arXiv subjects

Krzysztof Marciniak

Publications and source records attributed to Krzysztof Marciniak.

At least 19 recordsLinked to original sources

arXiv2026

From pencils of Novikov algebras of Stäckel type to soliton hierarchies

In this article we construct evolutionary soliton hierarchies from pencils of Novikov algebras of Stäckel type. We start by defining a special class of associative Novikov algebras, which we call Novikov algebras of Stäckel type, as they are associated with classical Stäckel metrics in Viète coordinates. We obtain sufficient conditions for pencils of these algebras so that the corresponding Dubrovin-Novikov Hamiltonian operators can be centrally extended, producing sets of pairwise compatible Poisson operators. These operators lead to coupled Korteweg-de~Vries (cKdV) and coupled Harry Dym (cHD) hierarchies, as well as to a triangular cKdV hierarchy and a triangular cHD hierarchy.

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arXiv2024

Stationary systems of the AKNS hierarchy

In this article we investigate stationary systems of the AKNS hierarchy. We prove that all these systems are classical separable Stäckel systems. The explicit transformation from jet variables to separation coordinates is obtained by comparing Lax representations in jet and separable coordinates.

nlin.SI↗

arXiv2024

Non-autonomous soliton hierarchies

A formalism of systematic construction of integrable non-autonomous deformations of soliton hierarchies is presented. The theory is formulated as an initial value problem (IVP) for an associated Frobenius integrability condition on a Lie algebra. It is showed that this IVP has a formal solution and within the framewrok of two particular subalgebras of the hereditary Lie algebra the explicit forms of this formal solution are presented. Finally, this formalism is applied to Korteveg-de Vries, dispersive water waves and Ablowitz-Kaup-Newell-Segur soliton hierarchies.

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arXiv2024

Stationary coupled KdV systems and their Stäckel representations

In this article we investigate stationary cKdV systems and prove that every $N$-field stationary cKdV system can be written, after a careful reparametrization of jet variables, as a classical separable Stäckel system on $N+1$ different ways. For each of these $N+1$ parametrizations we present an explicit map between the jet variables and the separation variables of the system. Finally, we show that each pair of Stäckel representations of the same stationary cKdV system, when considered in the phase space extended by Casimir variables, is connected by an appropriate finite-dimensional Miura map, which leads to an $(N+1)$-Hamiltonian formulation for the stationary cKdV system.

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arXiv2024

Algebraic curves as a source of separable multi-Hamiltonian systems

In this paper we systematically consider various ways of generating integrable and separable Hamiltonian systems in canonical and in non-canonical representations from algebraic curves on the plane. In particular, we consider Stäckel transform between two pairs of Stäckel systems, generated by 2n-parameter algebraic curves on the plane, as well as Miura maps between Stäckel systems generated by (n+N)-parameter algebraic curves, leading to multi-Hamiltonian representation of these systems.

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arXiv2023

Miura maps for Stäckel systems

We introduce the concept of Miura maps between parameter-dependent algebraic curves of hyperelliptic type. These Miura maps induce Miura maps between Stäckel systems defined (on the extended phase space) by the considered algebraic curves. This construction yields a new way of generating multi-Hamiltonian representations for Stäckel systems.

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arXiv2023

Stäckel representations of stationary KdV systems

In this article we study Stäckel representations of stationary KdV systems. Using Lax formalism we prove that these systems have two different representations as separable Stäckel systems of Benenti type, related with different foliations of the stationary manifold. We do it by constructing an explicit transformation between the jet coordinates of stationary KdV systems and separation variables of the corresponding Benenti systems for arbitrary number of degrees of freedom. Moreover, on the stationary manifold, we present the explicit form of Miura map between both representations of stationary KdV systems, which also yields their bi-Hamiltonian formulation.

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arXiv2022

Systematic construction of non-autonomous Hamiltonian equations of Painlevé-type. III. Quantization

This is the third article in our series of articles exploring connections between dynamical systems of Stäckel-type and of Painlevé-type. In this article we present a method of deforming of minimally quantized quasi-Stäckel Hamiltonians, considered in Part I to self-adjoint operators satisfying the quantum Frobenius condition, thus guaranteeing that the corresponding Schrödinger equations posses common, multi-time solutions. As in the classical case, we obtain here both magnetic and non-magnetic families of systems. We also show the existence of multitime-dependent quantum canonical maps between both classes of quantum systems.

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arXiv2022

Systematic construction of non-autonomous Hamiltonian equations of Painlevé-type. II. Isomonodromic Lax representation

This is the second article in a suite of articles investigating relations between Stäckel-type systems and Painlevé-type systems. In this article we construct isomonodromic Lax representations for Painlevé-type systems found in the previous paper by Frobenius integrable deformations of Stäckel-type systems. We first construct isomonodromic Lax representations for Painlevé-type systems in the so called magnetic representation and then, using a multitime-dependent canonical transformation, we also construct isomonodromic Lax representations for Painlevé-type systems in the non-magnetic representation. Thus, we prove that the Frobenius integrable systems constructed in Part I are indeed of Painlevé-type. We also present isomonodromic Lax representations for all one-, two- and three-dimensional Painlevé-type systems originating in our scheme. Based on these results we propose complete hierarchies of $P_{I}-P_{IV}$ that follow from our construction.

