine the affine Weyl symmetries of some two-dimensional birational maps known as QRT roots arising from Kahan--Hirota--Kimura discretisation of two different reduced Nahm systems. The main finding is that the symmetry types of these discrete systems are subgroups of the Weyl groups for Sakai's discrete Painlevé equations to which the QRT maps are the autonomous limits.

nlin.SI↗

arXiv2023

The $\mathfrak{sl}_{2}(\mathbb{R})$ coalgebra symmetry and the superintegrable discrete-time systems

In this paper, we classify all the variational discrete-time systems in quasi-standard form in $N$ degrees of freedom admitting coalgebra symmetry with respect to the generic realisation of the Lie-Poisson algebra $\mathfrak{sl}_{2}(\mathbb{R})$. This approach naturally yields several quasi-maximally and maximally superintegrable discrete-time systems, both known and new. We conjecture that this exhausts the (super)integrable cases associated with this algebraic construction.

nlin.SI↗

arXiv2023

Discrete integrable systems and random Lax matrices

We study properties of Hamiltonian integrable systems with random initial data by considering their Lax representation. Specifically, we investigate the spectral behaviour of the corresponding Lax matrices when the number $N$ of degrees of freedom of the system goes to infinity and the initial data is sampled according to a properly chosen Gibbs measure. We give an exact description of the limit density of states for the exponential Toda lattice and the Volterra lattice in terms of the Laguerre and antisymmetric Gaussian $β$-ensemble in the high temperature regime. For generalizations of the Volterra lattice to short range interactions, called INB additive and multiplicative lattices, the focusing Ablowitz--Ladik lattice and the focusing Schur flow, we derive numerically the density of states. For all these systems, we obtain explicitly the density of states in the ground states.

math-ph↗

arXiv2021

Algebraic entropy for face-centered quad equations

In this paper we define the algebraic entropy test for face-centered quad equations, which are equations defined on vertices of a quadrilateral plus an additional interior vertex. This notion of algebraic entropy is applied to a recently introduced class of these equations that satisfy a new form of multidimensional consistency called consistency-around-a-face-centered-cube (CAFCC), whereby the system of equations is consistent on a face-centered cubic unit cell. It is found that for certain arrangements of equations (or pairs of equations) in the square lattice, all known CAFCC equations pass the algebraic entropy test possessing either quadratic or linear growth.

nlin.SI↗

arXiv2019

Lagrangians and integrability for additive fourth-order difference equations

We use a recently found method to characterise all the invertible fourth-order difference equations linear in the extremal values based on the existence of a discrete Lagrangian. We also give some result on the integrability properties of the obtained family and we put it in relation with known classifications. Finally, we discuss the continuum limits of the integrable cases.

math-ph↗

arXiv2018

Darboux Integrability of Trapezoidal $H^{4}$ and $H^{6}$ Families of Lattice Equations II: General Solutions

In this paper we construct the general solutions of two families of quad-equations, namely the trapezoidal $H^{4}$ equations and the $H^{6}$ equations. These solutions are obtained exploiting the properties of the first integrals in the Darboux sense, which were derived in [Gubbiotti G., Yamilov R.I., J. Phys. A: Math. Theor. 50 (2017), 345205, 26 pages, arXiv:1608.03506]. These first integrals are used to reduce the problem to the solution of some linear or linearizable non-autonomous ordinary difference equations which can be formally solved.

nlin.SI↗