Singular Perturbation of Nonlinear Systems with Regular Singularity

Abstract

We extend Balser-Kostov method of studying summability properties of a singularly perturbed inhomogeneous linear system with regular singularity at origin to nonlinear systems of the form with a -valued function, holomorphic in a polydisc . We show that its unique formal solution in power series of , whose coefficients are holomorphic functions of , is -summable under a Siegel-type condition on the eigenvalues of . The estimates employed resemble the ones used in KAM theorem. A simple lemma is applied to tame convolutions that appear in the power series expansion of nonlinear equations. Applications to spherical Bessel functions and probability theory are indicated. The proposed summability method has certain advantages as it may be applied as well to (singularly perturbed) nonlinear partial differential equations of evolution type.

1. Introduction

We consider singularly perturbed nonlinear systems of the form ( means derivative of with respect to )

with and   -vector functions, holomorphic in a polydisc, say for some (here, denotes an open disc of radius , centered at , denotes its closure and ) such that the matrix is invertible, a condition that makes (1) possess a regular singularity at .

When (1) is linear, i.e., , where and are, respectively, a -vector and a matrix, whose entries are holomorphic in the polydisc , , such that exists, Balser and Kostov [1] have established the following: (a) there exists a unique formal solution in the ring of formal power series

in with coefficients in the ring of holomorphic functions on , continuous in its closure, satisfying

for some positive constants , and ; (b) provided the closed sector of opening angle about the bisecting direction and radius does not contain any ray on the direction of the eigenvalues ’s of ,

is the -Gevrey asymptotic expansion as tends to of a holomorphic function in ; (c) if is chosen so that (4) holds, then the formal series is, by an analogue of Borel-Ritt’s theorem for Gevrey asymptotic expansion (see, e.g., Section 3.2 of [2]), -summable in the direction and its sum equals .

Summability of formal solutions to singularly perturbed linear systems with irregular singularity at ,

with the Poincaré rank , has been investigated before in [3]. Contrarily to the previous case , the unique formal power series solution is always -summable, irrespective of whether (4), the additional condition satisfied by the eingenvalues of , holds or not. The case of , on the other hand, has been studied in [4] for and the summability of the formal series can be read from the properties of the initial data of (5). The case separates the two cases and we refer to [3] for an explanation on summability properties for each of distinct cases of simple examples in which and are a scalar and a scalar function depending only on . For recent investigations of the linear meromorphic system (5) with , , and , about summable-resurgent of the Borel transform of its highest level’s reduced formal solutions and connection-to-Stokes formulas, see [5] and references therein.

In the present article, all the three statements regarding the summability of the formal series , (a)–(c) above, will be extended for nonlinear differential equations (1).

The nonlinear extension of Balser-Kostov’s statements has been requested by our investigation of renormalization group (RG) flow equation over probability measures in , starting from a measure restricted to the sphere of radius , as goes to . Let be the unique extension in , with , of the meromorphic function

where is the Bessel function of order . This function is the logarithmic derivative of the Fourier-Stieltjes transform of the uniform measure on the -dimensional sphere of radius . In [6], is shown to satisfy a continued fraction of Gauss, convergent in uniformly in and this domain of analyticity is extended to the upper half-plane provided is real. In [7], given by (6) is the initial data whose summability properties are conjectured to be preserved under the RG dynamics. We refer to [6, 7] for the statistical mechanics context.

Retrospectively, the development of summability methods in probability started with the classical Wiener’s Tauberian theory (see [8, 9] for a concise and, respectively, extensive overviews). The most common types of summation methods called matrix methods may be used as a criterion for asymptotic distribution functions (mod ) of numerical sequences (see Section 7 in Chapter 1 of [10] for a review and [11] for necessary conditions on general dynamical systems, in connection with Gaussian processes). Summability methods are also used in the context of Lagrange interpolation of zeros of Jacobi polynomials and complete monotonicity of certain functions (see [12] and references therein). Related to the present work is the Borel summability of the expansion for the -vector statistical model at high temperatures proved in [13] (see also [14] for similar results on related models). Our investigation considers a hierarchical version of this model at the critical temperature, from a dynamical point of view.

The dependence in the argument of the r.h.s. of (6) is chosen in such way that attains, as goes to , a limit function

(see Proposition 2.1 of [6]). satisfies an ordinary differential equation of Riccati type

which, despite being nonlinear, can be dealt with by Balser-Kostov’s method. Equation (8) is of the form (1) with and (Statements (a)–(c) hold with in (8) replaced by for any 1-summable formal series in direction. In this case, the limit function (7) is replaced by )

Balser-Kostov summability proof in [1] of the formal series solution does not follow the usual route by which the (formal) Borel transform of is analytically continued along some sector of infinite radius (see, e.g., [2]). Their proof establishes instead Gevrey asymptotic expansion directly from (5), requiring for this an auxiliary lemma regarding an infinite system of linear equations of the same type whose coefficient matrix is independent of . Although (1) is nonlinear, the system of infinitely many equations obtained by taking derivatives of (1) with respect to is linear and Balser-Kostov’s method carries over to equation of the form (1). To prove these statements, suitable formulas and a simple but efficient way of estimating higher power of are provided.

