457 research outputs found
Differential systems with Fuchsian linear part: correction and linearization, normal forms and multiple orthogonal polynomials
Differential systems with a Fuchsian linear part are studied in regions
including all the singularities in the complex plane of these equations. Such
systems are not necessarily analytically equivalent to their linear part (they
are not linearizable) and obstructions are found as a unique nonlinear
correction after which the system becomes formally linearizable.
More generally, normal forms are found.
The corrections and the normal forms are found constructively. Expansions in
multiple orthogonal polynomials and their generalization to matrix-valued
polynomials are instrumental to these constructions.Comment: 24 page
Low Complexity Algorithms for Linear Recurrences
We consider two kinds of problems: the computation of polynomial and rational
solutions of linear recurrences with coefficients that are polynomials with
integer coefficients; indefinite and definite summation of sequences that are
hypergeometric over the rational numbers. The algorithms for these tasks all
involve as an intermediate quantity an integer (dispersion or root of an
indicial polynomial) that is potentially exponential in the bit size of their
input. Previous algorithms have a bit complexity that is at least quadratic in
. We revisit them and propose variants that exploit the structure of
solutions and avoid expanding polynomials of degree . We give two
algorithms: a probabilistic one that detects the existence or absence of
nonzero polynomial and rational solutions in bit
operations; a deterministic one that computes a compact representation of the
solution in bit operations. Similar speed-ups are obtained in
indefinite and definite hypergeometric summation. We describe the results of an
implementation.Comment: This is the author's version of the work. It is posted here by
permission of ACM for your personal use. Not for redistributio
Quasi-linear Stokes phenomenon for the second Painlev\'e transcendent
Using the Riemann-Hilbert approach, we study the quasi-linear Stokes
phenomenon for the second Painlev\'e equation . The
precise description of the exponentially small jump in the dominant solution
approaching as is given. For the asymptotic power
expansion of the dominant solution, the coefficient asymptotics is found.Comment: 19 pages, LaTe
Exactly solvable variable parametric Burgers type models
Exactly solvable variable parametric Burgers type equations in one-dimension
are introduced, and two different approaches for solving the corresponding
initial value problems are given. The first one is using the relationship
between the variable parametric models and their standard counterparts. The
second approach is a direct linearization of the variable parametric Burgers
model to a variable parametric parabolic model via a generalized Cole-Hopf
transform. Eventually, the problem of finding analytic and exact solutions of
the variable parametric models reduces to that of solving a corresponding
second order linear ODE with time dependent coefficients. This makes our
results applicable to a wide class of exactly solvable Burgers type equations
related with the classical Sturm-Liouville problems for the orthogonal
polynomials
A flexible error estimate for the application of centre manifold theory
In applications of centre manifold theory we need more flexible error estimates than that provided by, for example, the Approximation Theorem 3 by Carr [4, 6]. Here we extend the theory to cover the case where the order of approximation in parameters and that in dynamical variables may be completely different. This allows, for example, the effective evaluation of low-dimensional dynamical models at finite parameter values
- …