Doubly-resonant saddle-nodes in and the fixed singularity at infinity in Painlevé equations: analytic classification
In this work, we consider germs of analytic singular vector fields in with an isolated and doubly-resonant singularity of saddle-node type at the origin. Such vector fields come from irregular two-dimensional differential systems with two opposite non-zero eigenvalues, and appear for instance when studying the irregular singularity at infinity in Painlevé equations for generic values of the parameters. Under suitable assumptions, we prove a theorem of analytic normalization over sectorial domains, analogous to the classical one due to Hukuhara-Kimura-Matuda for saddle-nodes in . We also prove that these maps are in fact the Gevrey-1 sums of the formal normalizing map, the existence of which has been proved in a previous paper. Finally we provide an analytic classification under the action of fibered diffeomorphisms, based on the study of the so-called Stokes diffeomorphisms obtained by comparing consecutive sectorial normalizing maps à la Martinet-Ramis / Stolovitch for 1-resonant vector fields.
Key words and phrases:Painlevé equations, singular vector field, irregular singularity, resonant singularity, analytic classification, Stokes diffeomorphisms.
As in [Bit16b], we consider (germs of) singular vector fields in which can be written in appropriate coordinates as
where and are germs of holomorphic functions in of homogeneous valuation (order) at least two. They represent irregular two-dimensional differential systems having two opposite non-zero eigenvalues:
These we call doubly-resonant vector fields of saddle-node type (or simply doubly-resonant saddle-nodes). We will impose more (non-generic) conditions in the sequel. The motivation for studying such vector fields is at least of two types.
There are two independent resonance relations between the eigenvalues (here , and ): we generalize then the study in [MR82, MR83]. More generally, this work is aimed at understanding singularities of vector fields in . According to a theorem of resolution of singularities in dimension less than three in [MP13], there exists a list of “final models” for singularities (log-canonical) obtained after a finite procedure of weighted blow-ups for three dimensional singular analytic vector fields. In this list, we find in particular doubly-resonant saddles-nodes, as those we are interested in. In dimension , these final models have been intensively studied (for instance by Martinet, Ramis, Ecalle, Ilyashenko, Teyssier, …) from the view point of both formal and analytic classification (some important questions remain unsolved, though). In dimension , the problems of formal and analytic classification are still open questions, although Stolovitch has performed such a classification for 1-resonant vector fields in any dimension [Sto96]. The presence of two kinds of resonance relations brings new difficulties.
Our second main motivation is the study of the irregular singularity at infinity in Painlevé equations, for generic values of the parameters (cf. [Yos85]). These equations were discovered by Paul Painlevé [Pai02] because the only movable singularities of the solutions are poles (the so-called Painlevé property). Their study has become a rich domain of research since the important work of Okamoto [Oka77]. The fixed singularities of the Painlevé equations, and more particularly those at infinity, where notably investigated by Boutroux with his famous tritronquées solutions [Bou13]. Recently, several authors provided more complete information about such singularities, studying “quasi-linear Stokes phenomena” and also giving connection formulas; we refer to the following (non-exhaustive) sources [JK92, Kap04, KK93, JK01, CM82, CCH15]. Stokes coefficients are invariant under local changes of analytic coordinates, but do not form a complete invariant of the vector field. To the best of our knowledge there currently does not exist a general analytic classification for doubly-resonant saddle-nodes. Such a classification would provide a new framework allowing to analyze Stokes phenomena in that class of singularities.
In this paper we provide an analytic classification under the action of fibered diffeomorphisms for a specific (to be defined later on) class of doubly-resonant saddle-nodes which contains the Painlevé case. For this purpose, the main tool is a theorem of analytic normalization over sectorial domain (à la Hukuhara-Kimura-Matuda [HKM61] for saddle-nodes in ) for a specific class (to be defined later on) of doubly-resonant saddle-nodes which contains the Painlevé case. The analytic classification for this class of vector fields, inspired by the important works [MR82, MR83, Sto96] for 1-resonant vector fields, is based on the study of so-called Stokes diffeomorphisms, which are the transition maps between different sectorial domains for the normalization.
