22,122 research outputs found
Status of the differential transformation method
Further to a recent controversy on whether the differential transformation
method (DTM) for solving a differential equation is purely and solely the
traditional Taylor series method, it is emphasized that the DTM is currently
used, often only, as a technique for (analytically) calculating the power
series of the solution (in terms of the initial value parameters). Sometimes, a
piecewise analytic continuation process is implemented either in a numerical
routine (e.g., within a shooting method) or in a semi-analytical procedure
(e.g., to solve a boundary value problem). Emphasized also is the fact that, at
the time of its invention, the currently-used basic ingredients of the DTM
(that transform a differential equation into a difference equation of same
order that is iteratively solvable) were already known for a long time by the
"traditional"-Taylor-method users (notably in the elaboration of software
packages --numerical routines-- for automatically solving ordinary differential
equations). At now, the defenders of the DTM still ignore the, though much
better developed, studies of the "traditional"-Taylor-method users who, in
turn, seem to ignore similarly the existence of the DTM. The DTM has been given
an apparent strong formalization (set on the same footing as the Fourier,
Laplace or Mellin transformations). Though often used trivially, it is easily
attainable and easily adaptable to different kinds of differentiation
procedures. That has made it very attractive. Hence applications to various
problems of the Taylor method, and more generally of the power series method
(including noninteger powers) has been sketched. It seems that its potential
has not been exploited as it could be. After a discussion on the reasons of the
"misunderstandings" which have caused the controversy, the preceding topics are
concretely illustrated.Comment: To appear in Applied Mathematics and Computation, 29 pages,
references and further considerations adde
Galois differential algebras and categorical discretization of dynamical systems
A categorical theory for the discretization of a large class of dynamical
systems with variable coefficients is proposed. It is based on the existence of
covariant functors between the Rota category of Galois differential algebras
and suitable categories of abstract dynamical systems. The integrable maps
obtained share with their continuous counterparts a large class of solutions
and, in the linear case, the Picard-Vessiot group.Comment: 19 pages (examples added
Nonlinear Photonic Crystals: IV. Nonlinear Schrodinger Equation Regime
We study here the nonlinear Schrodinger Equation (NLS) as the first term in a
sequence of approximations for an electromagnetic (EM) wave propagating
according to the nonlinear Maxwell equations (NLM). The dielectric medium is
assumed to be periodic, with a cubic nonlinearity, and with its linear
background possessing inversion symmetric dispersion relations. The medium is
excited by a current producing an EM wave. The wave nonlinear
evolution is analyzed based on the modal decomposition and an expansion of the
exact solution to the NLM into an asymptotic series with respect to some three
small parameters , and . These parameters are
introduced through the excitation current to scale respectively
(i) its amplitude and consequently the magnitude of the nonlinearity; (ii) the
range of wavevectors involved in its modal composition, with
scaling its spatial extension; (iii) its frequency bandwidth, with scaling its time extension. We develop a consistent theory of
approximations of increasing accuracy for the NLM with its first term governed
by the NLS. We show that such NLS regime is the medium response to an almost
monochromatic excitation current . The developed approach not only
provides rigorous estimates of the approximation accuracy of the NLM with the
NLS in terms of powers of , and , but it also
produces new extended NLS (ENLS) equations providing better approximations.
Remarkably, quantitative estimates show that properly tailored ENLS can
significantly improve the approximation accuracy of the NLM compare with the
classical NLS
Separatrix splitting at a Hamiltonian bifurcation
We discuss the splitting of a separatrix in a generic unfolding of a
degenerate equilibrium in a Hamiltonian system with two degrees of freedom. We
assume that the unperturbed fixed point has two purely imaginary eigenvalues
and a double zero one. It is well known that an one-parametric unfolding of the
corresponding Hamiltonian can be described by an integrable normal form. The
normal form has a normally elliptic invariant manifold of dimension two. On
this manifold, the truncated normal form has a separatrix loop. This loop
shrinks to a point when the unfolding parameter vanishes. Unlike the normal
form, in the original system the stable and unstable trajectories of the
equilibrium do not coincide in general. The splitting of this loop is
exponentially small compared to the small parameter. This phenomenon implies
non-existence of single-round homoclinic orbits and divergence of series in the
normal form theory. We derive an asymptotic expression for the separatrix
splitting. We also discuss relations with behaviour of analytic continuation of
the system in a complex neighbourhood of the equilibrium
Source Galerkin Calculations in Scalar Field Theory
In this paper, we extend previous work on scalar theory using the
Source Galerkin method. This approach is based on finding solutions to
the lattice functional equations for field theories in the presence of an
external source . Using polynomial expansions for the generating functional
, we calculate propagators and mass-gaps for a number of systems. These
calculations are straightforward to perform and are executed rapidly compared
to Monte Carlo. The bulk of the computation involves a single matrix inversion.
The use of polynomial expansions illustrates in a clear and simple way the
ideas of the Source Galerkin method. But at the same time, this choice has
serious limitations. Even after exploiting symmetries, the size of calculations
become prohibitive except for small systems. The calculations in this paper
were made on a workstation of modest power using a fourth order polynomial
expansion for lattices of size ,, in , , and . In
addition, we present an alternative to the Galerkin procedure that results in
sparse matrices to invert.Comment: 31 pages, latex, figures separat
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