Formal Power Series Solutions Research Articles

A Note on the Convergence of Multivariate Formal Power Series Solutions of Meromorphic Pfaffian Systems

Here, we present some complements to theorems of R. Gerard and Y. Sibuya, on the convergence of multivariate formal power series solutions of nonlinear meromorphic Pfaffian systems. Their most known results concern completely integrable systems with non-degenerate linear parts, whereas we consider some cases of non-integrability and degeneracy.

Apparent singularities of D-finite systems

We generalize the notions of ordinary points and singularities from linear ordinary differential equations to D-finite systems. Ordinary points and apparent singularities of a D-finite system are characterized in terms of its formal power series solutions. We also show that apparent singularities can be removed like in the univariate case by adding suitable additional solutions to the system at hand. Several algorithms are presented for removing and detecting apparent singularities. In addition, an algorithm is given for computing formal power series solutions of a D-finite system at apparent singularities.

Gevrey Properties and Summability of Formal Power Series Solutions of Some Inhomogeneous Linear Cauchy-Goursat Problems

In this article, we investigate the Gevrey and summability properties of the formal power series solutions of some inhomogeneous linear Cauchy-Goursat problems with analytic coefficients in a neighborhood of $(0,0)\in \mathbb {C}^{2}$. In particular, we give necessary and sufficient conditions under which these solutions are convergent or are k-summable, for a convenient positive rational number k, in a given direction.

Solutions of quasianalytic equations

The article develops techniques for solving equations $$G(x,y)=0$$ , where $$G(x,y)=G(x_1,\ldots ,x_n,y)$$ is a function in a given quasianalytic class (for example, a quasianalytic Denjoy–Carleman class, or the class of $${\mathcal C}^\infty $$ functions definable in a polynomially-bounded o-minimal structure). We show that, if $$G(x,y)=0$$ has a formal power series solution $$y=H(x)$$ at some point a, then H is the Taylor expansion at a of a quasianalytic solution $$y=h(x)$$ , where h(x) is allowed to have a certain controlled loss of regularity, depending on G. Several important questions on quasianalytic functions, concerning division, factorization, Weierstrass preparation, etc., fall into the framework of this problem (or are closely related), and are also discussed.

Generalized Poincaré Condition and Convergence of Formal Solutions of Some Nonlinear Totally Characteristic Equations

This paper discusses a holomorphic nonlinear singular partial differential equation $(t \partial_t)^mu=F(t,x,\{(t \partial_t)^j \partial_x^{\alpha}u \}_{j+\alpha \leq m, j<m})$ that is of nonlinear totally characteristic type. The Newton Polygon at $x=0$ of the equation is defined, and by means of this polygon we define a generalized Poincare condition (GP) and a condition (R) that the equation has a regular singularity at $x=0$. Under these conditions, (GP) and (R), it is proved that every formal power series solution is convergent in a neighborhood of the origin.

Analytic and Summable Solutions of Inhomogeneous Moment Partial Differential Equations

We study the Cauchy problem for a general inhomogeneous linear moment partial differential equation of two complex variables with constant coefficients, where the inhomogeneity is given by the formal power series. We state sufficient conditions for the convergence, analytic continuation and summability of formal power series solutions in terms of properties of the inhomogeneity. We consider both the summability in one variable t (with coefficients belonging to some Banach space of Gevrey series with respect to the second variable z) and the summability in two variables (t,z).

Summable solutions of some partial differential equations and generalised integral means

We describe partial differential operators for which we can construct generalised integral means satisfying Pizzetti-type formulas. Using these formulas we give a new characterisation of summability of formal power series solutions to some multidimensional partial differential equations in terms of holomorphic properties of generalised integral means of the Cauchy data.

Fluid extraction across pumping and permeable walls in the viscous limit

In biological transport mechanisms such as insect respiration and renal filtration, fluid travels along a leaky channel allowing material exchange with systems exterior to the channel. The channels in these systems may undergo peristaltic pumping which is thought to enhance the material exchange. To date, little analytic work has been done to study the effect of pumping on material extraction across the channel walls. In this paper, we examine a fluid extraction model in which fluid flowing through a leaky channel is exchanged with fluid in a reservoir. The channel walls are allowed to contract and expand uniformly, simulating a pumping mechanism. In order to efficiently determine solutions of the model, we derive a formal power series solution for the Stokes equations in a finite channel with uniformly contracting/expanding permeable walls. This flow has been well studied in the case in which the normal velocity at the channel walls is proportional to the wall velocity. In contrast we do not assume flow that is proportional to the wall velocity, but flow that is driven by hydrostatic pressure, and we use Darcy’s law to close our system for normal wall velocity. We incorporate our flow solution into a model that tracks the material pressure exterior to the channel. We use this model to examine flux across the channel-reservoir barrier and demonstrate that pumping can either enhance or impede fluid extraction across channel walls. We find that associated with each set of physical flow and pumping parameters, there are optimal reservoir conditions that maximize the amount of material flowing from the channel into the reservoir.

A solution to Schröder's equation in several variables

Let ϕ be an analytic self-map of the n-ball, having 0 as the attracting fixed point and having full-rank near 0. We consider the generalized Schröder's equation, F∘ϕ=ϕ′(0)kF with k a positive integer and prove there is always a solution F with linearly independent component functions, but that such an F cannot have full rank except possibly when k=1. Furthermore, when k=1 (Schröder's equation), necessary and sufficient conditions on ϕ are given to ensure F has full rank near 0 without the added assumption of diagonalizability as needed in the 2003 Cowen/MacCluer paper. In response to Enoch's 2007 paper, it is proven that any formal power series solution indeed represents an analytic function on the whole unit ball. How exactly resonance can lead to an obstruction of a full rank solution is discussed as well as some consequences of having solutions to Schröder's equation.

