arXiv subjects

Marta Dell'Atti

Publications and source records attributed to Marta Dell'Atti.

16 recordsLinked to original sources

arXiv2026

Discrete differential-geometric Poisson brackets: general theory and explicit constructions

We review the discrete theory of differential-geometric Poisson brackets as introduced by B. A. Dubrovin, and we present a complete proof of their characterisation in the non-degenerate case. We use this characterisation to show several novel examples of such structures, providing a complete classification for non-degenerate four-dimensional differential-geometric Poisson brackets under the additional assumption of the Lie algebra to be of quasi-Frobenius type. We also present a complete classification of quasi-Frobenius filiform N-graded Lie algebras, including two families in arbitrary dimensions.

math-ph↗

arXiv2026

Matrix Dressing Beyond Pfaff-Toda

We develop a matrix pseudodifference operator framework that places theAdler-Pfaff and Pfaff-Toda hierarchies in a common algebraic setting. The bi-infinite Adler-Pfaff hierarchy then becomes a $2\times2$ matrix pseudodifference Lax hierarchy. We introduce the Matrix Pfaff-Toda hierarchy by dressing the matrix Laurent algebra $M_{2N}\bigl(\mathbb{C}[\mathcal{S},\mathcal{S}^{-1}]\bigr)$ using a Pfaff-Toda splitting of the matrix pseudodifference algebra. In the one-component case, its diagonal sector recovers the continuous Pfaff-Toda hierarchy, while the off-diagonal directions supply the missing odd flows, and the powers of a single bare operator reconstruct the full Adler-Pfaff hierarchy. For general $N$, on the regular factorization locus, the multicomponent Pfaff-Toda tau-functions of Savchenko and Zabrodin realize the commuting diagonal sector, with its dressing equations derived directly from the fermionic bilinear identity. Reductions to the block-diagonal and scalar cases recover multicomponent and scalar $2D$ Toda, identifying the even Pfaff hierarchy with an anti-diagonal Toda subhierarchy, and yielding the Krichever-Zabrodin C-Toda hierarchy.

nlin.SI↗

arXiv2026

Modified Painlevé systems with meromorphic solutions for polynomial Hamiltonians of all degrees

We review non-autonomous Hamiltonian systems, polynomial in two dependent variables, with the property that all of their solutions are meromorphic functions in the complex plane. These are related to known Hamiltonian systems with the Painlevé property, for which the solutions are single-valued outside a set of fixed singularities. Our systems are equivalent to them in the absence of fixed singularities, and give modified Painlevé equations otherwise. Using the geometric approach by computing the Okamoto's spaces of initial conditions for certain Hamiltonian systems with general coefficient functions, we obtain differential constraints on these functions for the systems to have only meromorphic solutions. Guided by the Newton polygon of the Hamiltonian function, we obtain all such systems with polynomial Hamiltonian of degree three, four, five, and seven, up to affine equivalence in the dependent variables, while there are none for degree six or degree higher than seven. We thus obtain a list of 12 standard polynomial Hamiltonians that can serve as reference for the Painlevé equivalence problem. This list contains also some new Hamiltonians not previously written down, such as quartic Hamiltonians for Painlevé I and II, quartic Hamiltonians for the modified Painlevé III and V equations, a quintic Hamiltonian for Painlevé IV and quintic and septic Hamiltonians for a modified Painlevé VI equation.

nlin.SI↗

arXiv2026

Random matrix ensembles and integrable differential identities

Integrable differential identities, together with ensemble-specific initial conditions, provide an effective approach for the characterisation of relevant observables and state functions in random matrix theory. We develop this approach for the unitary and orthogonal ensembles. In particular, we focus on a reduction where the probability measure is induced by a Hamiltonian expressed as a formal series of even interaction terms. We show that the order parameters for the unitary ensemble, that is associated with the Volterra lattice, provide a solution of the modified KP equation. The analogous reduction for the orthogonal ensemble, associated with the Pfaff lattice, leads to a new integrable chain. A key step for the calculation of order parameters for the orthogonal ensemble is the evaluation of the initial condition by using a map from orthogonal to skew-orthogonal polynomials. The thermodynamic limit leads to an integrable system (a chain for the orthogonal ensemble) of hydrodynamic type. Intriguingly, we find that the solution to the initial value problem for both the discrete system and its continuum limit are given by the very same semi-discrete dynamical chain.

