Small divisor problem for an analytic q-difference equation

Similar to the problem of linearization, the “small divisor problem” also arises in the discussion of invertible analytic solutions of a class of q-difference equations. In this paper we give the existence of such solutions under the Brjuno condition and prove that the equation may not have a nontrivial analytic solution when the Brjuno condition is violated. These results are applied to discussing a nonlinear iterative equation.     

Existence of Solutions for Nonlinear Fractional q-Difference Inclusions with Nonlocal Robin (Separated) Conditions

In this paper, we show the existence of solutions for nonlinear fractional q-difference inclusions involving convex as well as non-convex valued maps with nonlocal Robin (separated) conditions. Our results are new in the present configuration and are based on some standard principles for multivalued maps. A special case for q-difference inclusions in the given setting is also discussed. Some interesting observations are presented.     

First-order non-homogeneous q-difference equation for Stieltjes function characterizing q-orthogonal polynomials

In this paper, we give a characterization of some classical q-orthogonal polynomials in terms of a difference property of the associated Stieltjes function, i.e. this function solves a first-order non-homogeneous q-difference equation. The solutions of the aforementioned q-difference equation (given in terms of hypergeometric series) for some canonical cases, namely, q-Charlier, q-Kravchuk, q-Meixner and q-Hahn, are worked out.     

Some properties of meromorphic solutions of q-difference equations

We investigate the growth of transcendental meromorphic solutions of some complex q-difference equations and find lower bounds for Nevanlinna lower order for meromorphic solutions of such equations. We also obtain a q-difference version of Tumura-Clunie theorem.     

Existence of solutions of boundary value problems for singular fractional <Emphasis Type="Italic">q</Emphasis>-difference equations

In this paper, we study the existence and uniqueness of solutions for a class of singular three-point boundary value problems of fractional q-difference equations invovling fractional q-derivative of Riemann–Liouville type. Based on the generalization of Banach contraction principle, we obtain a sufficient condition for existence and uniqueness of solutions of the problem. By applying the Krasnoselskii’s fixed point theorem, we establish a sufficient condition for the existence of at least one solution of the problem. As applications, two examples are presented to illustrate our main results.     

Exact analytic solutions of Schrödinger linear and nonlinear equations

Exact analytic solutions of Schrödinger linear partial differential equations are obtained. Moreover, the cubic nonlinear Schrödinger equation is treated with the use of a well-known functional analytic method and the existence of convergent power series solutions is proved. From these solutions, under certain initial conditions, similar results as those presented in the literature are obtained.     

Asymptotic existence theorems for formal power series whose coefficients satisfy certain partial differential recursions

We study power series whose coefficients are holomorphic functions of another complex variable and a nonnegative real parameter s, and are given by a differential recursion equation. For positive integer s, series of this form naturally occur as formal solutions of some partial differential equations with constant coefficients, while for s=0 they satisfy certain perturbed linear ordinary differential equations. For arbitrary s?0, these series solve a differential-integral equation. Such power series, in general, are not multisummable. However, we shall prove existence of solutions of the same differential-integral equation that in sectors of, in general, maximal opening have the formal series as their asymptotic expansion. Furthermore, we shall indicate that the solutions so obtained can be related to one another in a fairly explicit manner, thus exhibiting a Stokes phenomenon.     

ON PROPERTIES OF q-DIFFERENCE EQUATIONS

In this article,we consider some type of q-difference equations,which have meromorphic solutions with Borel exceptional zeros and poles.We also give a precise result in the finite order case and some f...     

Kernel identities for van Diejen’s q-difference operators and transformation formulas for multiple basic hypergeometric series

The kernel function of Cauchy type for type BC is defined as a solution of linear q-difference equations. In this paper, we show that this kernel function intertwines the commuting family of van Diejen’s q-difference operators. This result gives rise to a transformation formula for certain multiple basic hypergeometric series of type BC. We also construct a new infinite family of commuting q-difference operators for which the Koornwinder polynomials are joint eigenfunctions.     

Transseries for a class of nonlinear difference equations *

Given a nonlinear analytic difference equation of level 1 with a formal power series solution ? ₀ we associate with it a stable manifold of solutions with asymptotic expansion ? ₀. This manifold can be represented by means of Borel summable series. All solutions with asymptotic expansion ? ₀ in some sector can be written as certain exponential series which are called transseries. Some of their properties are investigated: are resurgence properties and Stokes transition. Analogous problems for differential equations have been studied by Costin in [7]     

Legendre wavelet – quasilinearization technique for solving q-difference equations

Recently, various fixed point theorems have been used to prove the existence and uniqueness of the solutions for q-difference equations. In this paper, we obtain the existence and uniqueness theorems for a q-initial and a q-boundary value problem using the classical Newton’s method. Making use of the main theorems, a Legendre wavelet technique has been proposed to solve the q-difference equations numerically. The numerical simulation shows that the proposed scheme produces higher accuracy and is very straightforward to apply.     

