On Lie algebras responsible for integrability of (1+1)-dimensional scalar evolution PDEs
aa r X i v : . [ n li n . S I] F e b ON LIE ALGEBRAS RESPONSIBLE FOR INTEGRABILITYOF (1+1)-DIMENSIONAL SCALAR EVOLUTION PDES
SERGEI IGONIN
Center of Integrable Systems, P.G. Demidov Yaroslavl State University, Yaroslavl, Russia,
E-mail address: [email protected] MANNO
Dipartimento di Scienze Matematiche, Politecnico di Torino,Corso Duca degli Abruzzi 24, 10129 Torino, Italy
E-mail address: [email protected]
Abstract.
Zero-curvature representations (ZCRs) are one of the main tools in the theory of integrablePDEs. In particular, Lax pairs for (1+1)-dimensional PDEs can be interpreted as ZCRs.In [12], for any (1+1)-dimensional scalar evolution equation E , we defined a family of Lie algebras F ( E )which are responsible for all ZCRs of E in the following sense. Representations of the algebras F ( E )classify all ZCRs of the equation E up to local gauge transformations. In [13] we showed that, using thesealgebras, one obtains necessary conditions for existence of a B¨acklund transformation between two givenequations. The algebras F ( E ) are defined in terms of generators and relations.In this paper we show that, using the algebras F ( E ), one obtains some necessary conditions for in-tegrability of (1+1)-dimensional scalar evolution PDEs, where integrability is understood in the senseof soliton theory. Using these conditions, we prove non-integrability for some scalar evolution PDEs oforder 5. Also, we prove a result announced in [12] on the structure of the algebras F ( E ) for certainclasses of equations of orders 3, 5, 7, which include KdV, mKdV, Kaup-Kupershmidt, Sawada-Koteratype equations. Among the obtained algebras for equations considered in this paper and in [13], one findsinfinite-dimensional Lie algebras of certain polynomial matrix-valued functions on affine algebraic curvesof genus 1 and 0.In this approach, ZCRs may depend on partial derivatives of arbitrary order, which may be higher thanthe order of the equation E . The algebras F ( E ) generalize Wahlquist-Estabrook prolongation algebras,which are responsible for a much smaller class of ZCRs. Introduction
Zero-curvature representations (ZCRs) belong to the main tools in the theory of integrable nonlinearpartial differential equations (see, e.g., [40, 5]). In particular, Lax pairs for (1+1)-dimensional partialdifferential equations (PDEs) can be interpreted as ZCRs. This paper is a sequel of [12] and is part ofa research program on investigating the structure of ZCRs for PDEs of various types. (However, thepresent paper can be studied independently of [12].)Here we consider (1+1)-dimensional scalar evolution equations(1) u t = F ( x, t, u , u , . . . , u d ) , u = u ( x, t ) , Mathematics Subject Classification.
Key words and phrases.
Scalar evolution equations; zero-curvature representations; gauge transformations; normalforms for zero-curvature representations; infinite-dimensional Lie algebras; integrability conditions.
1N LIE ALGEBRAS RESPONSIBLE FOR INTEGRABILITY 2 where one uses the notation(2) u t = ∂u∂t , u = u, u k = ∂ k u∂x k , k ∈ Z ≥ . The number d ∈ Z > in (1) is such that the function F may depend only on x , t , u k for k ≤ d . Thesymbols Z > and Z ≥ denote the sets of positive and nonnegative integers respectively.The methods of this paper can be applied also to (1+1)-dimensional multicomponent evolution PDEs,which are discussed in Remark 7. Remark 1.
