A sufficient condition for a Rational Differential Operator to generate an Integrable System
AA sufficient condition for aRational Differential Operatorto generate an IntegrableSystem
Sylvain Carpentier*
November 6, 2018
Abstract
For a rational differential operator L = AB − , the Lenard-Magrischeme of integrability is a sequence of functions F n , n ≥ , such that(1) B ( F n +1 ) = A ( F n ) for all n ≥ and (2) the functions B ( F n ) pairwise commute. We show that, assuming that property (1) holdsand that the set of differential orders of B ( F n ) is unbounded, property (2) holds if and only if L belongs to a class of rational operators thatwe call integrable. If we assume moreover that the rational operator L is weakly non-local and preserves a certain splitting of the algebra offunctions into even and odd parts, we show that one can always findsuch a sequence ( F n ) starting from any function in Ker B. This resultgives some insight in the mechanism of recursion operators, whichencode the hierarchies of the corresponding integrable equations. * Department of Mathematics, Massachusetts Institute of Technology, Cam-bridge, MA 02139, USA a r X i v : . [ m a t h - ph ] M a y Introduction
In this paper, we work in the framework of an algebra of differen-tial functions, introduced in [BDSK09], which is a differential exten-sion of the algebra of differential polynomials in one variable, R = C [ u, u (cid:48) , u ” , . . . ] where the total derivative ∂ is defined by ∂ ( u ( n ) ) = u ( n +1) . More specifically, an algebra V is called an algebra of differen-tial functions if (1) it contains R , (2) it is endowed with commutingderivations ∂∂u ( n ) extending the partial derivatives on A in such a waythat, for all function f ∈ V , only finitely many of these extended par-tial derivatives of f are nonzero and (3) there is a derivation ∂∂x of V which is on R and commute with all the ∂∂u ( n ) . Then,(0.1) ∂ = (cid:88) n ≥ u ( n +1) ∂∂u ( n ) + ∂∂x is a derivation on V which we call again the total derivative . It extendsthe total derivative on R . The set C ≡ { v ∈ V , ∂ ( v ) = 0 } is a subalgebraof V called the subalgebra of constants. We assume that V is a domainand denote its field of fractions by K , which is endowed with a algebraof differential functions structure as well.Let V be an algebra of differential functions. Given F ∈ V , definethe associated evolutionnary vector field X F by :(0.2) X F = (cid:88) n ≥ F ( n ) ∂∂u ( n ) .X F is a derivation on V commuting with the total derivative ∂ andmapping u to F . Note that ∂ = X u (cid:48) . The commutator endows thespace of evolutionnary vector fields on V with a Lie algebra structure.This holds thanks to the identity(0.3) [ X F , X G ] = X X F ( G ) − X G ( F ) , satisfied for all F, G ∈ V . We can identify F ∈ V with X F and inparticular infer a Lie bracket { ., . } on V defined by(0.4) { F, G } = X F ( G ) − X G ( F ) . The differential order d ( F ) of a function F is defined as the greatestinteger n for which ∂F∂u ( n ) (cid:54) = 0 .A differential operator on V is an element of the algebra V [ ∂ ] , wherethe multiplication is defined by ∂F = F ∂ + F (cid:48) . If F, G ∈ V , observe that(0.5) X G ( F ) = (cid:88) n ≥ G ( n ) ∂F∂u ( n ) = D F ( G ) , where D F is the differential operator (cid:80) n ≥ ∂F∂u ( n ) ∂ n , called the Frechetderivative of F . Two functions F and G are said to commute, or to bea symetry of one another, if(0.6) { F, G } = 0 . In this paper, an integrable system is, by definition, an abelian Liesubalgebra of the Lie algebra V with bracket (0 . which contains func-tions of arbitrarily high order. Finally, F ∈ V (or rather the evolutionequation dudt = F ) is called integrable if F lies in an integrable system.It is known ([MS08], [IS80], [SS84]) that if F is integrable, then thereexists a so-called recursion operator L , defined by the property that itsLie derivative along F vanishes :(0.7) L F ( L ) : = X F ( L ) − [ D F , L ] = 0 . If L is a differential operator satisfying equation (0 . , it preserves thecentralizer of F in the Lie algebra V . However, in almost all interestingcases, L is not a differential operator but lies in an extension of V [ ∂ ] called the algebra of pseudodifferential operators , V (( ∂ − )) , where themultiplication is extended by letting for all a ∈ V (0.8) ∂ k a = (cid:88) n ≥ (cid:18) kn (cid:19) a ( n ) ∂ k − n , k ∈ Z . Most of the integrable systems encountered in the litterature aregenerated by such an operator L , namely they consist of the iterativeimages of a function F by L : { L n ( F ) } n ≥ . Often, the existence ofthis operator is used as a non-rigorous argument to claim that the cor-responding functions indeed define an abelian subalgebra of V .Of course, for an arbitrary pseudodifferential operator L , the expres-sion L ( F ) makes no sense. However, for the rational pseudodifferentialoperators, i.e. those that can be written as the ratio of two differentialoperators L = AB − , we can define an image of F by L if F lies in theimage of B . Namely, we say, as in [DSK13], that ( G, F ) are associatedthrough L if there exists a function H such that(0.9) ( G, F ) = ( A ( H ) , B ( H )) . In practice, we will just write G = L ( F ) , but we should always keep inmind that G is only defined modulo A ( KerB ) and in particular is notunique. We denote the algebra of rational pseudodifferential operators by V ( ∂ ) and simply call them rational operators . We refer to [CDSK12]and [CDSK14] for an in-depth study of the algebra V ( ∂ ) .In this paper we are interested in determining which pairs of differ-ential operators ( A, B ) produce integrable systems. More precisely, forwhich ( A, B ) ∈ V [ ∂ ] can we find a sequence of functions ( F n ) n ≥ ∈ V Z + such that : A ( F n ) = B ( F n +1 ) ∀ n ≥ , { B ( F n ) , B ( F m ) } = 0 ∀ n, m ≥ . (0.10)It is often said in the litterature that a sufficient condition is forthe rational operator L = AB − to be Nihenjuis , or hereditary , whichmeans that L satisfies the following identity for all function F ∈ V :(0.11) L A ( F ) ( L ) = L L B ( F ) ( L ) . Note that it follows from equations (0 . and (0 . that, if L = AB − is hereditary and recursion for B ( F ) , then it is recursion for A ( F ) .Although necessary, (0 . is not sufficient for the pair ( A, B ) to satisfy (0 . for some functions F n ∈ V . A counterexample is given by(0.12) L = ∂ − u (cid:48)(cid:48) ∂ , which is hereditary but does not generate commuting functions.We will see that, to satisfy (0 . for some functions F n ∈ V , therational operator L = AB − must lie in a finer subset of V ( ∂ ) which wecall the class of integrable rational operators. A differential operator A is called integrable if one can find a bidifferential skewsymmetricoperator M (i.e. an element of V [ ∂ , ∂ ] ) such that for all functions F and G one has(0.13) X A ( F ) ( G ) − X A ( G ) ( F ) = A ( M ( F, G )) . We then define a pair of operators ( A, B ) to be integrable if anyof their linear combination is integrable, more precisely if there existstwo skewsymmetric bidifferential operators M and N such that for allfunctions F, G and all constant λ , we have(0.14) X ( A + λB )( F ) ( G ) − X ( A + λB )( G ) ( F ) = ( A + λB )(( M + λN )( F, G )) . For example, any local Poisson structure H is an integrable opera-tor, and any compatible pair of local Poisson structures ( H, K ) is anintegrable pair.Finally, a rational operator L is called integrable if it can be writ-ten in the form AB − where ( A, B ) is an integrable pair of differentialoperators. This definition is natural for the following reason : Theorem 0.1.
Let A , B be two differential operators and ( H n ) n ≥ bea sequence of functions in V which spans an infinite dimensional spaceover C . Let us assume moreover that L is recursion for B ( H ) and thatfor all n ≥ , (0.15) B ( H n +1 ) = A ( H n ) . Then the functions B ( H n ) pairwise commute if and only if the pair ( A, B ) is integrable. In the first section of the paper, we recall elementary properties ofalgebras of differential functions, differential operators, Frechet deriva-tives and bidifferential operators. In the second section, we begin byrecalling and proving the result of Ibragimov and Shabat (stated in[IS80] and proved in [SS84]), which states that an integrable systemhas a recursion operator. Then, we move on studying hereditary op-erators and in particular their relation with the Lenard-Magri schemeof integrability (0 . . In the third section, we introduce integrableoperators and prove one of their key properties : Proposition 0.2.
Let L = AB − be an integrable rational operatorwhich is recursion for B ( F ) . Then A ( F ) and B ( F ) commute : (0.16) { A ( F ) , B ( F ) } = 0 . In the fourth section, we consider rational operators of a particularkind, namely, weakly non-local operators , which by definition are ratio-nal operators which can be written in the following form (see [MN01])(0.17) L = E ( ∂ ) + n (cid:88) i =1 p i ∂ − q i , where E is a differential operator, called the local part of L , and p i and q i are elements of K . It is shown in the paper that the space of weaklynon-local operators coincides with the space of rational operators AB − whose denominator B in the minimal fractional decomposition (i.e. degB is minimal, see [CDSK12]) has full kernel in K , i.e. :(0.18) degB = dim C Ker K B .
Differential operators with property (0 . have been studied in [DSKT15]and called strongly non-degenerate there. The majority of knownrecursion operators associated to integrable systems are weakly non-local. Differential operators with full kernel provide an easy way totest whether a function F lies in their image or not. Indeed, if KerB ∗ is spanned by q , . . . , q n , then(0.19) F ∈ ImB ⇐⇒ q i F ∈ ∂ K ∀ i = 1 . . . n . We study the structure of weakly non-local operators and examinewhat can be said when a rational operator L ∈ V ( ∂ ) is integrable andweakly non-local. In particular, we prove : Proposition 0.3.
Let L = E + (cid:80) ni =1 p i ∂ − q i be a weakly non-localoperator where the { p , . . . , p n } and { q , . . . , q n } are sets consisting oflinearly independent elements of V . Then L is integrable if and only ifit is hereditary and q i is a variational derivative for all i = 1 . . . n . Recall that the variational derivative of F ∈ V is defined by(0.20) δFδu : = (cid:88) n ≥ ( − ∂ ) n ( ∂F∂u ( n ) ) . In section , we will assume that the algebra of differential functions V can be split as a sum(0.21) V = V ¯0 ⊕ V ¯1 of eigenspaces of an involution σ of the algebra V , such that σ∂σ − = − ∂ and σ ∂∂u ( n ) σ − = ( − n ∂∂u ( n ) for all n ≥ . We call elements of V ¯0 even functions and elements of V ¯1 odd functions. It follows from thisdefinition that ∂ switches parity and that δδu preserves parity.We then give a sufficient condition for the existence of Lenard-Magrisequences for a subclass of rational operators : Theorem 0.4.
Let L ∈ ( V [ ∂ ]) ¯0 + V ¯1 ∂ − V ¯0 be integrable. If AB − isa right minimal fractional decomposition of L and F ∈ KerB , thenthere exists a sequence F n ∈ V , n ≥ such that(1) B ( F n +1 ) = A ( F n ) for all n ≥ ,(2) { B ( F n ) , B ( F m ) } = 0 for all n, m ≥ .Moreover, if d ( L ) > and d ( B ( F N )) is greater than the differentialorders of the coefficients of A and B for some N , then the sequence d ( B ( F n )) is unbounded. A more complete picture is given by :
Theorem 0.5.
Let L ∈ ( V [ ∂ ]) ¯0 + V ¯1 ∂ − V ¯0 be integrable. Then L k isweakly non-local and integrable for all k ≥ . If (0.22) L k = E k + n k (cid:88) i =1 p ki ∂ − q ki , where { p k , . . . , p kn k } and { q k , . . . , q kn k } are two sets of linearly inde-pendent functions and E k is a differential operator, then the functions p ki are odd, E k is even and q ki are even variational derivatives : q ki = δρ ki δu , with even ρ ki . Moreover, for all k, l ≥ and i, j ∈ { , n k }×{ , n l } , { p ki , p lj } = 0 , p ki q lj ∈ ∂ V ,ρ lj is a conserved density for the equation u t = p ki . (0.23)Theorems . and . also hold when the non-local part of L liesin V ¯0 ∂ − V ¯1 . Note also that by Proposition . one could replace theassumption L integrable by : L is hereditary and the q i ’s are variationalderivatives. Theorem . was proved in [EOR93] for the KdV equation.In section , we explain how Theorem . reproves the integrabilityof most of the integrable equations for which a recursion operator isknown. In particular, the Korteweg-de Vries (KdV) equation(0.24) u t = u (cid:48)(cid:48)(cid:48) + 3 uu (cid:48) . satisfies the hypothesis of Theorems . and . . Indeed, it admits thesimple recursion operator(0.25) L KdV = ∂ + 2 u + u (cid:48) ∂ − , which is known to be hereditary, as a ratio of two compatible local Pois-son structures. Furthermore, the algebra R of differential polynomialsin u admits a decomposition as in (0 . by declaring u to be even and ∂ to be odd. For this splitting, u (cid:48) is odd, ∂ + u is an even operator and is an even variational derivative, hence L KdV is integrable by Propos-tition . . Therefore we can apply Theorem . starting at F = 1 andobtain an integrable hierarchy containing KdV, which we get at thesecond step of the Lenard-Magri scheme. Here the differential ordersgo to + ∞ because d ( KdV ) = 3 is greater than the differential ordersof the coefficients of L KdV .Similarly, the integrability of the Krichever-Novikov (KN) equation(0.26) u t = u (cid:48)(cid:48)(cid:48) − u (cid:48)(cid:48) u (cid:48) + P ( u ) u (cid:48) , where d Pdu = 0 , follows from Theorems . and . . In [DS08], Demskoi and Sokolovexhibit a recursion operator for ( KN ) of the form(0.27) L KN = ∂ + a ∂ + a ∂ + a ∂ + a + G ∂ − δρ δu + u (cid:48) ∂ − δρ δu . The space of Laurent differential polynomials in u , A = C [ u ± , u (cid:48)± , ... ] admits a decomposition as in (0 . by declaring u to be even and ∂ tobe odd. From the explicit formulas given in [DS08], it is straighforwardto check that a i has the same parity as i for i = 1 , . . . , and that ρ i areeven for i = 1 , . Hence, the local part of L KN is even and so are thefunctions δρ i δu , since variational derivatives preserve parity. Moreover, G is the equation (KN) itself, which is odd. Finally, one checks that L KN is hereditary and that the condition on the degrees ( last part ofTheorem . ) is met, hence (0 . is integrable, by Theorem . .As we noted earlier, not all integrable systems admit a weakly non-local recursion operator. For instance, the Calogero-Degasperis-Ibragimov-Shabat (CDIS) equation,(0.28) dudt = u (cid:48)(cid:48)(cid:48) + 3 u u (cid:48)(cid:48) + 9 uu (cid:48) + 3 u u (cid:48) , has a rational integrable recursion operator which is rational, but notweakly non-local :(0.29) L CDIS = 1 u ∂ ( ∂ + 2 u ) − ( ∂ + u − u (cid:48) u ) ( ∂ + 2 u ) ∂ − u . For this particular equation, it is not hard to check that we canapply L CDIS infinitely many times to u (cid:48) and that L CDIS is integrable.Since
CDIS = L CDIS ( u (cid:48) ) , we conlude from Theorem . that CDIS is integrable. In the last section of the paper we show how to use ourtechniques to prove integrability of all equations from the classificationlist of [SW09].We conjecture that every integrable system comes with an integrablerational recursion operator :
Conjecture . Let
W ⊂ V be an integrable system. Then there existsan integrable rational operator L = AB − such that(0.30) L F ( L ) = 0 ∀ F ∈ W . Our results can be naturally extended to the case of several fieldvariables. We plan on tackling these extensions in a future publication.I would like to thank Victor Kac for suggesting the problem and formany helpful discussions. I am also very thankful to Alberto De Solefor his careful and patient reading of this paper. Preliminaries
We begin by recalling a few properties of algebras of differential func-tions, differential operators, Frechet derivatives and evolutionnary vec-tor fields. For a more complete introduction to these notions, we re-fer the reader to [BDSK09] (for algebras of differential functions) and[MS08] (for Frechet derivatives and evolutionnary vector fields).