A discrete and q asymptotic iteration method

ABSTRACT

We introduce a finite difference and q-difference analogues of the Asymptotic Iteration Method of Ciftci, Hall, and Saad. We give necessary, and sufficient condition for the existence of a polynomial solution to a general linear second-order difference or q-difference equation subject to a ‘terminating condition’, which is precisely defined. When a difference or q-difference equation has a polynomial solution, we show how to find the second solution.     

A Monomial-by-Monomial Method for Computing Regular Solutions of Systems of Pseudo-Linear Equations

This paper deals with the local analysis of systems of pseudo-linear equations. We define regular solutions and use this as a unifying theoretical framework for discussing the structure and existence of regular solutions of various systems of linear functional equations. We then give a generic and flexible algorithm for the computation of a basis of regular solutions. We have implemented this algorithm in the computer algebra system Maple in order to provide novel functionality for solving systems of linear differential, difference and q-difference equations given in various input formats.     

Développement asymptotique et sommabilité des solutions des équations linéaires aux q-différences

We describe q-analogs of asymptotic expansions and of multisummable series and we apply them to the local analytic classification of irregular linear q-difference equations, thus answering questions of G.D. Birkhoff. To cite this article: J.-P. Ramis et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Instabilities for supercritical Schrödinger equations in analytic manifolds

In this paper we consider supercritical nonlinear Schrödinger equations in an analytic Riemannian manifold (Md,g), where the metric g is analytic. Using an analytic WKB method, we are able to construct an Ansatz for the semiclassical equation for times independent of the small parameter. These approximate solutions will help to show two different types of instabilities. The first is in the energy space, and the second is an immediate loss of regularity in higher Sobolev norms.     

Bispectral quantum Knizhnik�CZamolodchikov equations for arbitrary root systems

The bispectral quantum Knizhnik–Zamolodchikov (BqKZ) equation corresponding to the affine Hecke algebra H of type A _N-1 is a consistent system of q-difference equations which in some sense contains two families of Cherednik’s quantum affine Knizhnik–Zamolodchikov equations for meromorphic functions with values in principal series representations of H. In this paper, we extend this construction of BqKZ to the case where H is the affine Hecke algebra associated with an arbitrary irreducible reduced root system. We construct explicit solutions of BqKZ and describe its correspondence to a bispectral problem involving Macdonald’s q-difference operators.     

Galois Theory of Difference Equations with Periodic Parameters

We develop a Galois theory for systems of linear difference equations with periodic parameters, for which we also introduce linear difference algebraic groups. We apply this to constructively test if solutions of linear q-difference equations, with q ∈ ?* and q not a root of unity, satisfy any polynomial ζ-difference equations with ζ ^t  = 1, t ≥ 1.     

Stability of the analytic and numerical solutions for impulsive differential equations

This paper deals with the stability analysis of the analytic and numerical solutions of impulsive differential equations. In particular, the linear equation with variable coefficients and the nonlinear equation are considered. The stability conditions of the analytic solutions of these impulsive differential equations and the numerical solutions of the θ-methods are obtained. Finally, some numerical experiments are given.     

Quicksilver Solutions of a q‐Difference First Painlevé Equation

In this paper, we present new, unstable solutions, which we call quicksilver solutions, of a q‐difference Painlevé equation in the limit as the independent variable approaches infinity. The specific equation we consider in this paper is a discrete version of the first Painlevé equation (qP_I), whose phase space (space of initial values) is a rational surface of type . We describe four families of almost stationary behaviors, but focus on the most complicated case, which is the vanishing solution. We derive this solution's formal power series expansion, describe the growth of its coefficients, and show that, while the series is divergent, there exist true analytic solutions asymptotic to such a series in a certain q‐domain. The method, while demonstrated for qP_I, is also applicable to other q‐difference Painlevé equations.     

La filtration canonique par les pentes d'un module aux q-différencesThe canonical slope filtration of a q-difference module

According to Adams' lemma, to the first slope of the Newton polygon of a q-difference equation is associated a full complement of convergent solutions. We draw from this the existence of a canonical filtration by the slopes of q-difference modules, such that the associated graded module functor is faithful, exact and tensor compatible. To cite this article: J. Sauloy, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 11–14