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We revisit the third fundamental theorem of Lie (Lie III) for finite dimensional Lie algebras in the context of infinite dimensional matrices.
We will study solvability of nonlinear second-order elliptic system of partial differential equations with nonlinear boundary conditions. We study the generalized Steklov Robin eigensystem (with possibly matrices weights) in which the…
The present paper is devoted to a new criterion for disconjugacy of a second order linear differential equation. Unlike most of the classical sufficient conditions for disconjugacy, our criterion does not involve assumptions on the…
The Calabi-Yau differential equations of degree 2 and 3 are listed. The idea is to bring some order into the "big table" into the "big table" (math. AG/0507430). The author has benefitted from an unpublished result by Yifan Yang, which…
The aim of this paper is to provide an effective framework for analysing bifurcations of equilibria in nonlinearly periodically forced delay differential equations. First, we establish the existence of a periodic smooth finite-dimensional…
We describe the sequences {x_n}_n given by the non-autonomous second order Lyness difference equations x_{n+2}=(a_n+x_{n+1})/x_n, where {a_n}_n is either a 2-periodic or a 3-periodic sequence of positive values and the initial conditions…
This paper is part of a series of papers in which the asymptotic theory and appropriate symbolic computer code are developed to compute the asymptotic expansion of the solution of an n-th order ordinary differential equation. The paper…
On the base of Lie algebraic and differential geometry methods, a wide class of multidimensional nonlinear integrable systems is obtained, and the integration scheme for such equations is proposed.
In general, high order splitting methods suffer from an order reduction phenomena when applied to the time integration of partial differential equations with non-periodic boundary conditions. In the last decade, there were introduced…
In this paper we study about the existence of solutions of certain kind of non-linear differential and differential-difference equations. We give partial answer to a problem which was asked by chen et al. in [13].
In this article we present first an algorithm for calculating the determining equations associated with so-called ``nonclassical method'' of symmetry reductions (a la Bluman and Cole) for systems of partial differentail equations. This…
This paper provides two results that are useful in the study of the existence and the stability properties of a periodic solution for a given dynamical system. The first result deals with scalar time-periodic systems and establishes the…
We study the large time behaviour of the solution of a linear dispersive PDEs posed on a finite interval, when the prescribed boundary conditions are time periodic. We use the approach pioneered in Fokas & Lenells 2012 for nonlinear…
This survey article is the outgrowth of two talks given at the Journ\'ees X-UPS "P\'eriodes et transcendance" at \'Ecole polytechnique. Periods are complex numbers whose real and imaginary parts can be written as integrals of rational…
The detailed construction of the general solution of a second order non-homogenous linear operatordifference equation is presented. The wide applicability of such an equation as well as the usefulness of its resolutive formula is shown by…
In a major advance and simplification of this field, we show that A Local Resolution of the Problem of Time - also viewable as A Local Theory of Background Independence - can at the classical level be described solely by of Lie's…
This paper presents the first in a series of results that allow us to develop a theory providing finer control over the complexity of normalisation, and in particular of cut elimination. By considering atoms as self-dual non-commutative…
Lie group theory states that knowledge of a $m$-parameters solvable group of symmetries of a system of ordinary differential equations allows to reduce by $m$ the number of equations. We apply this principle by finding some \emph{affine…
The problem of linearization by point transformations is solved for equations in the generalized Riccati and Abel chain of order not exceeding the fourth. It is shown in particular that nonlinear third order and fourth order equations from…
We describe a solving semi-decision method based on examination of the rational structures of the generalized integrating factors of first-order ODEs. We propose a conjecture that for some family of equations of the type…