In this paper we study analytic (linear or) nonlinear systems of ordinary differential equations, at an irregular singularity of rank one, under nonresonance conditions. It is shown that the formal asymptotic exponential series solutions (transseries solutions: countable linear combinations of formal power series multiplied by small exponentials) are Borel summable in a generalized sense along any direction in which the exponentials decay. Conversely, any solution that decreases along some direction is the Borel sum of a transseries. The summation procedure introduced is an extension of Borel summation which is linear, multiplicative, commutes with differentiation and complex conjugation. The summation algorithm uses the formal solutions alone (and not the differential equation that they solve). Along singular (Stokes) directions, the functions reconstructed by summation are shown to be given by Laplace integrals along special paths, a subset of Ecalle's median paths. The one-to-one correspondence established between actual solutions and generalized Borel sums of transseries is constant between Stokes lines and changes if a Stokes line is crossed (local Stokes phenomenon). We analyze the connection between local and classical Stokes phenomena.
Borel summation techniques are developed to obtain exact invariants from formal adiabatic invariants (given as divergent series in a small parameter) for a class of differential equations, under assumptions of analyticity of the coefficients; the method relies on the study of associated partial differential equations in the complex plane. The type and location of the singularities of these associated functions, important in determining exponentially small corrections to formal invariants are also briefly discussed.
For analytic nonlinear systems of ordinary differential equations, under some non-degeneracy and integrability conditions we prove that the formal exponential series solutions (trans-series) at an irregular singularity of rank one are Borel summable (in a sense similar to that of Ecalle). The functions obtained by re-summation of the trans-series are precisely the solutions of the differential equation that decay in a specified sector in the complex plane. We find the dependence of the correspondence between the solutions of the differential equation and trans-series as the ray in the complex …
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 transformations of convenient orders and convenient direction in order to construct a solution of the same equation that is meromorphic on $\mathbb{C}^{*}$. We use this theorem to construct explicitly an invertible matrix solution of a linear q-difference system with rational coefficients, of which entries are meromorphic on $\mathbb{C}^{*}$. Moreover, when the system is…
We consider a class of second order ordinary differential equations describing one-dimensional systems with a quasi-periodic analytic forcing term and in the presence of damping. As a physical application one can think of a resistor-inductor-varactor circuit with a periodic (or quasi-periodic) forcing function, even if the range of applicability of the theory is much wider. In the limit of large damping we look for quasi-periodic solutions which have the same frequency vector of the forcing term, and we study their analyticity properties in the inverse of the damping coefficient. We find that …
Integro-differential methods, currently exploited in calculus, provide an inexhaustible source of tools to be applied to a wide class of problems, involving the theory of special functions and other subjects. The use of integral transforms of the Borel type and the associated formalism is shown to be a very effective mean, constituting a solid bridge between umbral and operational methods. We merge these different points of view to obtain new and efficient analytical techniques for the derivation of integrals of special functions and the summation of associated generating functions as well.
This paper is divided in two parts. In the first part we consider irregular singular analytic q-difference equations, with q\in ]0,1[, and we show how the Borel sum of a divergent solution of a differential equation can be uniformly approximated on a convenient sector by a meromorphic solution of such a q-difference equation. In the second part, we work under the assumption q\in ]1,+\infty[. In this case, at least four different q-Borel sums of a divergent solution of an irregular singular analytic q-difference equations are spread in the literature: under convenient assumptions we clarify the…
We show connection formulae of local solutions of the Ramanujan equation between the origin and the infinity. These solutions are given by the Ramanujan function, the $q$-Airy function and the divergent basic hypergeometric series ${}_2φ_0(0,0;-;q,x)$. We use two different $q$-Borel-Laplace resummation methods to obtain our connection formulae. We also introduce the $q$-Borel-Laplace transformation of level $r-1$, which are higher order extension of these transformations. These methods are useful to obtain an asymptotic formula of a divergent series ${}_rφ_0(0,0,\dots ,0;-;q,x)$.
In this paper, we consider a q-analogue of the Borel-Laplace summation where q>1 is a real parameter. In particular, we show that the Borel-Laplace summation of a divergent power series solution of a linear differential equation can be uniformly approximated on a convenient sector, by a meromorphic solution of a corresponding family of linear q-difference equations. We perform the computations for the basic hypergeometric series. Following J. Sauloy, we prove how a fundamental set of solutions of a linear differential equation can be uniformly approximated on a convenient domain by a fundament…
In a companion paper, Grimshaw (Asymptotic Methods in Fluid Mechanics, 2010, pp. 71-120) has demonstrated how techniques of Borel summation can be used to elucidate the exponentially small terms that lie hidden beyond all orders of a divergent asymptotic expansion. Here, we provide an alternative derivation of the generalized solitary waves of the fifth-order Korteweg-de Vries equation. We will first optimally truncate the asymptotic series, and then smooth the Stokes line. Our method provides an explicit view of the switching-on mechanism, and thus increased understanding of the Stokes Phenom…
In the sixth chapter of his notebooks Ramanujan introduced a method of summing divergent series which assigns to the series the value of the associated Euler-MacLaurin constant that arises by applying the Euler-MacLaurin summation formula to the partial sums of the series. This method is now called the Ramanujan summation process. In this paper we calculate the Ramanujan sum of the exponential generating functions $\sum_{n\geq 1}\log n e^{nz}$ and $\sum_{n\geq 1}H_n^{(j)} e^{-nz}$ where $H_n^{(j)}=\sum_{m=1}^n \frac{1}{m^j}$. We find a surprising relation between the two sums when $j=1$ from w…
We compute all Voros coefficients of the third Painleve equation of the type D6 (in the sense of [OKSO]) and discuss the parametric Stokes phenomena occurring to formal transseries solutions. We derive connection formulas for parametric Stokes phenomena under an assumption for Borel summability of transseries solutions. Furthermore, we also compute the Voros coefficient of the degenerate third Painleve equation of the type D7 in Appendix D.
We prove that for divergent series solutions of nonlinear (or linear) differential systems near a generic irregular singularity, the common prescription of summation to the least term is, if properly interpreted, meaningful and correct, and we extend this method to transseries solutions. In every direction in the complex plane at the singularity (Stokes directions {\em not} excepted) there exists a nonempty set of solutions whose difference from the ``optimally'' (i.e., near the least term) truncated asymptotic series is of the same (exponentially small) order of magnitude as the least term of…
In a wide range of quantum theoretical settings -- from quantum mechanics to quantum field theory, from gauge theory to string theory -- singularities in the complex Borel plane, usually associated to instantons or renormalons, render perturbation theory ill-defined as they give rise to nonperturbative ambiguities. These ambiguities are associated to choices of an integration contour in the resummation of perturbation theory, along (singular) Stokes directions in the complex Borel plane (rendering perturbative expansions non-Borel summable along any Stokes line). More recently, it has been sho…
Let $\mathcal M$ be a holonomic algebraic $\mathcal D$-module on the affine line, regular everywhere including at infinity. Malgrange gave a complete description of the Fourier-Laplace transform $\widehat{\mathcal M}$, including its Stokes multipliers at infinity, in terms of the quiver of $\mathcal M$. Let $F$ be the perverse sheaf of holomorphic solutions to $\mathcal M$. By the irregular Riemann-Hilbert correspondence, $\widehat{\mathcal M}$ is determined by the enhanced Fourier-Sato transform $F^\curlywedge$ of $F$. Our aim here is to recover Malgrange's result in a purely topological way,…
