Jean-Pierre Magnot
email: magnot@math.cnrs.fr

Enrique G Reyes
email: enrique.reyes@usach.cl

AND Vladimir Rubtsov

ON (t 2 , t 3 )-ZAKHAROV-SHABAT EQUATIONS OF GENERALIZED KADOMTSEV-PETVIASHVILI HIERARCHIES

Keywords: differential associative algebra, formal pseudo-differential operators, Kadomtsev-Petviashvili hierarchy MSC (2020): 37K10, 37K20, 37K30, 47G30

We review the integration of the KP hierarchy in several nonstandard contexts. Specifically, we consider KP in the following associative differential algebras: an algebra equipped with a nilpotent derivation; an algebra of functions equipped with a derivation that generalizes the gradient operator; an algebra of quaternion-valued functions; a differential Lie algebra; an algebra of polynomials equipped with the Pincherle differential; a Moyal algebra. In all these cases we can formulate and solve the Cauchy problem of the KP hierarchy. Also, in each of these cases we derive different Zakharov-Shabat (t 2 , t 3 )-equations -that is, different Kadomtsev-Petviashvili equations-and we prove existence of solutions arising from solutions to the corresponding KP hierarchy.

Introduction

The Kadomtsev-Petviashvili (KP) hierarchy is an integrable system in an infinite number of independent variables (t 1 , t 2 , ..) that contains numerous integrable equations in two and three independent variables. Among them, we mention the Boussinesq equation, the KP equation, the KdV equations and the Gelfand-Dickey flows. Usually, these equations are derived from the equations of the KP hierarchy posed on formal pseudodifferential operators ΨDO(A) with coefficients in the algebra of smooth periodic functions A = C ∞ (S 1 , R), or in the algebra of rapidly decaying functions on the line, or in the Gelfand-Dickey algebraic framework explained in [1, Chapter 1]. Classical discussions of this system of equations can be found in [START_REF] Dickey | Soliton equations and Hamiltonian systems[END_REF][START_REF] Miwa | Solitons, Differential Equations, Symmetries and Infinite Dimensional Algebras[END_REF]. A solution to the KP hierarchy satisfies the so-called Zhakarov-Shabat (ZS) relations, as explained in [START_REF] Dickey | Soliton equations and Hamiltonian systems[END_REF][START_REF] Etigoff | Factorization of differential operators, quasideterminants, and nonabelian Toda field equations[END_REF]. The ZS relations are zero-curvature equations (and hence they provide one of the justifications for stating that the KP hierarchy is an integrable system), from which the standard KP equation in the t 2 and t 3 variables can be derived.

It is already well-known that it is possible to change the algebra A and define a similar KP system on ΨDO(A) even when A is not commutative. We mention the paper [START_REF] Mulase | Solvability of the super KP equation and a generalization of the Birkhoff decomposition[END_REF] by Mulase for a very general construction of a KP hierarchy on a differential associative algebra, the paper [START_REF] Etigoff | Factorization of differential operators, quasideterminants, and nonabelian Toda field equations[END_REF], and also Kupershmidt treatise [START_REF] Kuperschmidt | KP or mKP: Mathematics of Lagrangian, Hamiltonian and Integrable Systems[END_REF] for a thorough study of the Hamiltonian content of the non-commutative KP hierarchy. A particular example appears in [START_REF] Kuperschmidt | KP or mKP: Mathematics of Lagrangian, Hamiltonian and Integrable Systems[END_REF][START_REF] Mc Intosh | The quaternionic KP hierarchy and conformally immersed 2-tori on the 4sphere[END_REF], in which the algebra A is the algebra C ∞ (R, H), where H is the skew-field of quaternions. In these two references we can already see that the equations deduced from the (t 2 , t 3 )-ZS equations can be deeply different from the classical KP equation, see also Section 3.3 below. Other examples of "non-standard" KP hierarchies can be found in [START_REF] Strachan | Integrability of the Frobenius algebra-valued Kadomtsev-Petviashvili hierarchy[END_REF][START_REF] Magnot | On the Kadomtsev-Petviashvili hierarchy in an extended class of formal pseudo-differential operators[END_REF][START_REF] Mendoza | Massey products, A ∞ -algebras, differential equations, and Chekanov homology[END_REF]; we also mention [START_REF] Magnot | Well-posedness of the Kadomtsev-Petviashvili hierarchy, Mulase factorization, and Frölicher Lie groups[END_REF] for non-standard commutative examples, and [START_REF] Hamanaka | Commuting flows and conservation laws for noncommutative Lax hierarchies[END_REF][START_REF] Takasaki | Nonabelian KP hierarchy with Moyal algebraic coefficients[END_REF][START_REF] Sakakibara | Factorization methods for noncommutative KP and Toda hierarchy[END_REF] as representative works on a KP hierarchy posed on algebras equipped with a Moyal multiplication. Now, the construction of general solutions to the KP system for differential algebras A has been systematically studied in [START_REF] Eslami Rad | The Cauchy problem of the Kadomtsev-Petviashvili hierarchy with arbitrary coefficient algebra[END_REF][START_REF] Magnot | Well-posedness of the Kadomtsev-Petviashvili hierarchy, Mulase factorization, and Frölicher Lie groups[END_REF]: in these references, the issues of existence and uniqueness of solutions, and well-posedness of the corresponding Cauchy problem, are formulated and stated in frameworks as general as possible. But we remark that, strangely, the analogues of the KP equations and their solutions in generalized settings appear to have been mostly ignored, with very few exceptions (we mention just [START_REF] Kuperschmidt | KP or mKP: Mathematics of Lagrangian, Hamiltonian and Integrable Systems[END_REF][START_REF] Mc Intosh | The quaternionic KP hierarchy and conformally immersed 2-tori on the 4sphere[END_REF][START_REF] Hamanaka | Commuting flows and conservation laws for noncommutative Lax hierarchies[END_REF][START_REF] Magnot | Well-posedness of the Kadomtsev-Petviashvili hierarchy, Mulase factorization, and Frölicher Lie groups[END_REF]). Even some of the authors of the present work, in a first step of investigations, have been doubtful to deduce other (t 2 , t 3 )-ZS equations than those already highlighted by other authors. We show in the present work that we can deduce many types of equations: for example, we find first order nonlinear equations, Navier-Stokes-type equations, and many others. To the best of our knowledge, the equations that we present are new.

The paper is organized as follows. We recall old and not-so-old constructions for formal pseudo-differential operators and KP hierarchy on a unital differential (non-graded) associative algebra A in section 2, and then we exhibit the announced examples in section 3. We remark that the results of [START_REF] Eslami Rad | The Cauchy problem of the Kadomtsev-Petviashvili hierarchy with arbitrary coefficient algebra[END_REF] allow us to state theorems on the well-posedness our non-commutative KP hierarchies, and that these theorems imply that we can prove the existence of a full class of smooth solutions to our non-commutative (t 2 , t 3 )-ZS equations. We present these existence results also in Section 3.

2.

The KP hierarchy of a general differential associative algebra 2.1. Algebraic setting. Let K be a characteristic zero field and let (A, +, * , ∂) be a differential associative algebra over K. We fix the following framework. Definition 2.1. We define

ΨDO(A) = n∈Z a n ξ n : a n ∈ A; a n = 0 for n >> 0
where ξ is a formal variable.

Hereafter we assume that A is unital, that is, we assume that there exists 1 ∈ A such that ∀a ∈ A, 1 * a = a * 1 = a. We call the elements of ΨDO(A) "pseudodifferential operators" even though we are aware that we are incurring in a slight abuse of terminology. Definition 2.2. Let us define addition and multiplication on ΨDO(A).

+ : ΨDO(A) × ΨDO(A) -→ ΨDO(A) ( a n ξ n , b m ξ m ) -→ (a p + b p )ξ p and * : ΨDO(A) × ΨDO(A) -→ ΨDO(A) a(ξ) = a n ξ n , b(ξ) = b m ξ m -→ m,n α∈N 1 α! (a n ∂ α b m )(D α ξ ξ n )ξ m Theorem 2.3. (ΨDO(A), +, * ) is a unital (associative) K-algebra.
This theorem is well-known, see for example [START_REF] Dickey | Soliton equations and Hamiltonian systems[END_REF], or the more recent paper [START_REF] Eslami Rad | The Cauchy problem of the Kadomtsev-Petviashvili hierarchy with arbitrary coefficient algebra[END_REF]. Hereafter we omit the multiplication symbol * or we use simply " • " . We also point out that ΨDO(A) is a graded algebra with deg ξ ( a n ξ n ) = max{n ∈ Z : a n = 0} and (by convention) deg ξ (0) = -∞. This grading allows us to define the vector subspaces ΨDO k (A) = {P ∈ ΨDO(A) : deg ξ (P ) ≤ k}, which will appear below.

Proposition 2.4. The decomposition ΨDO(A) = DO(A) ⊕ IO(A), in which IO(A) = ΨDO -1 (A) is the algebra of integral operators and DO(A) = ΨDO(A) \ ΨDO -1 (A) is the subalgebra of differential operators, is a vector space splitting. We note that IO(A) may be a non-unital algebra.