Gevrey Order and Summability of Formal Series Solutions of some Classes of Inhomogeneous Linear Partial Differential Equations with Variable Coefficients

We investigate Gevrey order and summability properties of formal power series solutions of some classes of inhomogeneous linear partial differential equations with variable coefficients and analytic initial conditions. In particular, we give necessary and sufficient conditions under which these solutions are convergent or are k-summable, for a convenient k, in a given direction.

Q-analogue of summability of formal solutions of some linear q-difference-differential equations

Let \(q\gt 1\). The paper considers a linear \(q\)-difference-differential equation: it is a \(q\)-difference equation in the time variable \(t\), and a partial differential equation in the space variable \(z\). Under suitable conditions and by using \(q\)-Borel and \(q\)-Laplace transforms (introduced by J.-P. Ramis and C. Zhang), the authors show that if it has a formal power series solution \(\hat{X}(t,z)\) one can construct an actual holomorphic solution which admits \(\hat{X}(t,z)\) as a \(q\)-Gevrey asymptotic expansion of order \(1\).

Diagonal solutions to the 2-Toda hierarchy

Using a combinatorial description of the Bernstein operator and its action on Schur functions, we describe the formal power series solutions to a family of partial differential equations known as the 2-Toda hierarchy. We also characterize diagonal solutions and use this to prove that a special family of formal power series, called content-type series, are solutions to the 2-Toda hierarchy. As examples, we prove that various generating series for permutation factorization problems (including the double Hurwitz problem), correlators for the Schur measure on partitions and the formal character expansion of the HCIZ integral are all solutions to the 2-Toda hierarchy.

Building Meromorphic Solutions ofq-Difference Equations Using a Borel–Laplace Summation

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 put in the Birkhoff-Guenther normal form, we prove that the solutions we compute are exactly the same as the one constructed by Ramis, Sauloy and Zhang.

On Gevrey orders of formal power series solutions to the third and fifth Painlevé equations near infinity

The question under consideration is Gevrey summability of formal power series solutions to the third and fifth Painleve equations near infinity. We consider the fifth Painleve equation in two cases: when \(\alpha\beta\gamma\delta \neq 0\) and when \(\alpha\beta\gamma \neq 0\), \(\delta =0\) and the third Painleve equation when all the parameters of the equation are not equal to zero. In the paper we prove Gevrey summability of the formal solutions to the fifth Painleve equation and to the third Painleve equation, respectively.

The Borel summable solutions of heat operators on a real analytic manifold

We study heat type equations ∂tu=Δ˜u, where the operator Δ˜ is given by a sum of squares of commuting, real analytic vector fields acting on a real analytic manifold. We give necessary and sufficient conditions for convergence and Borel summability of formal power series solutions in terms of generalized integral means of the initial data. The results are also valid for affine perturbations of Δ˜.

Summability of formal solutions of linear partial differential equations with divergent initial data

We study the Cauchy problem for a general homogeneous linear partial differential equation in two complex variables with constant coefficients and with divergent initial data. We state necessary and sufficient conditions for the summability of formal power series solutions in terms of properties of divergent Cauchy data. We consider both the summability in one variable t (with coefficients belonging to some Banach space of Gevrey series with respect to the second variable z) and the summability in two variables (t,z). The results are presented in the general framework of moment-PDEs.

On q-asymptotics for linear q-difference-differential equations with Fuchsian and irregular singularities

We consider a Cauchy problem for some family of linear q-difference-differential equations with Fuchsian and irregular singularities, that admit a unique formal power series solution in two variables Xˆ(t,z) for given formal power series initial conditions. Under suitable conditions and by the application of certain q-Borel and Laplace transforms (introduced by J.-P. Ramis and C. Zhang), we are able to deal with the small divisors phenomenon caused by the Fuchsian singularity, and to construct actual holomorphic solutions of the Cauchy problem whose q-asymptotic expansion in t, uniformly for z in the compact sets of C, is Xˆ(t,z). The small divisorsʼ effect is an increase in the order of q-exponential growth and the appearance of a power of the factorial in the corresponding q-Gevrey bounds in the asymptotics.

Entire Solutions of f(kz) = kf(z)f′(z)

A colleague asked whether x, sin x and sinh x are essentially the only solutions of the non-linear differential equation f(2x) = 2f(x)f′ (x) on [0, +∞) with boundary condition f′(0) = 1. Here we study the formal power series solutions of the more general equation f(kz) = kf(z)f′(z), where k is a non-zero complex number, with f(0) = 0, and we show that a transcendental, formal power series solution exists for only a countable set of k, and that each such solution is an entire function.

Borel summability of a certain divergent formal power series solution for an initial value problem

In this paper we study the Borel summability of a certain divergent formal power series solution for an initial value problem. We show the Borel summability under the condition that an initial value function $\phi(x)$ is an entire function of exponential

On singularly perturbed q-difference-differential equations with irregular singularity

We study a q-analog of a singularly perturbed Cauchy problem with irregular singularity in the complex domain. We construct solutions of this problem that are holomorphic on open half-q-spirals. Using a version of a q-analog of the Malgrange---Sibuya theorem obtained by J.-P. Ramis, J. Sauloy, and C. Zhang, we show the existence of a formal power-series solution in the perturbation parameter which is the q-asymptotic expansion of these holomorphic solutions.