Algebras of differential functions. heyThe basic differential algebra that we consider here is the algebra ofdifferential polynomials in u , namely(1.1) R = C [ u, u (cid:48) , u (cid:48)(cid:48) , . . . ] . The total derivative ∂ on R is defined by letting ∂ ( u ( n ) ) = u ( n +1) forall n ≥ . An algebra of differential functions in u is an extension of R in the following sense : Definition 1.1.
An algebra of differential functions V in the variable u is an algebra extension of R endowed with derivations ( ∂∂u ( n ) ) n ≥ and ∂∂x such that(1) ( ∂∂u ( n ) ) n ≥ extend the partial derivatives in R ,(2) The derivations ( ∂∂u ( n ) ) n ≥ , ∂∂x pairwise commute,(3) For all f ∈ V , ∂f∂u ( n ) = 0 for all but finitely many n ,(4) ∂P∂x = 0 ∀ P ∈ R .
We define the total derivative ∂ on V by the following formula(1.2) ∂ = (cid:88) n ≥ u ( n +1) ∂∂u ( n ) + ∂∂x , which estends ∂ from R . For f ∈ V and n ≥ , we will often write f ( n ) instead of ∂ n ( f ) , and f, f (cid:48) , f (cid:48)(cid:48) , ... instead of f, ∂ ( f ) , ∂ ( f ) , ... .Typical examples of algebras of differential functions that we willconsider are: the algebra of differential polynomials itself, any local-ization of it by some element f ∈ R , or, more generally, by somemultiplicative subset S ⊂ R , such as the whole field of fractions Q = C ( u ( n ) | n ≥ , or any algebraic extension of the algebra R or of thefield Q obtained by adding a solution of certain polynomial equation. An example of the latter type is V = C [ √ u ± , u (cid:48) , u (cid:48)(cid:48) , ... ] , obtained bystarting from the algebra R , adding the square root of the element u ,and localizing by √ u .In all the sequel, let V be an algebra of differential functions. Definition 1.2.
Let F ∈ V . We call differential order of F , which willbe denoted by d ( F ) or d F , the following quantity :(1.3) d ( F ) = max { n ≥ | ∂F∂u ( n ) (cid:54) = 0 } . If the set on the RHS of (1 . is empty, we call F a quasiconstant ofthe algebra V . If F is a quasiconstant such that ∂F∂x = 0 , we say that F is a constant of V . We will denote the subset of quasiconstants (resp.constants) by Q V (resp. C V ), or when there is no confusion by Q (resp C ). We will always assume that C V is an algebraically closed field. Remark . For all F ∈ V such that d ( F ) ≥ , we have d ( F (cid:48) ) = d ( F ) + 1 . This follows immediately from (1 . and the fact that partialderivatives do not increase the differential order. The latter fact holdsbecause partial derivatives commute. In particular, F (cid:48) (cid:54) = 0 . Therefore(1.4) Ker ∂ = C V . We denote the canonical projection
V → V /∂ V by (cid:82) . This notationis justified by the integration by parts property(1.5) (cid:90) F (cid:48) G = − (cid:90) F G (cid:48) ∀ F, G ∈ V . Given a function F ∈ V of differential order N we call evolutionequation associated to F the following equation :(1.6) dudt = F ( u, u (cid:48) , . . . , u ( N ) ) . We also associate to F a derivation on V (1.7) X F ≡ (cid:88) n ≥ F ( n ) ∂∂u ( n ) , which is called the evolutionnary vector field associated to F . Con-versely, F is said to be the characteristic function of X F . Lemma 1.4. (1) For any F ∈ V , [ X F , ∂ ] = 0 .(2) For all F, G ∈ V , [ X F , X G ] = X X F ( G ) − X G ( F ) . Proof. (1) is a consequence of (1 . and the following identity(1.8) [ ∂∂u ( n +1) , ∂ ] = ∂∂u ( n ) ∀ n ≥ . which is immediate from (1 . . The terms involving products of par-tial derivatives vanishing in the commutator [ X F , X G ] , there are somefunctions G n ∈ V such that(1.9) [ X F , X G ] = (cid:88) n G n ∂∂u ( n ) . From (1) we know that ∂ commutes with [ X F , X G ] . Hence for all n ≥ , G n = G n ) and [ X F , X G ] = X G . Moreover, G = [ X F , X G ]( u ) = X F ( G ) − X G ( F ) , which completes the proof. (cid:3) From the second part of Lemma . , it follows that the followingbracket is a Lie bracket on V :(1.10) { F, G } ≡ X F ( G ) − X G ( F ) . Indeed (1 . is obviously bilinear over the constants and skewsym-metric. Furthermore, Jacobi identity holds because of (2) in Lemma . . Definition 1.5.
F, G ∈ V are said to be symmetries of one another,or to commute, if(1.11) { F, G } = 0 . Remark . The use of the word symmetry is an abuse of language.What (1 . means is that, if u is a solution of the equation (1 . , then u + (cid:15)G is a solution of (1 . as well modulo (cid:15) if and only if (1 . holds. A more rigorous denomination is generator of an infinitesimalsymmetry. Definition 1.7.
An integrable system on V is an abelian subalgebra g of ( V , { ., . } ) such that the set of differential orders of elements of g isunbounded. Definition 1.8.
Given F ∈ V , we say that ρ ∈ V is a conserved densityof the evolution equation dudt = F , or simply of F , if(1.12) (cid:90) X F ( ρ ) = 0 . When ρ ∈ ∂ V , (1 . holds because the derivation X F commutes with ∂ . In that case, we say that ρ is a trivial conserved density. By inte-gration by parts, (1 . is equivalent to(1.13) (cid:90) δρδu .F = 0 , where the function δρδu is called the variational derivative of ρ . It isgiven by(1.14) δρδu ≡ (cid:88) n ( − ∂ ) n ( ∂ρ∂u ( n ) ) . Definition 1.9.
For n ≥ let V n be the subset of functions in V whosedifferential order is at most n . We say that V is normal if ∂∂u ( n ) ( V n ) = V n for all n ≥ and ∂∂x ( Q V ) = Q V . Lemma 1.10. If V is normal, then (1.15) Ker δδu = ∂ V . Proof.
See Proposition . in [BDSK]. In our case Q V ⊂ ∂ V . (cid:3) Differential operators. hIn this subsection, we will be reviewing definitions and results from [ CDSK and [ CDSK . We refer the reader to these papers forthe proofs. Let K be a differential field with subfield of constants C . Definition 1.11.
A differential operator A ( ∂ ) on K is an element ofthe algebra K [ ∂ ] in which multiplication is defined by ∂a = a∂ + a (cid:48) for every a ∈ K . A pseudodifferential operator A ( ∂ ) on K is an el-ement of the algebra K (( ∂ − )) in which multiplication is defined by ∂ − a = (cid:80) n ≥ ( − n a ( n ) ∂ − n − for all a ∈ V . The algebra K (( ∂ − )) isa skewfield. A rational ( pseudodifferential ) operator on K A ( ∂ ) isa pseudodifferential operator which can be written as the ratio of twodifferential operators B ( ∂ ) and C ( ∂ ) : A = BC − . We denote theset of rational operators on K by K ( ∂ ) . The following inclusions areobvious(1.16) K [ ∂ ] ⊂ K ( ∂ ) ⊂ K (( ∂ − )) If L = (cid:80) N −∞ l n ∂ n ∈ K (( ∂ − )) with l N (cid:54) = 0 , we call N = d ( L ) the degreeof L . Lemma 1.12. K ( ∂ ) is a field, which can be described also in terms ofleft fractions : (1.17) K ( ∂ ) = { AB − | ( A, B ) ∈ K [ ∂ ] ×K [ ∂ ] × } = { D − C | ( C, D ) ∈ K [ ∂ ] ×K [ ∂ ] × } Proof.
Part (a) of Proposition . in [CDSK12]. (cid:3) Definition 1.13.
For L ∈ K ( ∂ ) we call right (resp. left) fractionaldecomposition a pair ( A, B ) ∈ K [ ∂ ] × K [ ∂ ] ∗ such that L = AB − (resp. L = B − A ). Lemma 1.14.
Let L ∈ K ( ∂ ) . Then :(1) There exists a right fractional decompostion ( A , B ) of L suchthat for any other right fractional decomposition ( A, B ) of L there exists a non-zero differential operator D such that A = A D and B = B D . We call ( A , B ) a minimal right frac-tional decomposition of L .(2) The analogous statement for left fractions holds.(3) Finally, if D − C is a minimal left fractional decomposition of L , then d ( D ) = d ( B ) .Proof. (1) and (2) follows from part (b) of Proposition . in [CDSK12]. (3) can be found in Remark . of [CDSK13]. (cid:3) Lemma 1.15.
Let A be a non-trivial differential operator on K . Then (1.18) dim C kerA ≤ d ( A ) . Moreover,
ImA is infinite-dimensional over C .Proof. The set of differential orders of ( A ( u ( n ) )) n ≥ is clearly unbounded,hence ImA is a infinite-dimensional space over C . As for the statement (1 . , see A. . (a) in [DSK13]. (cid:3) Definition 1.16.
The ring K [ ∂ ] is right and left principal ideal domain([CDSK13b]). If A and B are differential operators, we call left (resp.right) least common multiple of the pair ( A, B ) a generator of the left(resp. right) ideal of K [ ∂ ] generated by A and B . Lemma 1.17.
Let f , . . . , f n ∈ V be linearly independent over C . Thenthere exists a differential operator P ∈ V [ ∂ ] with degree n such that KerP = (cid:104) f , . . . , f n (cid:105) .Proof. Let us prove the claim by induction on n . If n = 1 , we define P = f ∂ − f (cid:48) . It is clear that P has degree and that its kernelis spanned over C by f . Let us consider n + 1 linearly independent functions f , . . . , f n +1 and let Q be a degree n differential operatorwhose kernel is spanned by f , . . . , f n . By construction of Q , Q ( f n +1 ) (cid:54) =0 , hence we can define R = ( Q ( f n +1 ) ∂ − Q ( f n +1 ) (cid:48) ) Q . We can see that f i ∈ KerR for i = 1 , . . . , n + 1 . We conclude that KerR is spanned bythe f i ’s using equation (1 . and noting that deg ( R ) = n + 1 . (cid:3) Frechet Derivatives. heyFrom now on, we consider an algebra of differential functions V over u . Furthermore, we assume that V is a domain and let K be its fieldof fraction. Recall that C is assumed to be algebraically closed. Fora complete discussion on Frechet derivatives, we send the interestedreader to [MS08]. Definition 1.18.
Let F ∈ V . We define the Frechet derivative of F tobe the differential operator D F such that(1.19) D F ( G ) = X G ( F ) ∀ G ∈ V . Consequentely, for F ∈ V we have(1.20) D F = (cid:88) m ∂F∂u ( m ) ∂ m . Definition 1.19.
Let P = (cid:80) Nk =0 p k ∂ k be a differential operator over V and F ∈ V . We define the differential operator ( D P ) F as follows(1.21) ( D P ) F = N (cid:88) k =1 F ( k ) D p k ( ∂ ) . If L = (cid:80) N −∞ l k ∂ k ∈ V (( ∂ − )) and F ∈ V , we let(1.22) X F ( L ) = N (cid:88) −∞ X F ( l k ) ∂ k . Lemma 1.20.
The following identities hold for all differential opera-tors
A, B ∈ V [ ∂ ] and functions F, G ∈ V : (1.23) D A ( F ) = AD F + ( D A ) F . (1.24) ( D AB ) F = ( D A ) B ( F ) + A ( D B ) F . (1.25) ( D A ) F ( G ) = X G ( A )( F ) . Proof. (1 . follows from two facts. First, D ab = aD b + bD a for allfunctions a, b ∈ V since partial derivatives are derivations of V . Sec-ond,we have D a (cid:48) = ∂D a for all function a ∈ V , which can be inferredfrom the identity (1 . .Let us consider the three following specializations of (1 . : D AB ( F ) = ABD F + ( D AB ) F ,D A ( B ( F )) = AD B ( F ) + ( D A ) B ( F ) ,D B ( F ) = BD F + ( D B ) F . (1.26)We find (1 . by multiplying the third line of (1 . on the left by A ,adding the second line and substracting the first. As for (1 . we firstnote that, since X G is a derivation of V commuting with ∂ , we have forall G ∈ V (1.27) X G ( A ( F )) = X G ( A )( F ) + A ( X G ( F )) . Then, using the definition (1 . of the Frechet derivative, we have(1.28) D A ( F ) ( G ) = X G ( A )( F ) + AD F ( G ) . We obtain (1 . comparing (1 . with (1 . . (cid:3) Definition 1.21.
Let ∗ be the anti-involution of V (( ∂ − )) such that ∂ ∗ = − ∂ and F ∗ = F for all F ∈ V . It is well defined since the relation (0 . is preserved. Lemma 1.22.
Let V be normal and F ∈ V . Then D F = D ∗ F if andonly if F is a variational derivative : (1.29) D F = D ∗ F ⇐⇒ ∃ ρ ∈ V , F = δρδu . Proof.
See Proposition . in [BDSK09]. (cid:3) Lemma 1.23.
Let
F, G ∈ V . Then (1.30) X F ( D G ) = [ D F , D G ] + X G ( D F ) + D { F,G } . Proof.
Since V is a Lie algebra for { ., . } , Jacobi identity holds for ( F, G, H ) for all H ∈ V . Therefore { F, { G, H }} + { G, { H, F }} + { H, { F, G }} = ( X F − D F )( { G, H } ) − ( X G − D G )( { F, H } ) − ( X { F,G } − D { F,G } )( H )= ( X F − D F )( X G − D G )( H ) − ( X G − D G )( X F − D F )( H ) − ( X { F,G } − D { F,G } )( H )= [ D F , D G ]( H ) + X G ( D F )( H ) − X F ( D G )( H )+ D { F,G } ( H ) + ([ X F , X G ] − X { F,G } )( H )= ([ D F , D G ] + X G ( D F ) − X F ( D G ) + D { F,G } )( H ) . (1.31)Since this is true for all H we obtain the desired result. (cid:3) Bidifferential operators.
