`Abstract and Applied AnalysisVolume 2012 (2012), Article ID 930385, 86 pageshttp://dx.doi.org/10.1155/2012/930385`
Research Article

On the Parametric Stokes Phenomenon for Solutions of Singularly Perturbed Linear Partial Differential Equations

UFR de Mathématiques, Université de Lille 1, 59655 Villeneuve d'Ascq Cedex, France

Received 27 March 2012; Accepted 8 July 2012

Copyright © 2012 Stéphane Malek. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study a family of singularly perturbed linear partial differential equations with irregular type in the complex domain. In a previous work, Malek (2012), we have given sufficient conditions under which the Borel transform of a formal solution to the above mentioned equation with respect to the perturbation parameter converges near the origin in and can be extended on a finite number of unbounded sectors with small opening and bisecting directions, say , for some integer . The proof rests on the construction of neighboring sectorial holomorphic solutions to the first mentioned equation whose differences have exponentially small bounds in the perturbation parameter (Stokes phenomenon) for which the classical Ramis-Sibuya theorem can be applied. In this paper, we introduce new conditions for the Borel transform to be analytically continued in the larger sectors , where it develops isolated singularities of logarithmic type lying on some half lattice. In the proof, we use a criterion of analytic continuation of the Borel transform described by Fruchard and Schäfke (2011) and is based on a more accurate description of the Stokes phenomenon for the sectorial solutions mentioned above.

1. Introduction

We consider a family of singularly perturbed linear partial differential equations of the form for given initial conditions where is a complex perturbation parameter, is some positive integer, is some positive integer larger than 2, and is a finite subset of with the property that there exists an integer with for all , and the coefficients belong to where denotes the space of holomorphic functions in near the origin in . In this work, we make the assumption that the coefficients of (1.1) factorize in the form where belong to . The initial data are assumed to be holomorphic functions on a product of two sectors , where is a fixed bounded sector centered at 0 and , , are sectors with opening larger than centered at the origin whose union form a covering of , where is some neighborhood of 0. For all , this family belongs to a class of partial differential equations which have a so-called irregular singularity at (in the sense of [1]).

In the previous work [2], we have given sufficient conditions on the initial data , for the existence of a formal series solution of (1.1), with holomorphic coefficients on for some disc , with , such that, for all , the solution of the problem (1.1), (1.2) defines a holomorphic function on which is the 1-sum of on . In other words, for all fixed , the Borel transform of with respect to defined as is holomorphic on some disc and can be analytically continued (with exponential growth) to sectors , centered at 0, with infinite radius and with the bisecting direction of the sector . But in general, due to the fact that the functions do not coincide on the intersections (known as the Stokes phenomenon), the Borel transform cannot be analytically extended to the whole sectors for all , where by convention , , and .

In this work, we address the question of the possibility of analytic continuation, location of singularities, and behaviour near these singularities of the Borel transform within the sector . More precisely, our goal is to give stronger conditions on the initial data under which the Borel transform can be analytically continued to the full-punctured sector except a half lattice of points , , depending on and some well-chosen complex number and moreover develop logarithmic singularities at (Theorem 5.8).

In a recent paper of Fruchard and Schäfke, see [3], an analogous study has been performed for formal WKB solutions to the singularly perturbed Schrödinger equation where is a formal series with holomorphic coefficients on some domain avoiding and . The authors show that the Borel transform of with respect to converges near the origin and can be analytically continued along any path avoiding some lattices of points depending on . We also mention that formal parametric Stokes phenomenon for 1-dimensional stationary linear Schrödinger equation , where is a polynomial, has been investigated by several other authors using WKB analysis, see [46]. In a more general framework, analytic continuation properties related with the Stokes phenomenon have been studied by several authors in different contexts. For nonlinear systems of ODEs with irregular singularity at of the form and for nonlinear systems of difference equations , under nonresonance conditions, we refer to [7, 8]. For linearizations procedures for holomorphic germs of in the resonant case, we make mention to [9, 10]. For analytic conjugation of vector fields in to normal forms, we indicate [11, 12]. For Hamiltonian nonlinear first-order partial differential equations, we notice [13].