The layout of this paper is as follows. In Section 2 (Proposition 3), we prove existence of a unique solution of (1) in power series of . In Section 3 (Proposition 6), we show that the formal power series in solution of (1) is Gevrey of order . In Section 4 (Proposition 7) Gevrey asymptotic is established. Our main result, the -summability of the formal solution of (1), is stated in Section 5 (Theorem 9) and proved using Propositions 3, 6, and 7 of the previous sections. The main ingredient (Lemma 4) is employed to tame arbitrarily large number of convolutions arisen in the expansion of in powers of . The advantage of the proposed summability method is that it can be applied to nonlinear partial differential equations of evolution type [7, 15].

2. Power Series in

Under the hypothesis on , the series

converges (in norm) absolutely in , uniformly in , with the coefficients , regarded as a multilinear operator,

endowed with an operator norm induced by the Euclidean space :

holomorphic in as a function of .

In (12) and from now on, denotes a -vector with -th component and Euclidean norm . The product is denoted by to distinguish from the components ’s of .

Since the left hand side of (1) vanishes for , and is assumed in (10), a solution of (1) in power series reads

Remark 1. Observe that admits a trivial solution for which is unique by the implicit function theorem. For the corresponding to example (9), does not vanish identically but and (9) may be replaced by and according to which and hold. Equations (1), under the hypothesis of invertible , can always be reduced to the same form with satisfying (10). For this, by the implicit function theorem, can be solved for and satisfies .

Substituting the power series (14) into (10) together with (1), we are led to a system of equations

with given by

for ; for any two sequences and , their convolution product is a sequence defined by and

The restriction in (18) results from the fact that our sequence starts with and a convolution involving sequences cannot have nonvanishing component if .

Consequently, for any arbitrary, (16) for forms a closed system of equations, involving unknown functions which can be solved by iteration starting from

If (16) for and have been solved, then

Regarding the inverse matrix , we have the following.

Lemma 2 (see Lemma 1 of [1]). Suppose (4) holds with and , , eigenvalues of . One can always find such that, if for some , the inverse matrix in (21), given by is bounded and satisfies , uniformly in . If , let , , the eigenvalues of , be so that their distances from every ray intercepting are bounded from below by a constant : Then, together with the formula for inverse of a matrix , where is the transposed of the cofactors matrix of (see, e.g., [16]) and the boundedness in of all cofactors of gives uniformly in for every .

Proposition 3. Let be given by (10) with the eigenvalues of obeying hypothesis (4). There exist , , and such that (1) has a solution holomorphic in . The solution converges, as in the sector , to the unique solution of in satisfying .

Proof. Since (14) solves (1), its coefficients satisfy the formal relations (16) whose solution depends on the existence of inverse matrix for every and . Assuming (4) holds for every eigenvalue of , let and be such that (23), and consequently (25), holds. Hence, given by (21) is bounded uniformly in , uniquely defined for every and, in view of these, holomorphic in .

Let and be the supremum in of and , respectively: By Cauchy formula and there exists (=, ) such thatNow, we prove that the majorant series converges and is bounded by for some . For this, the following lemma will play an important role here and in the further sections. See Lemma 3.1 of [17] for similar result and Lemma 2.1 of Treves [18].

. Let be given and let . Consider the sequence with or and Thenholds for every .

Proof. Since , and holds for any real numbers and , we have

It thus follows from (31) with that

holds for any and the sum is equal to for .

Let and be as in Lemmas 2 and 4. Let and in (29) be such that (see Remark 5) for some and . Supposeholds for , with the sequence in Lemma 4 with and .

Hence, by (20) together with (25), (27), and (36), we have and, by (21) and (18) together with (25), with denoting the sequence . Taking the sup over in both sides together with (27), (34), and (36), holds for , provided and With and satisfying these conditions, we conclude and is a sequence of holomorphic functions, uniformly bounded in by , whose sum is bounded (in norm) byprovided satisfies , by (41). Under this choice of , uniformly in and the solution we have obtained by the formal expansions (16) and (18) acquires sense. The power series solution (14) of (1) thus converges to a unique analytic function in . The proof of uniqueness will be omitted.

From the uniform convergence of (14) we conclude that, for any fixed , the solution tends to where is the unique solution of equation for , by the analytic implicit function theorem (see, e.g., Section 2.3 of [19]). Note that the solution is regular at since, by (14), it must satisfy and this concludes the proof of Proposition 3.  