Definition 1.24.
A bidifferential operator on V is an element M of V [ ∂ , ∂ ] :(1.32) M ( ∂ , ∂ ) = (cid:88) k,l M kl ∂ k ∂ l . where M kl are functions in V . It naturally defines a map from V ⊗ C V to V : for any F, G ∈ V , we let(1.33) M ( F, G ) = (cid:88) k,l M kl F ( k ) G ( l ) . For a bidifferential operator M and a function F ∈ V we define differ-ential operators M F and M F by(1.34) M F = (cid:88) k,l M kl F ( k ) ∂ l . (1.35) M F = (cid:88) k,l M kl F ( l ) ∂ k . By construction, we have for all
F, G ∈ V (1.36) M F ( G ) = M G ( F ) = M ( F, G ) . For a bidifferential operator M we let d ( M ) = sup F ∈V d ( M F ) ,d ( M ) = sup F ∈V d ( M F ) . (1.37) For a bidifferential operator M and a differential operator B we definethe bidifferential operators BM and M B by letting for all
F, G ∈ V : ( BM )( F, G ) = B ( M ( F, G )) , ( M B )( F, G ) = M ( F, B ( G )) . (1.38) Example . We constructed in Definition . a bidifferential op-erator D A given a differential operator A . We call D A the Frechetderivative of A . Lemma 1.26.
Let M be a bidifferential operator and B be a differentialoperator on V . Then there exists a unique pair ( N, P ) of bidifferentialoperators on K such that(1) M = BP + N ,(2) d ( N ) < d ( B ) .Similarly, there exists a unique pair ( Q, R ) of bidifferential operatorson K such that(1) M = QB + R ,(2) d ( R ) < d ( B ) .Proof. The uniqueness is obvious. Let us show the existence for theleft division. If d ( M ) < d ( B ) there is nothing to do. Otherwise, let b k ∂ k be the leading term of B and A ( F ) ∂ l be the leading term of M F .Then if we let (cid:102) M F = M F − B A ( F ) b k ∂ l − k , we have d ( (cid:102) M ) < d ( M ) . Weconclude by induction on d ( M ) . (cid:3) Lemma 1.27.
Let M and N be two bidifferential operators over V , A and B two differential operators such that for all F ∈ V (1.39) M A ( F ) = BN F . Then there exists a bidifferential operator P on K such that for all F ∈ V (1.40) N F = P A ( F ) . Moreover M = BP .Proof. Let k = d ( B ) . We prove the existence of P by induction on d ( N ) . If d ( N ) = 0 , then the leading term of (1 . reads M k ( A ( F )) = b k N ( F ) hence N is divisble on the right by A , i.e. (1 . is satisfiedfor some bidifferentila operator P . If d ( N ) = l > , the leading termof (1 . is M k + l ( A ( F )) = b k N l ( F ) hence there exists a differentialoperator Q such that N l = Q ( A ( F )) . Therefore we have(1.41) ( M − B ( Q∂ l )) A ( F ) = B ( N − N l ∂ l ) F , and we can use the induction hypothesis to conclude that (1 . holdsfor some bidifferential operator P . Combining (1 . with (1 . gives(1.42) M A ( F ) = BP A ( F ) Finally, since the image of A is infinite-dimensional by Lemma . ,and since the kernel of the map F (cid:55)→ Q F is finite dimensional for anybidifferential operator Q , we can remove A from (1 . and deduce that M = BP . (cid:3) Lemma 1.28.
Let M and N be two bidifferential operators and A and B be two differential operators on V such that AM F = N F B for all F ∈ V . Then there exists a bidifferential operator P on K such that M = P B and N = AP .Proof. Performing Euclidean divisions of M by B on the right we canreduce the problem to the case where d ( M ) < d ( B ) . Let us showthat in that case M = 0 . Let L be a differential field extension of K with C K = C L , where B has a full kernel. Such an extension existsby the Picard-Vessiot theory. In L , we have equality in (1 . . Let b ∈ Ker L B . We know that for all F ∈ V , A ( M F ( b )) = A ( M b ( F )) = 0 .In the language of differential operators, this means that AM b = 0 ,therefore that M b =0. Thus, for all F ∈ V the kernel of M F in L hasdimension over C at least d ( B ) . This implies that M F = 0 , because weare in the case where d ( B ) > d ( M ) . Hence M = 0 . (cid:3) Lemma 1.29.
Let A and B be two differential operators and AD = BC be their least right common multiple. Let M and N be bidifferential op-erators such that AM = BN . Then there exists a bidifferential operator P such that M = DP and N = CP .Proof. This lemma holds when M and N are differential operators bythe very definition of the least right common multiple. Hence, for all F ∈ V , we know that M F is divisible on the left by D . By Lemma . , this means that M is divisible on the left by D . (cid:3) Hereditary Operators
Let V be a normal algebra of differential functions in u . Let us assumethat V is a domain and let K be its field of fraction. Let C be thesubfield of constants, assumed to be algebraically closed. Recursion operators.
Definition 2.1.
The Lie derivative L F ( L ) of a pseudodifferential op-erator L along the function F is defined to be(2.1) L F ( L ) = X F ( L ) − [ D F , L ] . We say ([Olv93]) that L is recursion for F if(2.2) L F ( L ) = 0 . Remark . Every pseudodifferential operator L is recursion for and u (cid:48) . The constant operator is recursion for any function F . TheLie derivative L F is a derivation of the algebra of pseudodifferentialoperators V (( ∂ − )) , because both X F and [ D F , . ] are. Proposition 2.3.
Let F ∈ V , and H = AB − , K = CD − be two(non-local) Poisson structures on V (see [DSK13] for the definition)such that F = A ( X ) = C ( Y ) , where X, Y ∈ V and B ( X ) and D ( Y ) are variational derivatives. Then, Λ = HK − is a recursion operatorfor F .Proof. Since B ( X ) is a variational derivative, we have by (1 . : D B ( X ) = D ∗ B ( X ) . We specialize formula ( ii ) of Proposition . in [DSK13] for thepairs ( A, B ) and ( X, G ) , where G ∈ V , to get(2.3) A ∗ ( D B ( G ) ( F ) + D ∗ F ( B ( G ))) + B ∗ ( D A ( G ) ( F ) − D F ( A ( G ))) = 0 . Using (1 . , (1 . , and the fact that A ∗ B + B ∗ A = 0 (Poisson struc-tures are skewadjoint), we have for all G ∈ V (2.4) A ∗ ( X F ( B )( G ) + D ∗ F ( B ( G ))) + B ∗ ( X F ( A )( G ) − D F ( A ( G ))) = 0 . Therefore we obtain the following differential operator identity :(2.5) A ∗ X F ( B ) + A ∗ D ∗ F B + B ∗ X F ( A ) − B ∗ D F A = 0 . Multiplying (2 . on the left by ( B ∗ ) − , on the right by B − , and usingthe skewadjointness of H , we deduce that(2.6) X F ( H ) = D F H + HD ∗ F . Since the same argument holds for K , we conclude that(2.7) X F ( HK − ) = [ D F , HK − ] , which means that HK − is a recursion operator for F . (cid:3) Next, we address the problem of existence of a recursion operator foran integrable system (cf. [MS08], [IS80], [SS84]). Proposition 2.4.
Let W be an integrable system in V spanned bycountably many ( F n ) n ≥ such that their differential orders d F n are strictlyincreasing and d F ≥ . Let us assume moreover that the leading termof D F is invertible in V and admits a d F -th root in V . Then thereexists a pseudodifferential operator L in V (( ∂ − )) which is recursionfor all the F n . We will first state and prove some lemmas before proving Proposition . . Lemma 2.5.
Let A be a pseudodifferential operator of degree d A and F ∈ V . Then for all non-zero integer i ∈ Z , (2.8) d ( L F ( A i )) = d ( L F ( A )) + ( i − d ( A ) . Proof.
Let us first consider a positive integer i . As we noted in Remark . , L F is a derivation of V (( ∂ − )) , hence(2.9) L F ( A i ) = i − (cid:88) k =0 A k L F ( A ) A i − − k . In the RHS of (2 . , all terms have degree d ( L F ( A )) + ( i − d A andhave the same leading coefficient, therefore the result follows. Thesame holds if we replace A by its inverse A − . It remains to check that d ( L F ( A − )) = d ( L F ( A )) − d A . This follows from the identity(2.10) L F ( A − ) = − A − L F ( A ) A − . (cid:3) Lemma 2.6.
Let F ∈ V and G ∈ V be two functions with G notconstant and d F ≥ . Then (2.11) d ( L F ( G )) = d F − . Proof. X F ( G ) differential operator of degree or −∞ . Moreover, since G is not a constant, the degree of the commutator [ D F , G ] is d F − . (cid:3) Lemma 2.7.
Let F be a function of differential order d F ≥ and ( T, S ) be two pseudodifferential operators of degree such that d ( L F ( S )) = m < d ( L F ( T )) = n . Then, there exists a Laurent series φ ( z ) ∈ C (( z − )) of degree such that T − φ ( S ) has degree n − d F + 1 . Fur-thermore, d ( L F ( φ ( S ))) = d ( L F ( S )) .Proof. Let T = (cid:80) −∞
Let F be a function in V such that d F ≥ and T n ∈V (( ∂ − )) be a sequence of pseudodifferential operators of degree onewith invertible leading terms such that the sequence d ( L F ( T n )) is strictlydecreasing. Then there exists a sequence of complex laurent series φ n of degree such that the sequence φ n ( T n ) admits a limit L in V (( ∂ − )) .Proof. Let us denote d ( L F ( T n )) by d n and rename T n the sequenceof operators. Making use of Lemma . we can find Laurent seriesof degree ψ n ( z ) for n ≥ such that all ψ n ( T n ) − T have degree d − d F + 1 . We denote ψ n ( T n ) by T n for n ≥ and let T = T .Because we assumed that the leading term of the T n is invertible in V ,the sequence T n lies in V (( ∂ − )) . Moreover, since ψ n are Laurent seriesof degree in C , for all n ≥ we have by Lemma . :(2.18) d ( L F ( T n )) = d ( L F ( T n )) . Next, we use Lemma . one more time to obtain Laurent series ofdegree ψ n ( z ) for n ≥ such that ψ n ( T n ) − T has degree d − d F + 1 for all n ≥ . As previously, let us denote by T n the operator ψ n ( T n ) for n ≥ , and let T , = T , . Since d > d , for all n ≥ , T n − T =( T n − T ) + ( T − T ) has degree d − d F + 1 . Moreover, for all n ≥ we have(2.19) d ( L F ( T n )) = d n . Iterating the argument, we construct a sequences of pseudodifferentialoperators T kn with coefficients in V and Laurent series ( ψ kn ) n ≥ k +1 suchthat for all m, k > n ,(2.20) d ( T nn − T km ) = d n − d F + 1 , and such that for all k ≥ n (2.21) d ( L F ( T kn )) = d n . Specializing (2 . to k = m we get for n < m (2.22) d ( T nn − T mm ) = d n − d F + 1 . Therefore T nn = ψ n . . . ψ n − n ( T n ) = φ n ( T n ) admits a limit L in V (( ∂ − )) . (cid:3) Lemma 2.9.
Let F and G be two commuting functions, with d G ≥ .Then, if µ F denotes the degree of [ D F , ∂ ] , and provided that D G admitsa d G -th root in V (( ∂ − )) , (2.23) d ( L F ( D /d G G )) = µ F − d G + 1 . Proof.
By Lemma . , and since F and G commute, we have(2.24) X F ( D G ) = [ D F , D G ] + X G ( D F ) . In other words,(2.25) L F ( D G ) = X G ( D F ) . Therefore d ( L F ( D G )) = d ( X G ( D F )) . Note that, since d G ≥ , X G ( H ) (cid:54) =0 for any nonconstant function H . Hence the degree of X G ( D F ) is thehighest integer n for which ∂F∂u ( n ) is not a constant. This is the same asthe degree of [ D F , ∂ ] . Therefore(2.26) d ( L F ( D G )) = µ F . We complete the proof combining Lemma . with (2 . . (cid:3) Lemma 2.10.
Let W be an integrable system in V spanned by count-ably many functions ( F n ) n ≥ such that their differential orders d F n arestrictly increasing and d F ≥ . Let us assume moreover that ∂F ∂u ( dF admits a d F -th root in V . Then D F n admits a d F n -th root in V (( ∂ − )) for all n ≥ . Proof.
A pseudodifferential operator L = l N ∂ N + · · · ∈ V (( ∂ − )) admitsa N -th root if and only if l N does in V . Hence we want to show that ∂F n ∂u ( dFn ) admits a d F n -th root for all n . We know it is the case when n = 1 . Let us denote by X a d F -th root of ∂F ∂u ( dF in V . By degreeconsiderations in (2 . , we see that [ D F n , D F ] has degree at most d F n + d F − . Hence the coefficient of order d F n + d F − in thecommutator [ D F n , D F ] must vanish, i.e.(2.27) d F n ∂F n ∂u ( d Fn ) ( ∂F ∂u ( d F ) ) (cid:48) = d F ∂F ∂u ( d F ) ( ∂F n ∂u ( d Fn ) ) (cid:48) , which is to say that there is a contant α such that(2.28) ( ∂F n ∂u ( d Fn ) ) d F = α ( ∂F ∂u ( d F ) ) d Fn . and therefore(2.29) ∂F n ∂u ( d Fn ) = ( α /d F X ) d Fn . (cid:3) Proof. , of Proposition . . Let T n = D /d Fn F n , which are well-defined in V (( ∂ − )) thanks to Lemma . . Moreover, they have the same leading term which is invertiblein V because the leading term of D F is. By Lemma . , we have for n ≥ (2.30) d ( L F ( T n )) = µ F − d F n + 1 . These degrees are strictly decreasing, hence we can apply Lemma . to find Laurent series of degree φ n and a pseudodifferential operator L ∈ V (( ∂ − )) such that φ n ( T n ) converges to L . Taking the limit m →∞ in (2 . , we get(2.31) d ( φ n ( T n ) − L ) = µ F − d F + 2 − d F n . Let us check that L F k ( L ) = 0 for all k . First of all, using Lemma . ,we have(2.32) d ( L F k ( T n )) = µ F k − d F n + 1 . Since L F k is a derivation and φ n is a complex Laurent series of degree , the same holds if we replace T n by φ n ( T n ) (2.33) d ( L F k ( φ n ( T n ))) = µ F k − d F n + 1 . Moreover, by definition of the Lie derivative, it is clear that for anypseudodifferential operator M and function H ,(2.34) d ( L H ( M )) ≤ d ( M ) + d H . Finally, combining equations (2 . to (2 . : d ( L F k ( L )) ≤ d ( L F k ( φ n ( T n ))) + d ( L F k ( L − φ n ( T n ))) ≤ d ( L F k ( φ n ( T n ))) + d ( L − φ n ( T n )) + d F k ≤ µ F k − d F n + 1 + µ F − d F + 2 − d F n + d F k ≤ ( µ F + µ F k − d F + d F k + 3) − d F n . (2.35)The sequence of degrees d F n is strictly increasing. Therefore for all k ≥ L is recursion for F k :(2.36) L F k ( L ) = 0 . (cid:3) Remark . Let W be an integrable system in V . Then the quotientspace W / ( W ∩ V ) admits a countable basis. Indeed, it follows fromLemma . that if two functions F and G of differential order n ≥ commute, then ∂F∂u ( n ) and ∂G∂u ( n ) are proportional. Hereditary operators.