Proof. The proof relies on standard arguments:

• if (n, m) ∈ N 2 , then for all α ∈ N, we have that deg ξ (D α ξ ξ n )ξ m ∈ N ∪ {-∞}, which shows that DO(A) is stable under multiplication, • while for (n, m) ∈ (N * ) 2 , ∀α ∈ N, we have that deg ξ (D α ξ ξ -n )ξ -m < 0, which shows that DO(A) is stable under multiplication.
A consequence of this proposition is that any operator a ∈ ΨDO(A) decomposes uniquely under this direct sum as a = a D + a S for a D ∈ DO(A) and a S ∈ IO(A).

In the most classical framework, A is an algebra of functions equipped with a derivation ∂. Standard choices are A = C ∞ (S 1 , K) with K = R, C or, in [START_REF] Kuperschmidt | KP or mKP: Mathematics of Lagrangian, Hamiltonian and Integrable Systems[END_REF][START_REF] Mc Intosh | The quaternionic KP hierarchy and conformally immersed 2-tori on the 4sphere[END_REF], K = H, and ∂ = d dx . In these contexts, the algebras of functions A are Fréchet algebras equipped with natural notions of differentiability, and the operations of addition, multiplication and differentiation are smooth. It follows that addition and multiplication on ΨDO(A) are smooth in the following sense: if

a = n∈Z a n ξ n , b = n∈Z b n ξ n , we set a + b = c = n∈Z c n ξ n and ab = d = n∈Z d n ξ n .
Then, the map The foregoing set-up circumvent the use of the technical tools recently developed in [START_REF] Eslami Rad | The Cauchy problem of the Kadomtsev-Petviashvili hierarchy with arbitrary coefficient algebra[END_REF][START_REF] Magnot | Well-posedness of the Kadomtsev-Petviashvili hierarchy, Mulase factorization, and Frölicher Lie groups[END_REF], where a fully rigorous framework for smoothness on objects such as ΨDO(A), is described and used. The geometric approach to smoothness described therein is based on the theories of diffeologies and Frölicher spaces. We will not dwell on it here, we refer the reader to the mentioned papers [START_REF] Eslami Rad | The Cauchy problem of the Kadomtsev-Petviashvili hierarchy with arbitrary coefficient algebra[END_REF][START_REF] Magnot | Well-posedness of the Kadomtsev-Petviashvili hierarchy, Mulase factorization, and Frölicher Lie groups[END_REF] for details, but we do mention that, for instance, our point of view allows us to develop a fully rigorous approach to smoothness on Frobenius algebras -so as to justify all the formal calculations of [START_REF] Strachan | Integrability of the Frobenius algebra-valued Kadomtsev-Petviashvili hierarchy[END_REF][START_REF] Strachan | Frobenius manifolds and Frobenius algebra-valued integrable systems[END_REF][START_REF] Zhang | Hamiltonian structures and integrability of Frobenius algebra-valued (n,m)th KdV hierarchy[END_REF]-by adapting the considerations appearing in [START_REF] Eslami Rad | The Cauchy problem of the Kadomtsev-Petviashvili hierarchy with arbitrary coefficient algebra[END_REF].

((a n ) n∈Z , (b n ) n∈Z ) → ((c n ) n∈Z , (d n ) n∈Z ) is smooth as a map from Π Z A × Π Z A → Π Z A × Π Z A,
Remark 2.5. We comment on the relation of our approach with more standard (and also more restrictive) viewpoints. Our notion of smoothness restricts to the notion of smoothness in infinite dimensional geometry on manifolds (with atlases) modelled on complete locally convex topological vector spaces. This latter theory is discussed in [START_REF] Neeb | Towards a Lie theory of locally convex groups[END_REF]. In particular, see [START_REF] Magnot | Well-posedness of the Kadomtsev-Petviashvili hierarchy, Mulase factorization, and Frölicher Lie groups[END_REF], if we work with Fréchet manifolds of operators, being smooth in our sense is equivalent to being Fréchet smooth.

2.2.

Integration of the KP hierarchy on a differential algebra. Let T = {t n } n∈N * be an infinite set of formal (time) variables and let us consider the algebra of formal series ΨDO

(A[[T ]]) with infinite set of formal variables t 1 , t 2 , • with T -valuation val defined by val T (t n ) = n [13].
If we assume that the algebra operations on A are smooth, then so are the corresponding operations on A[[T ]], essentially because they are defined term by term using finite numbers of operations on A. This intuitive description can be fully formalized in the context of ultrametric completions, see [START_REF] Eslami Rad | The Cauchy problem of the Kadomtsev-Petviashvili hierarchy with arbitrary coefficient algebra[END_REF]. Now we extend this notion of smoothness to ΨDO(A[[T ]]) as in [START_REF] Eslami Rad | The Cauchy problem of the Kadomtsev-Petviashvili hierarchy with arbitrary coefficient algebra[END_REF][START_REF] Magnot | Well-posedness of the Kadomtsev-Petviashvili hierarchy, Mulase factorization, and Frölicher Lie groups[END_REF], and we are able to talk about smoothness at the level of pseudo-differential operators. We will make some further remarks on this theme after Theorem 2.6.

The Kadomtsev-Petviashvili (KP) hierarchy reads

(1)

dL dt k = (L k ) D , L , k ≥ 1 , with initial condition L(0) = L 0 ∈ ∂ + IO(A).
The dependent variable L is chosen to be of the form

L = ξ + α≤-1 u α ξ α ∈ ΨDO 1 (A[[T ]]) .
A standard reference on (1) is L.A. Dickey's treatise [START_REF] Dickey | Soliton equations and Hamiltonian systems[END_REF], see also [START_REF] Miwa | Solitons, Differential Equations, Symmetries and Infinite Dimensional Algebras[END_REF][START_REF] Mulase | Complete integrability of the Kadomtsev-Petvishvili equation[END_REF][START_REF] Mulase | Solvability of the super KP equation and a generalization of the Birkhoff decomposition[END_REF].

In order to solve the KP hierarchy, we need the following groups (see e.g. [START_REF] Eslami Rad | The Cauchy problem of the Kadomtsev-Petviashvili hierarchy with arbitrary coefficient algebra[END_REF][START_REF] Magnot | Well-posedness of the Kadomtsev-Petviashvili hierarchy, Mulase factorization, and Frölicher Lie groups[END_REF] for the latest adaptations of Mulase's constructions appearing in [START_REF] Mulase | Complete integrability of the Kadomtsev-Petvishvili equation[END_REF][START_REF] Mulase | Solvability of the super KP equation and a generalization of the Birkhoff decomposition[END_REF]):

Ḡ = 1 + IO(A[[T ]]) , Ψ = P = α∈Z a α ξ α : a α ∈ A[[T ]] , val T (a α ) ≥ α and P | t=0 ∈ 1 + ΨDO -1 (A) ,
where P | t=0 is the equivalence class P mod I, and

I is the ideal of A[[T ]] generated by {t 1 , t 2 , • • • }, and
D = P = α∈Z a α ξ α : P ∈ Ψ and a α = 0 for α < 0 .
We have a matched pair (see for example [START_REF] Majid | Matched pairs of Lie groups associated to solutions of the Yang-Baxter equations[END_REF])

Ψ = Ḡ D
which is smooth under the terminology we gave before. The following result, from [START_REF] Eslami Rad | The Cauchy problem of the Kadomtsev-Petviashvili hierarchy with arbitrary coefficient algebra[END_REF], gives a synthesized statement of our results on existence and uniqueness of solutions to the KP hierarchy [START_REF] Dickey | Soliton equations and Hamiltonian systems[END_REF], and on smooth dependence on the initial conditions.

Theorem 2.6.

Consider the KP hierarchy 1 with initial condition L(0) = L 0 . Then,

(1) There exists a pair (S, Y ) ∈ Ḡ×D such that the unique solution to Equation (1) with L| t=0 = L 0 is

L(t 1 , t 2 , • • • ) = Y L 0 Y -1 = SL 0 S -1 .
(2) The pair (S, Y ) is uniquely determined by the smooth decomposition problem

exp k∈N τ k L k 0 = S -1 Y ,
and the solution L depends smoothly on the initial condition L 0 .

(3) The solution operator L is smoothly dependent on the initial value L 0 .

In this theorem the expression "smoothly dependent" is to be understood essentially as explained in the two paragraphs before Remark 2.5, namely, it means that each coefficient of the solution L = L = ξ + α≤-1 u α ξ α , understood as a series in the T variables, depends smoothly on a finite number of the coefficients of the initial value L 0 . Indeed, if one reads the computations given in e.g. [START_REF] Mulase | Solvability of the super KP equation and a generalization of the Birkhoff decomposition[END_REF][START_REF] Eslami Rad | The Cauchy problem of the Kadomtsev-Petviashvili hierarchy with arbitrary coefficient algebra[END_REF][START_REF] Magnot | Well-posedness of the Kadomtsev-Petviashvili hierarchy, Mulase factorization, and Frölicher Lie groups[END_REF] where the computations of coefficients are sketched, it is possible to check that each coefficient of L is obtained via a large but finite number of smooth operations (addition, multiplication, inversion, derivation) in A[[T ]] from the coefficients of L 0 .

