201 research outputs found
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.Comment: 6 page
An integrating factor matrix method to find first integrals
In this paper we developed an integrating factor matrix method to derive
conditions for the existence of first integrals. We use this novel method to
obtain first integrals, along with the conditions for their existence, for two
and three dimensional Lotka-Volterra systems with constant terms. The results
are compared to previous results obtained by other methods
An orbit-preserving discretization of the classical Kepler problem
We present a remarkable discretization of the classical Kepler problem which
preserves its trajectories and all integrals of motion. The points of any
discrete orbit belong to an appropriate continuous trajectory.Comment: 7 page
On a two-parameter extension of the lattice KdV system associated with an elliptic curve
A general structure is developed from which a system of integrable partial
difference equations is derived generalising the lattice KdV equation. The
construction is based on an infinite matrix scheme with as key ingredient a
(formal) elliptic Cauchy kernel. The consistency and integrability of the
lattice system is discussed as well as special solutions and associated
continuum equations.Comment: Submitted to the proceedings of the Oeresund PDE-symposium, 23-25 May
2002; 17 pages LaTeX, style-file include
Long-time behaviour of discretizations of the simple pendulum equation
We compare the performance of several discretizations of the simple pendulum
equation in a series of numerical experiments. The stress is put on the
long-time behaviour. We choose for the comparison numerical schemes which
preserve the qualitative features of solutions (like periodicity). All these
schemes are either symplectic maps or integrable (preserving the energy
integral) maps, or both. We describe and explain systematic errors (produced by
any method) in numerical computations of the period and the amplitude of
oscillations. We propose a new numerical scheme which is a modification of the
discrete gradient method. This discretization preserves (almost exactly) the
period of small oscillations for any time step.Comment: 41 pages, including 18 figures and 4 table
Invariant varieties of periodic points for some higher dimensional integrable maps
By studying various rational integrable maps on with
invariants, we show that periodic points form an invariant variety of dimension
for each period, in contrast to the case of nonintegrable maps in which
they are isolated. We prove the theorem: {\it `If there is an invariant variety
of periodic points of some period, there is no set of isolated periodic points
of other period in the map.'}Comment: 24 page
B\"acklund Transformations of MKdV and Painlev\'e Equations
For there are and actions on the space of solutions of
the first nontrivial equation in the Z_2$ actions on the space of solutions of the standard MKdV equation.
These actions survive scaling reduction, and give rise to transformation groups
for certain (systems of) ODEs, including the second, fourth and fifth
Painlev\'e equations.Comment: 8 pages, plain te
A note on the integrable discretization of the nonlinear Schr\"odinger equation
We revisit integrable discretizations for the nonlinear Schr\"odinger
equation due to Ablowitz and Ladik. We demonstrate how their main drawback, the
non-locality, can be overcome. Namely, we factorize the non-local difference
scheme into the product of local ones. This must improve the performance of the
scheme in the numerical computations dramatically. Using the equivalence of the
Ablowitz--Ladik and the relativistic Toda hierarchies, we find the
interpolating Hamiltonians for the local schemes and show how to solve them in
terms of matrix factorizations.Comment: 24 pages, LaTeX, revised and extended versio
An integrable multicomponent quad equation and its Lagrangian formulation
We present a hierarchy of discrete systems whose first members are the
lattice modified Korteweg-de Vries equation, and the lattice modified
Boussinesq equation. The N-th member in the hierarchy is an N-component system
defined on an elementary plaquette in the 2-dimensional lattice. The system is
multidimensionally consistent and a Lagrangian which respects this feature,
i.e., which has the desirable closure property, is obtained.Comment: 10 page
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