When we consider a function Q = Q ( x, t, u , u , . . . , u l ) for some l ∈ Z ≥ , we always assumethat this function is analytic on an open subset of the space V with the coordinates x, t, u , u , . . . , u l .For example, Q may be a meromorphic function, because a meromorphic function is analytic on someopen subset. If we say that Q is defined on a neighborhood of a point a ∈ V , we assume that thefunction Q is analytic on this neighborhood.PDEs of the form (1) have attracted a lot of attention in the last 50 years and have been a source ofmany remarkable results on integrability. In particular, some types of equations (1) possessing higher-order symmetries and conservation laws have been classified (see, e.g., [24, 25, 31] and references therein).However, the problem of complete understanding of all integrability properties for equations (1) is stillfar from being solved.Examples of integrable PDEs of the form (1) include the Korteweg-de Vries (KdV), Krichever-Novikov [20, 38], Kaup-Kupershmidt [16], Sawada-Kotera [32] (Caudrey-Dodd-Gibbon [1]) equations(these equations are discussed below). Many more examples can be found in [24, 25, 31] and referencestherein.In the present paper, integrability is understood in the sense of soliton theory and the inverse scat-tering method, relying on the use of ZCRs. (This is sometimes called S-integrability.) As discussed inRemark 16, this approach to integrability is not equivalent to the approach of symmetries and conser-vation laws. Definition 1.
Let g be a finite-dimensional Lie algebra. For an equation of the form (1), a zero-curvaturerepresentation ( ZCR ) with values in g is given by g -valued functions(3) A = A ( x, t, u , u , . . . , u p ) , B = B ( x, t, u , u , . . . , u p + d − )satisfying(4) D x ( B ) − D t ( A ) + [ A, B ] = 0 . The total derivative operators D x , D t in (4) are(5) D x = ∂∂x + X k ≥ u k +1 ∂∂u k , D t = ∂∂t + X k ≥ D kx (cid:0) F ( x, t, u , u , . . . , u d ) (cid:1) ∂∂u k . The number p in (3) is such that the function A may depend only on the variables x , t , u k for k ≤ p .Then equation (4) implies that the function B may depend only on x , t , u k ′ for k ′ ≤ p + d − of order ≤ p . In other words, a ZCR given by A , B is of order ≤ p iff ∂∂u l ( A ) = 0 for all l > p .The right-hand side F = F ( x, t, u , . . . , u d ) of (1) appears in condition (4), because F appears in theformula for the operator D t in (5). Note that (4) can be written as [ D x + A, D t + B ] = 0, because[ D x , D t ] = 0. Remark 2.
The methods of this paper are applicable also to ZCRs with values in infinite-dimensionalLie algebras. Such ZCRs are discussed in Section 5 and in Section 6.3.In [12] and in this paper we study the following problem. How to describe all ZCRs (3), (4) for agiven equation (1)?
N LIE ALGEBRAS RESPONSIBLE FOR INTEGRABILITY 3
In the case when p = 0 and the functions F , A , B do not depend on x , t , a partial answer to thisquestion is provided by the Wahlquist-Estabrook prolongation method (WE method for short). Namely,for a given equation of the form u t = F ( u , u , . . . , u d ), the WE method constructs a Lie algebra so thatZCRs of the form(6) A = A ( u ) , B = B ( u , u , . . . , u d − ) , D x ( B ) − D t ( A ) + [ A, B ] = 0correspond to representations of this algebra (see, e.g., [39, 17, 2, 11]). It is called the
Wahlquist-Estabrook prolongation algebra . Note that in (6) the function A = A ( u ) depends only on u .To study the general case of ZCRs (3), (4) with arbitrary p for any equation (1), we need to considergauge transformations, which are defined below.Without loss of generality, one can assume that g is a Lie subalgebra of gl N for some N ∈ Z > , where gl N is the algebra of N × N matrices with entries from R or C . So our considerations are applicableto both cases gl N = gl N ( R ) and gl N = gl N ( C ). And we denote by GL N the group of invertible N × N matrices.Let K be either C or R . Then gl N = gl N ( K ) and GL N = GL N ( K ). In this paper, all algebras aresupposed to be over the field K . Definition 2.