2.3. ZS-Equations deduced from the hierarchy. It is well-known that the zero curvature equations

(2) ∂ ∂t m L n + - ∂ ∂t n L m + = [L m + , L n + ]
are a consequence of the KP hierarchy, see for instance [1, Proposition 5.1.4] or [START_REF] Etigoff | Factorization of differential operators, quasideterminants, and nonabelian Toda field equations[END_REF], and that these equations are a system of non-linear equations in three independent variables, namely, t n , t m and the "space variable" implicit in the derivation ∂ of the differential algebra A. Our Theorem 2.6 implies the following result (in the proposition below, we put t k = 0 for k = n, m):

Proposition 2.7. Let L = ξ + α≤-1 u α ξ α ∈ Ψ 1 (A[[T ]]
) and assume that the zero curvature equation (2) is a system of equations for N dependent variables

u 1 , • • • , u N . Fix u 1,0 , • • • , u N,0 ∈ A. Then, the system of equations (2) has a unique smooth solution L with u 1 (0) = u 1,0 , • • • , u N (0) = u N,0
, and this solution is smooth with respect to initial conditions.

Proof. We define

L(0) = L 0 ∈ ∂ + IO(A) as L(0) = ξ + u 1,0 ξ -1 + • • • + u N,0 ξ -N ,
we solve the KP hierarchy with initial condition L 0 , and we apply [1, Proposition 5.1.4].

We note that this proposition does not solve the full Cauchy problem for (2), since our initial condition fixes only the "space" dependence of the solution, while the dependence on t m , t n is fixed by the solution to the complete KP hierarchy with initial data L 0 defined as in the above proof. Nonetheless we think it is a relevant result on solutions to very general zero curvature equations.

We consider the set of independent variables {t 1 , t 2 , t 3 } and write down explicitly the corresponding zero curvature equations; we are mostly interested in the equations arising from the set {t 2 , t 3 }, but we also try the {t 1 , t 2 } and {t 1 , t 3 } easy cases for completeness.

With the notation

L = k≤1 u k ξ k
with u 1 = 1 and u 0 = 0, we have

L 2 L 3 σ 3 - ξ 3 σ 2 ξ 2 0 σ 1 0 3u -1 ξ σ 0 2u -1 ξ 0 (3u -2 + 3∂u -1 )ξ 0
in which σ k , k = 0, 1, 2, 3, indicates higher symbols (that is, homogeneous parts of degree k) of the corresponding pseudo-differential operator, and therefore

L 2 + , L 3 + = ξ 2 , ξ 3 + 3 ξ 2 , u -1 ξ + 3 ξ 2 , u -2 + ∂u -1 +2 u -1 , ξ 3 + 6 [u -1 , u -1 ξ] + 6 [u -1 , u -2 + ∂u -1 ] = 0 + (6∂u -1 ξ 2 + 3∂ 2 u -1 ξ) + (6∂u -2 ξ + 3∂ 2 u -2 + 6∂ 2 u -1 ξ + 3∂ 3 u -1 ) +(-6∂u -1 ξ 2 -6∂ 2 u -1 ξ -2∂ 3 u -1 ) + (6[u -1 , u -2 ] -6∂u -1 u -1 ) = (3∂ 2 u -1 + 6∂u -2 )ξ + ([u -1 , u -2 ] -6∂u -1 u -1 + ∂ 3 u -1 + 3∂ 2 u -2 ) .
The ZS-equations for the pairs (t 1 , t 2 ) and (t 1 , t 3 ) read respectively as

(3) dL 2 + dt 1 - dL + dt 2 = L + , L 2 + which gives: (4) du -1 dt 1 = ∂u -1 ,

and

(5)

dL 3 + dt 1 - dL + dt 3 = L + , L 3 + which gives: (6) du-1 dt1 = ∂u -1 du-2 dt1 = ∂u -2
which corresponds to the first equations of the hierarchy

du -k dt1 = du -k dx when A = (C ∞ (R), • , d/dx).
Here, ∂ is an arbitrary derivation so that, instead of the usual identification d/dt 1 = d/dx, we get for example the formal integration formula

u -1 = exp(t 1 ∂)u 0 for an initial value u -1 | t1=0 = u 0 .
We now go to the ZS-equation for the pair (t 2 , t 3 ), ( 7)

dL 3 + dt 2 - dL 2 + dt 3 = L 2 + , L 3 + .
We have:

L 2 + = ξ 2 + 2 u -1 , L 3 + = ξ 3 + 3 u -1 ξ + 3 ∂u -1 + 3 u -2 and [L 2 + , L 3 + ] = (3∂ 2 u -1 + 6∂u -2 )ξ + 6[u -1 , u -2 ] -6∂u -1 u -1 + ∂ 3 u -1 + 3∂ 2 u -2 . Equation (7) yields du -1 dt 2 = ∂ 2 u -1 + 2∂u -2 (8) d(u -2 + ∂u -1 ) dt 2 - 2 3
du -1 dt 3 = 2[u -1 , u -2 ] -2∂u -1 u -1 + 1 3 ∂ 3 u -1 + ∂ 2 u -2 . ( 9
)
If we set 2u -1 = u, t 2 = y and t 3 = t, system (8), [START_REF] Magnot | On the Kadomtsev-Petviashvili hierarchy in an extended class of formal pseudo-differential operators[END_REF] becomes

u y = ∂ 2 u + 4∂u -2 (10) ∂u y + 2u -2,y - 2 3 u t = 2[u, u -2 ] -∂u u + 1 3 ∂ 3 u + 2∂ 2 u -2 . ( 11
)
The system [START_REF] Magnot | Well-posedness of the Kadomtsev-Petviashvili hierarchy, Mulase factorization, and Frölicher Lie groups[END_REF], ( 11) reduces to the system appearing in [1, Exercise 5.

1.8] if u -1 , u -2 commute.
We remark that this system of equations is obtained directly from our PDOs calculus without further assumptions on the derivation ∂. The classical KP-II equation on C ∞ (S 1 , R) can be deduced from it by an adequate substitution procedure, as explained in [START_REF] Dickey | Soliton equations and Hamiltonian systems[END_REF]. In order to work in generalized settings, we stop our computations at system (10)- [START_REF] Mendoza | Massey products, A ∞ -algebras, differential equations, and Chekanov homology[END_REF], because the substitution procedure just mentioned may not be available in them.

KP equations and hierarchies in various contexts

3.1. When ∂ is an exterior derivation. We say that ∂ is an exterior derivation if ∂ 2 = 0. In this case, uξ m • vξ n = (uv)ξ m+n + (m u ∂v) ξ m+n-1 , and therefore the zero-curvature equations ( 8) and ( 9) read ( 12)

du-1 dt2 -2∂u -2 = 0 du-2 dt2 -2 3 du-1 dt3 = -2∂u -1 u -1 + 2[u -1 , u -2 ] ,
in which we have used that the first equation implies that ∂u -1,t2 = 0. Now we set 2u -1 = u, t 2 = y and t 3 = t. Equation ( 10) yields ( 13) u = 4∂ -1 y ∂u -2 , and therefore equation [START_REF] Mendoza | Massey products, A ∞ -algebras, differential equations, and Chekanov homology[END_REF] becomes

∂ y u -2 - 4 3 ∂ -1 y ∂ t ∂u -2 = 4[∂ -1 y ∂u -2 , u -2 ] ,
this is

(14) ∂ 2 y u -2 - 4 3 ∂ t ∂u -2 = 4∂ y [∂ -1 y ∂u -2 , u -2 ] ,
an equation that reduces to a linear wave equation in the commutative case. Proposition 2.7 yields the result Proposition 3.1. Let us fix u ∈ A. Equation ( 14) has a solution u -2 (y, t) with u -2 (0, 0) = u, and this solution is smooth with respect to the initial condition u.

Proof. We choose u(y) such that ∂ y u(y) = 4∂u. Such a u(y) exists because the derivation ∂ y is surjective on A[[y, t]]. Now we apply Proposition 2.7 with u 1,0 = (1/2) u(0) and u 2,0 = u, thereby obtaining a solution u -1 (y, t) and u -2 (y, t) to ( 12), or, equivalently, to the system ( 13)- [START_REF] Mulase | Solvability of the super KP equation and a generalization of the Birkhoff decomposition[END_REF]. By construction, the solution u -2 (y, t) satisfies u -2 (0, 0) = u and our general theory implies that it is smooth with respect to the initial condition u.

3.2.