Remark 5. As observed at end of Section 1 of [1], the estimate for the (uniformly in ) radius of convergence of (14) can be much smaller than the radius of the largest disk in which , the radius of convergence for the solution of the linear system at . The Cauchy majorant method (see Section 3.3 of [19]) applied to (45) yields a majorant of , holomorphic in a disc , where depends on , with a constant defined in (29), and . Another (not sharp) method of this type, exploiting Lemma 4 to eliminate convolutions, yields a majorant holomorphic in a disc of radius possibly smaller than . Despite this, since the latter method is undeniably practical and more suitable for extensions, we shall apply it in all further sections.

3. Formal Power Series in

As in (10), the double series

converges (in norm) absolutely in , uniformly in , with the coefficients regarded as a multilinear operator

By consistency, but may not be identically zero. From here on, when no ambiguity arises, we drop the dot that precedes the power of in , introduced to distinguish from its components .

Proposition 6. Suppose the formal power series (2) satisfies (1), formally, with obeying the hypotheses stated after (1). Then, the coefficients of (2) are analytic functions of in the open disc and there exist positive constants and such that holds for all and , with . In other words, the formal power series is of Gevrey order 1; that is, .

Proof. Substituting the power series (2) into (47), we are thus led to the following equations: for , we have which has already been solved for . Recall that solves and the coefficients of the power series in satisfy as the estimates for in the previous section, uniformly in , hold for . However, together with (50) can be used, since the do not depend on , to improve the disk of convergence of the series of . See Remark 5.
For , we have Observe that the sum over has no limit as the sequence starts from and the convolution product, now defined by for any two sequences and , imposes no restriction on their number.
To isolate , the largest index term in (53), we have to show that the matrix (recall ) is invertible for every for some . For this, we take so small that and, consequently, holds uniformly in .
It follows from (53) and (55) that and this relation determines uniquely in terms of earlier coefficients. Note that is holomorphic in and, by (51) and (43) for any , by letting small enough. Now, to obtain an estimate on the growth rate of , let denote the -th Nagumo norm of and let the supremum in of . The properties we shall use on Nagumo’s norms is proved in [1] and references therein and are here summarized:
;
;
;
,
for any two functions and holomorphic in and nonnegative integers .
Let us assume that holds for with , for some positive constants and to be determined. Similar to (29) and (36), holds for some and large enough. Then, it follows by (57), (61), and (34) and the properties of Nagumo normswhere the last inequality holds provided and and this completes the induction: with and fixed so that (58) and (63) hold.
By definition (59) of Nagumo norm, holds for all uniformly in for some , with and , which concludes the proof of Proposition 6.

4. Gevrey Asymptotic

In order to set up an equation involving derivatives of with respect to , we write

and for the sequence of those functions defined on ; analogously to (10) and (47), we write

where stands for the -th derivative of with respect to the first argument divided by . The -th “total derivative” of with respect to ( depends on explicitly and implicitly through ) can thus be written as

where

is a linear operator (matrix) and depends only on derivatives of with respect to of order lower than .

Differentiating (1) times with respect to , dividing by , we have

for , where

may be considered as inhomogeneous holomorphic function of in , and for simply (1):

Proposition 7. Let be the unique holomorphic solution of (1) on with , , and as in Proposition 3. There exist , , and positive constants and such that holds for all and every point in .

Proof. The case follows straightforwardly from Proposition 3. (70) which is a linear singular perturbation equation with regular singularity which can be dealt with in the following auxiliary result due to Balser-Kostov [1] (see Lemma 3 therein). For this, we drop temporarily all subindices in (70).
Let be expanded in power of ,and consider a sequence satisfying the systemBy (74) and linearity, the sum over all equations in (75) yields an equation of the form (70) satisfying by the sum . We assume that admits an expansionabsolutely convergent for , uniformly in . For given by (68) and (71) this will actually be proven by induction when we resume the proof of Proposition 7. We write, in addition, if is majorized by , i.e., if holds for all . If is a -vector or a matrix means majorized relation for each component. For any , let denote the vector and analogous notation for the matrix .
. There exists a unique sequence of functions , holomorphic in , satisfying (75). Each has a zero of order at : , and satisfies where holds for some and small enough. is, in addition, the unique analytic solution in of with .
Proof. Let us assume that , as a function of and for every , has a zero of order at and is represented by We will show that these assumptions lead to the actual solution of (75). Plugging (81) into (75) yields for and . Observe that, by (69) and (74), together with the fact that (recall ), is invertible for every if we take so small that and holds uniformly in .
From these relations, we have Defining it follows, by (25), (78), and (79), that for . Since for and for hold for all , we conclude (77) provided the geometric series converges for some . By (74) and (69) if is chosen small enough and, thence, is a uniformly convergent series of analytic functions in which solves (80). Since no other solution of (80), regular at , exists, the proof of Lemma 8 is concluded.
We continue the proof of Proposition 7. It remains to show that the series (76) is uniformly convergent in . This follows by induction. Clearly, is holomorphic in . Suppose that is holomorphic in for each . Then, by (71), is holomorphic in the same domain. By Lemma 8, is holomorphic in and, by (71), we conclude it also holds for , justifying its representation as a convergent series (76), uniformly in . By induction, is holomorphic in for each and where depends on the with . For , by (78), holds for all and . For , we consider the modification of Nagumo norms: with . Recalling (71), (76), and (78), it follows from (89) thatwhere, by (68),and, together with the properties of Nagumo norms, for From these, together with (92), a recursive relation of the same type studied in Section 3 may be derived for the (see (60)–(64)) and one may conclude that (the details for this estimate are left to the reader)holds for all and some suitable constants and . Picking together with the property of Nagumo norms yields for all uniformly in , with and . We choose in order to include the case. This concludes the proof of Proposition 7.