Let
G ⊂ GL N be the connected matrix Lie group corresponding to the Lie algebra g ⊂ gl N . (That is, G is the connected immersed Lie subgroup of GL N corresponding to the Lie subalgebra g ⊂ gl N .) A gauge transformation is given by a matrix-function G = G ( x, t, u , u , . . . , u l ) with valuesin G . Here l can be any nonnegative integer.For any ZCR (3), (4) and any gauge transformation G = G ( x, t, u , . . . , u l ), the functions(7) ˜ A = GAG − − D x ( G ) · G − , ˜ B = GBG − − D t ( G ) · G − satisfy D x ( ˜ B ) − D t ( ˜ A ) + [ ˜ A, ˜ B ] = 0 and, therefore, form a ZCR. (This is explained in Remark 8.)Since A , B take values in g and G takes values in G , the functions ˜ A , ˜ B take values in g .The ZCR (7) is said to be gauge equivalent to the ZCR (3), (4). For a given equation (1), formulas (7)determine an action of the group of gauge transformations on the set of ZCRs of this equation. Remark 3.
According to Definition 2, we study gauge transformations with values in G . Alternatively,one can take some other Lie group ˜ G ⊂ GL N whose Lie algebra is g and consider gauge transformationswith values in ˜ G . The results of this paper will remain valid, if one replaces G by ˜ G everywhere.We would like to emphasize that equation (1) remains fixed and does not change under the actionof gauge transformations. Also, we do not have any action on solutions u ( x, t ) of equation (1). Inthe literature on integrable PDEs, other authors sometimes consider transformations of different na-ture with different properties and call them gauge transformations. So, when one speaks about gaugetransformations, one should carefully define what they are.The WE method does not use gauge transformations in a systematic way. In the classification ofZCRs (6) this is acceptable, because the class of ZCRs (6) is relatively small.The class of ZCRs (3), (4) is much larger than that of (6). Gauge transformations play a veryimportant role in the classification of ZCRs (3), (4). Because of this, the classical WE method does notproduce satisfactory results for (3), (4), especially in the case p > F p for each p ∈ Z ≥ so that the following property holds.For every finite-dimensional Lie algebra g , any g -valued ZCR (3), (4) of order ≤ p is locally gaugeequivalent to the ZCR arising from a homomorphism F p → g .More precisely, as discussed below, in [12] we defined a Lie algebra F p for each p ∈ Z ≥ and eachpoint a of the infinite prolongation E of equation (1). So the full notation for the algebra is F p ( E , a ).The definition of F p ( E , a ) from [12] is recalled in Section 3 of the present paper.The family of Lie algebras F ( E ) mentioned in the abstract of this paper consists of the algebras F p ( E , a ) for all p ∈ Z ≥ , a ∈ E . N LIE ALGEBRAS RESPONSIBLE FOR INTEGRABILITY 4
Recall that the infinite prolongation E of equation (1) is an infinite-dimensional manifold with thecoordinates x , t , u k for k ∈ Z ≥ . The precise definitions of the manifold E and the algebras F p ( E , a )for any equation (1) are presented in Section 3. For every p ∈ Z ≥ and a ∈ E , the algebra F p ( E , a ) isdefined in terms of generators and relations. (To clarify the main idea, in Example 1 we consider thecase p = 1.)For every finite-dimensional Lie algebra g , homomorphisms F p ( E , a ) → g classify (up to gauge equiv-alence) all g -valued ZCRs (3), (4) of order ≤ p , where functions A , B are defined on a neighborhood ofthe point a ∈ E . See Section 3 for more details.According to Section 3, the algebras F p ( E , a ) for p ∈ Z ≥ are arranged in a sequence of surjectivehomomorphisms(8) · · · → F p ( E , a ) → F p − ( E , a ) → · · · → F ( E , a ) → F ( E , a ) . According to Remark 14, for each p ∈ Z > , the algebra F p ( E , a ) is responsible for ZCRs of order ≤ p ,and the algebra F p − ( E , a ) is responsible for ZCRs of order ≤ p −