On the case when ∂ is the gradient. Let n ∈ N * and let us consider A = C ∞ (R n , R n ) with component-wise operations (this is an instance of a "Hadamard product"). A is therefore a commutative and associative algebra, but it not an integral domain. The algebra A can be equipped with the topology of uniform convergence of derivatives at any order on any compact subset of R n . For the first part of the computations, we can also consider A = C ∞ c (R n , R n ) (functions with compact support) equipped with the Whitney topology (see. e.g. [START_REF] Kriegl | The convenient setting for global analysis Math[END_REF]) or

A = S(R n , R n ) (Schwartz space).
We denote by x = (x 1 , ...

x n ) t a point in R n , by u = (f 1 (x), • • • , f n (x)
) t a point in A, and we define the vector derivation ∇ by ( 15)

∇u = ∇(f 1 (x), ...f n (x)) t = ∂f 1 ∂x 1 (x), ..., ∂f n ∂x n (x) t .
Our notation is justified by the fact that "on the diagonal",

f 1 = f 2 = • • • = f n , our derivation ∇ is indeed standard gradient.
The fact that ∇ is a derivation is clear; let us check the Leibniz rule for benefit of the reader:

We take u = (f

1 (x), • • • , f n (x)) t and v = (g 1 (x), • • • , g n (x)) t in A.
Then (we omit the argument x for brevity),

∇(u • v) = ∂(f 1 g 1 ) ∂x 1 , ..., ∂(f n g n ) ∂x n t = ∂f 1 ∂x 1 g 1 + f 1 ∂g 1 ∂x 1 , ..., ∂f n ∂x n g n + f n ∂g n ∂x n t = ∂f 1 ∂x 1 , • • • , ∂f n ∂x n t (g 1 , • • • , g n ) t + (f 1 , • • • f n ) t ∂g 1 ∂x 1 , ..., ∂g n ∂x n t = ∇(u) v + u ∇(v) .
We note that this computation shows that we can replace the partial derivatives [START_REF] Neeb | Towards a Lie theory of locally convex groups[END_REF] for any n-tuple of smooth vector fields X 1 , • • • X n on R n and still obtain a meaningful derivation on A. We stick with our original choice in view of Remark 3.2 below. Equations ( 7) read as follows (for any of the three choices of A mentioned at the beginning of this subsection):

∂ ∂x1 , • • • , ∂ ∂xn appearing in
(16) du-1 dt2 = ∇ 2 u -1 + 2∇u -2 -2 du-1 dt3 + 6u -1 ∇u -1 -3∇ 2 u -1 = -2∇ 3 u -1 -3∇ 2 (u -2 + u -1 ) -3 du-2 dt2 . Assuming that A = C ∞ (R n , R n ) equipped with the smooth compact-open topol- ogy, we can choose v ∈ C ∞ (R n , R n )[[t 2 , t 3 ]] such that du-2 dt2 = ∇(v). Setting h 1 = 2∇ 2 u -1 -3v -3 dv dt2 -3∇u -1 , we get: (17)
       d 2 u-1 dt 2 2 = ∇ 2 du-1 dt2 + 2v ∇h 1 = -2 du-1 dt3 + 6u -1 ∇u -1 -3∇ 2 u -1 3 dv dt2 + 3v = -h 1 + 2∇ 2 u -1 -3∇u -1
where t 2 and t 3 are external R-variables. We note that we obtain a "1D Navier-Stokes-like" equation in the second line. In general we can make the following Remark 3.2. We can get the Navier-Stokes equation at any dimension on

A = C ∞ (R n , R n ) if we impose the additional constraint div(u -1 ) = n k=1 ∂u -1,k ∂x k = 0. Indeed, let h be a solution of the equation ∇h = ∇h 1 + 3∇ 2 u -1 -ν∆u -1
we get a four lines system:

(18)              d 2 u-1 dt 2 2 = ∇ 2 du-1 dt2 + 2v ∇h = -2 du-1 dt3 + 6u -1 ∇u -1 + ν∆u -1 3 dv dt2 + 3v = -h 1 + 2∇ 2 u -1 -3∇u -1 ∇h = ∇h 1 + 3∇ 2 u -1 -ν∆u -1
The added line has a particular solution

h = h 1 + 3∇u -1 -ν x1 0 ∆u -1 (z 1 , ...z n )dz 1 , ..., ν xn 0 ∆u -1 (z 1 ; ..., z n )dz n t only valid in C ∞ (R n , R n )[[t 2 , t 3 ]]. 3.3. On C ∞ (R, H).
Let H be the field of quaternions. We give here explicitly the equations derived from the quaternionic KP hierarchy [START_REF] Mc Intosh | The quaternionic KP hierarchy and conformally immersed 2-tori on the 4sphere[END_REF][START_REF] Kuperschmidt | KP or mKP: Mathematics of Lagrangian, Hamiltonian and Integrable Systems[END_REF], obtained for ∂ = d dx and for A = C ∞ (S 1 , H) or A = C ∞ (R, H). Equations ( 8) and ( 9) become:

(19) du-1 dt2 = ∂ 2 u -1 + 2∂u -2 3 du-2 dt2 -2 du-1 dt3 = [u -1 , u -2 ] -6∂u -1 u -1 -2∂ 3 u -1 -3∂ 2 u -2
where multiplication is the H-multiplication, while the system (10), [START_REF] Mendoza | Massey products, A ∞ -algebras, differential equations, and Chekanov homology[END_REF] become

u y = ∂ 2 u + 4u -2
and the quaternionic KP equation Therefore we get the following system:

3u yy = ∂ 4u t -6∂u u -∂ 3 u + 3∂ u, ∂ -1 u y -∂u or (20) 3u yy = ∂ 4u t -3(∂u u + u ∂u) -∂ 3 u + 3∂ u, ∂ -1 u y .
(21)                            d dt2 a = ∂ 2 a + 2∂a , d dt2 b = ∂ 2 b + 2∂b , d dt2 c = ∂ 2 c + 2∂c , d dt2 d = ∂ 2 d + 2∂d 3 da dt2 -2 da dt3 = -6 (a∂a + b∂b + c∂c + d∂d) -2∂ 3 a -3∂ 2 a 3 d(b +∂b) dt2 -2 db dt3 = c c d d -6 (a∂b + b∂a + d∂c -c∂d) -2∂ 3 b -3∂ 2 b 3 dc dt2 -2 dc dt3 = d d b b -6 (a∂c + c∂a + c∂d -d∂c) -2∂ 3 c -3∂ 2 c 3 d(d +∂d) dt2 -2 dd dt3 = b b c c -6 (a∂d + d∂a + c∂b -b∂c) -2∂ 3 u -1 -3∂ 2 d
Proposition 2.7 tells us that given u 0 -1 , u 0 -2 ∈ H, this system has a unique smooth solution with u -1 (0) = u 0 -1 , u -2 (0) = u 0 -2 , and that this solution is smooth with respect to initial conditions. We also have: Proof. We choose u 0 -2 = (1/4)(u -∂ 2 u) and we apply Proposition 2.7 with u 1,0 = (1/2)u and u 2,0 = u 0 -2 . By construction, the solution u(y, t) = 2u -1 (y, t) solves [START_REF] Strachan | Integrability of the Frobenius algebra-valued Kadomtsev-Petviashvili hierarchy[END_REF], it satisfies u(0, 0) = u, and our general theory implies that this solution is smooth with respect to the initial condition u.

3.4. On the case when ∂ is a derivation on a Lie algebra. Let g be a Lie algebra and let T g be its tensor algebra, namely

T g = ⊕ k∈N T k (g) , T k (g) = g ⊗k = g ⊗ • • • ⊗ g , (T 0 (g) = K) .
A derivation D ∈ Der(g) induces a derivation ∂ on T g by the formulas:

∀λ ∈ K, ∂λ = 0, ∂(u 1 ⊗• • •⊗u n ) = (Du 1 )⊗u 2 ⊗• • •⊗u n +u 1 ⊗(Du 2 )⊗u 3 • • • u n +• • •+u 1 ⊗• • •⊗u n-1 ⊗(Du n ) for all u 1 , • • • , u n ∈ g.
Equations ( 7) read as:

                   du (0) -1 dt2 = 0 du (0) -1 dt3 = 3 2 du (0) -2 dt3 du (1) -1 dt2 = D 2 u (1) -1 + 2Du (1) -2 3 du (1) -2 dt2 -2 du (1) -1 dt3 = -2D 3 u (1) -1 -3D 2 u (1) -2

• • •

Let us specialize our computations when D is an inner derivative. Let a ∈ g. We set D = [a, .] and we examine the equations obtained by considering the grading of the tensor algebra,

u -k = u (0) -k + u (1)
-k + u

-k + • • • Equation [START_REF] Kuperschmidt | KP or mKP: Mathematics of Lagrangian, Hamiltonian and Integrable Systems[END_REF], in its lowest tensor orders, reads:

                   du (0) -1 dt2 = 0 du (0) -1 dt3 = 3 2 du (0) -2 dt3 du (1) -1 dt2 = a, a, u (1) -1 + [2a, u (1)
-2 ] 3 du (1) -2 dt2 -2 du (1) -1 dt3 = -2 a, a, a, u (1) -1 -3 a, a, u (1) -2
• • • Proposition 2.7 allow us to conclude that the Cauchy problem of this infinite system of equations can be solved in our algebraic setting. We omit the details.