In [Yos84, Yos85] Yoshida shows that doubly-resonant saddle-nodes arising from the compactification of Painlevé equations (for generic values for the parameters) are conjugate to vector fields of the form:
with and such that . One should notice straight away that this “conjugacy” does not agree with what is traditionally (in particular in this paper) meant by conjugacy, for Yoshida’s transform takes the form
where each is formal power series although appears with negative exponents. This expansion may not even be a formal Laurent series. It is, though, the asymptotic expansion along of a function analytic in a domain
for some small , where is a sector of opening greater than with vertex at the origin and is a polydisc of small poly-radius . Moreover the are actually Gevrey-1 power series. The drawback here is that the transforms are convergent on regions so small that taken together they cannot cover an entire neighborhood of the origin in (which seems to be problematic to obtain an analytic classification à la Martinet-Ramis).
Several authors studied the problem of convergence of formal transformations putting vector fields as in into “normal forms”. Shimomura, improving on a result of Iwano [Iwa80], shows in [Shi83] that analytic doubly-resonant saddle-nodes satisfying more restrictive conditions are conjugate (formally and over sectors) to vector fields of the form
via a diffeomorphism whose coefficients have asymptotic expansions as in sectors of opening greater than .
Stolovitch then generalized this result to any dimension in [Sto96]. More precisely, Stolovitch’s work offers an analytic classification of vector fields in with an irregular singular point, without further hypothesis on eventual additional resonance relations between eigenvalues of the linear part. However, as Iwano and Shimomura did, he needed to impose other assumptions, among which the condition that the restriction of the vector field to the invariant hypersurface is a linear vector field. In [BDM08], the authors obtain a Gevrey-1 summable “normal form”, though not as simple as Stolovitch’s one and not unique a priori, but for more general kind of vector field with one zero eigenvalue. However, the same assumption on hypersurface is required (the restriction is a linear vector field). Yet from [Yos85] (and later [Bit16b]) stems the fact that this condition is not met in the case of Painlevé equations .
In comparison, we merely ask here that the restricted vector field be orbitally linearizable (see Definition 1.7), i.e. the foliation induced by on (and not the vector field itself) be linearizable. The fact that this condition is fulfilled by the singularities of Painlevé equations formerly described is well-known. As discussed in Remark 1.17, the more general context also introduces new phenomena and technical difficulties as compared to prior classification results.
1.1. Scope of the paper
The action of local analytic / formal diffeomorphisms fixing the origin on local holomorphic vector fields of type by change of coordinates is given by
In [Bit16b] we performed the formal classification of such vector fields by exhibiting an explicit universal family of vector fields for the action of formal changes of coordinates at (called a family of normal forms). Such a result seems currently out of reach in the analytic category: it is unlikely that an explicit universal family for the action of local analytic changes of coordinates be described anytime soon. If we want to describe the space of equivalent classes (of germs of a doubly-resonant saddle-node under local analytic changes of coordinates) with same formal normal form, we therefore need to find a complete set of invariants which is of a different nature. We call moduli space this quotient space and would like to give it a (non-trivial) presentation based on functional invariants à la Martinet-Ramis [MR82, MR83].
We only deal here with -fibered local analytic conjugacies acting on vector fields of the form with some additional assumptions detailed further down (see Definitions 1.1, 1.3 and 1.7). Importantly, these hypothesis are met in the case of Painlevé equations mentioned above. The classification under the action of general (not necessarily fibered) diffeomorphisms can be found in [Bit16a]).
First we prove a theorem of analytic sectorial normalizing map (over a pair of opposite “wide” sectors of opening greater than whose union covers a full punctured neighborhood of ). Then we attach to each vector field a complete set of invariants given as transition maps (over “narrow” sectors of opening less than ) between the sectorial normalizing maps. Although this viewpoint has become classical since the work of Martinet and Ramis, and has latter been generalized by Stolovitch as already mentioned, our approach has some geometric flavor. For instance, we avoid the use of fixed-point methods altogether to establish the existence of the normalizing maps, and generalize instead the approach of Teyssier [Tey04, Tey03] relying on path-integration of well-chosen -forms (following Arnold’s method of characteristics [Arn74]).