5. Summability of the Formal Series

Theorem 9. Let (1) be considered with given by (10) where the eigenvalues of obey hypothesis (4) for in a domain with . Then, there exists a radius such that for the formal solution is 1-summable in direction.

Proof. By Taylor’s theoremwhere the integral is along a path from to inside . This, together with Proposition 7, implies for every and , with any proper subsector of . In addition, Proposition 6 states that , a formal solution of (1), is an element of and therefore is an element of for any . Take now and sufficiently small. Hence, by definition (see Section 1.5 of [2]), is an asymptotic expansion of order , as in the sector , of , which by Proposition 3 is an analytic solution of (1) in the domain . Then, as , by hypothesis, is the only Gevrey order asymptotic expandable function in which has as its asymptotic expansion, and is -summable in direction (see, e.g., Section 3.2 of [2]).

6. Summary of Results

Let be a singularly perturbed nonlinear system with holomorphic in a polydisc for some and the matrix invertible. In Sections 3, 4, and 5, we have proven the following:(a)There exists a unique formal solution in the ring of formal power series in with coefficients in the ring of holomorphic functions on , continuous in its closure, satisfying , for some positive constants , , and .(b)Provided the closed sector of opening angle about the bisecting direction and radius does not contain any ray on the direction of the eigenvalues ’s of (the Siegel-type condition (4)), is the -Gevrey asymptotic expansion as tends to of a holomorphic function in .(c)If is chosen so that the eigenvalues ’s of are Siegel-type, then the formal series is -summable in the direction and its sum equals .

To prove (a)–(c) we have extended Balser-Kostov summability method [1], establishing in Section 4 Gevrey asymptotic expansion directly from the equation. The ingredient for this is Lemma 8 (analogous to Lemma 3 of [1]), regarding an infinite system of linear equations of type (5) with and independent of . Although (1) is nonlinear, a system of infinitely many linear equations is obtained by taking as many derivatives of (1) with respect to as one wishes. Lemma 4 is the additional key ingredient that has provided to all the three statements suitable formulas and simple but efficient way of estimating higher power of .

The nonlinear extension of Balser-Kostov summability method has been requested by our investigation of renormalization group (RG) flow equation over probability measures in as goes to . In terms of its Fourier transform the initial value problem of the RG flow equation reads

with , the Fourier transform of the uniform measure on the -dimensional sphere of radius . We observe that the Gaussian measure is one stationary solution of the flow equation (99). The problem is to find so that the initial uniform measure is driven by RG flow to the Gaussian stationary point, interpreted in probability theory as the Gaussian scaling limit of nonlinear process (99), starting from the uniform measure on the -sphere. The paper [6] deals with the case, for which the critical value is . The large case is approached through resummation method [7] writing the Legendre transform of the RG flow and in formal power series expansion in .

We have used the radial coordinate to describe the flow: . As explained in the introduction, the result of the present paper applies to the initial condition . Let be the unique extension in , with and , of the meromorphic function

Then, satisfies the Riccati type differential equation (8), which is of the forma (1) with and given by (9), so the above summarized statements apply in this case. The usefulness of Balser-Kostov summability method combined with Lemma 4 is illustrated in [7, 15]. The method when applied to the RG flow equation (99) yields existence of formal series in together with Gevrey estimates to the coefficients, uniformly in (global existence) at the critical value .

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

Domingos H. U. Marchetti thanks Gordon Slade and David Brydges for their hospitality at the University of British Columbia, at the time the present paper was written. William R. P. Conti was supported by FAPESP under Grant no. 07/59739-4.