1. The surjective homomorphism F p ( E , a ) → F p − ( E , a ) in (8) reflects the fact that any ZCR of order ≤ p − ≤ p . The homomorphism F p ( E , a ) → F p − ( E , a ) is defined by formulas (57), using generators ofthe algebras F p ( E , a ), F p − ( E , a ).Using F p ( E , a ), we obtain some necessary conditions for integrability of equations (1) and necessaryconditions for existence of a B¨acklund transformation between two given equations. To get such results,one needs to study certain properties of ZCRs (3), (4) with arbitrary p , and we do this by means of thealgebras F p ( E , a ). As explained above, the classical WE method (which studies ZCRs of the form (6))is not sufficient for this.Applications of F p ( E , a ) to obtaining necessary conditions for integrability of equations (1) are pre-sented in Section 6. Examples of the use of these conditions in proving non-integrability for someequations of order 5 are given in Section 6 as well. Applications of F p ( E , a ) to the theory of B¨acklundtransformations are described in [13]. See also Remark 6 below.Furthermore, we present a number of results on the structure of the algebras F p ( E , a ) for some classesof scalar evolution PDEs of orders 3, 5, 7 and concrete examples. The KdV equation is consideredin [12] and in Example 2 below. The Krichever-Novikov equation is discussed in Proposition 2, which isproved in [13]. In Section 6.2 we study the algebras F p ( E , a ) and integrability properties for a parameter-dependent 5th-order scalar evolution equation, which was considered by A. P. Fordy [6] in connectionwith the H´enon-Heiles system. The problem to study this equation was suggested to us by A. P. Fordy.In the theory of integrable (1+1)-dimensional PDEs, one is especially interested in ZCRs dependingon a parameter. That is, one studies ZCRs of the form A = A ( λ, x, t, u , . . . , u p ) , B = B ( λ, x, t, u , . . . , u p + d − ) , D x ( B ) − D t ( A ) + [ A, B ] = 0 , where g -valued functions A , B depend on x , t , u k and a parameter λ . For a given equation (1),existence of a nontrivial parameter-dependent ZCR is reflected in the structure of the algebras F p ( E , a )of equation (1). This is illustrated by Examples 2, 3.In this paper we mostly study equations of the form(9) u t = u q +1 + f ( x, t, u , u , . . . , u q − ) , q ∈ { , , } , where f is an arbitrary function. Examples of such PDEs include • the KdV equation u t = u + u u , • the modified KdV (mKdV) equation u t = u + u u , • the Kaup-Kupershmidt equation [16] u t = u + 10 u u + 25 u u + 20 u u , • the Sawada-Kotera equation [32] u t = u + 5 u u + 5 u u + 5 u u , which is sometimes called theCaudrey-Dodd-Gibbon equation [1].Many more examples of integrable PDEs of this type can be found in [24, 25] and references therein. N LIE ALGEBRAS RESPONSIBLE FOR INTEGRABILITY 5
Remark 4.
A classification of equations of the form u t = u + g ( x, u , u , u ) , u t = u + g ( u , u , u , u , u )satisfying certain integrability conditions related to generalized symmetries and conservation laws ispresented in [24]. We study the problem of describing all ZCRs (3), (4) for a given equation (1). Thisproblem is very different from the description of generalized symmetries and conservation laws.Let L , L , L be Lie algebras. One says that L is obtained from L by central extension if there is anideal I ⊂ L such that I is contained in the center of L and L / I ∼ = L . Note that I may be of arbitrarydimension.We say that L is obtained from L by applying several times the operation of central extension if thereis a finite collection of Lie algebras g , g , . . . , g k such that g ∼ = L , g k ∼ = L and g i is obtained from g i − by central extension for each i = 1 , . . . , k .Equations of the form (9) are considered in Theorem 3. Some consequences of Theorem 3 are sum-marized in Remark 5. Remark 5.