math-ph↗

arXiv2026

Skew-orthogonal polynomials for a quartic Freud weight: two classes of quasi-orthogonal polynomials

This work is a thorough investigation of skew-orthogonal polynomials with respect to a quartic Freud weight. We provide an explicit method to evaluate skew-orthogonal polynomials of any degree as linear combinations of orthogonal polynomials. The coefficients of these combinations can be evaluated via novel recursive relations. Moreover, we observe that skew-orthogonal polynomials with even and odd degree constitute two families of quasi-orthogonal polynomials with respect to two different semi-classical Laguerre weights, and we provide the first instance of closed recursive relations involving skew-orthogonal polynomials only.

math.CA↗

arXiv2026

Geometric aspects of non-homogeneous 1+0 operators

Led by the key example of the Korteweg-de Vries equation, we study pairs of Hamiltonian operators which are non-homogeneous and are given by the sum of a first-order operator and an ultralocal structure. We present a complete classification of the Casimir functions associated with the degenerate operators in two and three components. We define tensorial criteria to establish the compatibility of two non-homogeneous operators and show a classification of pairs for systems in two components, with some preliminary results for three components as well. Lastly, we study pairs composed of non-degenerate operators only, introducing the definition of bi-pencils. First results show that the considered operators can be related to Nijenhuis geometry, proving a compatibility result in this direction in the framework of Lie algebras.

math-ph↗

arXiv2026

Quadratic Bureau-Guillot systems with the first and second Painlevé transcendents in the coefficients. Part I: geometric approach and birational equivalence

Bureau proposed a classification of systems of quadratic differential equations in two variables which are free of movable critical points, which was recently revised by Guillot. We revisit the quadratic Bureau-Guillot systems with the first and second Painlevé transcendent in the coefficients. We explain their birational equivalence by using the geometric approach of Okamoto's spaces of initial conditions and the method of iterative polynomial regularisation, solving the Painlevé equivalence problem for the Bureau-Guillot systems with non-rational meromorphic coefficients. We also find that one of the systems related to the second Painlevé equation can be transformed into a Hamiltonian system (which we call the cubic Bureau Hamiltonian system) via the iterative polynomial regularisation.

math-ph↗

arXiv2026

Generalisation of Bureau-Guillot systems with Painlevé transcendents in the coefficients

We construct a generalisation of what we call Bureau-Guillot systems, i.e. systems of first order equations with coefficient functions being Painlevé transcendents. The same Painlevé equation is related to the system and it appears as regularising condition in the regularisation process. The systems considered are birationally equivalent to the Okamoto polynomial Hamiltonian systems with rational coefficients for Painlevé equations, hence they possess the Painlevé property. This work extends the results of Bureau-Guillot in a two-fold way. On one side, we consider polynomial systems with degree larger than 2 that are free of movable critical points. These systems contain not only transcendents $\text{P}_{\text{I}}$ and $\text{P}_{\text{II}}$ in the coefficients, but also transcendents $\text{P}_{\text{III}}$, $\text{P}_{\text{IV}}$, $\text{P}_{\text{V}}$ and $\text{P}_{\text{VI}}$ (and/or their derivatives). On the other side, we explore examples of rational systems with the Painlevé transcendents in the coefficients birationally equivalent to the Okamoto polynomial systems. Lastly, we present a simpler version of the change of variables to obtain the analogues of the Bureau-Guillot systems. In this framework we discuss generalisations including the mixed case: systems related to the equation $(\text{P}_{\text{J}})$, with ${\text{J}=\text{I}, \dots, \text{VI}}$, containing coefficient functions that are solutions to $(\text{P}_{\text{K}})$ with $\text{K}\neq \text{J}$. In the latter, the equation $(\text{P}_{\text{K}})$ appears as a regularising condition during the regularisation process. Although we are primarily interested in systems possessing the Painlevé property, we also briefly discuss an analogous construction for systems including coefficient functions solving quasi-Painlevé equation.