Let us apply this on g = V ect(R n ) and a = ∂ 1 + • • • ∂ n , the same formulas apply and with the same notations and definitions as in section 3.2 we have [a, .] = ∇ as in section 3.2 but here the same equations read as:

                   du (0) -1 dt2 = 0 du (0) -1 dt3 = 3 2 du (0) -2 dt3 du (1) -1 dt2 = ∇ 2 u (1) -1 + 2∇u (1) -2 3 du (1) -2 dt2 -2 du (1) -1 dt3 = -2∇ 3 u (1) -1 -3∇ 2 u (1) -2 • • •
∀T ∈ End(A), ∂ ξ T = T X -XT = [-X, T ].
Following its fundamental properties [START_REF] Pincherle | Operatori lineari e coefficienti di fattoriali[END_REF][START_REF] Rota | On the Foundations of Combinatorial Theory. VIII: Finite Operator Calculus[END_REF], since ∂ ξ X = 0 and ∂ ξ d dX = 1, it follows that ∂ ξ is also known as a derivation on (polynomial) differential operators A d dX . Proposition 3.4. Let A = K((X)) or A = K((X -1 )). i.e. Then:

(1) In the sequel, the derivation in the formal X-variable will be noted by ∂ X .

∂ ξ A = 0 (
Proof. (1) and ( 2) are straightforward from the definitions. In order to prove (3), we get by induction that ∂ ξ ξ -n = -nξ n+1 , which defines it by A -linearity, and then we can check that ∂ ξ satisties the Leibnitz rule on ΨDO(A ).

We now consider ΨDO(B) =

k∈Z a k ζ k : a k ∈ B, a k = 0, k >> 0 . If P ∈ ΨDO(B), we can write P = k∈Z a k ζ k (or "P = k∈Z a k ∂ k ξ ") that is, ζ is the formal variable that corresponds to the derivation ∂ ξ on B.

Thus, we have a "hybrid" system, a Lie algebra bracket and a commutator in an associative algebra:

du-1 dt2 = [X, [X, u -1 ] -2[X, u -2 ] 3 du-2 dt2 -2 du-1 dt3 = [u -1 , u -2 ] + 6[X, u -1 ]u -1 + 2[X, [X, [X, u -1 ]]] -3[X, [X, u -2 ]
] . Once again, Proposition 2.7 allow us to conclude that the Cauchy problem of this system of equations can be solved in our algebraic setting.

3.6. Moyal KP equation. We work in the framework of Hamanaka's paper [START_REF] Hamanaka | Commuting flows and conservation laws for noncommutative Lax hierarchies[END_REF], see also [START_REF] Sakakibara | Factorization methods for noncommutative KP and Toda hierarchy[END_REF][START_REF] Takasaki | Nonabelian KP hierarchy with Moyal algebraic coefficients[END_REF]. We take independent variables x k equipped with non-commutative multiplication [x k , x l ] = i θ kl and we consider the induced multiplication on functions f (x 1 , x 2 , • • • ) given by

f (x) g(x) = f (x) • g(x) + i 2 θ kl ∂ x k f (x)∂ x l g(x) + O(θ 2 ) .
In this context [START_REF] Magnot | Well-posedness of the Kadomtsev-Petviashvili hierarchy, Mulase factorization, and Frölicher Lie groups[END_REF], [START_REF] Mendoza | Massey products, A ∞ -algebras, differential equations, and Chekanov homology[END_REF] become

u y = ∂ 2 u + 4∂u -2
and the Moyal KP equation [START_REF] Strachan | Frobenius manifolds and Frobenius algebra-valued integrable systems[END_REF] 3u yy = ∂ 4u t -3(∂u u + u ∂u) -

∂ 3 u + 3∂ u, ∂ -1 u y ,
where the commutator is [f, g] = f g -g f . This last equation is exactly Equation (3.17) in [START_REF] Hamanaka | Commuting flows and conservation laws for noncommutative Lax hierarchies[END_REF]. Soliton solutions to [START_REF] Strachan | Frobenius manifolds and Frobenius algebra-valued integrable systems[END_REF] have been found by Etigoff, Gelfand and Retakh, see [START_REF] Etigoff | Factorization of differential operators, quasideterminants, and nonabelian Toda field equations[END_REF], and also by Paniak in [START_REF] Paniak | Exact Noncommutative KP and KdV Multi-solitons[END_REF]. We finish this paper with a general result on the initial value problem for [START_REF] Strachan | Frobenius manifolds and Frobenius algebra-valued integrable systems[END_REF]. We omit its proof.

  namely, each coefficient c n and d n depend smoothly (in the usual way) on finite numbers of coefficients a n and b n .

Let us expand u - 1 =- 1 u - 1 =

 111 a + ib + jc + kd and u -2 = a + ib + jc + kd where (a, b, c, d, a , b , c , d ) ∈ A 8 [[T ]]. We compute: [u -1 , u -2 ] = a∂a + b∂b + c∂c + d∂d + (a∂b + b∂a + d∂c -c∂d) i + (a∂c + c∂a + c∂d -d∂c) j + (a∂d + d∂a + c∂b -b∂c) k

Proposition 3 . 3 .

 33 Let us fix u ∈ A. Equation (20) has a solution u(y, t) ∈ A[[y, t]] with u(0, 0) = u, and this solution is smooth with respect to the initial condition u.

3. 5 .

 5 On Pincherle derivative. Let us consider A = K[X], with K = R or C. The Pincherle derivative is defined as follows:

2 )

 2 ∂ ξ extends to a derivation on A d dX . (3) ∂ ξ extends to a derivation on the algebra B = ΨDO(A ) = k∈Z a k ξ k : a k ∈ A , a k = 0, k >> 0 by the relation ∂ ξ ξ = 1.

Proposition 3 . 5 .

 35 Let us fix u ∈ (A, ). Equation[START_REF] Strachan | Frobenius manifolds and Frobenius algebra-valued integrable systems[END_REF] has a solution u(y, t) ∈ A[[y, t]] with u(0, 0) = u, and this solution is smooth with respect to the initial condition u.

Acknowledgements: V.R. thanks IHES for hospitality; E.G.R.'s research is partially supported by the FONDECYT grant #1201894.


Marie-Dominique Heusse

Guillaume Cabanac
email: guillaume.cabanac@univ-tlse3.fr

ORCID growth and field-wise dynamics of adoption: A case study of the Toulouse scientific area

Keywords: bibliography, disambiguation, ORCID, researcher identifier, multidisciplinarity

Research-focused information systems harvest and promote the scientific output of researchers. Disambiguating author identities is key when disentangling homonyms to avoid merging several persons' records. ORCID offers an identifier to link one's identity, affiliations, and bibliography. While funding agencies and scholarly publishers promote ORCID, little is known about its adoption rate. We introduce a method to quantify ORCID adoption according to researchers' discipline and occupation in a higher-education organisation. We semi-automatically matched the 6,607 staff members affiliated to the 145 labs of the Toulouse scientific area with the 7.3 million profiles at orcid.org. The observed ORCID adoption of 41.8% comes with disciplinewise disparities. Unexpectedly, only 48.3% of all profiles listed at least one work and profiles with no works might just have been created to get an identifier. Those 'empty' profiles are of little interest for the entity disambiguation task. To our knowledge, this is the first study of ORCID adoption at the scale of a multidisciplinary scientific metropole. This method is replicable and future studies can target other cases to contrast the dynamics of ORCID adoption worldwide.

INTRODUCTION

Scholarly publications feature the names of the contributing authors in the byline of the articles. With an increasing number of scholarly works published each year, the number of homonyms is growing too: "Of the more than 6 million authors in a major journal citations and abstracts database, more than two-thirds of them share a last name and single initial with another author, and an ambiguous name in the same database refers on average to eight people." [START_REF] Sabine | You know who you are, but do others? Share: the newsletter of the Australian National Data Service[END_REF] Such ambiguity has several implications, a critical one being identity theft. The recent Surgisphere scandal a.k.a. LancetGate stressed this issue of homonymy detection [START_REF] Piller | Who's to blame? These three scientists are at the heart of the Surgisphere COVID-19 scandal[END_REF]. A US-based tenured faculty padded his CV with two-thirds of publications not written by him but by homonyms.

Bibliometric studies also suffer from the 'namesake' problem and [START_REF] Harzing | Health warning: might contain multiple personalities-the problem of homonyms in Thomson Reuters Essential Science Indicators[END_REF] stressed the case of author Y. Wang publishing 9 papers a day as per Web of Science data. In reality, this identity amalgamates thousands of academics whose production is considered as one.

Research organisations worldwide use Current Research Information Systems (CRIS) to collect, analyse, and share their research output [START_REF] Fabre | The visibility of authority records, researcher identifiers, academic social networking profiles, and related faculty publications in search engine results[END_REF][START_REF] Sivertsen | Developing current research information systems (CRIS) as data sources for studies of research[END_REF]. They rely on document identifiers (e.g., DOI) and author identifiers, such as VIAF, ISNI, and later ORCID in October 2012. 1 ORCID stands for 'Open Researcher and Contributor ID' and is operated by a non-profit organisation that mints identifiers for authors of scholarly works [START_REF] Haak | ORCID: A system to uniquely identify researchers[END_REF]. As of May 2021, 2 there were 11 million ORCIDs created, a number greater than 7.8 million researchers, a UNESCO estimate of the worldwide population of researchers (Soete, Schneegans, Eröcal, Angathevar, & Rasiah 2015, p. 33). This discrepancy may partly come from non-researchers creating ORCID profiles, such as support staff and librarians. Organisations and publishers also adopted and promoted ORCID: funding agencies and editorial managers request an ORCID to submit a proposal, a manuscript or a report during peer-review [START_REF] Haak | Using ORCID, DOI, and other open identifiers in research evaluation[END_REF][START_REF] Hanson | Early adopters of ORCID functionality enabling recognition of peer review: Two brief case studies[END_REF]. ORCID has also become a key component for open archives [START_REF] Brown | Open access in context: Connecting authors, publications and workflows using ORCID identifiers[END_REF].