Theorem 3 implies that, for any equation of the form (9) with q ∈ { , , } , • for every p ≥ q + δ q, the algebra F p ( E , a ) is obtained from F p − ( E , a ) by central extension, • for every p ≥ q + δ q, the algebra F p ( E , a ) is obtained from F q − δ q, ( E , a ) by applying severaltimes the operation of central extension.Here δ q, is the Kronecker delta. So δ , = 1, and δ q, = 0 if q = 3.Theorem 3 was announced without proof in [12]. In Section 4 we give a detailed proof for it.Applications of Theorem 3 to obtaining necessary conditions for integrability of equations (9) arepresented in Section 6. Results similar to Theorem 3 can be proved for many other evolution equationsas well. See, e.g., Proposition 2 about the Krichever-Novikov equation.Other approaches to the study of the action of gauge transformations on ZCRs can be found in [21,22, 23, 29, 30, 33] and references therein. For a given ZCR with values in a matrix Lie algebra g , thepapers [21, 22, 29] define certain g -valued functions, which transform by conjugation when the ZCRtransforms by gauge. Applications of these functions to construction and classification of some types ofZCRs are described in [21, 22, 23, 29, 30, 33].To our knowledge, the theory of [21, 22, 23, 29, 30, 33] does not produce any infinite-dimensional Liealgebras responsible for ZCRs. So this theory does not contain the algebras F p ( E , a ).2. Preliminaries
We continue to use the notations introduced in Section 1. In particular, E is the infinite prolongationof equation (1). According to Definition 3 in Section 3, E is an infinite-dimensional manifold with thecoordinates x , t , u k for k ∈ Z ≥ .We suppose that the variables x , t , u k take values in K , where K is either C or R . A point a ∈ E isdetermined by the values of the coordinates x , t , u k at a . Let a = ( x = x a , t = t a , u k = a k ) ∈ E , x a , t a , a k ∈ K , k ∈ Z ≥ , be a point of E . In other words, the constants x a , t a , a k are the coordinates of the point a ∈ E in thecoordinate system x , t , u k .The general theory of the Lie algebras F p ( E , a ), p ∈ Z ≥ , is presented in Section 3. Before describingthe general theory, we would like to discuss some examples and applications. Example 1.
To clarify the definition of F p ( E , a ) presented in Section 3, let us consider the case p = 1.To this end, we fix an equation (1) and study ZCRs of order ≤ ≤ A = A ( x, t, u , u ) , B = B ( x, t, u , u , . . . , u d ) , D x ( B ) − D t ( A ) + [ A, B ] = 0
N LIE ALGEBRAS RESPONSIBLE FOR INTEGRABILITY 6 on a neighborhood of a ∈ E is gauge equivalent to a ZCR of the form˜ A = ˜ A ( x, t, u , u ) , ˜ B = ˜ B ( x, t, u , u , . . . , u d ) , (11) D x ( ˜ B ) − D t ( ˜ A ) + [ ˜ A, ˜ B ] = 0 , (12) ∂ ˜ A∂u ( x, t, u , a ) = 0 , ˜ A ( x, t, a , a ) = 0 , ˜ B ( x a , t, a , a , . . . , a d ) = 0 . (13)Moreover, according to Theorem 1, for any given ZCR of the form (10), on a neighborhoodof a ∈ E there is a unique gauge transformation G = G ( x, t, u , . . . , u l ) such that the func-tions ˜ A = GAG − − D x ( G ) · G − , ˜ B = GBG − − D t ( G ) · G − satisfy (11), (12), (13) and G ( x a , t a , a , . . . , a l ) = Id, where Id is the identity matrix.In the case of ZCRs of order ≤
