97,217 research outputs found
On solutions of Linear Ordinary Difference Equations in their Coefficient Field
We extend the notion of monomial extensions of differential fields, i.e. simp- le transcendental extensions in which the polynomials are closed under differentiation, to difference fields. The structure of such extensions provides an algebraic framework for solving generalized linear difference equations with coefficients in such fields. We then describe algorithms for finding the denominator of any solution of those equations in an important subclass of monomial extensions that includes transcendental indefinite sums and products. This reduces the general problem of finding the solutions of such equations in their coefficient fields to bounding their degrees. In the base case, this yields in particular a new algorithm for computing the rational solutions of q-difference equations with polynomial coefficients
Continuous Symmetries of Difference Equations
Lie group theory was originally created more than 100 years ago as a tool for
solving ordinary and partial differential equations. In this article we review
the results of a much more recent program: the use of Lie groups to study
difference equations. We show that the mismatch between continuous symmetries
and discrete equations can be resolved in at least two manners. One is to use
generalized symmetries acting on solutions of difference equations, but leaving
the lattice invariant. The other is to restrict to point symmetries, but to
allow them to also transform the lattice.Comment: Review articl
Differential operators on the superline, Berezinians, and Darboux transformations
We consider differential operators on a supermanifold of dimension . We
define non-degenerate operators as those with an invertible top coefficient in
the expansion in the "superderivative" (which is the square root of the
shift generator, the partial derivative in an even variable, with the help of
an odd indeterminate). They are remarkably similar to ordinary differential
operators. We show that every non-degenerate operator can be written in terms
of `super Wronskians' (which are certain Berezinians). We apply this to Darboux
transformations (DTs), proving that every DT of an arbitrary non-degenerate
operator is the composition of elementary first order transformations. Hence
every DT corresponds to an invariant subspace of the source operator and, upon
a choice of basis in this subspace, is expressed by a super-Wronskian formula.
We consider also dressing transformations, i.e., the effect of a DT on the
coefficients of the non-degenerate operator. We calculate these transformations
in examples and make some general statements.Comment: 24 pages, LaTeX, some editorial changes (as compared with the earlier
version
Spectral methods in general relativistic astrophysics
We present spectral methods developed in our group to solve three-dimensional
partial differential equations. The emphasis is put on equations arising from
astrophysical problems in the framework of general relativity.Comment: 51 pages, elsart (Elsevier Preprint), 19 PostScript figures,
submitted to Journal of Computational & Applied Mathematic
Spherical Structures in Conformal Gravity and its Scalar-Tensor Extension
We study spherically-symmetric structures in Conformal Gravity and in a
scalar-tensor extension and gain some more insight about these gravitational
theories. In both cases we analyze solutions in two systems: perfect fluid
solutions and boson stars of a self-interacting complex scalar field. In the
purely tensorial (original) theory we find in a certain domain of parameter
space finite mass solutions with a linear gravitational potential but without a
Newtonian contribution. The scalar-tensor theory exhibits a very rich structure
of solutions whose main properties are discussed. Among them, solutions with a
finite radial extension, open solutions with a linear potential and logarithmic
modifications and also a (scalar-tensor) gravitational soliton. This may also
be viewed as a static self-gravitating boson star in purely tensorial Conformal
Gravity.Comment: 24 pages, revised version, accepted for publication in Phys. Rev.
Dynamics of two-component electromagnetic and acoustic extremely short pulses
The distinctive features of passing the two-component extremely short pulses
through the nonlinear media are discussed. The equations considered describe
the propagation in the two-level anisotropic medium of the electromagnetic
pulses consisting of ordinary and extraordinary components and an evolution of
the transverse-longitudinal acoustic pulses in a crystal containing the
paramagnetic impurities with effective spin S=1/2. The solutions decreasing
exponentially and algebraically are studied.Comment: LaTeX, 11 pages, 6 figure
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