In the proof of our main result, we will use a criterion for the analytic continuation of the Borel transform described by Fruchard and Schäfke in [3] (Theorem (FS) in Theorem 5.8). Following this criterion, in order to prove the analytic continuation of the Borel transform , say, on the sector , for any fixed , we need to have a complete description of the Stokes relation between the solutions and of the form for all , for some integer , where is a set of aligned complex numbers such that with (for some ) and , , are the -sums of some formal series on . If the relation (1.6) holds, then can be analytically continued along any path in the punctured sector and has logarithmic growth as tends to in a sector. Actually, under suitable conditions on the initial data , we have shown that such a relation holds for , for some well-chosen and for all , see (5.145) in Theorem 5.8. In order to establish such a Stokes relation (1.6), we proceed in several steps.

In the first step, following the same strategy as in [2], using the linear map , we transform the problem (1.1) into an auxiliary regularly perturbed singular linear partial differential equation which has an irregular singularity at and whose coefficients have poles with respect to at the origin, see (4.9). Then, for , we construct a formal transseries expansion of the form solution of the problem (4.9), (4.10), where each is a formal series in with coefficients , which are holomorphic on a punctured polydisc . We show that the Borel transform of each with respect to , defined by , satisfies an integrodifferential Cauchy problem with rational coefficients in , holomorphic with respect to near the origin and meromorphic in with a pole at zero, see (4.20), (4.21). For well-chosen and suitable initial data, we show that each defines a holomorphic function near the origin with respect to and on a punctured disc with respect to and can be analytically continued to functions defined on the products , where , are suitable open sectors with small opening and infinite radius. Moreover, the functions have exponential growth rate with respect to , namely, there exist such that for all in their domain of definition and all (Proposition 4.12). In order to get these estimates, we use the Banach spaces depending on two parameters and with norms of functions bounded by for some bounded sequence already introduced in [2]. If one expands the functions with respect to , we show that the generating function can be majorized by a series which satisfies a Cauchy problem of Kowalevski type (4.47), (4.48) and is therefore convergent near the origin in .

We construct a sequence of actual functions , , as Laplace transform of the functions with respect to along a halfline . We show that the functions are holomorphic functions on the domain and that the functions are exponentially flat as tends to 0 on as -valued functions. In the proof, we use, as in [2], a deformation of the integration's path in and the estimates (1.8). Using the Ramis-Sibuya theorem (Theorem (RS) in Proposition 4.15), we deduce that each is the -sum of a formal series on , for (Proposition 4.15). We notice that the functions actually coincide with the functions mentioned above solving the problem (1.1), (1.2). We deduce that, for a suitable choice of , the function solves (1.1) on the domain .

In the second part of the proof, we establish the connection formula which is exactly the Stokes relation (1.6) on (Proposition 5.2). The strategy we follow consists in expressing both functions and as Laplace transforms of objects that are no longer functions in general but distributions supported on which are called staircase distributions in the terminology of [8]. We stress the fact that such representations of transseries expansions as generalized Laplace transforms were introduced for the first time by Costin in the paper [8]. Notice that similar arguments have been used in the work [14] to study the Stokes phenomenon for sectorial holomorphic solutions to linear integro-differential equations with irregular singularity.

In Lemma 5.5, we show that can be written as a generalized Laplace transform in the direction of a staircase distribution , which is a convergent series in on with coefficients in some Banach spaces of staircase distributions on depending on the parameters and (see Definition 2.3). We observe that the distribution solves moreover an integro-differential Cauchy problem with rational coefficients in , holomorphic with respect to near the origin and meromorphic with respect to at zero, see (5.80), (5.81). The idea of proof consists in showing that each function can be expressed as a Laplace transform in a sequence of directions tending to of a sequence of staircase distributions (which are actually convergent series in with coefficients that are functions in on with exponential growth). Moreover, each distribution solves an integro-differential Cauchy problem (5.37), (5.38), whose coefficients tend to the coefficients of an integro-differential equation (5.39), (5.40), as tends to , having a unique staircase distribution solution . Under the hypothesis that the initial data (5.38) converge to (5.40) as , we show that the sequence converges to in the Banach space with precise norm estimates with respect to and (Lemma 5.3). In order to show this convergence, we use a majorazing series method together with a version of the classical Cauchy-Kowalevski theorem (Proposition 2.22) in some spaces of analytic functions near the origin in with dependence on initial conditions and coefficients applied to the auxiliary problem (5.66), (5.68). Using a continuity property of the Laplace transform (3.5), we show that each function can be actually expressed as the Laplace transform of in the direction and finally that itself is the Laplace transform of some staircase distribution solving (5.80), (5.81).