ORCID PROFILE: CREATION, UPDATE, AND VISIBILITY SETTINGS

There were two ways to create an ORCID profile. On the one hand, up until 2016 an institution could create an ID for its employees, such as the University of Colorado 3 . As of March 2021, ORCID listed 11 million profiles and 1,230 member organisations, 80% of which were of type 'research institute.' 4 There were 6 French institutional members (4 research institutes and 2 publishers) at the end of 2019. Meanwhile many French institutions joined a consortium agreement with ORCID in December 2019. 5 As of March 2021, ORCID listed 44 French members (40 research institutes, 2 repository/profile organisations, 1 funder, 1 intergovernmental economic organisation), 91% of which being research institutes. To the best of our knowledge, none of the French members created ORCIDs on behalf of their staff, though.

On the other hand, authors are free to sign up with their email and obtain an ID. They are asked to pick a visibility setting for their profile: public, restricted to trusted parties only, or private (Fig. 1). Several sections make an ORCID profile, as shown in Fig. 2: Person identifiers, Employment, Education and qualifications, Invited positions and distinctions, Membership and service, Funding, and Works. These are completed by the profile owner and trusted parties upon permission. Other sections are updated by trusted parties only with the owner's explicit permission: Peer review and Research resources. 6 Profile owners can set the visibility of each item in all sections of their profiles. For instance, a publicly available profile may contain items that are available to trusted parties only, or even visible to the owner only (e.g., some confidential works). ORCID operates no control over the contents of the profiles. As a result, many profiles are void of information: all sections appear to be empty, and even the owner's identify is undisclosed (we elaborate on this in Sect. 5). There is no way for readers to know if owners entered no data at all or if owners chose to mask some entries (picking a visibility different from 'Everyone'). While some authors prefer not to disclose information legitimately, Teixeira da Silva (2021) reported some abusive profile creations by papers mills to fool publishers at paper submission time.

ORCID owners can contribute data themselves or use built-in import features. For some sections, such as Funding and Works, owners provide an identifier (e.g., DOI, PubMed ID, ArXiv ID) or a search query and the system fetches all relevant metadata. Haak et al. (2019, p. 15) reported that 67% of the 49 million works in ORCID profiles were added through an API provided by Scopus, ResearcherId, PubMed Central, and CrossRef mostly. According to an OECD survey of corresponding authors of Scopus-indexed peer-reviewed journals, 68% of the 6,476 corresponding authors from a representative sample of territories and fields of science had an ORCID (Bello & Galindo-Rueda 2020, p. 74). This figure might overestimate the adoption of ORCID, as it concerns selective venues and selected scientists (corresponding authors) who are likely to have created an ORCID during the submission process.

To overcome these biases, we designed a study focusing on a large scientific area in France (Heusse & Cabanac 2020) gathering 6,471 research staff (a sample of equivalent size compared to the OECD study). We found an adoption rate of about 40% in the Toulouse area, to be compared with the 17% in Caen University, a smaller site accounting for 1,047 research staff members [START_REF] Boudry | Use of author identifier services (ORCID, ResearcherID) and academic social networks (Academia.edu, ResearchGate) by the researchers of the University of Caen Normandy (France): A case study[END_REF]. To the best of our knowledge, there is no other estimate of ORCID adoption in the general research population of a scientific metropole. The present paper extends our previous study published in French [START_REF] Heusse | Adoption de l'identifiant chercheur ORCID : le cas des universités toulousaines[END_REF] to report the ORCID adoption in the Toulouse scientific area for the period 2012-2020.

CASE STUDY: THE TOULOUSE SCIENTIFIC AREA

This article aims to assess the dynamics of ORCID adoption in one of the leading academic site in France.

About twenty years ago, European countries have harmonised the structure of training at university. The organisation of research, however, is not uniform across Europe. This article focusing on a scientific area in France, let us briefly summarise the higher-education landscape in France [START_REF] Angermuller | Academic careers and the valuation of academics. A discursive perspective on status categories and academic salaries in France as compared to the U.S., Germany and Great Britain[END_REF][START_REF] Chevaillier | French academics: Between the professions and the civil service[END_REF][START_REF] Grossetti | Spatial organisation of french research from the scholarly publication standpoint (1999-2017): Long-standing dynamics and policy-induced disorder[END_REF]. This context is important to bear in mind when interpreting the results presented in this paper.

France had 115,308 teaching and research staff (Meuric 2020, p. 91) in 2017 working as tenured civil servants or contract workers:

• Academics employed by universities and higher schools (grandes écoles) share their agenda between research and teaching (192 hours a year of face-to-face teaching). Nine-tenth are tenured staff (Meuric 2020, p. 100).

• Full-time research staff employed by national institutes (by decreasing number of researchers: CNRS, INRAE, INSERM, IRD, INRIA and a few smaller institutes with less than 200 researchers) and to a lesser extent by universities and higher schools, dedicate most of their time to research. They are not expected to teach yet some do teach a few hours a year.

Whatever their employer, all staff are affiliated to a public research laboratory (laboratoire de recherche) which provides office space and research facilities (Chevaillier 2001, p. 57). A laboratory is not usually run by a single PI (as in some countries like in the US) but gather many teams (équipes) and host from 30 to 500+ members.

The Toulouse scientific area is third after Paris and Lyon in terms of scientific output [START_REF] Grossetti | Spatial organisation of french research from the scholarly publication standpoint (1999-2017): Long-standing dynamics and policy-induced disorder[END_REF]). More than 100,000 students a year are trained in all disciplines, 45% of which are taught in a masters or doctoral programme. More than 9,000 researchers whose disciplinary breakdown is detailed in Sect. 4.1 work in 145 public research laboratories. The Toulouse scientific area is known best for its contributions in health, astronomy, universe science, and economics.

METHOD AND DATA

This section introduces the data collection protocol we used. We detail how the demographic data retrieved from the institutions were matched to ORCID profiles and manually validated. The resulting dataset supports the longitudinal analysis of ORCID adoption at the scale of a French scientific conurbation. To the best of our knowledge, this is the first study of ORCID adoption at this geographic scale.

Demographics and bibliographic data collection

Demographic data of the 6,607 persons affiliated to the research centres in the Toulouse scientific area were collected in a previous study [START_REF] Heusse | Faire parler les données sur la recherche grâce au web sémantique : le projet VIVO[END_REF]. This staff registry tabulates 5,029 faculty and researchers as well as 1,578 support staff involved in research activities: engineers, technicians, clinical research fellows. The 4,284 PhD students enrolled in universities and schools (écoles) were not included. Each individual included in the registry is characterised by his/her identify (first name, last name), year of birth, sex, institution, job category, rank, laboratory, and scientific domain. The staff registry lists individuals (Fig. 3) with the following job categories:

• Faculty (enseignants-chercheurs) teach and conduct research activities as junior faculty (maître de conférences) or senior faculty (professeur des universités).

• Researchers are involved in research and can teach without it being mandatory. Four ranks were considered: postdocs (post-doctorant), junior (chargé de recherche), senior (directeur de recherche), and undefined (e.g., visitors).

• Other Staff refers to support personnel, such as engineers (ingénieur d'étude and ingénieur de recherche), high-school teachers affiliated to university (enseignant du second degré), hospital practitioners (praticien hospitalier).

Demographic data were collated by each research structure (145 laboratories). We merged these data into the staff registry of the Toulouse scientific area and tagged each person with the discipline of the lab he/she belongs to. Each lab is associated to one of the six following discipline groups delineated by the research council of the Toulouse scientific area to reflect its major scientific highlights: Figure 4 shows how the 6,607 distribute among job categories and ranks. The AHS and LEM disciplines have a larger number of faculty compared to researchers and other staff. The situation is reversed in AAE and HBA where most of the personnel is no faculty but researchers and other staff. The large number of other staff in HBA corresponds to clinical research personnel who are active in teaching hospitals (centre hospitalier universitaire). The sex ratio is imbalanced across disciplines (Fig. 5), with more males in all disciplines but Art, Humanities, Social Sciences and Health, Biology, Agronomy.

We aimed to retrieve the ORCID profile of the 6,607 individuals by querying the ORCID public API 7 with their firstname and lastname. Results were overabundant for some identities such as "Philippe Durand" that yielded 4,534 profiles. The top-ranked profiles were perfect homonyms and the remaining were partial matches such as "Philippe S. Durand" and "Romain Durand." 8 We considered the top 20 results per query only. Each retrieved profile was downloaded as an XML file 9 recording the profile creation date, the date of last update, and the person's identity, biography, affiliations, and works. Part of these data appear on the online version of ORCID profiles (Fig. 2).

