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After introducing q-analogues of the Borel and Laplace transformations, we prove that to every formal power series solution of a linear q-difference equation with rational coefficients, we may apply several q-Borel and Laplace…
Algorithms for computing rational generating functions of solutions of one-dimensional difference equations are well-known and easy to implement. We propose an algorithm for computing rational generating functions of solutions of…
In the first part of planned series of papers the formal general solutions to selection of 80 examples of different types of second order nonlinear PDEs in two independent variables with constant parameters are given. The main goal here is…
We consider a three dimensional complex polynomial, or rational, vector field (equivalently, a two-form in three variables) which admits a Liouvillian first integral. We prove that there exists a first integral whose differential is the…
The paper develops the method for construction of the families of particular solutions to the nonlinear Partial Differential Equations (PDE) without relation to the complete integrability. Method is based on the specific link between…
We develop some calculation schemes to determine dynamics of a wide class of integrable quantum-optical models using their symmetry adapted reformulation in terms of polynomial Lie algebras $su_{pd}(2)$. These schemes, based on "diagonal"…
In this paper, we propose a procedure for constructing an infinite number of families of solutions of given linear differential equations with partial derivatives with constant coefficients. We use monogenic functions that are defined on…
When a system of first order linear ordinary differential equations has eigenvalues of large magnitude, its solutions exhibit complicated behaviour, such as high-frequency oscillations, rapid growth or rapid decay. The cost of representing…
We consider regular polynomial interpolation algorithms on recursively defined sets of interpolation points which approximate global solutions of arbitrary well-posed systems of linear partial differential equations. Convergence of the…
In this short note we discuss ordinary differential equations which linearize upon one (or more) differentiations. Although the subject is fairly elementary, equations of this type arise naturally in the context of integrable systems.
An infinite family of irreducible homogeneous free divisors in $K[x, y, z]$ is constructed. Indeed, we identify sets of monomials $X$ such that the general polynomial supported on $X$ is a free divisor.
In this paper we propose a geometric approach to study Painlev\'e equations appearing as constrained systems of three first-order ordinary differential equations. We illustrate this approach on a system of three first-order differential…
We determine a considerable class of nonlinear partial differential equation systems which have global regular solutions. Uniqueness is not a direct general consequence of this method. The scheme can be applied to the incompressible Navier…
In this paper, we present a modular strategy which describes key properties of the absolute primary decomposition of an equidimensional polynomial ideal defined by polynomials with rational coefficients. The algorithm we design is based on…
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…
The purpose of this paper is to study a class of ill-posed differential equations. In some settings, these differential equations exhibit uniqueness but not existence, while in others they exhibit existence but not uniqueness. An example of…
In this paper, we provide a complete Plancherel-Rotach asymptotic analysis of polynomials that satisfy a second-order difference equation with linear coefficients. According to the signs of the parameters, we classify the difference…
Cubic and quartic non-autonomous differential equations with continuous piecewise linear coefficients are considered. The main concern is to find the maximum possible multiplicity of periodic solutions. For many classes, we show that the…
A systematic study of the discrete second order projective system is presented, complemented by the integrability analysis of the associated multilinear mapping. Moreover, we show how we can obtain third order integrable equations as the…
This paper presents the non-linear generalization of a previous work on matrix differential models. It focusses on the construction of approximate solutions of first-order matrix differential equations Y'(x)=f(x,Y(x)) using matrix-cubic…