math-ph↗

arXiv2025

Spaces of initial conditions for quartic Hamiltonian systems of Painlevé and quasi-Painlevé type

The geometric approach for Painlevé and quasi-Painlevé differential equations in the complex plane is applied to non-autonomous Hamiltonian systems, quartic in the dependent variables. By computing their defining manifolds (analogue of the Okamoto's space of initial conditions in the quasi-Painlevé case), we provide a classification of such systems. We distinguish the various cases by the local behaviour at the movable singularities of the solutions, which are algebraic poles or ordinary poles. The principal cases are categorised by the initial base points of the system in the extended phase space $\mathbb{CP}^2$ and their multiplicities, arising from the coalescence of $4$ simple base points in the generic case. Through the mechanisms of coalescence of base points and degeneration (by setting certain coefficient functions in the Hamiltonian to $0$), all possible sub-cases of quartic Hamiltonian systems with the quasi-Painlevé property are obtained, and are characterised by their corresponding Newton polygons. As particular sub-cases we recover certain systems equivalent to known Painlevé equations, or variants thereof. The resulting picture is a multi-faceted description of each case: the local behaviour around singularities, the surface type, and the Newton polygon.

nlin.SI↗

arXiv2025

Geometric approach for the identification of Hamiltonian systems of quasi-Painlevé type

Some new Hamiltonian systems of quasi-Painlevé type are presented and the analogue of Okamoto's space of initial conditions computed. Using the geometric approach that was introduced originally for the identification problem of Painlevé equations, comparing the irreducible components of the inaccessible divisors arising in the blow-up process, we find bi-rational coordinate changes between some of these systems that give rise to the same global Hamiltonian structure. This scheme thus gives a method for identifying Hamiltonian systems up to bi-rational maps, which is performed in this article for systems of quasi-Painlevé type having singularities that are either square-root type algebraic poles or ordinary poles.

math.CA↗

arXiv2025

r-Matrices for integrable systems

We consider some algebraic and geometric aspects of the theory of integrable systems in finite dimensions, associated with the existence of a classical $r$-matrix, first introduced by Sklyanin as the classical analogue of the quantum version. The importance of the notion of the $r$-matrix in this context relies on the fact that it connects the Hamiltonian structure of integrable equations with the factorisation problem which provides their explicit solution. In this framework, the Lax matrix is interpreted as the coadjoint orbit of a Lie algebra $\mathfrak{g}$, and the existence of a non-dynamical $r$-matrix allows the introduction of a second Lie algebra structure on $\mathfrak{g}$. Depending on the properties of the $r$-matrix associated with the specific system, we distinguish between bialgebras and dialgebras. Bialgebras are associated with a skew-symmetric $r$-matrix, were introduced by Drinfeld, and connected to the interplay between the two Lie algebras structures on $\mathfrak{g}$ and its dual $\mathfrak{g}^*$ respectively. Dialgebras refer to a larger class of $r$-matrix and are related to the factorisation properties of the system, were introduced by Semenov-Tian-Shansky and consist in two Lie algebras $\mathfrak{g}$ and $\mathfrak{g}_R$ coexisting on the same vector space.

math-ph↗

arXiv2025

Iterative polynomial regularisation of PIV-type: Hamiltonian systems and Newton polygons

We study several polynomial Hamiltonian systems of PIV-type (including the mixed case quasi-PIV), and show that via the iterative process of polynomial regularisation, it is possible to identify the "minimal" Hamiltonian system. The selected Hamiltonian function is associated with the Newton polygon with minimal area and smallest highest total degree.