Definition 2.12.
A rational operator L = AB − ∈ V ( ∂ ) is called hereditary (cf. [O93]) if the following identity holds for all F ∈ V (2.37) L A ( F ) ( L ) = L L B ( F ) ( L ) . Remark . Using, (1 . , we can rewrite the preceding definitionas follows. An operator L = AB − is hereditary if and only if for all F ∈ V ,(2.38) X A ( F ) ( L ) − [( D A ) F , L ] = L ( X B ( F ) ( L ) − [( D B ) F , L ]) . Indeed, [ AD F , AB − ] = AB − [ BD F , AB − ] for all F . Remark . In the definition of hereditariness, the choice of the frac-tional decomposition does not matter. If A = A X , B = B X andequation (2 . holds for ( A, B ) , it also holds for ( A , B ) . Indeed,every coefficient of (2 . as a pseudodifferential operator on V is adifferential operator on F . Since they vanish on the image of X , whichis infinite dimensional over C , they are identically zero.The following lemma gives a reason for the name hereditary : if L isrecursion for G ∈ V , it is also recursion for L ( G ) . Lemma 2.15. If L = AB − is a rational operator recursion for B ( F ) and hereditary, it follows from Definition . that L is recursion for A ( F ) as well. Proof.
Obvious. (cid:3)
Example . Here are a few hereditary operators :(a) any matrix rational operator with constant coefficients. In thatcase, both sides of (2 . vanish.(b) L = ∂ ( ∂ + u ) ∂ − = ∂ + u + u (cid:48) ∂ − . In that case, A = ∂ ( ∂ + u ) and B = ∂ . Moreover, we have ( D A ) F = ∂F ( D B ) F = 0 X A ( F ) ( L ) = ∂A ( F ) ∂ − X B ( F ) ( L ) = ∂F (cid:48) ∂ − . (2.39) Therefore, for L , ∂ − (2 . ∂ is equivalent to(2.40) A ( F ) − F ∂ ( ∂ + u ) + ( ∂ + u ) F ∂ = ( ∂ + u ) F (cid:48) . The coefficient of ∂ is on both sides of (2 . . The constantcoefficients also agree, as(2.41) ( F (cid:48) + F u ) (cid:48) − F u (cid:48) = F (cid:48)(cid:48) + uF (cid:48) . It remains to check that the coefficients in front of ∂ are thesame in both sides of (2 . . This follows from(2.42) − F u + F (cid:48) + uF = F (cid:48) . (c) L = ∂ + 2 u + u (cid:48) ∂ − . It can be checked by a direct computationas in ( b ) . Lemma 2.17.
Let L = AB − be a hereditary rational operator. Then L N is hereditary for any N ∈ Z .Proof. First, let us show that L − is itself hereditary. Note that L − = BA − . Since L F is a derivation we have for F ∈ V X B ( F ) ( L − ) = − L − X B ( F ) ( L ) L − = − L − ( LX B ( F ) ( L )) L − = L − ( − L − X A ( F ) ( L ) L − )= L − X A ( F ) ( L − ) . (2.43)It remains to consider the case N ≥ . Let A = A , B = B , C = 0 and D = 0 . Assuming to have defined A k , B k , C k − and D k − for k ≥ we let C k and D k such that BC k = A k D k and let A k +1 = AC k and B k +1 = B k D k . It is easy to show by induction that L k = A k B k − .Let us show by induction on k that(2.44) L A k ( F ) ( L ) = L k L B k ( F ) ( L ) . By definition of hereditariness this is true for k = 1 . As for the induc-tion step, L A k +1 ( F ) ( L ) = L AC k ( F ) ( L )= L L BC k ( F ) ( L )= L L A k D k ( F ) ( L )= L k +1 L B k D k ( F ) ( L )= L k +1 L B k +1 ( F ) ( L ) . (2.45)To conclude note that L A k ( F ) ( L k ) = (cid:80) k − i =0 L i L A k ( F ) ( L ) L k − − i . Hence L A k ( F ) ( L k ) = L k k − (cid:88) i =0 L i L B k ( F ) ( L ) L k − − i L k L B k ( F ) ( L k ) . (2.46) (cid:3) The following proposition says that a rational operator must beheredtary in order to generate an integrable system.
Proposition 2.18.
Let L = AB − be a rational operator and ( H n ) n ≥ a sequence of functions, which span an infinite-dimensional space over C , such that(1) B ( H n +1 ) = A ( H n ) for all n ≥ ,(2) { B ( H n ) , B ( H m ) } = 0 for all n, m ≥ .Then L is recursion for all the B ( H n ) and hereditary.Proof. Let F = B H n for some n ≥ . We know that F commutes with A ( H m ) for all m ≥ . This means by definition that(2.47) X F ( A ( H m )) = D F ( A ( H m )) ∀ m ≥ . If we denote − X F ( H m ) by x m , we can rewrite (2 . as(2.48) ( X F ( A ) − D F A )( H m ) = A ( x m ) ∀ m ≥ . Similarly, because F commutes with B ( H m ) for all m ≥ , we see that(2.49) ( X F ( B ) − D F B )( H m ) = B ( x m ) ∀ m ≥ . Let C and D be differential operators such that CA = DB . Then,multiplying on the left (2 . by C and (2 . by D , we get that(2.50) C ( X F ( A ) − D F A )( H m ) = D ( X F ( B ) − D F B )( H m ) ∀ m ≥ . Since the sequence H m spans an infinite-dimensional space over C , (2 . implies that(2.51) C ( X F ( A ) − D F A ) = D ( X F ( B ) − D F B ) , which rewrites into(2.52) ( X F ( A ) − D F A ) = AB − ( X F ( B ) − D F B ) . because L = AB − = C − D . From there, we can conclude that L isrecursion for F . Indeed X F ( AB − ) = X F ( A ) B − − AB − X F ( B ) B − = ( D F A + AB − ( X F ( B ) − D F B )) B − − AB − X F ( B ) B − = D F AB − − AB − D F = [ D F , AB − ] , (2.53)where we used (2 . to obtain the second line of (2 . .Since L is recursion for A ( H n ) and B ( H n ) for all n ≥ , equation (2 . holds for F = H n for all n ≥ . Moreover, since the H n ’s spanan infinite-dimensional space over C , we get that (2 . holds for all H ∈ V and that L is hereditary. (cid:3) Proposition 2.19.
Let A ( ∂ ) ∈ V [ ∂ ] be a hereditary differential opera-tor, and F ∈ V be such that A is recursion for F . Then (2.54) { A n ( F ) , A m ( F ) } = 0 ∀ n, m ≥ . Proof.
Since A is recursion for F , it follows from iterating Lemma . that A is recursion for A n ( F ) for all n ≥ . Let n ≥ . We want toprove by induction on m ≥ that A n ( F ) and A n + m ( F ) commute. Itis obviously true when m = 0 . Let us assume that, for some m ≥ , A n ( F ) and A n + m ( F ) commute. Since A is recursion for A n ( F ) , we have { A n ( F ) , A m +1 ( F ) } = X A n ( F ) ( A m +1 ( F )) − X A m +1 ( F ) ( A n ( F ))= ( X A n ( F ) ( A ) − [ D A n ( F ) , A ])( A m ( F ))+ A ( { A n ( F ) , A m ( F ) } )= 0 + 0 . (2.55) (cid:3) Proposition 2.20.
Let A = (cid:80) Nk =0 a k ∂ k be a hereditary differentialoperator on V . Then for all ≤ k ≤ N , (2.56) d ( a k ) ≤ N + 1 . Proof. (2 . for B = 1 means that for all F ∈ V ,(2.57) X A ( F ) ( A ) − [( D A ) F , A ] = AX F ( A ) . Both X A ( F ) ( A ) and AX F ( A ) have degree in ∂ at most N hence(2.58) d ∂ ([( D A ) F , A ]) ≤ N ∀ F ∈ V . Let m = max k d ( a k ) . Then the coefficient of ∂ N + m − in [( D A ) F , A ] is(2.59) m ( (cid:88) k F ( k ) ∂a k ∂u ( m ) ) a (cid:48) N − N a N ( (cid:88) k F ( k ) ∂a k ∂u ( m ) ) (cid:48) , which is non-zero for F ∈ V with sufficiently large differential order.Therefore(2.60) m ≤ N + 1 . (cid:3) Integrable Operators
Hereditariness is not a sufficient condition for a rational operator togenerate an integrable system. Let us consider the operator(3.1) L = ∂ − u (cid:48)(cid:48) ∂ . We begin by finding the minimal right fractional decomposition of L .Since ∂ ( u (cid:48)(cid:48) u (cid:48)(cid:48)(cid:48) ∂ −
1) = u (cid:48)(cid:48) ∂ u (cid:48)(cid:48)(cid:48) ∂ we have(3.2) L = ( u (cid:48)(cid:48) u (cid:48)(cid:48)(cid:48) ∂ − u (cid:48)(cid:48)(cid:48) ∂ ) − = AB − . Therefore we can compute D A and D B . Namely, for all F ( D A ) F = F (cid:48) u (cid:48)(cid:48)(cid:48) ( − u (cid:48)(cid:48) u (cid:48)(cid:48)(cid:48) ∂ + 1) ∂ , ( D B ) F = − F (cid:48) u (cid:48)(cid:48)(cid:48) ∂ . (3.3)We want to check that L is hereditary, i.e. that (2 . holds for all F ∈ V :(3.4) X A ( F ) ( L ) − [( D A ) F , L ] = L ( X B ( F ) ( L ) − [( D B ) F , L ]) . Let us plug in (3 . and (3 . into (3 . , multiply on the left by ∂ andon the right by ∂ − . (3 . is equivalent to the following(3.5) A ( F ) (cid:48)(cid:48) − [ ∂ F (cid:48) u (cid:48)(cid:48)(cid:48) ( − u (cid:48)(cid:48) u (cid:48)(cid:48)(cid:48) ∂ + 1) ∂, u (cid:48)(cid:48) ] = u (cid:48)(cid:48) B ( F ) (cid:48)(cid:48) − u (cid:48)(cid:48) [ − ∂ F (cid:48) u (cid:48)(cid:48)(cid:48) ∂ , u (cid:48)(cid:48) ] . Putting u (cid:48)(cid:48) inside the commutator and rearranging the terms yields A ( F ) (cid:48)(cid:48) − u (cid:48)(cid:48) B ( F ) (cid:48)(cid:48) = [( ∂ F (cid:48) u (cid:48)(cid:48)(cid:48) ( − u (cid:48)(cid:48) u (cid:48)(cid:48)(cid:48) ∂ + 1) + u (cid:48)(cid:48) ∂ F (cid:48) u (cid:48)(cid:48)(cid:48) ∂ ) ∂, u (cid:48)(cid:48) ]= [( ∂ F (cid:48) u (cid:48)(cid:48)(cid:48) − F (cid:48) u (cid:48)(cid:48)(cid:48) ∂ ) ∂, u (cid:48)(cid:48) ]= ( F (cid:48) u (cid:48)(cid:48)(cid:48) ) (cid:48) u (cid:48)(cid:48)(cid:48) = u (cid:48)(cid:48)(cid:48) B ( F ) (cid:48) . (3.6)Hence to check that L is hereditary amounts to check that the followingidentity holds for all F :(3.7) A ( F ) (cid:48) = u (cid:48)(cid:48) B ( F ) (cid:48) , which is satisfied. Therefore L is hereditary.However, L is recursion for a function F if and only if F lies in the C -span of and u (cid:48) . Indeed, recall that L is recursion for a function F if and only if(3.8) X F ( ∂ − u (cid:48)(cid:48) ∂ ) = [ D F , ∂ − u (cid:48)(cid:48) ∂ ] . Taking the conjugation of (3 . by ∂ gives(3.9) F (cid:48)(cid:48) = [ ∂D F ∂ − , u (cid:48)(cid:48) ] . Since the degree of [ M, G ] is d ( M ) − when M is a pseudodifferentialoperator and G is a non-constant function, the degree of D F has to beat most . From there it is easy to conclude that the only functions forwhich L is a recursion operator are linear combination of u (cid:48) and .This example shows that to be hereditary for a rational operatordoes not imply that it can generate infinitely many functions. A ra-tional operator which generates an integrable system falls into a finnersubclass of operators which we call integrables and study in detail inthis section.We now proceed to define integrable operators. We will show thatany positive power of integrable L is integrable, and that integrabilityis a necessary condition to generate infinitely many commuting func-tions. We will also state and prove one of the key properties of theseoperators which is that if integrable L = AB − is recursion for B ( F ) ,then A ( F ) and B ( F ) commute. Definition 3.1.
A differential operator A ∈ K [ ∂ ] is called integrableif there exists a bidifferential operator M on K such that for all F ∈ V (3.10) X A ( F ) ( A ) = D A ( F ) A + A ( M F − D F A ) . Or equivalently,(3.11) X A ( F ) ( A ) − ( D A ) F A = AM F . Example . A Poisson differential operator H is integrable. Indeedrecall from [BDSK09] (1 . that H is a Poisson differential operatorif and only if it is skewadjoint H ∗ = − H and satisfies for all ( F, G ) (3.12) H ( D G H ( F ) + D ∗ H ( F ) ( G ) − D F H ( G ) + D ∗ F H ( G )) = { H ( F ) , H ( G ) } . Moreover, since H ∗ = − H , we have(3.13) D ∗ H ( F ) ( G ) + D ∗ F H ( G ) = ( D H ) ∗ F ( G ) , hence for all ( F, G ) , the following equation holds(3.14) X H ( F ) ( H )( G ) + HD F ( G ) − D H ( F ) H ( G ) = H ( D H ) ∗ F ( G ) . Rearranging the LHS of (3 . using (1 . , we get for all F :(3.15) X H ( F ) ( H ) − ( D H ) F H = H ( D H ) ∗ F . Not all integrable operators are Poisson :
Example . The operator A = ∂ ( ∂ + u ) is integrable. Indeed (3 . holds for A and all function F ∈ V (3.16) ∂ ( F (cid:48)(cid:48) + ( uF ) (cid:48) ) − ∂F ∂ ( ∂ + u ) = ∂ ( ∂ + u )( − F ∂ + F (cid:48) ) . Remark . The bidifferential operator M in (3 . has to be skewsym-metric. Indeed, thanks to (1 . , (3 . and (3 . are equivalent to(3.17) X A ( F ) ( A )( G ) − X A ( G ) ( A )( F ) = A ( M ( F, G )) ∀ F, G ∈ V . Definition 3.5.
