arXiv subjects

Maciej Błaszak

Publications and source records attributed to Maciej Błaszak.

14 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.

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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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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

Nonhomogeneous Dispersive Water Waves and Painlevé equations

In this letter we consider three nonhomogeneous deformations of Dispersive Water Wave (DWW) soliton equation and prove that their stationary flows are equivalent to three famous Painlevé equations, i.e. $P_{II}$, $P_{III}$ and $P_{IV},$ respectively.

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arXiv2019

Lax Representations for Separable Systems from Benenti Class

In this paper we construct Lax pairs for Stäckel systems with separation curves from so-called Benenti class. For each system of considered family we present an infinite family of Lax representations, parameterized by smooth functions of spectral parameter.

math-ph↗

arXiv2018

Deformation quantization with minimal length

We develop a complete theory of non-formal deformation quantization exhibiting a nonzero minimal uncertainty in position. An appropriate integral formula for the star-product is introduced together with a suitable space of functions on which the star-product is well defined. Basic properties of the star-product are proved and the extension of the star-product to a certain Hilbert space and an algebra of distributions is given. A C*-algebra of observables and a space of states are constructed. Moreover, an operator representation in momentum space is presented. Finally, examples of position eigenvectors and states of maximal localization are given.

math-ph↗

arXiv2016

Coherence and squeezing along quantum trajectories

We perform a detailed analysis of the behavior of coherent and squeezed states undergoing time evolution. We calculate time dependence of expectation values of position and momentum in coherent and squeezed states (which can be interpreted as quantum trajectories in coherent and squeezed states) and examine how coherence and squeezing is affected during time development, calculating time dependence of position and momentum uncertainty. We focus our investigations on two quantum systems. First we consider quantum linear system with Hamiltonian quadratic in $q$ and $p$ variables. As the second system we consider the simplest quantum nonlinear system with Hamiltonian quartic in $q$ and $p$ variables. We calculate the explicit formulas for the time development of expectation values and uncertainties of position and momentum in an initial coherent state.

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arXiv2009

Bi-presymplectic chains of co-rank one and related Liouville integrable systems

Bi-presymplectic chains of one-forms of co-rank one are considered. The conditions in which such chains represent some Liouville integrable systems and the conditions in which there exist related bi-Hamiltonian chains of vector fields are derived. To present the construction of bi-presymplectic chains, the notion of dual Poisson-presymplectic pair is used and the concept of d-compatibility of Poisson bivectors and d-compatibility of presymplectic forms is introduced. It is shown that bi-presymplectic representation of related flow leads directly to the construction of separation coordinates in purely algorithmic way. As an illustration bi-presymplectic and bi-Hamiltonian chains in ${\mathbb R}^3$ are considered in detail.

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