math-ph↗

arXiv2024

Lagrangian multiforms on coadjoint orbits for finite-dimensional integrable systems

Lagrangian multiforms provide a variational framework to describe integrable hierarchies. The case of Lagrangian $1$-forms covers finite-dimensional integrable systems. We use the theory of Lie dialgebras introduced by Semenov-Tian-Shansky to construct a Lagrangian $1$-form. Given a Lie dialgebra associated with a Lie algebra $\mathfrak{g}$ and a collection $H_k$, $k=1,\dots,N$, of invariant functions on $\mathfrak{g}^*$, we give a formula for a Lagrangian multiform describing the commuting flows for $H_k$ on a coadjoint orbit in $\mathfrak{g}^*$. We show that the Euler-Lagrange equations for our multiform produce the set of compatible equations in Lax form associated with the underlying $r$-matrix of the Lie dialgebra. We establish a structural result which relates the closure relation for our multiform to the Poisson involutivity of the Hamiltonians $H_k$ and the so-called ``double zero'' on the Euler-Lagrange equations. The construction is extended to a general coadjoint orbit by using reduction from the free motion of the cotangent bundle of a Lie group. We illustrate the dialgebra construction of a Lagrangian multiform with the open Toda chain and the rational Gaudin model. The open Toda chain is built using two different Lie dialgebra structures on $\mathfrak{sl}(N+1)$. The first one possesses a non-skew-symmetric $r$-matrix and falls within the Adler-Kostant-Symes scheme. The second one possesses a skew-symmetric $r$-matrix. In both cases, the connection with the well-known descriptions of the chain in Flaschka and canonical coordinates is provided.

math-ph↗

arXiv2023

Classification of degenerate non-homogeneous Hamiltonian operators

In this paper, the authors investigate non-homogeneous Hamiltonian operators composed of a first-order Dubrovin-Novikov operator and an ultralocal one. The study of such operators turns out to be fundamental for the inverted system of equations associated with a class of Hamiltonian scalar equations. Often, the involved operators are degenerate in the first-order term. For this reason, a complete classification of the operators with degenerate leading coefficient in systems of 2 and 3 components is presented.

math-ph↗

arXiv2021

Symmetric Matrix Ensemble and Integrable Hydrodynamic Chains

The partition function of the Symmetric Matrix Ensemble is identified with the $τ-$function of a particular solution of the Pfaff Lattice. We show that in the case of even power interactions, in the thermodynamic limit, the $τ-$function corresponds to the solution of an integrable chain of hydrodynamic type. We prove that the hydrodynamic chain so obtained is diagonalisable and admits hydrodynamic reductions in Riemann invariants in an arbitrary number of components.

math-ph↗

arXiv2015

Neutrino and Photon Lensing by Black Holes: Radiative Lens Equations and Post-Newtonian Contributions

We extend a previous phenomenological analysis of photon lensing in an external gravitational background to the case of a massless neutrino, and propose a method to incorporate radiative effects in the classical lens equations of neutrinos and photons. The study is performed for a Schwarzschild metric, generated by a point-like source, and expanded in the Newtonian potential at first order. We use a semiclassical approach, where the perturbative corrections to neutrino scattering, evaluated at one-loop in the Standard Model, are compared with the Einstein formula for the deflection using an impact parameter formulation. For this purpose, we use the renormalized expression of the graviton/fermion/fermion vertex presented in previous studies. We show the agreement between the classical and the semiclassical formulations, for values of the impact parameter $b_h$ of the neutrinos of the order of $b_h\sim 20$, measured in units of the Schwarzschild radius. The analysis is then extended with the inclusion of the post Newtonian corrections in the external gravity field, showing that this extension finds application in the case of the scattering of a neutrino/photon off a primordial black hole. The energy dependence of the deflection, generated by the quantum corrections, is then combined with the standard formulation of the classical lens equations. We illustrate our approach by detailed numerical studies, using as a reference both the thin lens and the nonlinear Virbhadra-Ellis lens.

hep-ph↗