We say that a pair of differential operators ( A, B ) on K is integrable if there exists two bidifferential operators M and N such that for all F ∈ V and for all λ ∈ C ,(3.18) X ( A + λB )( F ) ( A + λB ) − ( D A + λB ) F ( A + λB ) = ( A + λB )( M + λN ) F . We say that a rational operator L ∈ K ( ∂ ) is integrable if there existsan integrable pair ( A, B ) such that L = AB − . Note that (3 . isstronger than asking for ( A + λB ) to be integrable for all λ . (3 . is equivalent to the following three equations, symmetric in A and B , forall F ∈ V : AM F = X A ( F ) ( A ) − ( D A ) F ABN F = X B ( F ) ( B ) − ( D B ) F BAN F + BM F = X A ( F ) ( B ) + X B ( F ) ( A ) − ( D A ) F B − ( D B ) F A. (3.19) Remark . A compatible pair of local Poisson structures ( H, K ) isintegrable. Indeed, (3 . holds with H + λK for any constant λ .Comparing with (3 . , we obtain the claim. Lemma 3.7.
Let
A, B, C be differential operators such that A = BC is integrable. Then B is integrable.Proof. We know that there exists a bidifferential operator M such thatfor all F ∈ V (3.20) X A ( F ) ( A ) = D A ( F ) A + A ( M F − D F A ) . Since A = BC and evolutionnary vector fields act via (1 . as deriva-tions of the ring of differential operators, we obtain for all F ∈ V (3.21) X BC ( F ) ( B ) C + BX BC ( F ) ( C ) − D BC ( F ) BC + BCD F BC = BCM F . Let us start by simplifying the two last terms of the LHS using (1 . twice : D BC ( F ) = BD C ( F ) + ( D B ) C ( F ) = BCD F + B ( D C ) F + ( D B ) C ( F ) . (3.22)Combining (3 . and (3 . , we obtain(3.23) B ( CM F + ( D C ) F BC − X BC ( F ) ( C )) = ( X BC ( F ) ( B ) − ( D B ) C ( F ) B ) C. We are in the situation where BQ F = P C ( F ) C for some bidifferentialoperator Q and(3.24) P F = X B ( F ) ( B ) − ( D B ) F B. Lemma . implies that there exists a bidifferential operator R suchthat P = BR . This precisely means that (3 . holds, i.e. that B isan integrable operator. (cid:3) Next, we make a connection with the previous notion of hereditaryoperator by showing that integrability implies hereditariness.
Lemma 3.8.
Let L be an integrable rational operator. Then L is hered-itary. Conversely if L is hereditary, AB − is a minimal right fractionaldecompoisition of L and B is integrable, then the pair ( A, B ) is inte-grable. Proof.
Let ( A, B ) be a pair of operators satisfying (3 . and let M, N be the bidifferential operators corresponding to A and B . Let us con-dider the LHS of equation (2 . : LHS = X A ( F ) ( AB − ) − [( D A ) F , AB − ]= ( X A ( F ) ( A ) − ( D A ) F A ) B − − AB − ( X A ( F ) ( B ) − ( D A ) F B ) B − = AB − ( BM F − X A ( F ) ( B ) + ( D A ) F B ) B − = AB − ( X B ( F ) ( A ) − ( D B ) F A − AN F ) B − = AB − ( X B ( F ) ( A ) − ( D B ) F A − AB − ( X B ( F ) ( B ) − ( D B ) F B )) B − = AB − ( X B ( F ) ( AB − ) − [( D B ) F , AB − ])= RHS. (3.25)We used the first line of (3 . to go from line to line , the thirdline of (3 . to deduce line from line , and finally the second lineof (3 . to go from line to line .If we now simply assume that L is hereditary, that AB − is minimaland that B integrable with corresponding matrix bidifferential operator N , we can say from the four last lines of (3 . that(3.26) RHS = AB − ( X B ( F ) ( A ) − ( D B ) F A − AN F ) B − , where RHS denotes the right hand side of (2 . . Similarly we canrewrite the LHS of (2 . as we did in (3 . (3.27) LHS = ( X A ( F ) ( A ) − ( D A ) F A ) B − − AB − ( X A ( F ) ( B ) − ( D A ) F B ) B − . By hereditariness of L , the quantities in (3 . and (3 . are equal. Inother words, we have the equation(3.28) P = AB − Q, where for all F ∈ V , P F = X A ( F ) ( A ) − ( D A ) F AQ F = X A ( F ) ( B ) + X B ( F ) ( A ) − ( D A ) F B − ( D B ) F A − AN F . (3.29)Let CA = DB be the left least common multiple of the pair ( A, B ) . Itis also the right least common multiple of the pair ( C, D ) . Moreover (3 . rewrites as CP = DQ . Applying Lemma . provides us witha bidifferential operator M such that P = AMQ = BM . (3.30)Therefore the pair ( A, B ) satisfies (3 . and is integrable. (cid:3) Lemma 3.9.
Let A and B be integrable differential operators such that AB − is hereditary. Then the pair ( A, B ) is integrable.Proof. If we look carefully at the second part of the proof of Lemma . ,we notice that we did not use the minimality of the pair ( A, B ) to reach (3 . . From there, if we know that A is integrable, we can concludeas well the integrability of ( A, B ) . Indeed, since A is integrable, P isdivisible on the left by A by (3 . and we can cancel by A in (3 . . (cid:3) Corollary 3.10.
Let A , B and C be differential operators. If the pair ( AC, BC ) is integrable, then so is ( A, B ) . In particular, for an inte-grable operator L with right minimal fractional decomposition A B − ,we have that ( A , B ) is integrable.Proof. By Lemma . AB − = ( AC )( BC ) − is hereditary and byLemma . , both A and B are integrable since AC and BC are. Weconclude using Lemma . . If L is integrable there exist an integrablepair of differential operators ( A, B ) such that L = AB − . If A B − isa minimal right fractional decomposition of L , we can find an operator E such that A = A E and B = B E . Therefore the pair ( A , B ) isintegrable as well. (cid:3) Example . The operator L = ∂ − u (cid:48)(cid:48) ∂ is not integrable. Indeed,from the previous remark and by (3 . we should have that u (cid:48)(cid:48)(cid:48) ∂ isintegrable. However, this is not the case, which will follow from Propo-sition . . Therefore hereditariness does not imply integrability.The two following propositions aim at showing that the set of inte-grable operators is stable under taking integer powers. This propertywill turn out to be essential in proving the commutativity of the func-tions generated by the rational operator. Proposition 3.12.
Let A and B be two differential operators such that ( A, B ) is integrable, and let AD = BC be their right least commonmultiple (see Definition . ). Then ( AC, BD ) is integrable as well.Proof. Let M and N be the bidifferential operators corresponding to A and B . By (3 . for B and (1 . we can write for all functions F ∈ V (3.31) X BD ( F ) ( B ) = ( D BD ) F B − B ( D D ) F B + B ( N ) D ( F ) . Multiplying on the right by D we have(3.32) X BD ( F ) ( BD ) − ( D BD ) F BD = B ( X BD ( F ) ( D ) − ( D D ) F BD + N D ( F ) D ) . To say that BD is integrable is therefore the same as saying that(3.33) Y = X BD ( F ) ( D ) − ( D D ) F BD + N D ( F ) D , is divisible on the left by D as a bidifferential operator. By constructionof D and by Lemma . it will be enough to show that AY = BP forsome bidifferential operator P . From now on we will work modulo theleft ideal of V [ ∂ ] generated by B .Let us apply the derivation X BD ( F ) to the identity AD = BC .(3.34) AX BD ( F ) ( D ) + X BD ( F ) ( A ) D ≡ X BD ( F ) ( B ) C. We now want to rewrite the RHS of (3 . . Projecting (3 . modulo B , we have(3.35) X BD ( F ) ( B ) ≡ ( D BD ) F B. Using equations (3 . to (3 . we get AY ≡ ( D BD ) F .BC − X BD ( F ) ( A ) D − A ( D D ) F BD + A ( N ) D ( F ) D. (3.36)Replacing F by D ( F ) in the third line of (3 . yields(3.37) X A ( D ( F )) ( B ) + X B ( D ( F )) ( A ) − ( D A ) D ( F ) B − ( D B ) D ( F ) A ≡ AN D ( F ) . After comparing (3 . with (3 . and recalling that AD = BC weget AY ≡ ( D BD ) F AD − A ( D D ) F BD + X AD ( F ) ( B ) D − ( D B ) D ( F ) AD − ( D A ) D ( F ) BD . (3.38)Moreover, by (1 . , we have ( D BD ) F AD − ( D B ) D ( F ) AD = B ( D D ) F AD ≡ A ( D D ) F BD + ( D A ) D ( F ) BD = ( D AD ) F BD . (3.39)Therefore (3 . can be simplified to(3.40) AY ≡ [ X AD ( F ) ( B ) − ( D AD ) F B ] D. After replacing AD by BC and using (1 . one more time we get AY ≡ [ X BC ( F ) ( B ) − ( D BC ) F B ] D ≡ [ X BC ( F ) ( B ) − ( D B ) C ( F ) B ] D ≡ B ( N ) C ( F ) D ≡ . (3.41)We proved that BD is an integrable operator. Since ( B, A ) is inte-grable as well by symmetry of (3 . , we infer that AC is an integrableoperator as well. Moreover L = AB − is hereditary by Lemma . andhence L = ( AC )( BD ) − is hereditary too. This enables us to concludethat the pair ( AC, BD ) is integrable thanks to Lemma . . (cid:3) Proposition 3.13.
Let L be an integrable operator. Then L n is inte-grable for all n ∈ Z + Proof.
Let ( A, B ) be an integrable pair of differential operators suchthat L = AB − . We know by Lemma . that L is hereditary, andtherefore so is L k for all k ≥ by Lemma . .Let us begin by defining sequences of differential operators A n , B n , C n and D n . We let A = A , B = B , C = 1 and D = 1 . Assumingthat we defined A n , B n , C n − and D n − , we let(3.42) B n C n = AD n . be the least right common multiple of the pair ( A, B n ) . We then define(3.43) A n +1 = A n C n , B n +1 = BD n . It is easy to check by induction that L n = A n B n − for all n ≥ .Let us prove by induction that D n divides D n +1 on the left for all n ≥ , i.e. that there exists a differential operator E n such that D n +1 = D n E n .We know that(3.44) AD = B C = BD C . Hence, since AD = BC is the right least common multiple of the pair ( A, B ) , we can find an operator E such that(3.45) D = D E , D C = C E . Let us rename D by E . Assume that we constructed n operators E , . . . , E n such that for all ≤ k ≤ n (3.46) D k = E . . . E k , E k C k +1 = C k E k +1 . Combining equations (3 . , (3 . and (3 . , we obtain AD n +1 = B n +1 C n +1 = BD n C n +1 = BD n − E n C n +1 = B n ( E n C n +1 ) . (3.47)By (3 . , (3 . and by definition of the least right common multiple,one can find an operator E n +1 such that(3.48) D n +1 = D n E n +1 , E n C n +1 = C n E n +1 . Therefore (3 . holds for all k ≥ . Note that E k and C k are rightcoprime for all k ≥ . Indeed, D k and C k are right coprime and E k divides D k on the right. Moreover, we have for all k ≥ (3.49) A k = AC . . . C k − . We now prove by induction on n that ( A n , B n ) is integrable for all n ≥ . The case n = 0 is true by hypothesis. Let us assume that it istrue for k = 2 n and define operators K and L such that(3.50) A k K = B k L is the least right common multiple of the pair ( A k , B k ) . We are goingto show that for some operator M ,(3.51) A k M = A k L, B k M = B k K. Combining (3 . with (3 . gives for all k ≥ (3.52) A ( C . . . C k − K ) = B k L. Comparing (3 . with (3 . we deduce the existence of an operator H such that(3.53) L = C k H, C . . . C k − K = D k H = E . . . E k H. Using (3 . and the fact that C i .E i +1 = E i .C i +1 is the least rightcommon multiple of the pair ( E i , C i ) for all i ≥ , we find that thereexists a differential operator (cid:101) H such that(3.54) H = C k +1 . . . C k − (cid:101) H, K = E k +1 . . . E k (cid:101) H. Combining (3 . and (3 . gives (3 . with M = (cid:101) H . By Proposition . , the pair ( A k L, B k K ) is integrable, hence so is ( A k , B k ) by (3 . and Corollary . .Since by equations (3 . and (3 . A m (resp. B m ) divides A l (resp. B l ) on the left whenever m ≥ l and all the A n ( resp. B n )are integrable, we know by Lemma . that the operators A m (resp. B m ) are integrable for all m ≥ . Moreover, since L m = A m B m − ishereditary for all m ≥ , Lemma . enables us to claim that the pairs ( A m , B m ) are integrable for all m ≥ . In particular, L m is integrablefor all m ∈ Z + . (cid:3) As it was the case with hereditariness, integrability is a necessarycondition for a rational operator to produce commuting functions.More specifically, we prove :
Proposition 3.14.
Let L be a rational operator where L = AB − is aminimal fractional decomposition and let ( H n ) n ≥ be a sequence in V ,which spans an infinite-dimensional space over C , such that(1) B ( H n +1 ) = A ( H n ) for all n ≥ ,(2) { B ( H n ) , B ( H m ) } = 0 for all n, m ≥ . Then the pair ( A, B ) is integrable.Proof. Let us begin by defining two bidifferential operators, letting M F := X A ( F ) ( A ) − ( D A ) F AN F := X B ( F ) ( B ) − ( D B ) F B, (3.55)for F ∈ V . By commutativity of the functions A ( H n ) , we have for all n, m ≥ (3.56) X A ( H n ) ( A ( H m )) = X A ( H m ) ( A ( H n )) . Since X A ( H n ) is a derivation of V commuting with ∂ and using (3 . we get X A ( H n ) ( A ( H m )) = X A ( H n ) ( A )( H m ) + A ( X A ( H n ) ( H m ))= ( M ) H n ( H m ) + A ( X A ( H n ) ( H m )) + ( D A ) H n ( H m ) . (3.57)On the other hand, by (1 . and (1 . we have X A ( H m ) ( A ( H n )) = D A ( H n ) ( A ( H m ))= ( D A ) H n ( H m ) + A ( D H n ( A ( H m ))) . (3.58)Combining equations (3 . , (3 . and (3 . we get(3.59) ( M H n + A ( X A ( H n ) − D H n A ))( H m ) = 0 . Similarly, replacing A with B and M with N , we also have for all n, m ≥ (3.60) ( N H n + B ( X B ( H n ) − D H n B ))( H m ) = 0 . Let CA = DB be the least left common multiple of the pair ( A, B ) .Recall that for all n ≥ , B ( H n +1 ) = A ( H n ) . Therefore, multiplying (3 . on the left by C and (3 . for n + 1 on the left by D , we obtainfor all n, m ≥ :(3.61) ( C ( M H n − AD H n A ))( H m ) = ( D ( N H n +1 − BD H n +1 B ))( H m ) . Since this is true for all m ≥ and the span of the H m is infinitedimensional over C we have an operator identity for all n ≥ :(3.62) C ( M H n − AD H n A ) = D ( N H n +1 − BD H n +1 B ) . Since CA = DB is also the right least common multiple of the pair ( C, D ) because A.B − is a minimal fractionnal decomposition, we de-duce from (3 . the existence of differential operators P n , such thatfor all n ≥ ,(3.63) N H n +1 − BD H n +1 B = BP n . Let Q and R be two bidifferential operators such that for all F ∈ V (a) N F = BQ F + R F ,(b) d ( R F ) < d ( B ) ,which we can find using Lemma . . It follows from (3 . that forall n ≥ , R H n +1 = 0 . Since the span of the H n is infinite dimensionalover C we have R = 0 . Therefore B divides N on the left, i.e. B isintegrable. Similarly, we can prove that A is integrable. Finally, recallthat L = AB − is hereditary by Proposition . . Therefore the pair ( A, B ) is integrable by Lemma . . (cid:3) The following lemma states that when an integrable operator L isrecursion for a function, the latter commutes with its image under L . Lemma 3.15.