As a by-product of this normalization we deduce that the normalizing sectorial diffeomorphisms are Gevrey- asymptotic to the normalizing formal power series of [Bit16b], retrospectively proving their -summability (with respect to the coordinate). When the vector field additionally supports a symplectic transverse structure (which is again the case of Painlevé equations) we prove that the (essentially unique) sectorial normalizing map is performed by a transversally symplectic diffeomorphism. We deduce from this a theorem of analytic classification under the action of transversally symplectic diffeomorphisms.
1.2. Definitions and main results
To state our main results we need to introduce some notations and nomenclature.
For , we denote by an (arbitrary small) open neighborhood of the origin in .
We denote by , with , the -algebra of germs of holomorphic functions at the origin of , and by the group of invertible elements for the multiplication (also called units), i.e. elements such that .
is the Lie algebra of germs of singular holomorphic vector fields at the origin . Any vector field in can be written as
with vanishing at the origin.
is the group of germs of a holomorphic diffeomorphism fixing the origin of . It acts on by conjugacy: for all
we define the push-forward of by by
where is the Jacobian matrix of .
is the subgroup of of fibered diffeomorphisms preserving the -coordinate, i.e. of the form .
We denote by the subgroup of formed by diffeomorphisms tangent to the identity.
All these concepts have formal analogues, where we only suppose that the objects are defined with formal power series, not necessarily convergent near the origin.
A diagonal doubly-resonant saddle-node is a vector field of the form
with and of order at least two. We denote by the set of such vector fields.
One can also define the foliation associate to a diagonal doubly-resonant saddle-node in a geometric way. A vector field is orbitally equivalent to a diagonal doubly-resonant saddle-node i.e. is conjugate to some , where and if and only if the following conditions hold:
there exists a germ of irreducible analytic hypersurface which is transverse to the eigenspace (corresponding to the zero eigenvalue) at the origin, and which is stable under the flow of ;
, where is the Lie derivative of and .
By Taylor expansion up to order with respect to , given a vector field written as in (1.1) we can consider the associate 2-dimensional system:
with , such that the following conditions hold:
, with and .
Based on this expression, we state:
The residue of is the complex number
We say that is non-degenerate (resp. strictly non-degenerate) if (resp. ).
It is obvious that there is an action of on . The residue is an invariant of each orbit of under the action of by conjugacy (see [Bit16b]).
The main result of [Bit16b] can now be stated as follows:
[Bit16b] Let be non-degenerate. Then there exists a unique formal fibered diffeomorphism tangent to the identity such that:
where , are formal power series in without constant term and are such that .
The vector field obtained in (1.6) is called the formal normal form of . The formal fibered diffeomorphism is called the formal normalizing map of .
The above result is valid for formal objects, without considering problems of convergence. The first main result in this work states that this formal normalizing map is analytic in sectorial domains, under some additional assumptions that we are now going to precise.
We say that a germ of a vector field in is orbitally linear if
for some and .
We say that a germ of vector field in is analytically (resp. formally) orbitally linearizable if is analytically (resp. formally) conjugate to an orbitally linear vector field.
We say that a diagonal doubly-resonant saddle-node is div-integrable if is (analytically) orbitally linearizable.
Alternatively we could say that the foliation associated to is linearizable. Since is analytic at the origin of and has two opposite eigenvalues, it follows from a classical result of Brjuno (see [Mar81]), that is analytically orbitally linearizable if and only if it is formally orbitally linearizable.
We denote by the set of strictly non-degenerate diagonal doubly-resonant saddle-nodes which are div-integrable.
