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Jan L. Cieslinski

Publications and source records attributed to Jan L. Cieslinski.

14 recordsLinked to original sources
arXiv2010

New definitions of exponential, hyperbolic and trigonometric functions on time scales

We propose two new definitions of the exponential function on time scales. The first definition is based on the Cayley transformation while the second one is a natural extension of exact discretizations. Our eponential functions map the imaginary axis into the unit circle. Therefore, it is possible to define hyperbolic and trigonometric functions on time scales in a standard way. The resulting functions preserve most of the qualitative properties of the corresponding continuous functions. In particular, Pythagorean trigonometric identities hold exactly on any time scale. Dynamic equations satisfied by Cayley-motivated functions have a natural similarity to the corresponding diferential equations. The exact discretization is less convenient as far as dynamic equations and differentiation is concerned.

math.CA↗
arXiv2009

A direct approach to the construction of standard and non-standard Lagrangians for dissipative dynamical systems with variable coefficients

We present a direct approach to the construction of Lagrangians for a large class of one-dimensional dynamical systems with a simple dependence (monomial or polynomial) on the velocity. We rederive and generalize some recent results and find Lagrangian formulations which seem to be new. Some of the considered systems (e.g., motions with the friction proportional to the velocity and to the square of the velocity) admit infinite families of different Lagrangian formulations.

math-ph↗
arXiv2009

Comment on `conservative discretizations of the Kepler motion'

We show that the exact integrator for the classical Kepler motion, recently found by Kozlov ({\it J. Phys. A: Math. Theor.\} {\bf 40} (2007) 4529-4539), can be derived in a simple natural way (using well known exact discretization of the harmonic oscillator). We also turn attention on important earlier references, where the exact discretization of the 4-dimensional isotropic harmonic oscillator has been applied to the perturbed Kepler problem.

math-ph↗
arXiv2009

Algebraic construction of the Darboux matrix revisited

We present algebraic construction of Darboux matrices for 1+1-dimensional integrable systems of nonlinear partial differential equations with a special stress on the nonisospectral case. We discuss different approaches to the Darboux-Backlund transformation, based on different lambda-dependencies of the Darboux matrix: polynomial, sum of partial fractions, or the transfer matrix form. We derive symmetric N-soliton formulas in the general case. The matrix spectral parameter and dressing actions in loop groups are also discussed. We describe reductions to twisted loop groups, unitary reductions, the matrix Lax pair for the KdV equation and reductions of chiral models (harmonic maps) to SU(n) and to Grassmann spaces. We show that in the KdV case the nilpotent Darboux matrix generates the binary Darboux transformation. The paper is intended as a review of known results (usually presented in a novel context) but some new results are included as well, e.g., general compact formulas for N-soliton surfaces and linear and bilinear constraints on the nonisospectral Lax pair matrices which are preserved by Darboux transformations.

nlin.SI↗
arXiv2007

Pseudospherical surfaces on time scales: a geometric definition and the spectral approach

We define and discuss the notion of pseudospherical surfaces in asymptotic coordinates on time scales. Thus we extend well known notions of discrete pseudospherical surfaces and smooth pseudosperical surfaces on more exotic domains (e.g, the Cantor set). In particular, we present a new expression for the discrete Gaussian curvature which turns out to be valid for asymptotic nets on any time scale. We show that asymptotic Chebyshev nets on an arbitrary time scale have constant negative Gaussian curvature. We present also the quaternion-valued spectral problem (the Lax pair) and the Darboux-Backlund transformation for pseudospherical surfaces (in asymptotic coordinates) on arbitrary time scales.

math.DG↗
arXiv2007

Pseudospherical surfaces on time scales

We define and discuss the notion of pseudospherical surfaces in asymptotic coordinates on time scales. Two special cases, namely dicrete pseudospherical surfaces and smooth pseudosperical surfaces are consistent with this description. In particular, we define the Gaussian curvature in the discrete case.

math.DG↗
arXiv2006

Discretization of multidimensional submanifolds associated with Spin-valued spectral problems

We present a large family of Spin(p,q)-valued discrete spectral problems. The associated discrete nets generated by the so called Sym-Tafel formula are circular nets (i.e., all elementary quadrilaterals are inscribed into circles). These nets are discrete analogues of smooth multidimensional immersions in R^m including isothermic surfaces, Guichard nets, and some other families of orthogonal nets.

nlin.SI↗
arXiv2006

A geometric interpretation of the spectral parameter for surfaces of constant mean curvature

Considering the kinematics of the moving frame associated with a constant mean curvature surface immersed in S^3 we derive a linear problem with the spectral parameter corresponding to elliptic sinh-Gordon equation. The spectral parameter is related to the radius R of the sphere S^3. The application of the Sym formula to this linear problem yields constant mean curvature surfaces in E^3. Independently, we show that the Sym formula itself can be derived by an appropriate limiting process R -> infinity.

nlin.SI↗
arXiv2005

On simulations of the classical harmonic oscillator equation by difference equations

We show that any second order linear ordinary diffrential equation with constant coefficients (including the damped and undumped harmonic oscillator equation) admits an exact discretization, i.e., there exists a difference equation whose solutions exactly coincide with solutions of the corresponding differential equation evaluated at a discrete sequence of points (a lattice). Such exact discretization is found for an arbitrary lattice spacing.

physics.pop-ph↗
arXiv2002

Darboux Covariant Equations of von Neumann Type and their Generalizations

Lax pairs with operator valued coefficients, which are explicitly connected by means of an additional condition, are considered. This condition is proved to be covariant with respect to the Darboux transformation of a general form. Nonlinear equations arising from the compatibility condition of the Lax pairs in the matrix case include, in particular, Nahm equations, Volterra, Bogoyavlenskii and Toda lattices. The examples of another one-, two- and multi-field lattice equations are also presented.

nlin.SI↗