Let F ∈ V and ( A, B ) be an integrable pair of differen-tial operators such that L = AB − is recursion for B ( F ) . Then A ( F ) and B ( F ) commute.Proof. We know that B is integrable, meaning by (3 . that thereexists a bidifferential operator M , such that for all G ∈ V ,(3.64) X B ( G ) ( B ) = D B ( G ) B + B ( M G − D G B ) . Since M is skewsymmetric, we have(3.65) M G ( G ) = 0 ∀ G ∈ V . Let us now recall that L being recursion for B ( F ) means(3.66) X B ( F ) ( AB − ) = [ D B ( F ) , AB − ] . Rearranging the terms of (3 . and multiplying on the right by B , weget(3.67) X B ( F ) ( A ) − D B ( F ) A = AB − ( X B ( F ) ( B ) − D B ( F ) B ) . Let CA = DB be the left common multiple of the pair ( A, B ) . Wededuce from (3 . that(3.68) C ( X B ( F ) ( A ) − D B ( F ) A ) = D ( X B ( F ) ( B ) − D B ( F ) B ) . Let CA = DB be the right least common multiple of the pair ( C, D ) .Let E be such that A = A E and B = B E . From (3 . it followsthat there exists a differential operator H such that(3.69) X B ( F ) ( A ) − D B ( F ) A = A H, X B ( F ) ( B ) − D B ( F ) B = B H. Comparing (3 . with the second line of (3 . , we have(3.70) H = E ( M F − D F B ) . Therefore, by (3 . and (3 . ,(3.71) H ( F ) = − E ( D F ( B ( F ))) . Applying the first line of (3 . to F and using (3 . we get(3.72) X B ( F ) ( A ( F )) = D B ( F ) ( A ( F )) , proving the claim. (cid:3) The next proposition provides us with the first sufficient conditionfor a rational operator L to generate an infinte sequence of commutingfunctions, providing that L n F is defined for all n and some F . Proposition 3.16.
Let L = AB − be a rational operator with ( A, B ) integrable and let ( H n ) n ≥ be a sequence in V such that(1) L is recursion for B ( H ) ,(2) A ( H n ) = B ( H n +1 ) for all n ≥ .Then the functions B ( H n ) pairwise commute.Proof. L is in particular hereditary by Lemma . . Therefore, since L is recursion for B ( G ) and by Lemma . , L is recursion for all the B ( H n ) . We know that L k is integrable for all k ≥ by Proposition . .Let L k = A k B − k be the right minimal fractional decomposition of L k .By Theorem . in [CDSK14] we know that for any n ≥ there existsa function F n,k such that B ( H n ) = B k ( F n,k ) and B ( H n + k ) = A k ( F n + k ) .It follows directly from Lemma . that B ( H n ) and B ( H n + k ) commutein V . This holds for all n, k ≥ . (cid:3) Remark . The first condition in Proposition . is met whenever G lies in the kernel of B . Corollary 3.18.
Let A , B be two differential operators and let ( H n ) n ≥ be a sequence of functions in V which spans an infinite-dimensionalspace over C . Let us assume moreover that L is recursion for B ( H ) and that for all n ≥ , (3.73) B ( H n +1 ) = A ( H n ) . Then the functions B ( H n ) pairwise commute if and only if the pair ( A, B ) is integrable.Proof. It follows immediately from Propositions . and . . (cid:3) Example . The rational operator L = ∂ ( ∂ + u ) ∂ − is integrable.It follows from Example . after performing the change of variables u → u + λ . On the other hand, if we let A = ∂ ( ∂ + u ) , B = ∂ , and H n = ( ∂ + u ) n (1) , it is clear that L is recursion for B ( H ) = 0 and that B ( H n +1 ) = A ( H n ) for all n ≥ . Therefore the functions H (cid:48) n pairwisecommute. Note that u t = H (cid:48) is the Burgers equation.The following proposition says that hereditary operators are not farfrom being integrable. Proposition 3.20.
Let L be a hereditary rational operator with rightminimal fractional decomposition AB − and left minimal fractional de-composition C − D . Assume moreover that ( C, D ) are right coprime.Then the pair ( A, B ) is integrable.Proof. Recall that thanks to Lemma . it is enough to show that B is integrable. If we look carefully at the proof of Lemma . , morespecifically if we equate the second and the fifth line of (3 . , we seethat the hereditariness of L is equivalent to the equation(3.74) P F = LQ F − L R F for all F ∈ V , where the bidifferential operators P, Q, R are given by P F = X A ( F ) ( A ) − ( D A ) F AQ F = X A ( F ) ( B ) + X B ( F ) ( A ) − ( D A ) F B − ( D B ) F AR F = X B ( F ) ( B ) − ( D B ) F B. (3.75)Asking for the integrability of B amounts to ask for R to be divisibleon the left by B . We will show that fact. Let us first use the leftpresentation L = C − D in (3 . :(3.76) P F = C − DQ F + C − DC − DR F . Rearranging (3 . , we get for all F ∈ V (3.77) CP F − DQ F = DC − DR F . Now we use the right minimality of the fraction DC − to deduce that DR is divisible on the left by C . Given that CA = DB is the leastright common multiple of the pair ( C, D ) since AB − is a right minimaldecomposition of L , it follows from Lemma . that R is divisible onthe left by B . (cid:3) Example . In Example . , we saw that the rational operator L = ∂ − u (cid:48)(cid:48) ∂ is not integrable. Note that L does not meet the hypothesisof Proposition . . Indeed, ∂ and u (cid:48)(cid:48) ∂ are not right coprime. Weakly non-local Operators
In this section we study weakly non-local rational operators. Let V be a normal algebra of differential functions, and let K be its field offractions. Definition 4.1.
A weakly non-local operator L is a rational operatorwhich can be written in the following form(4.1) L = E ( ∂ ) + n (cid:88) i =1 p i ∂ − q i , where E is a differential operator and p i and q i are elements of V . Wedenote the space of weakly non-local operators with coefficients in V by W V , or simply W when there is no confusion on the algebra V . Definition 4.2.
Let A be a differential algebra with the subfield ofconstants C and let P ∈ A [ ∂ ] . We say that P has a full kernel in A if(4.2) dim C Ker A P = d ( P ) . As we show next, weakly non-local operators can be characterizedby their denominators in a minimal fractional decomposition AB − . Lemma 4.3.
Let L = E ( ∂ ) + (cid:80) ni =1 p i ∂ − q i be a weakly non-local op-erator, and A and B be two differential operators. Then, we have : AL = n (cid:88) i =1 A ( p i ) ∂ − q i mod V [ ∂ ] ,LB = n (cid:88) i =1 p i ∂ − B ∗ ( q i ) mod V [ ∂ ] . (4.3) Proof.
This follows directly from the fact that, for all i = 1 , . . . , n wecan find by the Euclidean division two differential operators C i and D i such that(4.4) Ap i = C i ∂ + A ( p i ) , q i B = ∂D i + B ∗ ( q i ) . (cid:3) Lemma 4.4.
The vector space W V is isomorphic to V [ ∂ ] ⊕ ( V ⊗ C V ) under the map E + (cid:80) i p i ∂ − q i → E ⊕ ( (cid:80) i p i ⊗ q i ) .Proof. Let Z = span { p∂ − q | ( p, q ) ∈ V } . We need to prove that Z isisomorphic to V ⊗ C V . Let φ be the map from V ⊗ C V to Z sending thetensor f ⊗ g to the operator f ∂ − g . φ is surjective by definition of Z . To check that it is an injective map let us take two sets consisting of linearlyindependent functions { f , . . . , f n } and { g , . . . , g n } and assume that(4.5) L = n (cid:88) i =1 f i ∂ − g i = 0 . If we expand L as a Laurent series in ∂ and equal its coefficients to we get that for all nonnegative integer k , (cid:80) ni =1 f i g i ( k ) = 0 which isthe same as saying that for all differential operator P , (cid:80) i f i P ( g i ) = 0 .Since the g j are linearly independent functions and for a given i , wecan pick by Lemma . an operator P i annihilating all the g j ’s except g i . Therefore f i should be trivial, which is a contradiction. (cid:3) Lemma 4.5.
Let L ∈ K ( ∂ ) be a rational operator with minimal rightfractional decomposition L = AB − . Then the following statements areequivalent(1) L ∈ W K .(2) B has a full kernel in K .(3) B ∗ has a full kernel in K .Moreover if L = E ( ∂ ) + (cid:80) ni =1 p i ∂ − q i where both the p i ’s and q i ’s arelinearly independent elements of K , then d ( B ) = n and B is a rightleast common multiple of the differential operators q i ∂ . Finally, KerB ∗ is spanned by the q i ’s.Proof. Let us first prove that (1) = ⇒ (3) . Let E ∈ K [ ∂ ] and p i , q i ∈ K be such that(4.6) L = AB − = E + n (cid:88) i =1 p i ∂ − q i , where the { p , . . . , p n } and { q , . . . , q n } are sets consisting of linearlyindependent functions. Then, multiplying (4 . on the right by B andusing Lemma . we obtain(4.7) (cid:88) i p i ∂ − B ∗ ( q i ) = 0 . By Lemma . , (4 . implies that { q , . . . , q n } ⊂ KerB ∗ . Hence(4.8) n ≤ dim C KerB ∗ . Let C be a common multiple of the differential operators q i ∂ , i.e. a dif-ferential operator such that for all i = 1 , . . . , n we can find a differential operator M i satisfying(4.9) C = 1 q i ∂M i . From equations (4 . and (4 . we have AB − = E + n (cid:88) i =1 p i ( 1 q i ∂ ) − = E + n (cid:88) i =1 p i M i M − i ( 1 q i ∂ ) − = ( EC + n (cid:88) i =1 p i M i ) C − . (4.10)Since AB − is a right minimal fractional decomposition of L , thereexists by Lemma . an operator D such that A = ( EC + n (cid:88) i =1 p i M i ) DB = CD. (4.11)In particular,(4.12) d ( B ) ≤ d ( C ) = n. Recall that d ( B ) = d ( B ∗ ) . Hence, by (4 . , (4 . and Lemma . ,(4.13) d ( B ∗ ) = dim C KerB ∗ = n, which means that B ∗ has a full kernel in K spanned by the functions q i .We also get from d ( B ) = d ( C ) that B is a least right common multipleof the differential operators q i ∂ , and that(4.14) A = EB + n (cid:88) i =1 p i M i . We now prove that (2) = ⇒ (1) by induction on the degree of B .We only have to prove that B − ∈ W K since it follows from Lemma . that W K is stable by left or right multiplication by elements of V [ ∂ ] .If B is a degree operator such that dim C KerB = 1 then it can bewritten in the form f ∂g . Therefore B − = g ∂ − f is weakly nonlocal.Let B be an operator of degree n + 1 with full kernel in K and f anelement in its kernel. Then we can find an operator E such that(4.15) B = E∂ f . The map
Φ :
KerB → KerEg (cid:55)→ ( g/f ) (cid:48) . (4.16)has a one dimensional kernel spanned by f . Since by hypothesis KerB is ( n + 1) − dimensional, we get that KerE is at least n -dimensional.We also know that d ( E ) = n . Therefore by Lemma . , E has a fullkernel in K . Moreover, Φ is surjective, i.e.(4.17) KerE = Im Φ ⊂ ∂K. By the induction hypothesis, E − is weakly non-local. Let { a i } and { b i } be two sets of linearly independent functions such that(4.18) E − = n (cid:88) i =1 a i ∂ − b i . Multiplying (4 . on the left by E we have from Lemma . :(4.19) n (cid:88) i =1 E ( a i ) ∂ − b i = 0 . Therefore, by Lemma . , E ( a i ) = 0 for i = 1 , . . . , n . Moreover, weknow that KerE is a subset of ∂ K thanks to (4 . . For i = 1 . . . n , let d i ∈ K be such that a i = d (cid:48) i . Then, we have B − = f ∂ − E − = n (cid:88) i =1 f ∂ − d (cid:48) i ∂ − b i = n (cid:88) i =1 f d i ∂ − b i − f ∂ − d i b i . (4.20)where we used the identity ∂h − h∂ = h (cid:48) , valid for all h ∈ K . Therefore, B − is weakly non-local.Finally, let us check that (3) = ⇒ (2) . From the second step of theproof, we know that B ∗− is weakly non-local. Using (1) = ⇒ (3) , wededuce that B has a full kernel in K . (cid:3) Remark . It follows from the proof of Lemma . that a rationaloperator L ∈ V ( ∂ ) lies in W V if and only if both B ∗ and C have a fullkernel in V , where AB − (resp. C − D ) is a right (resp. left) minimalfractional decomposition of L .Full kernel operators admit an interesting characterization in termsof integrability. Proposition 4.7.
Let B ∈ V [ ∂ ] be a differential operator with fullkernel in V . Then it is integrable if and only if KerB ∗ is spanned byvariational derivatives.Proof. We keep the notations of Lemma . . Namely : KerB ∗ isspanned by the linearly independent elements q i , i = 1 , . . . , n of V and B = q i ∂M i for all i for some differential operators M i ∈ K [ ∂ ] . By (3 . , B is integrable if and only if B divides on the left the differ-ential operator X B ( F ) ( B ) − D B ( F ) B for all F ∈ V . Since B is a rightleast common multiple of the operators q i ∂ , it is equivalent to say thatfor all i = 1 , . . . , n and for all F ∈ V , X B ( F ) ( B ) − ( D B ) F B is divisibleon the left by q i ∂ which itself amounts to say that for all F ∈ V and i = 1 , . . . , n (4.21) X B ( F ) ( B ∗ )( q i ) = B ∗ ( D B ( F ) ∗ ( q i )) . As we have B ∗ ( q i ) = 0 by definition, the LHS of (4 . can be rewrittenas − B ∗ ( X B ( F ) ( q i )) . Taking the adjoint of (1 . and applying to q i yields(4.22) D B ( F ) ∗ ( q i ) = ( D B ) ∗ F ( q i ) . Both X B ( F ) ( q i ) and ( D B ) ∗ F ( q i ) are differential operators applied to F .Therefore the integrability of B is equivalent to the following set ofequations for i = 1 . . . n and all F ∈ V (4.23) X B ( F ) ( q i ) = − D B ( F ) ∗ ( q i ) . The RHS of (4 . can be rewritten as follows : D B ( F ) ∗ ( q i ) = D Mi ( F ) (cid:48) qi ∗ ( q i )= ( M i ( F ) (cid:48) D qi + 1 q i ∂D M i ( F ) ) ∗ ( q i )= ( − q i M i ( F ) (cid:48) D q i ) ∗ ( q i )= − D ∗ q i ( B ( F )) . (4.24)Plugging (4 . in (4 . , we see that B is integrable if and only if forall F ∈ V and i = 1 , . . . , n , we have(4.25) D q i ( B ( F )) = D ∗ q i ( B ( F )) . Since the image of B is infinite-dimensional over C , we can simplify (4 . into(4.26) D q i = D ∗ q i , i = 1 . . . n. Using Lemma . , we conclude that B is integrable if and only if(4.27) KerB ∗ ⊂ δ V δu . (cid:3) In the next lemma we give some technical yet useful result on thevector space structure of some extension of the space of weakly non-local operators.