The vector field corresponding to the irregular singularity at infinity in the Painlevé equations is orbitally equivalent to an element of , for generic values of the parameters (see [Yos85]).
We can now state the first main result of our paper (we refer to section for more details on 1-summability).
Let and let (given by Theorem 1.5) be the unique formal fibered diffeomorphism tangent to the identity such that
where and are formal power series without constant term. Then:
the normal form is analytic (i.e. ), and it also is div-integrable, i.e. ;
the formal normalizing map is 1-summable (with respect to ) in every direction .
there exist analytic sectorial fibered diffeomorphisms and , (asymptotically) tangent to the identity, defined in sectorial domains of the form and respectively, where
(for any and some small enough), which admit as weak Gevrey-1 asymptotic expansion in these respective domains, and which conjugate to . Moreover and are the unique such germs of analytic functions in sectorial domains (see Definition 2.2).
Although item 3 above is a straightforward consequence of the 1-summability of (item 2 above), we will in fact start by proving item 3 in Corollary 4.2, and establish the 1-summability of item 2 in a second step (see Proposition 5.6). What we will obtain at first directly is only the weak 1-summability (see subsection 2.3) of (see Proposition 4.18), and not immediately the 1-summability. To obtain the “true” 1-summability, we will need to prove that the transition maps between and are exponentially close to the identity (see Proposition 5.2), and then to use a fundamental theorem of Martinet and Ramis (see Theorem 2.22).
We call and the sectorial normalizing maps of .
They are the 1-sums of along the respective directions and . Notice that and are germs of analytic sectorial fibered diffeomorphisms, i.e. they are of the form
(see section 2. for a precise definition of germ of analytic sectorial fibered diffeomorphism). The fact that they are also (asymptotically) tangent to the identity means that we have:
In fact, we can prove the uniqueness of the sectorial normalizing maps under weaker assumptions.
Let and be two germs of sectorial fibered diffeomorphisms in and respectively, where and are as in Theorem 1.10, which are (asymptotically) tangent to the identity and such that
Then, they necessarily coincide with the the sectorial normalizing maps and defined above.
Since two analytically conjugate vector fields are also formally conjugate, we fix now a normal form
with , and vanishing at the origin.
We denote by the set of germs of holomorphic doubly-resonant saddle-nodes in which are formally conjugate to by formal fibered diffeomorphisms tangent to the identity, and denote by the set of orbits of the elements in this set under the action of .
According to Theorem 1.10, to any we can associate two sectorial normalizing maps , which can in fact extend analytically in domains and , where is an asymptotic sector in the direction with opening (see Definition 2.3):
Then, we consider two germs of sectorial fibered diffeomorphisms analytic in , with
Notice that are isotropies of , i.e. they satisfy:
With the above notations, we define as the group of germs of sectorial fibered isotropies of , tangent to the identity, and admitting the identity as Gevrey-1 asymptotic expansion (see Definition 2.4) in sectorial domains of the form , with .
The two sectorial isotropies and defined above are called the Stokes diffeomorphisms associate to .
Our second main result gives the moduli space for the analytic classification that we are looking for.
is well-defined and bijective.
In particular, the result states that Stokes diffeomorphisms only depend on the class of in the quotient . We will give a description of this set of invariants in terms of power series in the space of leaves in section 5.
In this paper we start by proving a theorem of sectorial normalizing map analogous to the classical one due to Hukuhara-Kimura-Matuda for saddle-nodes in [HKM61], generalized later by Stolovitch in any dimension in [Sto96]. Unlike the method based on a fixed point theorem used by these authors, we have used a more geometric approach (following the works of Teyssier [Tey03, Tey04]) based on the resolution of an homological equation, by integrating a well chosen 1-form along asymptotic paths. This latter approach turned out to be more efficient to deal with the fact that is not necessarily linearizable. Indeed, if we look at [Sto96] in details, one of the first problem is that in the irregular systems that needs to be solved by a fixed point method (for instance equation in the cited paper), the non-linear terms would not be divisible by the “time” variable in our situation. This would complicate the different estimations that are done later in the cited work. This is the first main new phenomena we have met.