On the other hand, in Lemma 5.7, under suitable conditions on , , we can also write as a generalized Laplace transform in the direction of the staircase distribution mentioned above solving (5.80), (5.81). Therefore, the equality holds on . The method of proof consists again in showing that can be written as Laplace transform in a sequence of directions tending to of a sequence of staircase distributions (which are actually convergent series in with coefficients that are functions in on with exponential growth). Moreover, each distribution solves an integro-differential Cauchy problem (5.98), (5.99), whose coefficients tend to the coefficients of the integro-differential equation (5.80). Under the assumption that the initial data (5.99) converge to the initial data (5.81), we show that the sequence converges to the solution of (5.80), (5.81) (i.e., ) in the Banach space , as , see Lemma 5.6. This convergence result is obtained again by using a majorazing series technique which reduces the problem to the study of some linear differential equation (5.106), (5.109), whose coefficients and initial data tend to zero as . Finally, by continuity of the Laplace transform, can be written as the Laplace transform of in direction .

After Theorem 5.8, we give an application to the construction of solutions to some specific singular linear partial differential equations in having logarithmic singularities at the points , for . We show that under the hypothesis that the coefficients are polynomials in , the Borel transform turns out to solve the linear partial differential equation (5.149). We would like to mention that there exists a huge literature on the study of complex singularities and analytic continuation of solutions to linear partial differential equations starting from the fundamental contributions of Leray in [15]. Several authors have considered Cauchy problems , where is a differential operator of some order , for initial data , . Under specific hypotheses on the symbol , precise descriptions of the solutions of these problems are given near the singular locus of the initial data . For meromorphic initial data, we may refer to [1618] and for more general ramified multivalued initial data, we may cite [1923].

The layout of this work is as follows.

In Section 2, we introduce Banach spaces of formal series whose coefficients belong to spaces of staircase distributions and we study continuity properties for the actions of multiplication by functions and integro-differential operators on these spaces. In this section, we also exhibit a Cauchy Kowalevski theorem for linear partial differential problems in some space of analytic functions near the origin in with dependence of their solutions on the coefficients and initial data which will be useful to show the connection formula (5.28) stated in Section 5.

In Section 3, we recall the definition of a Laplace transform of a staircase distribution as introduced in [8] and we give useful commutation formulas with respect to multiplication by polynomials, exponential functions, and derivation.

In Section 4, we construct formal and analytic transseries solutions to the singularly perturbed partial differential equation with irregular singularity (1.1).

In Section 5, we establish the crucial connection formula relying on the analytic transseries solution and the solution of (1.1). Finally, we state the main result of the paper which asserts that the Borel transform in the perturbation parameter of the formal solution of (1.1) can be analytically continued along any path in the punctured sector and has logarithmic growth as tends to in a sector for all .

2. Banach Spaces of Formal Series with Coefficients in Spaces of Staircase Distributions: A Cauchy Problem in Spaces of Analytic Functions

2.1. Weighted Banach Spaces of Distributions

We define to be the space of complex valued -functions with compact support in , where is the set of the positive real numbers . We also denote by the space of distributions on . For , we write the -derivative of in the sense of distribution, for , with the convention .

Definition 2.1. A distribution is called staircase if can be written in the form for unique integrable functions such that the support of is in for all .

Remark 2.2. Given a compact set , a general distribution can always be written as a -derivative of a continuous function on restricted to the test functions with support in , where depends on , see [24].

Definition 2.3. Let be a real number, an integer and let for all integers . Let be an open sector centered at 0 and let . We denote by the vector space of all locally integrable functions such that is finite. We denote by the vector space of staircase distributions such that is finite.

Remark 2.4. Let such that . If , then for all and we have that is a decreasing sequence on . Likewise, if , then for all and we have that is a decreasing sequence on .

Let be the Heaviside one-step function defined by , if and , if . Let the operator defined on distributions by . For a subset , we denote by the function which is equal to 1 on and 0 elsewhere.

The proofs of the following Lemmas 2.5 and 2.6, Propositions 2.7, 2.8, 2.9, and Corollary 2.10 are given in the appendix of [25], see also [8].

Lemma 2.5. Let and , where and . Then is a staircase distribution and the decomposition of has the following terms , and for , where and .