We tagged the profiles mentioning the named entities relevant to the Toulouse scientific area: 12 cities (the largest ones being Auch, Castres, Foix, Rodez, Tarbes, and Toulouse), university names (Champollion, Capitole, Mirail and its new name Jean Jaurès, and Paul Sabatier) and 13 school acronyms (e.g., ENSEEIHT, SUPAERO, TBS). This tag with the count of matching entities proved helpful to the visual inspection performed, that we detail in the next section.

Matching the staff registry to ORCID profiles: semi-automated approach

With the ORCID data collected, we faced three cases for a given person's identity used as query:

1. No matching ORCID profile.

7 https://orcid.org/organizations/integrators/API 8 We noticed that some identities (as listed by the laboratories) differed from the identities authors use in their publications. Consider the case birth name vs. maiden name, for instance. As a result, we did not use phrasal search (e.g., "Philippe Durand" with quotes) to query ORCID profiles. 2. One matching ORCID profile.

3. Between two and 20 matching ORCID profiles.

The first author (MDH) performed a visual inspection of all matches. Candidate profiles were validated based on the biography and affiliations (listed in the profiles and on the publications appearing as works). The major pitfall concerned 'empty' profiles registered for a given identity but showing no works. Such profiles were validated with a quasi certainty for discriminant identities (rare associations of first and last names), using Google Scholar to check the absence of homonyms.

Each pair of matching identity-profile pair was annotated with one of these codes:

• code '0' when the looked-up identity and the ORCID profile did not match (e.g., different first names).

• code '1' when the looked-up identity and the ORCID profile matched and the profile contents featured evidence of the Toulouse scientific area.

• For empty ORCID profiles:

code '?' when no homonyms were detected via Google Scholar searches for the looked-up identity, suggesting that there is only one known author with this identity.

code '??' when homonyms were detected via Google Scholar searches.

We considered as ORCID adopters the 2,789 identities with code '1' or '?.' This is a conservative estimate of the number of ORCID adopters in the Toulouse scientific area since some identities coded '??' were discarded because of lacking evidence.

QUANTITATIVE RESULTS

As of September 2020, 41.8% of the considered population had an ORCID profile. Setting aside the 'Other staff' job category, the ORCID adoption amounts to 46.9%. These two figures reflect some disparities when broken down by job category or discipline, as discussed in this section.

ORCID adoption by job category

Researchers were the most likely (54.7%) to register an ORCID profile, with a varying adoption per rank (Fig. 6). Junior and senior tenured researchers registered more ORCID profiles (64.0% on average) than postdocs (36.4%). This imbalance is all the more surprising as postdocs usually have an intense research activity and they are expected to care for the visibility of their research [START_REF] Nicholas | Does the scholarly communication system satisfy the beliefs and aspirations of new researchers? Summarizing the Harbingers research[END_REF].

Faculty members were less likely to register an ORCID (42.5%) and there is little difference with respect to seniority. This lesser adoption can result from two factors. On the one hand, faculty members are overrepresented in disciplines with a lesser use of ORCIDs: Art, Humanities, Social Sciences and Law, Economics, Management. On the second hand, part of the faculty members are more active in teaching and management roles than in research per se.

The Other Staff category has the lowest ORCID adoption rate (25.5%). They are engineers, technicians, and assistants part of a research lab. Their work is acknowledged in the papers and they are sometimes co-authors of their lab's publications.

ORCID adoption by discipline and sex

The average ORCID adoption per discipline is 39.2%, ranging from 21.7% for Law, Economics, Management to 49.8% for Physics, Chemistry (Fig. 7). This result is in line with the publication habits of these disciplines and the use of identifiers that is well established in the hard sciences. The sex imbalance of adopters reflects the one we observed in the population (Fig. 5) except for Health, Biology, Agronomy where women adopters are 5% less than women versus men in this discipline.

Figure 8 shows the adoption of each discipline per job category. Researchers are top adopters for all disciplines but Health, Biology, Agronomy. The promotion of ORCID by research institutes10 might be a favouring factor here. In Law, Economics, Management researchers and faculty members adopted ORCID at a lower rate than average. Other Staff have a high adoption rate in AAE, HBA, and PC; these are disciplines whose labs were more likely to list support personnel as co-authors (see Sect. 6.2).

ORCID adoption through time

ORCID was launched on October 16, 2012. Each profile retrieved via the ORCID API includes its creation date. We used this information to plot the dynamics of ORCID adoption. The first ORCID profile created by a researcher in the Toulouse area dates back from October 20, 2012. They were 51 early adopters in 2012 from the Toulouse area: 10 in October, 28 in November and 13 in December.

The number of newly created ORCID profiles between October 2012 and September 2020 is shown in Fig. 9. Creations follow a steady rate with peaks in October, which coincides with grant applications to the French National Research Agency (Agence Nationale de la Recherche). This agency requires consortium members to be identified with an ORCID: usually the PI and at least one partner per institution involved. 11We split the cumulated red line of Fig. 9 to plot the cumulated number of profile creation for each discipline (Fig. 10). Disciplines gather into two groups characterised by different growth types. Throughout the 2012-2020 period, AAE-EMC-HBA-PC showed a greater adoption rate compared to AHS-LEM. For the fist group, HBA adopted ORCID early, then AAE became a forefront adopter. Despite a slow start, PC is nowadays the leading adopting discipline. For the second group with lower adoption, LEM was quicker to create ORCID profiles compared to AHS until 2018 when AHS showed the largest adoption rate among all six disciplines. The rationale behind the changing adoption speed is unknown. We can only speculate on the different publishing houses active in EMC-AAE-HBA-PC (i.e., STEM -Science, Technology, Engineering, and Mathematics) versus AHS-LEM (i.e., SSH -Social Sciences and Humanities) and their use of ORCID :

• STEM publishers are generally large-sized firms (e.g., Elsevier, Springer, Wiley). Most of them are members of the ORCID consortium and they integrated ORCID into their peer-review and production system. Authors and reviewers are encouraged to create and link their ORCID to their profile on the submission/review platform (Johnson, Watkinson, & Mabe 2018, p. 162).

• SSH publishers are more diverse and smaller publishing houses (e.g., local University Presses). Many are small-sized companies publishing works on a specific topic, such as Dalloz in Law, Vrain in Philosophy, Presses Universitaires de Rennes in the social sciences and humanities. Most of these publishers are not members of the ORCID consortium and their authors are not asked to provide an ORCID at submission stage.

QUALITATIVE RESULTS

Diversity of ORCID profile (mis)uses

Let us recall that researchers from the Toulouse area register an ORCID themselves (no institutional automated registration is performed) and fill the associated sections, such as identity, employment biography, and works (Sect. 2). The ideal ORCID profile as illustrated in Fig. 2 lists comprehensive data about the author and his/her publications. Looking at the ORCID profile in our corpus, we found that many are far from such thoroughly completed profile with up-to-date data. Most profiles appear to be incomplete and even empty: Fig. 11 only shows 'philippe Durand' as the author's identity (there were 5 such empty profiles for this name at the time of data extraction). This homonym issue is even worse with surnames such as Wang [START_REF] Youtie | Tracking researchers and their outputs: New insights from ORCIDs[END_REF].

For our corpus, Fig. 12 shows the percentage of ORCID profiles with at least one publication listed. All disciplines considered, only 48.3% of the profiles feature one or more works. This is larger than the 26.8% expected from the ORCID statistics page.12 Using a per discipline breakdown, Health, Biology, Agronomy has the largest rate of completed profiles (57.5%). Arts, Humanities, Social Sciences is the least adopting discipline with only 26.2% of completed profiles (but certainly not completed comprehensively). There is no direct way to check how comprehensive an ORCID profile is. One should consider the works listed with care and do not infer any selection process from the owner of the profile: the listed works might just have been pushed by a few publishers. Another point relates to the default visibility setting for new data added. When profile owners decide 'private by default' (Fig. 1) and never connect to ORCID and change the visibility to public, the works are attached to the profiles but invisible to the public.

Based on the visual inspection of 100+ profiles, we noted the following intriguing characteristics regarding two aspects:

• The Biography and Employment sections are supposed to list the successive affiliations of the profile owner. Yet, some users list foreign universities that never employed them. It appeared that these universities are affiliations of colleagues with whom the profile owner collaborated. There is also a propension to list prestigious institutions. we noted that some of them mention the Paris headquarters (e.g., CNRS Paris) instead of the regional lab (e.g., CNRS IRIT Toulouse). And yet, they use their 'local' affiliation in the published papers.

• The Works section do not seem comprehensive for most ORCID profiles that contain only a fraction of the publications of the profile owner. We noticed several profiles of highly productive researchers missing hundreds of bibliographic records when compared to the Web of Science. We also raised quality concerns: some bibliographic records did not show any publication date, some titles and authors were misspelled. These errors suggest that profile owners entered the bibliographic data manually themselves. They might have ignored that they could have automatically retrieved metadata by providing the DOIs or other identifiers they wanted to add to their profiles.