Lemma 4.8.
Let U be the following subspace of rational operators (4.28) U = { E ( ∂ ) + (cid:88) p α ∂ − q α + (cid:88) a β ∂ − b β ∂ − c β } . Then (4.29)
V ⊗ C V /∂ V ⊗ C V (cid:39)
U/W V via the morphism φ sending a ⊗ (cid:82) b ⊗ c to a∂ − b∂ − c .Proof. Let φ be the map from V ⊗ C V /∂ V ⊗ C V to U/W V sending thetensor f ⊗ (cid:82) g ⊗ h to the image of the nonlocal operator f ∂ − g∂ − h in U/W V . It is well defined because f ∂ − g (cid:48) ∂ − h = 0 in U/W V for anytriple ( f, g, h ) thanks to the identity g (cid:48) = ∂g − g∂ . It is surjective sinceany class in U/W V contains a sum of elements of the form f ∂ − g∂ − h .It remains to check the injectivity of φ . We will do so by proving byinduction on n the following statement :If { f , . . . , f n } and { (cid:82) g , . . . , (cid:82) g m } are two sets of linearly indepen-dent elements in V and V /∂ V , and { h ij } are functions in V then L = (cid:80) i,j f i ∂ − g j ∂ − h ij ∈ W V if and only if h ij = 0 for all ( i, j ) .If n = 1 , we have to see why L = (cid:80) mj =1 f ∂ − g j ∂ − h j ∈ W implies h j = 0 for j = 1 , . . . , m . Let a l , b l ∈ V such that(4.30) m (cid:88) j =1 f ∂ − g j ∂ − h j = k (cid:88) l =1 a l ∂ − b l . Multiplying on the left by ∂ f we get, thanks to Lemma . ,(4.31) m (cid:88) j =1 g j ∂ − h j = k (cid:88) l =1 ( a l /f ) (cid:48) ∂ − b l . Let e , . . . , e s be a basis over C of the space spanned by the h j ’s and b l ’s. Let α jr ∈ C (resp. β lr ) be the coordinates of h j (resp. b l ) in thisbasis. Then equation (4 . rewrites into(4.32) s (cid:88) r =1 ( m (cid:88) j =1 α jr g j − k (cid:88) l =1 β lr ( a l /f ) (cid:48) ) ∂ − e r = 0 . From Lemma . we conclude that, for r = 1 , . . . , s :(4.33) m (cid:88) j =1 α jr g j = k (cid:88) l =1 β lr ( a l /f ) (cid:48) . Projecting (4 . into V /∂ V , we get(4.34) m (cid:88) j =1 α jr (cid:90) g j = 0 . Since we assume the functionals (cid:82) g j to be linearly independent, weget that all the coordinates α jr must be , hence that h j = 0 for all j ,proving the statement for n = 1 . Assume that our statement holds for n − ≥ and let L be such that(4.35) L = n (cid:88) i =1 m (cid:88) j =1 f i ∂ − g j ∂ − h ij ∈ W V , where { f , . . . , f n } and { (cid:82) g , . . . , (cid:82) g m } are two sets of linearly inde-pendent elements in V and V /∂ V . Let i ∈ [1 , n ] . Multiplying on theleft equation (4 . by ∂ f i we have(4.36) (cid:88) i (cid:54) = i ,j ( f i f i ) (cid:48) ∂ − g j ∂ − h ij ∈ W K . Given that the functions ( f i f i ) (cid:48) are linearly independent over C , weuse the induction hypothesis to deduce that h ij = 0 for i (cid:54) = 0 and j = 1 , . . . , m . Since this is true for all i = 1 , ..., n , we conclude that h ij = 0 for all i = 1 , ..., n and all j = 1 , ..., m . (cid:3) Lemma 4.9.
Let L = E + (cid:80) ni =1 p i ∂ − q i be a weekly non-local operatorand AB − be its right minimal fractional decomposition, with d ( B ) = n .Then the following are equivalent (1) A ( KerB ) ⊂ ImB. (2) p i q j ∈ ∂ V ∀ ( i, j ) ∈ { , n } . (3) L is weakly non-local . (4.37) Proof.
By Lemma . , we see that(4.38) L = n (cid:88) i =1 m (cid:88) j =1 p i ∂ − q i p j ∂ − q j mod W. Therefore (2) ⇐⇒ (3) is a direct application of Lemma . . B is aright least common multiple of the operators q i ∂ , which implies that p ∈ ImB if and only if q i p is a total derivative for all i . In other words, (2) is saying that p i ∈ ImB for all i . To prove that (1) ⇐⇒ (2) , letus check that A ( KerB ) = (cid:104) p , . . . , p n (cid:105) . Recall that B = 1 q i ∂M i , i = 1 , . . . , n,A = EB + n (cid:88) i =1 p i M i . (4.39)From the first line of (4 . , we get that M i ( x ) is a constant for all x ∈ KerB and all i . Using the second line of (4 . , we see that A ( KerB ) ⊂(cid:104) p , . . . , p n (cid:105) . Since A and B are right coprime, KerA ∩ KerB = { } and dimA ( KerB ) = dimKerB = n . Therefore, A ( KerB ) = (cid:104) p , . . . , p n (cid:105) . (cid:3) In the two next propositions, we examine what can be said of rationaloperators which are both integrable and weakly non-local.
Proposition 4.10.
Let L = E + (cid:80) ni =1 p i ∂ − q i be an integrable weaklynon-local operator where { p i } and { q i } are sets consisting of linearly in-dependent functions over C . Then the space V = (cid:104) q , . . . , q n (cid:105) is spannedby variational derivatives and δδu ( p i q j ) ∈ V for all ( i, j ) ∈ { , n } Proof.
We already know from Lemma . that the kernel of B ∗ isspanned by the q i ’s where AB − is a right minimal fractional decom-position of L . Proposition . says that the q i ’s must be variationalderivatives. Since L is integrable it is hereditary in particular (Lemma . ). In particular, AB − satisfies equation (2 . . The LHS of (2 . lies in W for all F ∈ V . Indeed, X A ( F ) preserves W and so does mul-tiplication by a differential operator. Therefore, the RHS of (2 . liesin W for all F ∈ V :(4.40) ( E + n (cid:88) i =1 p i ∂ − q i )( L B ( F ) ( E ) + L B ( F ) ( n (cid:88) j =1 p j ∂ − q j )) ∈ W V . For the same reasons as above, we have for all F ∈ V :(4.41) n (cid:88) i =1 p i ∂ − q i L B ( F ) ( n (cid:88) j =1 p j ∂ − q j ) ∈ W V . In other words, for all F ∈ V we have(4.42) (cid:88) i,j p i ∂ − q i ( X B ( F ) ( p j ∂ − q j ) − [ D B ( F ) , p j ∂ − q j ]) ∈ W V . By Lemma . , we simplify (4 . into, for all F ∈ V , (cid:88) i,j p i ∂ − q i p j ∂ − ( X B ( F ) ( q j ) + D ∗ B ( F ) ( q j ))+ (cid:88) i,j p i ∂ − q i ( X B ( F ) ( p j ) − D B ( F ) ( p j )) ∂ − q j ∈ W V . (4.43)Recall by (4 . and (4 . that for all F ∈ V and for all i = 1 , . . . , n :(4.44) D ∗ B ( F ) ( q i ) = − D ∗ q i ( B ( F )) = − D q i ( B ( F )) = − X B ( F ) ( q i ) . Hence the first term in (4 . vanishes and we get that for all F ∈ V ,(4.45) (cid:88) i,j p i ∂ − q i ( X B ( F ) ( p j ) − D B ( F ) ( p j )) ∂ − q j ∈ W V . Since both the p i ’s and the q i ’s are linearly independent, it follows fromLemma . and equation (4 . that for all F ∈ V :(4.46) q i X B ( F ) ( p j ) − q i D B ( F ) ( p j ) ∈ ∂ V . Let us work in the quotient space V /∂ V . For all F ∈ V (4.47) q i D p j ( B ( F )) ≡ D B ( F ) ( p j ) q i . By (1 . and (4 . , (4 . simplifies into(4.48) B ( F ) D ∗ p j ( q i ) ≡ p j D ∗ B ( F ) ( q i ) ≡ − p j D q i ( B ( F )) ≡ − B ( F ) D ∗ q i ( p j ) . Hence, for all F ∈ V :(4.49) B ∗ ( D ∗ p j ( q i ) + D ∗ q i ( p j )) F ≡ . Since (4 . holds for all F ∈ V and that ∂ V (cid:54) = V ([BDSK09]), we have(4.50) B ∗ ( D ∗ p j ( q i ) + D ∗ q i ( p j )) = B ∗ ( δδu ( p i q j )) = 0 . Therefore, δδu ( p i q j ) ∈ V for all ( i, j ) ∈ [1 , n ] . (cid:3) For a weakly-non-local operator with minimal fractional decomposi-tion L = AB − , we only need to check that L is hereditary and thatthe functions appearing on the right of the reduced non-local part of L are variational derivatives to claim that L is integrable. Proposition 4.11.
Let L = E + (cid:80) ni =1 p i ∂ − q i be a weakly non-local ra-tional operator, where the p i ’s and the q i ’s are linearly independent over C . Then L is integrable if and only if the functions q i are variationalderivatives and L is hereditary. Proof.
Let AB − be a minimal fractional decomposition of L . By corol-lary . , L is integrable if and only if the pair ( A, B ) is integrable.Moreover, by Lemma . , the pair ( A, B ) is integrable if and only if L is hereditary and B is integrable. Finally, by Proposition . , B isintegrable if and only if the functions q i are variational derivatives. (cid:3) A sufficient condition of integrability
In this section we will prove that integrability is a necessary and suffi-cient condition to generate infinitely many commuting functions for aweakly non-local operator "preserving" a certain decomposition of V .Let σ be an involution of V . Let(5.1) V = V ¯0 ⊕ V ¯1 be the eigenspace decomposition of V for σ . We call elements of V ¯0 evenfunctions and elements of V ¯1 odd functions. Note that, since σ is analgebra morphism, we have for all ¯ i, ¯ j ∈ Z / Z , V ¯ i . V ¯ j ⊂ V ¯ i +¯ j . Assumefurthermore that σ∂σ − = − ∂ and that for all n ≥ , σ ∂∂u ( n ) σ − =( − n ∂∂u ( n ) . In other words, ∂ switches parity and δδu preserves it. Example . The algebra of differential polynomials R admits such adecomposition by declaring u to be even and ∂ to be odd. However, itis not the only way to decompose this algebra with the constraints wejust defined.As usual, an endomorphism of V is called even if it preserves thedecomposition (5 . , and odd if it switches parity. Here is the decom-position of V [ ∂ ] into even and odd parts :(5.2) V [ ∂ ] = ( V ¯0 [ ∂ ] + V ¯1 [ ∂ ] ∂ ) ⊕ ( V ¯1 [ ∂ ] + V ¯0 [ ∂ ] ∂ ) . We extend this decomposition to pseudodifferential operators bydeclaring that ∂ − is odd. Lemma 5.2.
Let L = E ( ∂ ) + (cid:80) ni =1 p i ∂ − q i be a weakly non-local inte-grable rational operator, where E is an even differential operator, q i ’sare linearly independent elements of V ¯0 and p i ’s are linearly indepen-dent elements of V ¯1 . Let p ∈ V ¯1 be such that L is recursion for p .Finally, let B be a right least common multiple of the operators q i ∂ and A be such that L = AB − . Then the following statements hold(1) p lies in the image of B . (2) { p, p i } = 0 for i = 1 . . . n .(3) pq i is a total derivative for i = 1 . . . n .Moreover, if F ∈ V is such that B ( F ) = p , then A ( F ) ∈ V ¯1 and L isrecursion for A ( F ) .Proof. By Proposition . , the functions q i are variational derivatives.In particular, for i = 1 , ..., n , we have(5.3) D q i = D ∗ q i . Therefore, for any function f ∈ V and any i = 1 . . . n , by definition ofthe variational derivative and of the adjoint action, we obtain δδu ( f q i ) = D ∗ f ( q i ) + D ∗ q i ( f )= D ∗ f ( q i ) + D q i ( f )= D ∗ f ( q i ) + X f ( q i ) . (5.4)We know that L is recursion for p , meaning that(5.5) X p ( L ) = [ D p , L ] . By Lemma . , the non-local part of equation (5 . is:(5.6) X p ( n (cid:88) i =1 p i ∂ − q i ) = n (cid:88) i =1 D p ( p i ) ∂ − q i − p i ∂ − D ∗ p ( q i ) . Since X p is a derivation, (5 . rewrites into(5.7) n (cid:88) i =1 ( D p ( p i ) − X p ( p i )) ∂ − q i = n (cid:88) i =1 p i ∂ − ( X p ( q i ) + D ∗ p ( q i )) . Remembering (1 . , (1 . and (5 . , we deduce from (5 . that(5.8) n (cid:88) i =1 { p i , p } ∂ − q i = n (cid:88) i =1 p i ∂ − δδu ( pq i ) . Since the variational derivative preserves the parity and since pq i isodd, we have for all i = 1 , ..., n (5.9) δδu ( pq i ) ∈ V ¯1 . By Lemma . , we deduce from equations (5 . and (5 . that for all i = 1 . . . n (5.10) { p, p i } = δδu ( pq i ) = 0 . Therefore statement (2) holds, and so does (3) by Lemma . . Finally,note that (1) and (3) are equivalent since B is a right least commonmultiple of the operators q i ∂ .Let F ∈ V be such that p = B ( F ) . By Lemma . and hereditari-ness of L , it follows that L is recursion for A ( F ) . We are left to checkthat A ( F ) is an odd function. From the first line of (4 . , we see that M i ( F ) is even for all i = 1 , . . . , n . Therefore, using the second line of (4 . , we get that A ( F ) ∈ V ¯1 . (cid:3) Theorem 5.3.