Lemma 2.6. Let be as in Lemma 2.5 and such that . Then, one has if and for ,

Proposition 2.7. Let and such that . Then belongs to and the decomposition (2.1) of has the following terms with and , for . Moreover, there exists a universal constant such that .

Proposition 2.8. The set of -functions with compact support in is dense in for all , and .

Proposition 2.9. Let such that . For all , we have . Moreover, there exists a universal constant such that for all .

In this paper, for all integers , we will denote the convolution for all where stands for the convolution product of with itself times for and with the convention that . From Propositions 2.7 and 2.9, we deduce the following.

Corollary 2.10. Let be such that and let be an integer. For all , one has . Moreover, there exists a universal constant such that for all .

In the next proposition, we study norm estimates for the multiplication by bounded analytic functions.

Proposition 2.11. Let and such that and let be a -function on such that there exist constants and such that for all . Then, for all , we have . Moreover, there exists a constant (depending on ) such that for all .

Proof. The proof can be found in [14] and is inspired from [25, Lemma ], but for the sake of completeness, we sketch it below. Without loss of generality, we can assume that has the following form , where with , for . Put . Then, .
From the Leibniz formula, we get the identity On the other hand, one can rewrite , where and denotes the th iteration of .
Due to Lemma 2.5, can be written , with and .
Therefore, we get the following identity
First of all, we have where is given in (2.9). From Lemma 2.6, we have the estimates for and all . Now, we give estimates for .
Using the Taylor formula with integral remainder and the hypothesis (2.9), we get Hence, from the Fubini theorem and the identity we deduce and hence From (2.14) and (2.18), we obtain for , all , where Now, we need to estimate . Due to the Stirling formula, as tends to infinity, there exists a universal constant such that for all . Using the hypothesis , we have Using again the Stirling formula, we get a constant (depending on ) such that for all . Moreover, Hence, for all . Finally, we obtain a constant depending only on such that for all . From (2.14) and (2.18), we have where
Now, we show that , , is a bounded sequence. Again, by the Stirling formula, we get a universal constant such that From the assumption (2.8) and the estimates that for all two real numbers, we have we get a constant such that for all .
Finally, from the equality (2.12) and estimates (2.13), (2.19), (2.26), (2.27) and (2.31), we get a constant depending only on such that for all . It remains to consider the case .
When , let , with . By definition, we can write

In the next proposition, we study norm estimates for the multiplication by polynomials.

Proposition 2.12. Let and such that and let be integers. Then, for all , one has . Moreover, there exists a constant (depending on ,) such that for all .

Proof. The proof is an adaptation of Proposition 2.11. Without loss of generality, we can assume that has the following form where with , for . We also put . Let . Then, . From the Leibniz formula, we get the identity On the other hand, one can rewrite , where and denotes the th iteration of .
Due to Lemma 2.5, can be written , with , and . Therefore, we get the following identity as:
(1) We first give estimates for . We write where Now, we gives estimates for . We write where for all . From the Taylor formula applied to on , we get that for all . Now, we recall that for all two real numbers, we have From (2.39), (2.40) and (2.41), we deduce that for all . From (2.37) and (2.42), we deduce that
(2) We give estimates for for all and all . From Lemma 2.6, we have the estimates for and all . Now, we give estimates for . Using the Taylor formula with integral remainder, we have that and from the classical identity we get from the Fubini theorem that Again, we write From the expression of , we have that for all , if , and , if . Using (2.49) in the right-hand side of the equality (2.48), we deduce from (2.47) that if , and if .
(3) We give estimates for , for . From the estimates (2.44) and (2.50), we get that From (2.37) and (2.42), we deduce from (2.51) that where for all , and . Now, we show that , , is a bounded sequence. We have for all . From the Stirling formula which asserts that as , we get a universal constant and a constant (depending on , ) such that for all . From (2.54), (2.55), we get a constant (depending on ) such that for all .
(4) We give estimates for . From the estimates (2.44) and (2.50), we get that Again from (2.37) and (2.42), we deduce from (2.57) that where and for all . Now, we remind from (2.31) that is a bounded sequence.
Finally, from (2.31), (2.36), (2.43), (2.52), (2.56), and (2.58), we deduce a constant (depending on , ) such that which gives the result. It remains to consider the case .
When , let , with . By definition, we can write