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arXiv2021

Systematic construction of non-autonomous Hamiltonian equations of Painlevé-type. I. Frobenius integrability

This article is the first one in a suite of three articles exploring connections between dynamical systems of Stäckel-type and of Painlevé- type. In this article we present a deformation of autonomous Stäckel-type systems to non-autonomous Frobenius integrable systems. First, we consider quasi-Stäckel systems with quadratic in momenta Hamiltonians containing separable potentials with time dependent coefficients and then we present a procedure of deforming these equations to non-autonomous Frobenius integrable systems. Then, we present a procedure of deforming quasi-Stäckel systems with so called magnetic separable potentials to non-autonomous Frobenius integrable systems. We also provide a complete list of all $2$- and $3\,$-dimensional Frobenius integrable systems, both with ordinary and with magnetic potentials, that originate in our construction. Further, we prove the equivalence between both classes of systems. Finally we show how Painlevé equations $P_{I}-P_{IV}$ can be derived from our scheme.

math-ph↗

arXiv2021

Transforming Stäckel Hamiltonians of Benenti type to polynomial form

In this paper we discuss two canonical transformations that turn Stäckel separable Hamiltonians of Benenti type into polynomial form: transformation to Viète coordinates and transformation to Newton coordinates. Transformation to Newton coordinates has been applied to these systems only very recently and in this paper we present a new proof that this transformation indeed leads to polynomial form of Stäckel Hamiltonians of Benenti type. Moreover we present all geometric ingredients of these Hamiltonians in both Viète and Newton coordinates.

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arXiv2020

Deforming Lie algebras to Frobenius integrable non-autonomous Hamiltonian systems

Motivated by the theory of Painlevé equations and associated hierarchies, we study non-autonomous Hamiltonian systems that are Frobenius integrable. We establish sufficient conditions under which a given finite-dimensional Lie algebra of Hamiltonian vector fields can be deformed to a time-dependent Lie algebra of Frobenius integrable vector fields spanning the same distribution as the original algebra. The results are applied to quasi-Stäckel systems.

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arXiv2019

Stäckel transform of Lax equations

We construct Lax pairs for a wide class of Stäckel systems by applying the multi-parameter Stäckel transform to Lax pairs of a suitably chosen systems from the seed class. For a given Stäckel system, the obtained set of non-equivalent Lax pairs is parametrized by an arbitrary function.

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arXiv2018

On reciprocal equivalence of Stäckel systems

In this paper we ivestigate Stäckel transforms between different classes of parameter-dependent Stäckel separable systems of the same dimension. We show that the set of all Stäckel systems of the same dimension splits to equivalence classes so that all members within the same class can be connected by a single Stäckel transform. We also give an explicit formula relating solutions of two Stäckel-related systems. These results show in particular that any two geodesic Stäckel systems are Stäckel equivalent in the sense that it is possible to transform one into another by a single Stäckel transform. We also simplify proofs of some known statements about multiparameter Stäckel transform.

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arXiv2017

Non-Homogeneous Hydrodynamic Systems and Quasi-Stäckel Hamiltonians

In this paper we present a novel construction of non-homogeneous hydrodynamic equations from what we call quasi-Stäckel systems, that is non-commutatively integrable systems constructed from appropriate maximally superintegrable Stäckel systems. We describe the relations between Poisson algebras generated by quasi-Stäckel Hamiltonians and the corresponding Lie algebras of vector fields of non-homogeneous hydrodynamic systems. We also apply Stäckel transform to obtain new non-homogeneous equations of considered type.

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arXiv2017

Classical and Quantum Superintegrability of Stäckel Systems

In this paper we discuss maximal superintegrability of both classical and quantum Stäckel systems. We prove a sufficient condition for a flat or constant curvature Stäckel system to be maximally superintegrable. Further, we prove a sufficient condition for a Stäckel transform to preserve maximal superintegrability and we apply this condition to our class of Stäckel systems, which yields new maximally superintegrable systems as conformal deformations of the original systems. Further, we demonstrate how to perform the procedure of minimal quantization to considered systems in order to produce quantum superintegrable and quantum separable systems.

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arXiv2016

Separable quantizations of Stäckel systems

In this article we prove that many Hamiltonian systems that can not be separably quantized in the classical approach of Robertson and Eisenhardt can be separably quantized if we extend the class of admissible quantizations through a suitable choice of Riemann space adapted to the Poisson geometry of the system. Actually, in this article we prove that for every quadratic in momenta Stäckel system (defined on a 2n-dimensional Poisson manifold) for which the Stäckel matrix consists of monomials in position coordinates there exist infinitely many quantizations - parametrized by n arbitrary functions - that turn this system into a quantum separable Stäckel system.

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arXiv2014

Flat coordinates for flat Stäckel systems

In this article we explicitely construct transformation bewteen separable and flat coordinates for flat Stäckel systems and exploit the structre of these systems in flat coordinates. In the elliptic case these coordinates become well known generalized elliptical coordinates of Jacobi.

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