70 citations · 573 across the 66 of their papers we have counts for
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From pencils of Novikov algebras of Stäckel type to soliton hierarchies
Maciej Błaszak, Krzysztof Marciniak, Błażej M. Szablikowski
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…
Stationary systems of the AKNS hierarchy
Maciej Błaszak, Krzysztof Marciniak, Błażej M. Szablikowski
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 j…
Non-autonomous soliton hierarchies
Maciej Błaszak, Krzysztof Marciniak, Błażej M. Szablikowski
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…
Algebraic curves as a source of separable multi-Hamiltonian systems
Maciej Blaszak, Krzysztof Marciniak
In this paper we systematically consider various ways of generating integrable and separable Hamiltonian systems in canonical and in non-canonical representations from algebraic cu…
Miura maps for Stäckel systems
Krzysztof Marciniak, Maciej Błaszak
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 th…
Stationary coupled KdV systems and their Stäckel representations
Błażej M. Szablikowski, Maciej Błaszak, Krzysztof Marciniak
In this article we investigate stationary cKdV systems and prove that every -field stationary cKdV system can be written, after a careful reparametrization of jet variables, as…
Multi-component Painlevé ODE's and related non-autonomous KdV stationary hierarchies
M. Błaszak
First, starting from two hierarchies of autonomous Stäckel ODE's, we reconstruct the hierarchy of KdV stationary systems. Next, we deform considered autonomous Stäckel systems to n…
Systematic construction of non-autonomous Hamiltonian equations of Painlevé-type. III. Quantization
Maciej Błaszak, Krzysztof Marciniak
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 def…
Stäckel representations of stationary KdV systems
Maciej Błaszak, Błażej M. Szablikowski, Krzysztof Marciniak
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…
Systematic construction of non-autonomous Hamiltonian equations of Painlevé-type. II. Isomonodromic Lax representation
Maciej Błaszak, Ziemowit Domański, Krzysztof Marciniak
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 rep…
Nonhomogeneous Dispersive Water Waves and Painlevé equations
Maciej Błaszak
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 Pa…
Contact Lax pairs and associated (3+1)-dimensional integrable dispersionless systems
M. Blaszak, A. Sergyeyev
We review the recent approach to the construction of (3+1)-dimensional integrable dispersionless partial differential systems based on their contact Lax pairs and the related -m…
Stäckel transform of Lax equations
Maciej Blaszak, Krzysztof Marciniak
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 gi…
Deforming Lie algebras to Frobenius integrable non-autonomous Hamiltonian systems
Maciej Blaszak, Krzysztof Marciniak, Artur Sergyeyev
Motivated by the theory of Painlevé equations and associated hierarchies, we study non-autonomous Hamiltonian systems that are Frobenius integrable. We establish sufficient conditi…
Non-Homogeneous Hydrodynamic Systems and Quasi-Stäckel Hamiltonians
Krzysztof Marciniak, Maciej Blaszak
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 constr…
Classical and Quantum Superintegrability of Stäckel Systems
Maciej Blaszak, Krzysztof Marciniak
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 t…
Dispersionless (3+1)-dimensional integrable hierarchies
Maciej Blaszak, Artur Sergyeyev
In the present paper we introduce a multi-dimensional version of the R-matrix approach to the construction of integrable hierarchies. Applying this method to the case of the Lie al…
Separable quantizations of Stäckel systems
Maciej Blaszak, Krzysztof Marciniak, Ziemowit Domanski
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 ex…
Flat coordinates for flat Stäckel systems
Krzysztof Marciniak, Maciej Blaszak
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 coordinate…
Flat minimal quantizations of Stackel systems and quantum separability
Maciej Blaszak, Ziemowit Domanski, Burcu Silidir
In this paper, we consider the problem of quantization of classical Stäckel systems and the problem of separability of related quantum Hamiltonians. First, using the concept of Stä…
Integrable quantum Stäckel systems
Maciej Blaszak, Ziemowit Domanski, Artur Sergyeyev +1
The Stäckel separability of a Hamiltonian system is well known to ensure existence of a complete set of Poisson commuting integrals of motion quadratic in the momenta. In the prese…
Invertible coupled KdV and coupled Harry Dym hierarchies
Maciej Blaszak, Krzysztof Marciniak
In this paper we discuss the conditions under which the coupled KdV and coupled Harry Dym hierarchies possess inverse (negative) parts. We further investigate the structure of nonl…
Construction and separability of nonlinear soliton integrable couplings
Maciej Blaszak, Blazej M. Szablikowski, Burcu Silindir
A very natural construction of integrable extensions of soliton systems is presented. The extension is made on the level of evolution equations by a modification of the algebra of…
On reciprocal equivalence of Stäckel systems
Maciej Blaszak, Krzysztof Marciniak
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…