M. Blaszak - Academia.edu (original) (raw)

Papers by M. Blaszak

Applied Mathematics Letters, 1990

arXiv: Exactly Solvable and Integrable Systems, 2017

Motivated by the theory of Painlev\'e equations and associated hierarchies, we study non-auto... more Motivated by the theory of Painlev\'e equations and associated hierarchies, we study non-autonomous Hamiltonian systems that are Frobenius integrable. In particular, we establish sufficient conditions under which the Hamiltonians for these systems can be obtained as polynomial-in-times deformations of explicitly time-independent Hamiltonians constituting a finite-dimensional Lie algebra with respect to the Poisson bracket. The results are illustrated by several examples.

Research Reports in Physics, 1990

By use of mastersymmetries we construct the action/angle variables for multi-soliton systems in t... more By use of mastersymmetries we construct the action/angle variables for multi-soliton systems in terms of the field variable u. Furthermore, an interpretation in terms of asymptotic data is given.

Journal of Mathematical Physics, 2008

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

Motivated by the theory of Painlev\'e equations and associated hierarchies, we study non-auto... more Motivated by the theory of Painlev\'e equations and associated hierarchies, we study non-autonomous Hamiltonian systems that are Frobenius integrable in the sense defined below. We establish sufficient conditions under which Hamiltonian vector fields forming a finite-dimensional Lie algebra can be deformed to time-dependent Frobenius integrable Hamiltonian vector fields spanning the same distribution as the original vector fields. The results are illustrated by several examples.

Journal of Physics A: Mathematical and General, 2004

For a class of Hamiltonian systems naturally arising in the modern theory of separation of variab... more For a class of Hamiltonian systems naturally arising in the modern theory of separation of variables, we establish their maximal superintegrability by explicitly constructing the additional integrals of motion.

Arxiv preprint nlin/0607057, Jul 25, 2006

Abstract: We consider the St\" ackel transform, also known as the coupling-constant metamorp... more Abstract: We consider the St\" ackel 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 ...

Arxiv preprint arXiv:0706.1473, Jun 13, 2007

We present a multiparameter generalization of the St\" ackel transform, also known as the co... more We present a multiparameter generalization of the St\" ackel transform, also known as the coupling-constant metamorphosis. We show that under certain conditions this transformation preserves the Liouville 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 ...

Acta Physica Polonica a, 1988

On considere le probleme de la decomposition de solutions N-soliton de diverses equations d'e... more On considere le probleme de la decomposition de solutions N-soliton de diverses equations d'evolution non lineaires en une somme de N solitons interactifs, ou chaque soliton peut etre considere comme modele de particule etendue. On analyse la deformation pendant l'interaction

Journal of Physics A: Mathematical and Theoretical, 2009

Bi-presymplectic chains of 1-forms of co-rank 1 are considered. The conditions under which such c... more Bi-presymplectic chains of 1-forms of co-rank 1 are considered. The conditions under which such chains represent some Liouville integrable systems and the conditions under which there exist related bi-Hamiltonian chains of vector fields are derived. To present the construction of bi-presymplectic chains, the notion of a 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 a related flow leads directly to the construction of separation coordinates in a purely algorithmic way. As an illustration, bi-presymplectic and bi-Hamiltonian chains in R 3 are considered in detail.

Journal of Physics A: Mathematical and Theoretical, 2008

Nonlinear Systems and Their Remarkable Mathematical Structures, Dec 6, 2019

We review the recent approach to the construction of (3+1)-dimensional integrable dispersionless ... more 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 R-matrix theory for the Lie algebra of functions with respect to the contact bracket. We discuss various kinds of Lax representations for such systems, in particular, linear nonisospectral contact Lax pairs and nonlinear contact Lax pairs as well as the relations among the two. Finally, we present a large number of examples with finite and infinite number of dependent variables, as well as the reductions of these examples to lower-dimensional integrable dispersionless systems.

Physica A: Statistical Mechanics and its Applications, 1996

We show that each stationary flow of the KdV and Harry Dym hierarchies of soliton equations, whic... more We show that each stationary flow of the KdV and Harry Dym hierarchies of soliton equations, which are (2m+1)-st order ODEs (m=0,1…), has two parametrisations as a set of Newton equations with velocity-independent forces. Forces are potential and these Newton equations follow from a Lagrangian function with an inefinite kinetic energy term. These two parametrisations are canonically inequivalent and give

Теоретическая и математическая физика, 2000

Reports on Mathematical Physics, 1999

The so-called central extension approach is applied t,o 'R.-matrix formalism to extend it from (l... more The so-called central extension approach is applied t,o 'R.-matrix formalism to extend it from (l+l) to (2+1)-dimensions. Two-dimensional integrable field and lattice-field systems are constructed.

Progress of Theoretical Physics, 1991

Using the algebra of symmetries/mastersymmetries the action/angle scalar fields and related vecto... more Using the algebra of symmetries/mastersymmetries the action/angle scalar fields and related vector fields for complex multisolitons are constructed in explicit form in terms of the field variable. *) The example of NLS in I fits to the scheme developed there only under some special reduction which is presented at the end of § 2.

