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Maciej Blaszak

Publications and source records attributed to Maciej Blaszak.

At least 37 records · Page 2Linked to original sources
arXiv2012

Construction and separability of nonlinear soliton integrable couplings

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 dynamical fields. The paper is motivated by recent works of Wen-Xiu Ma et al. (Comp. Math. Appl. 60 (2010) 2601, Appl. Math. Comp. 217 (2011) 7238), where new class of soliton systems, being nonlinear integrable couplings, was introduced. The general form of solutions of the considered class of coupled systems is described. Moreover, the decoupling procedure is derived, which is also applicable to several other coupling systems from the literature.

nlin.SI↗
arXiv2011

Phase Space Quantum Mechanics

The paper develop the alternative formulation of quantum mechanics known as the phase space quantum mechanics or deformation quantization. It is shown that the quantization naturally arises as an appropriate deformation of the classical Hamiltonian mechanics. More precisely, the deformation of the point-wise product of observables to an appropriate noncommutative $\star$-product and the deformation of the Poisson bracket to an appropriate Lie bracket is the key element in introducing the quantization of classical Hamiltonian systems. The considered class of deformations and the corresponding $\star$-products contains as a special cases all deformations which can be found in the literature devoted to the subject of the phase space quantum mechanics. Fundamental properties of $\star$-products of observables, associated with the considered deformations are presented as well. Moreover, a space of states containing all admissible states is introduced, where the admissible states are appropriate pseudo-probability distributions defined on the phase space. It is proved that the space of states is endowed with a structure of a Hilbert algebra with respect to the $\star$-multiplication. The most important result of the paper shows that developed formalism is more fundamental then the axiomatic ordinary quantum mechanics which appears in the presented approach as the intrinsic element of the general formalism. The equivalence of two formulations of quantum mechanics is proved by observing that the Wigner-Moyal transform has all properties of the tensor product. This observation allows writing many previous results found in the literature in a transparent way, from which the equivalence of the two formulations of quantum mechanics follows naturally.

math-ph↗
arXiv2010

Bi-presymplectic representation of Liouville integrable systems and related separability theory

Bi-presymplectic chains of one-forms of arbitrary co-rank 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 presented. In order to derived the construction of bi-presymplectic chains, the notions of dual Poisson-presymplectic pair, d-compatibility of presymplectic forms and d-compatibility of Poisson bivectors is used. The completely algorithmic construction of separation coordinates is demonstrated. It is also proved that Stäckel separable systems have bi-inverse-Hamiltonian representation, i.e. are represented by bi-presymplectic chains of closed one-forms. The co-rank of related structures depends on the explicit form of separation relations.

nlin.SI↗
arXiv2009

Bi-Hamiltonian representation of Stäckel systems

It is shown that a linear separation relations are fundamental objects for integration by quadratures of Stäckel separable Liouville integrable systems (the so-called Stäckel systems). These relations are further employed for the classification of Stäckel systems. Moreover, we prove that {\em any} Stäckel separable Liouville integrable system can be lifted to a bi-Hamiltonian system of Gel'fand-Zakharevich type. In conjunction with other known result this implies that the existence of bi-Hamiltonian representation of Liouville integrable systems is a necessary condition for Stäckel separability.

math-ph↗
arXiv2009

Classical R-matrix theory for bi-Hamiltonian field systems

The R-matrix formalism for the construction of integrable systems with infinitely many degrees of freedom is reviewed. Its application to Poisson, noncommutative and loop algebras as well as central extension procedure are presented. The theory is developed for (1+1)-dimensional case where the space variable belongs either to R or to various discrete sets. Then, the extension onto (2+1)-dimensional case is made, when the second space variable belongs to R. The formalism presented contains many proofs and important details to make it self-contained and complete. The general theory is applied to several infinite dimensional Lie algebras in order to construct both dispersionless and dispersive (soliton) integrable field systems.

nlin.SI↗
arXiv2009

Stackel systems: bi-Hamiltonian property and systematic construction

It is shown that separation conditions (separation curves) are fundamental objects of separability theory. They are used for the classification of certain clases of separable systems, for the proof of bi-Hamiltonian property and finally they allow to construct new separable systems from known ones.

