Krzysztof Marciniak

R-matrix approach to lattice integrable systems

An r‐matrix formalism is applied to the construction of the integrable lattice systems and their bi‐Hamiltonian structure. Miura‐like gauge transformations between the hierarchies are also investigated. In the end the ladder of linear maps between generated hierarchies is established and described.

Quasi-Lagrangian systems of Newton equations

Systems of Newton equations of the form q=−12A−1(q)∇k with an integral of motion quadratic in velocities are studied. These equations generalize the potential case (when A=I, the identity matrix) and they admit a curious quasi-Lagrangian formulation which differs from the standard Lagrange equations by the plus sign between terms. A theory of such quasi-Lagrangian Newton (qLN) systems having two functionally independent integrals of motion is developed with focus on two-dimensional systems. Such systems admit a bi-Hamiltonian formulation and are proved to be completely integrable by embedding into five-dimensional integrable systems. They are characterized by a linear, second-order partial differential equation PDE which we call the fundamental equation. Fundamental equations are classified through linear pencils of matrices associated with qLN systems. The theory is illustrated by two classes of systems: separable potential systems and driven systems. New separation variables for driven systems are foun...

Separation of variables in quasi-potential systems of bi-cofactor form

We perform variable separation in the quasi-potential systems of equations of the form ¨ q =− A −1 ∇k =− ˜ A −1 ∇ ˜ k ,w hereA and ˜ A are Killing tensors, by embedding these systems into a bi-Hamiltonian chain and by calculating the corresponding Darboux–Nijenhuis coordinates on the symplectic leaves of one of the Hamiltonian structures of the system. We also present examples of the corresponding separation coordinates in two and three dimensions.

Two newton decompositions of staionary flows of KdV and Harry Dym hierarchies

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 rise to new bihamiltonian structures in classical mechanics. Lax representations for these Newton equations are found.

Two families of nonstandard Poisson structures for Newton equations

Two families of nonstandard two-dimensional Poisson structures for systems of Newton equations are studied. They are closely related either with separable systems or with the so-called quasi-Lagrangian systems. A theorem characterizing the general form of bi-Hamiltonian formulation for separable systems in two and in n dimensions is formulated and proved.

From Stäckel systems to integrable hierarchies of PDE’s: Benenti class of separation relations

We propose a general scheme of constructing of soliton hierarchies from finite dimensional Stackel systems and related separation relations. In particular, we concentrate on the simplest class of separation relations, called Benenti class, i.e., certain Stackel systems with quadratic in momenta integrals of motion.

On Reciprocal Equivalence of Stäckel Systems

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

Stackel systems generating coupled KdV hierarchies and their finite-gap and rational solutions

We show how to generate coupled KdV hierarchies from Stackel separable systems of Benenti type. We further show that the solutions of these Stackel systems generate a large class of finite-gap and rational solutions for cKdV hierarchies. Most of these solutions are new.

Dirac reduction of dual Poisson–presymplectic pairs

A new notion of a dual Poisson–presymplectic pair is introduced and its properties are examined. The procedure of Dirac reduction of Poisson operators onto submanifolds proposed by Dirac is in this paper embedded in a geometric procedure of reduction of dual Poisson–presymplectic pairs. The method presented generalizes those introduced by J Marsden and T Ratiu for reductions of Poisson manifolds. Two examples are given.

Dirac Reduction Revisited

Abstract The procedure of Dirac reduction of Poisson operators on submanifolds is discussed within a particularly useful special realization of the general Marsden-Ratiu reduction procedure. The Dirac classification of constraints on ‘first-class’ constraints and ‘second-class’ constraints is reexamined.