Then we will see that the sectorial normalizing maps in the corollary above admit in fact the unique formal normalizing map given by Theorem 1.5 as “true” Gevrey-1 asymptotic expansion in and respectively. This will be proved by studying in (and more generally any germ of sectorial fibered isotropy of in “narrow” sectorial neighborhoods which admits the identity as weak Gevrey-1 asymptotic expansion). The main difficulty is to prove that such a sectorial isotropy of over the “narrow” sectors described above is necessarily exponentially close to the identity (see Lemma 5.20). This will be done via a detailed analysis of these maps in the space of leaves (see Definition 5.10). In fact, this is the second main new difficulty we have met, which is due to the presence of the “resonant” term
in the exponential expression of the first integrals of the vector field (see ). In [Sto96], similar computations are done in Subsection . In this part of the paper, infinitely many irregular differential equations appear when identifying terms of same homogeneous degree. The fact that is linear implies that these differential equations are all linear and independent of each others (i.e. they are not mixed together). In our situation, this is not the case and then more complicated.
1.3. Painlevé equations: the transversally Hamiltonian case
In [Yos85] Yoshida shows that a vector field in the class naturally appears after a suitable compactification (given by the so-called Boutroux coordinates [Bou13]) of the phase space of Painlevé equations , for generic values of the parameters. In these cases the vector field presents an additional transverse Hamiltonian structure. Let us illustrate these computations in the case of the first Painlevé equation:
As is well known since Okamoto [Oka80], can be seen as a non-autonomous Hamiltonian system
More precisely, if we consider the standard symplectic form and the vector field
induced by , then the Lie derivative
belongs to the ideal generated by in the exterior algebra of differential forms in variables . Equivalently, for any the flow of at time acts as a symplectomorphism between fibers and .
The weighted compactification given by the Boutroux coordinates [Bou13] defines a chart near as follows:
In the coordinates , the vector field is transformed, up to a translation with , to the vector field
where denotes the ideal generated by in the algebra of holomorphic forms in . We finally obtain
Therefore, both and are differential forms who lie in the ideal , in the algebra of germs of meromorphic 1-forms in with poles only in . This motivates the following:
Consider the rational 1-form
We say that vector field is transversally Hamiltonian (with respect to and dx) if
For any open sector , we say that a germ of sectorial fibered diffeomorphism in is transversally symplectic (with respect to and ) if
(Here denotes the pull-back of by ).
We denote by the group of transversally symplectic diffeomorphisms which are tangent to the identity.
The flow of a transversally Hamiltonian vector field defines a map between fibers and which sends onto , since
A fibered sectorial diffeomorphism is transversally symplectic if and only if .
A transversally Hamiltonian doubly-resonant saddle-node is a transversally Hamiltonian vector field which is conjugate, via a transversally symplectic diffeomorphism, to one of the form
with and analytic in and of order at least .
Notice that a transversally Hamiltonian doubly-resonant saddle-node is necessarily strictly non-degenerate (since its residue is always equal to ), and also div-integrable (see section 3). It follows from Yoshida’s work [Yos85] that the doubly-resonant saddle-nodes at infinity in Painlevé equations (for generic values of the parameters) all are transversally Hamiltonian.
We recall the second main result from [Bit16b].
Let be a diagonal doubly-resonant saddle-node which is supposed to be transversally Hamiltonian. Then, there exists a unique formal fibered transversally symplectic diffeomorphism , tangent to identity, such that:
where , a formal power series in without constant term and are such that .
Let be a transversally Hamiltonian doubly-resonant saddle-node and let be the unique formal normalizing map given by Theorem 1.21. Then the associate sectorial normalizing maps and are also transversally symplectic.
Since is 1-summable in , the formal power series is also 1-summable in , and its asymptotic expansion has to be the constant . By uniqueness of the 1-sum, we thus have . ∎
are also transversally symplectic.