A researcher can create multiple ORCID profiles linked to a several of his/her email addresses. One can only speculate that it was easier and faster to create a new ORCID instead of searching in one's archives for the ID and associated password created a long time ago. We found 18 individuals from our corpus who created multiple ORCID profiles: 8 cases of a completed vs empty profile, 7 cases of two empty profiles, and 3 cases of completed vs completed profile (with a varying degree of completion). The profile created last was not always the one completed by the researchers. We do not suspect any fraud related to fake profile creation as reported in (Teixeira da Silva 2021).

One may wonder: why don't people create ORCID profiles and, when they own one, why don't they fill it properly? ORCID is not the only platform allowing the creation of a profile for an author to list his/her publications [START_REF] Boudry | Use of author identifier services (ORCID, ResearcherID) and academic social networks (Academia.edu, ResearchGate) by the researchers of the University of Caen Normandy (France): A case study[END_REF]French & Fagan 2019;[START_REF] Tran | Faculty use of author identifiers and researcher networking tools[END_REF]. Some authors might own such a profile and view ORCID as a platform providing identifiers only, disregarding its bibliography curation capability. These authors might invest more efforts into Google Scholar and the likes at the expense of their ORCID profiles. We tested this hypothesis by tabulating (Tab. 1) the use of ORCID and Google Scholar for 15 individuals from the staff registry who were listed as 'Highly Cited Researchers' by Clarivate Analytics. 13No clear conclusions emerge: a majority of the most productive researchers in the Toulouse area failed to complete their ORCID profiles although 80% of them created an ORCID. Surprisingly, one of these Highly Cited Researchers in health sciences has no online presence at all, neither on ORCID nor on Google Scholar.

Focus on the ORCID profiles by 'other staff'

Slightly more than 25% of the 'other staff' have created an ORCID profile (Sect. 5.1). Including these staff as co-authors or not varies according to the disciplines. Other staff were included as co-authors in 7 out of the 30 labs in Heath, Biology, Agronomy.

CONCLUSION

Introduced in 2012, ORCID has become a key global infrastructure to disambiguate scholarly authors' identities. This study reported on the adoption of ORCID in one of the largest scientific areas in France. Contrary to [START_REF] Youtie | Tracking researchers and their outputs: New insights from ORCIDs[END_REF] who focused on Wang as a single surname of interest, we considered the entire multidisciplinary research workforce of a French scientific metropole. Matching the 6,607 staff members affiliated to the 145 labs based in the Toulouse area in 2016 with the ORCID registry, we showed a steadily increase for all research disciplines. The overall adoption is 41.8% with varying percentages among job ranks and disciplines. Faculty members in Health, Biology, Agronomy are leading adopters (62.0%) as opposed to those in Law, Economics, Management with 20.4% ORCID profiles. Such differences might result from various contexts: discipline-related incentives enacted by research institutions, funding agencies, and the publishing industry. For the 60% ORCID profiles with contents provided, a qualitative analysis reveals they were not comprehensively filled. Profile sections documenting the biography, employment history, other identifiers, awards and grants received are frequently lacking. The same goes for publications with incomplete lists of works. We acknowledge that the people hired between 2017 and 2021 are not considered in this study. Likewise the people who left Toulouse since 2016 are included in this study performed in 2020. All in all, our results suggest that a majority of researchers in the Toulouse area have little use of ORCIDs or understanding (or interest?) for identifiers.

There are only a few comparable studies of ORCID adoption worldwide. Dasler, Deane-Pratt, Lavasa, Rueda, and Dallmeier-Tiessen (2017) analysed the ORCID profiles created over 2012-2016, stressing breakdowns by discipline and location. This study at the global scale lacks a reference population, contrary to two studies on French scientific areas published later. The adoption in Toulouse of 41.8% (the present study) is higher than the 17.1% reported by [START_REF] Boudry | Use of author identifier services (ORCID, ResearcherID) and academic social networks (Academia.edu, ResearchGate) by the researchers of the University of Caen Normandy (France): A case study[END_REF] for the Caen University which is smaller in research workforce. We have no clue about this threefold observed difference and more research is needed to decipher this imbalance. The limited adoption of ORCID-and related misuses-calls for a better education to the issue of identity management for academics and the use of ORCID to tackle the issue of homonyms among authors worldwide, as stressed by the Surgisphere affair [START_REF] Piller | Who's to blame? These three scientists are at the heart of the Surgisphere COVID-19 scandal[END_REF].

FIGURE 2

 2 FIGURE 2 ORCID demo profile for the fake author Josiah Carberry.

FIGURE 6

 6 FIGURE 6 ORCID adoption by job category and rank.

FIGURE 9

 9 FIGURE 9 ORCID profile creation through time.

FIGURE 10

 10 FIGURE 10 Cumulated share of ORCID profiles by discipline.

FIGURE 11

 11 FIGURE 11Example of an 'empty' ORCID profile.

  Share of the 6,607 persons by job category and rank.

		2,000
		1,800
		1,600
		1,400
	Number of persons	600 800 1,000 1,200
		400
		200
		0
		Rank	Junior	Senior	Postdoc	Junior	Senior	Undefined	Other Staff
		Category	Faculty			Researcher		Other
		N	3,227			1,805			1,575
		%	48.8			27.3			23.8
			FIGURE 3

• AAE: Astronomy, Astrophysics, Environment • AHS: Arts, Humanities, Social Sciences • CP: Chemistry, Physics • EMC: Engineering, Mathematics, Computing • HBA: Health, Biology, Agronomy • LEM: Law, Economics, Management

  Share of the 6,607 persons by discipline and position. Share of the 6,607 persons by discipline and sex.

		1,200
			Faculty	Researcher	Other Staff
		1,000
		800
	Number of persons	600
		400
		200
		0
		Astronomy, Astrophysics,	Arts, Humanities, Social Sciences	Chemistry, Physics		Engineering, Mathematics,	Health, Biology, Agronomy	Law, Economics, Management
		Environment	(AHS, 16.5%)	(CP, 9.3%)			Computing			(HBA, 31.1%)		(LEM, 6.1%)
		(AAE, 10.6%)					(EMC, 26.3%)
			FIGURE 4 0% Women Men	10%	20%	30%	40%	50%	60%	70%	80%	90%	100%
		Astronomy, Astrophysics, Environment (AAE, 10.6%)
		Arts, Humanities, Social Sciences (AHS, 16.5%)
			Chemistry, Physics (CP, 9.3%)
		Engineering, Mathematics, Computing (EMC, 26.3%)
		Health, Biology, Agronomy (HBA, 31.1%)
		Law, Economics, Management (LEM, 6.1%)
			FIGURE 5

9 https://members.orcid.org/api/tutorial/reading-xml

  For researchers employed in French national research organisms (e.g., CNRS and INSERM) and affiliated to a lab in the Toulouse area,

	60%
	50%
	40%
	30%
	20%
	10%
	0%
	Astronomy, Astrophysics,	Arts, Humanities, Social Sciences	Chemistry, Physics	Engineering, Mathematics, Computing	Health, Biology, Agronomy	Law, Economics, Management
	Environment	(AHS, 16.5%)	(CP, 9.3%)	(EMC, 26.3%)	(HBA, 31.1%)	(LEM, 6.1%)
	(AAE, 10.6%)
		FIGURE 12 Share of ORCID profiles with at least one publication.

TABLE 1

 1 Online presence of 15 Highly Cited Researchers (HCR) as per Clarivate Analytics with their ORCID and Google Scholar profiles, if any. ORCID owners (with a Profile) might have a Biography, an Affiliation, and Publications listed or not. Being listed as co-authors is an incentive for the creation of ORCID profiles among the 'other staff' job category.

	ID Discipline		ORCID		Google
		Profile Biography Affiliation Publications	Scholar
	1 Agronomy	×	×	×	×
	2 Biology	×	×
	3 Ecology				×
	4 Ecology	×	×		×
	5 Economics				×
	6 Engineering	×	×	×	×
	7 Health	×			×
	8 Health	×		×	×
	9 Health	×	×	×
	10 Health
	11 Health	×
	12 Health	×
	13 Health	×	×	×
	14 Health	×		×
	15 Astronomy	×			×

It also happened in Astronomy, Astrophysics, Environment and Physics, Chemistry. Some extra cases occurred in Engineering, Mathematics, Computing. This virtually did not happened in Law, Economics, Management and Art, Humanities, Social Sciences.

This article is an extended version of a conference paper given in French[START_REF] Heusse | Adoption de l'identifiant chercheur ORCID : le cas des universités toulousaines[END_REF]. It covers a longer time period, includes the data for one overlooked lab, includes an analysis of adoption by sex, and features a revised discussion.

See for instance https://doranum.fr/wp-content/uploads/DoRANum-fiche-ORCID-maj-2020-10-08.pdf.

See page 20 in https://anr.fr/fileadmin/aap/2019/aapg-anr-2019-Guide.pdf

https://orcid.org/statistics

https://recognition.webofsciencegroup.com/awards/highly-cited/2019/

ACKNOWLEDGEMENTS

We are grateful to the two referees and the Editor-in-Chief for the invaluable remarks and comments that improved the manuscript.