nlin.SI↗
arXiv2008

Bi-Hamiltonian structures for integrable systems on regular time scales

A construction of the bi-Hamiltonian structures for integrable systems on regular time scales is presented. The trace functional on an algebra of $δ$-pseudo-differential operators, valid on an arbitrary regular time scale, is introduced. The linear Poisson tensors and the related Hamiltonians are derived. The quadratic Poisson tensors is given by the use of the recursion operators of the Lax hierarchies. The theory is illustrated by $Δ$-differential counterparts of Ablowitz-Kaup-Newell-Segur and Kaup-Broer hierarchies.

nlin.SI↗
arXiv2008

A Coordinate-Free Construction for a Class of Integrable Hydrodynamic-Type Systems

Using a (1,1)-tensor L with zero Nijenhuis torsion and maximal possible number (equal to the number of dependent variables) of distinct, functionally independent eigenvalues we define, in a coordinate-free fashion, the seed systems which are weakly nonlinear semi-Hamiltonian systems of a special form, and an infinite set of conservation laws for the seed systems. The reciprocal transformations constructed from these conservation laws yield a considerably larger class of hydrodynamic-type systems from the seed systems, and we show that these new systems are again defined in a coordinate-free manner, using the tensor L alone, and, moreover, are weakly nonlinear and semi-Hamiltonian, so their general solution can be obtained by means of the generalized hodograph method of Tsarev.

nlin.SI↗
arXiv2008

R-matrix approach to integrable systems on time scales

A general unifying framework for integrable soliton-like systems on time scales is introduced. The $R$-matrix formalism is applied to the algebra of $δ$-differential operators in terms of which one can construct infinite hierarchy of commuting vector fields. The theory is illustrated by two infinite-field integrable hierarchies on time scales which are difference counterparts of KP and mKP. The difference counterparts of AKNS and Kaup-Broer soliton systems are constructed as related finite-field restrictions.

nlin.SI↗
arXiv2007

Generalized Stäckel Transform and Reciprocal Transformations for Finite-Dimensional Integrable Systems

We present a multiparameter generalization of the Stäckel transform (the latter is also known as the coupling-constant metamorphosis) and show that under certain conditions this generalized Stäckel transform preserves the Liouville integrability, noncommutative integrability and superintegrability. The corresponding transformation for the equations of motion proves to be nothing but a reciprocal transformation of a special form, and we investigate the properties of this reciprocal transformation. Finally, we show that the Hamiltonians of the systems possessing separation curves of apparently very different form can be related through a suitably chosen generalized Stäckel transform.

nlin.SI↗
arXiv2007

Non-Hamiltonian systems separable by Hamilton-Jacobi method

We show that with every separable calssical Stackel system of Benenti type on a Riemannian space one can associate, by a proper deformation of the metric tensor, a multi-parameter family of non-Hamiltonian systems on the same space, sharing the same trajectories and related to the seed system by appropriate reciprocal transformations. These system are known as bi-cofactor systems and are integrable in quadratures as the seed Hamiltonian system is. We show that with each class of bi-cofactor systems a pair of separation curves can be related. We also investigate conditions under which a given flat bi-cofactor system can be deformed to a family of geodesically equivalent flat bi-cofactor systems.

nlin.SI↗
arXiv2007

Integrable discrete systems on R and related dispersionless systems

The general framework for integrable discrete systems on R in particular containing lattice soliton systems and their q-deformed analogues is presented. The concept of regular grain structures on R, generated by discrete one-parameter groups of diffeomorphisms, through which one can define algebras of shift operators is introduced. Two integrable hierarchies of discrete chains together with bi-Hamiltonian structures are constructed. Their continuous limit and the inverse problem based on the deformation quantization scheme are considered.

nlin.SI↗
arXiv2007

Dispersionful analogue of the Whitham hierarchy

The dispersionful analogue, by means of Lax formalism, of the zero-genus universal Whitham hierarchy together with its algebraic orbit finite-field reductions is considered. The theory is illustrated by several significant examples.

nlin.SI↗
arXiv2007

Natural coordinates for a class of Benenti systems

We present explicit formulas for the coordinates in which the Hamiltonians of the Benenti systems with flat metrics take natural form and the metrics in question are represented by constant diagonal matrices.

nlin.SI↗
arXiv2006

Reciprocal transformations for Stackel-related Liouville integrable systems

We consider the Stäckel transform, also known as the coupling-constant metamorphosis, which under certain conditions turns a Hamiltonian dynamical system into another such system and preserves the Liouville integrability. We show that the corresponding transformation for the equations of motion is nothing but the reciprocal transformation of a special form and we investigate the properties of this transformation. This result is further applied for the study of the $k$-hole deformations of the Benenti systems or more general seed systems.

nlin.SI↗