Let L ∈ ( V [ ∂ ]) ¯0 + V ¯1 ∂ − V ¯0 be an integrable rationaloperator. If AB − is a right minimal fractional decomposition of L and F ∈ KerB , then there exists a sequence F n ∈ V , n ≥ such that(1) B ( F n +1 ) = A ( F n ) for all n ≥ .(2) { B ( F n ) , B ( F m ) } = 0 for all n, m ≥ .Let m = max { d ( e k ) , d ( p i ) , d ( q i ) } , where L = (cid:80) k ≥ e k ∂ k + (cid:80) li =1 p i ∂ − q i and where both the p i ’s and the q i ’s are linearly independent over C .Then, if for some N ≥ , d ( B ( F N )) > m , the sequence of differentialorders d ( B ( F l )) goes to + ∞ .Proof. We begin by constructing the sequence ( F n ) using Lemma . .Recall that, if(5.11) L = E ( ∂ ) + l (cid:88) i =1 p i ∂ − q i , where l = d ( B ) , then A ( KerB ) is spanned by the functions p i . Sincewe assume that L ∈ ( V [ ∂ ]) ¯0 + V ¯1 ∂ − V ¯0 , we get that A ( KerB ) ⊂ V ¯1 . L ishereditary and recursion for B ( F ) , hence it is recursion for A ( F ) by Lemma . . One can apply Lemma . with L and p = A ( F ) , tofind a function F such that A ( F ) = B ( F ) and A ( F ) is odd. Iteratingthe argument, we construct F n , n ≥ such that B ( F n +1 ) = A ( F n ) forall n ≥ . The fact that the functions B ( F n ) pairwise commute followsfrom Corollary . .We now prove the second part of the Theorem, more precisely thatif for some N ≥ , d ( B ( F N )) > m , then for all n ≥ N , d ( B ( F n +1 )) = d ( B ( F n )) + d ( L ) . This follows by induction on n from (4 . . Indeed,let us assume that for some n ≥ N , d ( B ( F n )) > m . Then, fromthe first line of (4 . , we deduce that d ( M i ( F n )) = d ( B ( F n )) − .Hence, from the second line of (4 . , we deduce that d ( A ( F n )) = d ( E ( B ( F n )) = d ( B ( F n )) + d ( L ) , which is what we wanted to show,since A ( F n ) = B ( F n +1 ) by construction. (cid:3) Theorem 5.4.
Let L = E ( ∂ ) + (cid:80) ni =1 p i ∂ − q i be a hereditary rationaloperator where E is a even differential operator, p i ’s are linearly inde-pendent odd functions and q i ’s are linearly independent even variationalderivatives. Then L k is weakly non-local and integrable for all k ≥ .Moreover if (5.12) L k = E k ( ∂ ) + n k (cid:88) i =1 p ki ∂ − q ki . where { p k , . . . , p kn k } and { q k , . . . , q kn k } are two sets of linearly inde-pendent functions and E k is a differential operator, then the functions p ki are odd, E k is even and q ki are even variational derivatives. Finally,for all k, l ≥ and for all i, j ∈ [1 , n k ] × [1 , n l ] ,(a) { p ki , p lj } = 0 .(b) p ki .q lj ∈ ∂ V .(c) ρ lj is a conserved density of u t = p ki where q lj = δρ lj δu .Proof. Let us define A , B and M i for i = 1 . . . n as in (4 . . ByProposition . , L is integrable. Moreover, L is recursion for all the p i ’s. Indeed it is obviously recursion for and p i ∈ A ( KerB ) . Henceby (2 . L p i ( L ) = 0 . Therefore one can iterate Lemma . startingfrom any of the odd functions p i ’s. More specifically, for all i = 1 . . . n there exists a sequence of functions F im ∈ V such that p i = A ( F i ) and B ( F im +1 ) = A ( F im ) for all m ≥ . Note that A ( F im ) ∈ V ¯1 for all m ≥ and i = 1 . . . n . For all integer k ≥ we define the subspace W k ⊂ V ¯1 to be the span of the functions A ( F im ) for m ≤ k and i = 1 , ..., n . Thespaces W k do not depend on the choices of the sequence F im . Indeed,given F im , F im +1 is uniquely defined up to an element of KerB . But wehave A ( KerB ) = W . Note that W k ⊂ V ¯1 for all k ≥ .Let us prove by induction on k ≥ the following statements : First, L k is weakly non-local. Secondly, if { p k , . . . , p kn k } and { q k , . . . , q kn k } are two sets of linearly independent functions and E k is a differentialoperator such that(5.13) L k = E k + n k (cid:88) i =1 p ki ∂ − q ki , then the functions p ki lie in W k − for all i , E k is even and q ki are evenvariational derivatives. The statements hold for k = 1 . Let us assumethat they do for k ≥ . By definition of W k − , the functions p ki lie inthe image of B , and in particular for all i = 1 , . . . , n k and j = 1 , . . . , n there exists a function v ij ∈ V such that q j p ki = ( v ij ) (cid:48) ∈ ∂ V . Hence,(5.14) n k (cid:88) i =1 m (cid:88) j =1 p j ∂ − q j p ki ∂ − q ki ∈ W. By (5 . and (5 . , L k +1 is weakly non-local. Let us compute thenon-local part of L k +1 . L k +1 ≡ ( E + n (cid:88) j =1 p j ∂ − q j )( E k + n k (cid:88) i =1 p ki ∂ − q ki ) ≡ n k (cid:88) i =1 E ( p ki ) ∂ − q ki + (cid:88) i,j p j ∂ − ( v ij ) (cid:48) ∂ − q ki + n (cid:88) j =1 p j ∂ − E ∗ k ( q j ) ≡ n k (cid:88) i =1 ( E ( p ki ) + n (cid:88) j =1 p j v ij ) ∂ − q ki + n (cid:88) j =1 p j ∂ − ( E ∗ k ( q j ) − n k (cid:88) i =1 v ij q ki ) . (5.15)Let F ∈ V be such that p ki = B ( F ) (recall that W k ⊂ ImB ). Then bythe first line of equation (4 . and by construction of v ij we have forall ( i, j ) ∈ [1 , n k ] × [1 , n ] (5.16) v (cid:48) ij = ( M j ( F )) (cid:48) . Therefore, by the second line of (4 . , for all i = 1 . . . n k ,(5.17) E ( p ki ) + (cid:88) j p j v ij = A ( F ) mod W . Hence(5.18) ( L k +1 ) nlc ∈ W k ∂ − V .L is an even pseudodifferential operator, and so is L m for any m ≥ . Inparticular, the local part of L k +1 , E k +1 , is an even differential operator.For the same reason, and since W k ⊂ V ¯1 , we get that(5.19) ( L k +1 ) nlc ∈ W k ∂ − V ¯0 .L k is integrable, therefore by Proposition . and (5 . , the functions q ki are even variational derivatives.Let k ≥ and i ∈ [1 , n k ] . We know that p ki ∈ W k − . This implies bydefinition of W k − that L is recursion for p ki . Therefore L m is recursionfor p ki for all m ≥ . A direct application of Lemma . then gives that { p ki , p mj } = 0 and p ki q mj ∈ ∂ V for all m ≥ and j ∈ [1 , n m ] . We are left to prove ( c ) . Let l ≥ , j ∈ [1 , n l ] , and ρ lj be such that q lj = δρ lj δu .Then we have(5.20) (cid:90) p ki δρ lj δu = 0 . Recalling (1 . , this precisely means that ρ lj is a conserved density ofthe equation u t = p ki . (cid:3) Remark . The same statements hold if we switch the parity of the p i ’s and the q i ’s, but not the parity of E . Remark . The ratio of two compatible local Poisson structures isintegrable (Remark . ). If furthermore, one assumes that both H and K are odd, and that K has a full kernel spanned by even variationalderivatives, then L = H.K − satisfies the hypothesis of Theorems . and . , therefore it generates an integrable hierarchy of equations,under some assumption on differential orders of the coefficients of H and K . Examples λ -homogeneous equations with linear leading term. In [SW09], Wang and Sanders give a classification of λ -homogeneousdifferential polynomials with linear leading term, i.e. equations of theform(6.1) u t = F = u ( n ) + P, where P is a polynomial in u, . . . , u ( n − . Let λ and µ be some con-stants, then F is called λ − homogeneous of weight µ if it admits theone parameter group of scaling symmetries(6.2) ( x, t, u ) (cid:55)→ ( a − x, a − µ t, a λ u ) , a ∈ R + . Every λ -homogeneous equation of the form (6 . , modulo homogeneoustransformations in u , is an equation lying in the hierarchy of one of the equations displayed in [SW09] (p.103-104). Their corresponding re-cursion operators can be found in [W02].For the following equations, u t = u (cid:48)(cid:48)(cid:48) + 3 uu (cid:48) (Korteweg-de Vries) u t = u (5) + 10 uu (cid:48)(cid:48)(cid:48) + 25 u (cid:48) u (cid:48)(cid:48) + 20 u u (cid:48)(cid:48) (Kaup-Kuperschmidt) u t = u (5) + 10 uu (cid:48)(cid:48)(cid:48) + 10 u (cid:48) u (cid:48)(cid:48) + 20 u u (cid:48)(cid:48) (Sawada-Kotera) u t = u (cid:48)(cid:48)(cid:48) + u (cid:48) (Modified KdV) u t = u (cid:48)(cid:48)(cid:48) + u (cid:48) (Potential modified KdV) u t = u (5) + 5 u (cid:48)(cid:48) u (cid:48)(cid:48)(cid:48) − u (cid:48) u (cid:48)(cid:48)(cid:48) − u (cid:48) u (cid:48)(cid:48) + u (cid:48) (Potential Kuperschmidt) , one can check directly using [W02] that their recursion operators liein ( R [ ∂ ]) ¯0 + R ¯1 ∂ − δR ¯0 δu , where R + R ¯0 ⊕ R ¯1 is the decomposition ofthe space of differential polynomials into even and odd parts whichone obtains by declaring u to be even and ∂ to be odd. Moreover,each of these equations is odd. Therefore, to apply Theorems . and . , we are left to check that the weakly non-local operators arehereditary, which is a tedious but straightforward computation. Sincethe above equations are odd, we can apply Lemma . and initiate theLenard-Magri scheme with their corresponding recursion operators atthemselves. Thus, we conclude that they lie in integrable hierarchies.These abelian subalgebras of R are infinite-dimensional since the orderscondition in Theorem . is met for N = 1 or N = 2 in each case.As for the next four equations in the list, u t = u (cid:48)(cid:48)(cid:48) + u (cid:48) (Potential KdV) u t = u (5) + 10 uu (cid:48)(cid:48)(cid:48) + 152 u (cid:48)(cid:48) + 203 u (cid:48) (Potential Kaup-Kuperschmidt) u t = u (5) + 10 uu (cid:48)(cid:48)(cid:48) + 203 u (cid:48) (Potential Sawada-Kotera) u t = u (5) + 5 u (cid:48) u (cid:48)(cid:48)(cid:48) + 5 u (cid:48)(cid:48) − u u (cid:48)(cid:48)(cid:48) − uu (cid:48) u (cid:48)(cid:48) − u (cid:48) + 5 u u (cid:48) (Kuperschmidt) , one can check directly using [W02] that their recursion operators liein ( R [ ∂ ]) ¯0 + R ¯0 ∂ − δR ¯1 δu , where R + R ¯0 ⊕ R ¯1 is the decomposition ofthe space of differential polynomials into even and odd parts whichone obtains by declaring u and ∂ to be odd. Moreover, each of these equations is even. Therefore, to apply Theorems . and . , we are leftto check that the weakly non-local operators are hereditary. Similarly,these abelian subalgebras of R are infinite-dimensional since the orderscondition in Theorem . is met for N = 1 or N = 2 in each case.The Burgers and the Potential Burgers equations, u t = u (cid:48)(cid:48) + uu (cid:48) (Burgers) u t = u (cid:48)(cid:48) + u (cid:48) (Potential Burgers) , admit the following recursion operators L B = ∂ ( ∂ + u ) ∂ − L P B = ∂ + u (cid:48) . (6.3)Both of these operators are integrable and recursion for u (cid:48) . Moreover,it is clear that they can be applied infinitely many times to u (cid:48) . ByCorollary . the functions L nB ( u (cid:48) ) , n ≥ (resp. L nP B ( u (cid:48) ) , n ≥ ) definean integrable system.As for the last equation from the list of [SW09], the Calogero-Degasperis-Ibragimov-Shabat equation : u t = u (cid:48)(cid:48)(cid:48) + 3 u u (cid:48)(cid:48) + 9 uu (cid:48) + 3 u u (cid:48) (CDIS) , it admits a rational recursion operator, which is not weakly non-local :(6.4) L CDIS = 1 u ∂ ( ∂ + 2 u ) − ( ∂ + u − u (cid:48) u ) ( ∂ + 2 u ) ∂ − u. Moreover, it is not hard to check that L nCDIS ( u (cid:48) ) is well-defined for all n ≥ Indeed, if in the differential algebra extension of V , (cid:101) V = V [ w ] ,where w (cid:48) w = u , L nCDIS rewrites as follows for all n ≥ :(6.5) L nCDIS = 1 u ∂ w ∂ − uw∂ n ( ∂uw + ( uw ) (cid:48) ) 1 u ∂ − u. Hence,(6.6) L nCDIS ( u (cid:48) ) = 1 u ( 1 w (cid:90) ( uw∂ n +1 uw )(1)) (cid:48) , which lies in (cid:101) V since the differential operator uw∂ n +1 uw is skewadjointfor all n ≥ (the constant coefficient of a skewadjoint differentialoperator H is a total derivative since (cid:82) H (1) = (cid:82) H ∗ (1) = − (cid:82) H (1) ).It is clear from (6 . that L nCDIS ( u (cid:48) ) ∈ V ⊂ (cid:101) V for all n ≥ . Moreprecisely, (cid:82) ( uw∂ n +1 uw )(1) ∈ w V . Finally, one checks that L CDIS isan integrable rational operator and conclude that
CDIS is an integrableequation by Corollary . . Krichever-Novikov hierarchy. heyIn [DS08], Demskoi and Sokolov give a degree weakly non-local re-cursion operator L KN for the Krichever-Novikov equation, dudt = u (cid:48)(cid:48)(cid:48) − u (cid:48)(cid:48) u (cid:48) + P ( u ) u (cid:48) (Krichever-Novikov) , where P is a polynomial of degree at most . The recursion operatoris of the form L KN = ∂ + a ∂ + a ∂ + a ∂ + a + G ∂ − δρ δu + u (cid:48) ∂ − δρ δu . The space of Laurent differential polynomials in u , A = C [ u ± , u (cid:48)± , ... ] admits a decomposition into even and odd parts A = A ¯0 ⊕ A ¯1 bydeclaring u to be even and ∂ to be odd. From the explicit formulas givenin [DS08], it is straighforward to check that a i has the same parity as i for i = 1 , . . . , and that ρ i are even for i = 1 , . Hence, the local partof L KN is even and so are the functions δρ i δu , since variational derivativespreserve parity. Moreover, G is the equation (KN) itself, which is oddand so is u (cid:48) . Finally, one checks that L KN is hereditary and applyTheorem . (after checking the orders consition in Theorem . ) toconclude that ( KN ) lies in an infinite dimensional abelian subalgebraof ( A , { ., . } ) . References [BDSK09] A.Barakat, A. De Sole, V.G. Kac,
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