Physics Letters A, 1998

ABSTRACT

physica status solidi (b), 1988

Applied Mathematics Letters, 1990

arXiv: Exactly Solvable and Integrable Systems, 2017

Motivated by the theory of Painlev\'e equations and associated hierarchies, we study non-auto... more Motivated by the theory of Painlev\'e equations and associated hierarchies, we study non-autonomous Hamiltonian systems that are Frobenius integrable. In particular, we establish sufficient conditions under which the Hamiltonians for these systems can be obtained as polynomial-in-times deformations of explicitly time-independent Hamiltonians constituting a finite-dimensional Lie algebra with respect to the Poisson bracket. The results are illustrated by several examples.

Research Reports in Physics, 1990

By use of mastersymmetries we construct the action/angle variables for multi-soliton systems in t... more By use of mastersymmetries we construct the action/angle variables for multi-soliton systems in terms of the field variable u. Furthermore, an interpretation in terms of asymptotic data is given.

Journal of Mathematical Physics, 2008

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

Motivated by the theory of Painlev\'e equations and associated hierarchies, we study non-auto... more Motivated by the theory of Painlev\'e equations and associated hierarchies, we study non-autonomous Hamiltonian systems that are Frobenius integrable in the sense defined below. We establish sufficient conditions under which Hamiltonian vector fields forming a finite-dimensional Lie algebra can be deformed to time-dependent Frobenius integrable Hamiltonian vector fields spanning the same distribution as the original vector fields. The results are illustrated by several examples.

Journal of Physics A: Mathematical and General, 2004

For a class of Hamiltonian systems naturally arising in the modern theory of separation of variab... more For a class of Hamiltonian systems naturally arising in the modern theory of separation of variables, we establish their maximal superintegrability by explicitly constructing the additional integrals of motion.

Arxiv preprint nlin/0607057, Jul 25, 2006

Abstract: We consider the St\" ackel transform, also known as the coupling-constant metamorp... more Abstract: We consider the St\" ackel 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 ...

Arxiv preprint arXiv:0706.1473, Jun 13, 2007

We present a multiparameter generalization of the St\" ackel transform, also known as the co... more We present a multiparameter generalization of the St\" ackel transform, also known as the coupling-constant metamorphosis. We show that under certain conditions this transformation preserves the Liouville 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 ...

Acta Physica Polonica a, 1988

On considere le probleme de la decomposition de solutions N-soliton de diverses equations d'e... more On considere le probleme de la decomposition de solutions N-soliton de diverses equations d'evolution non lineaires en une somme de N solitons interactifs, ou chaque soliton peut etre considere comme modele de particule etendue. On analyse la deformation pendant l'interaction

Journal of Physics A: Mathematical and Theoretical, 2009

Bi-presymplectic chains of 1-forms of co-rank 1 are considered. The conditions under which such c... more Bi-presymplectic chains of 1-forms of co-rank 1 are considered. The conditions under which such chains represent some Liouville integrable systems and the conditions under which there exist related bi-Hamiltonian chains of vector fields are derived. To present the construction of bi-presymplectic chains, the notion of a 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 a related flow leads directly to the construction of separation coordinates in a purely algorithmic way. As an illustration, bi-presymplectic and bi-Hamiltonian chains in R 3 are considered in detail.

Journal of Physics A: Mathematical and Theoretical, 2008

Nonlinear Systems and Their Remarkable Mathematical Structures, Dec 6, 2019

We review the recent approach to the construction of (3+1)-dimensional integrable dispersionless ... more 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 R-matrix theory for the Lie algebra of functions with respect to the contact bracket. We discuss various kinds of Lax representations for such systems, in particular, linear nonisospectral contact Lax pairs and nonlinear contact Lax pairs as well as the relations among the two. Finally, we present a large number of examples with finite and infinite number of dependent variables, as well as the reductions of these examples to lower-dimensional integrable dispersionless systems.

Physica A: Statistical Mechanics and its Applications, 1996

We show that each stationary flow of the KdV and Harry Dym hierarchies of soliton equations, whic... more We show that each stationary flow of the KdV and Harry Dym hierarchies of soliton equations, which are (2m+1)-st order ODEs (m=0,1…), has two parametrisations as a set of Newton equations with velocity-independent forces. Forces are potential and these Newton equations follow from a Lagrangian function with an inefinite kinetic energy term. These two parametrisations are canonically inequivalent and give

Теоретическая и математическая физика, 2000

Reports on Mathematical Physics, 1999

The so-called central extension approach is applied t,o 'R.-matrix formalism to extend it from (l... more The so-called central extension approach is applied t,o 'R.-matrix formalism to extend it from (l+l) to (2+1)-dimensions. Two-dimensional integrable field and lattice-field systems are constructed.

Progress of Theoretical Physics, 1991

Using the algebra of symmetries/mastersymmetries the action/angle scalar fields and related vecto... more Using the algebra of symmetries/mastersymmetries the action/angle scalar fields and related vector fields for complex multisolitons are constructed in explicit form in terms of the field variable. *) The example of NLS in I fits to the scheme developed there only under some special reduction which is presented at the end of § 2.

Physics Letters A, 1998

ABSTRACT

physica status solidi (b), 1988