We denote by resp. the group of germs of sectorial fibered isotropies of , admitting the identity as Gevrey-1 asymptotic expansion in sectorial domains of the form , and which are transversally symplectic.
Let us denote by the set of germs of vector fields which are formally conjugate to via (formal) transversally symplectic diffeomorphisms tangent to the identity. As a consequence of Theorems (1.16) and (1.22), we can now state the following result.
is a well-defined bijection.
1.4. Outline of the paper
In section 2, we introduce the different tools we need concerning asymptotic expansion, Gevrey-1 series and 1-summability. We will in particular introduce a notion of “weak” 1-summability.
In section 3, we prove Proposition 3.1, which states that we can always formally conjugate a non-degenerate doubly-resonant saddle-node which is also div-integrable to its normal form up to remaining terms of order , for all , and the conjugacy is actually -summable.
In section 4, we prove that for all , there exists a unique pair of sectorial normalizing maps tangent to the identity which conjugates to its normal form over sectors with opening greater than and arbitrarily close to . The existence is given by Corollary 4.2, while the uniqueness clause stated in Proposition 1.13 is proved thanks to Proposition 4.16. Moreover, we will see that and both admit the unique formal normalizing map given by Theorem 1.5 as weak Gevrey-1 asymptotic expansion (see Proposition 4.18).
In section 5, we show that the Stokes diffeomorphisms and , which admit a priori the identity only as weak Gevrey-1 asymptotic expansion, admit in fact the identity as “true” Gevrey-1 asymptotic expansion. This will be done by studying more generally the germs of sectorial isotropies of the normal form in sectorial domains with “narrow” opening (see Corollary 5.2). Using a theorem by Martinet and Ramis [MR82] reformulated in Theorem 2.22, which is a “non-abelian” version of the Ramis-Sibuya theorem, we will obtain the fact that is 1-summable in every direction , of 1-sums and respectively in the corresponding domains (see Corollary 5.6). We then give a short proof of Theorem 1.10, just by using the different lemmas and propositions needed and proved earlier in this paper. After that, we will once again use Theorem 2.22 in order to obtain both Theorems 1.16 and 1.24. We give in Proposition 5.24 a description of the moduli space of analytic classification in terms of some spaces of power series in the space of leaves.
- 1 Introduction
- 2.1 Sectorial germs
- 2.2 Gevrey-1 power series and 1-summability
- 2.3 Weak Gevrey-1 power series and weak 1-summability
- 2.4 Weak 1-summability versus 1-summability
- 2.5 Some useful tools on 1-summability of solutions of singular linear differential equations
3 1-summable preparation up to any order
- 3.1 Analytic normalization on the hyperplane
- 3.2 1-summable simplification of the “dependent” affine part
- 3.3 1-summable straightening of two invariant hypersurfaces
- 3.4 1-summable normal form up to arbitrary order
- 3.5 Proof of Proposition 3.1
4 Sectorial analytic normalization
- 4.1 Proof of Proposition 4.1.
- 4.2 Proof of Propositions 4.3 and 4.5
- 4.3 Proof of Lemma 4.7
- 4.4 Resolution of the homological equation: proof of Lemma 4.6
- 4.5 Sectorial isotropies in “wide” sectors and uniqueness of the normalizations: proof of Proposition 1.13.
- 4.6 Weak 1-summability of the normalizing map
5 Analytic classification
- 5.1 Proofs of the main results (assuming Proposition 5.2 )
5.2 Sectorial isotropies in narrow
sectors and space of leaves: proof of Proposition 5.2.
- 5.2.1 Sectorial first integrals and the space of leaves
- 5.2.2 Sectorial isotropies in the space of leaves
- 5.2.3 Action on the resonant monomial in the space of leaves
- 5.2.4 Action on the resonant monomial
- 5.2.5 Power series expansion of sectorial isotropies in the space of leaves
- 5.2.6 Sectorial isotropies: proof of Proposition 5.2
- 5.3 Description of the moduli space and some applications