1, this gauge transformation G depends on x , t , u , so G = G ( x, t, u ).In a similar result about ZCRs of order ≤ p , which is described in Theorem 1, the corresponding gaugetransformation depends on x , t , u , . . . , u p − .Therefore, we can say that properties (13) determine a normal form for ZCRs (10) with respect tothe action of the group of gauge transformations on a neighborhood of a ∈ E .A similar normal form for ZCRs (3), (4) with arbitrary p is described in Theorem 1 and Remark 10.Since the functions ˜ A , ˜ B from (11), (12), (13) are analytic on a neighborhood of a ∈ E , these functionsare represented as absolutely convergent power series˜ A = X l ,l ,i ,i ≥ ( x − x a ) l ( t − t a ) l ( u − a ) i ( u − a ) i · ˜ A l ,l i ,i , (14) ˜ B = X l ,l ,j ,...,j d ≥ ( x − x a ) l ( t − t a ) l ( u − a ) j . . . ( u d − a d ) j d · ˜ B l ,l j ...j d . (15)Here ˜ A l ,l i ,i and ˜ B l ,l j ...j d are elements of a Lie algebra, which we do not specify yet.Using formulas (14), (15), we see that properties (13) are equivalent to(16) ˜ A l ,l i , = ˜ A l ,l , = ˜ B ,l ... = 0 ∀ l , l , i ∈ Z ≥ . To define F ( E , a ) in terms of generators and relations, we regard ˜ A l ,l i ,i , ˜ B l ,l j ...j d from (14), (15) asabstract symbols. By definition, the Lie algebra F ( E , a ) is generated by the symbols ˜ A l ,l i ,i , ˜ B l ,l j ...j d for l , l , i , i , j , . . . , j d ∈ Z ≥ so that relations for these generators are provided by equations (12), (16).That is, in order to get relations for the generators ˜ A l ,l i ,i , ˜ B l ,l j ...j d of the algebra F ( E , a ), we substi-tute (14), (15) in (12), taking into account (16). A more detailed description of this construction isgiven in Section 3 and in [12] (with a slightly different notation for the generators). Example 2.
It is well known that the KdV equation u t = u + u u possesses an sl ( K )-valued ZCRdepending polynomially on a parameter λ . This is reflected in the structure of the algebras F p ( E , a ) forKdV as follows.Consider the infinite-dimensional Lie algebra sl ( K [ λ ]) ∼ = sl ( K ) ⊗ K K [ λ ], where K [ λ ] is the algebraof polynomials in λ . (If we regard K as a rational algebraic curve with coordinate λ , the elementsof sl ( K [ λ ]) can be identified with polynomial sl ( K )-valued functions on this rational curve.)According to [12], the algebra sl ( K [ λ ]) plays the main role in the description of F p ( E , a ) for the KdVequation. Namely, it is shown in [12] that, for KdV, the algebras F p ( E , a ) are obtained from sl ( K [ λ ])by applying several times the operation of central extension. In particular, F ( E , a ) is isomorphic to thedirect sum of sl ( K [ λ ]) and a 3-dimensional abelian Lie algebra. (In the computation of F ( E , a ) in [12]we use the fact that the structure of the Wahlquist-Estabrook prolongation algebra for KdV is knownand contains sl ( K [ λ ]) [3, 4].)Also, one can prove similar results on the structure of F p ( E , a ) for many other evolution equationspossessing parameter-dependent ZCRs. N LIE ALGEBRAS RESPONSIBLE FOR INTEGRABILITY 7
Example 3.
For any constants e , e , e ∈ C , one has the Krichever-Novikov equation [20, 38](17) KN( e , e , e ) = (cid:26) u t = u −
32 ( u ) u + ( u − e )( u − e )( u − e ) u (cid:27) . We denote by so ( C ) the 3-dimensional orthogonal Lie algebra over C . According to [20, 26], if e = e = e = e then the Krichever-Novikov equation (17) has an so ( C )-valued ZCR with ellip-tic parameter. One can see this in the structure of the algebras F p ( E , a ) as follows.Suppose that e = e = e = e . According to Proposition 2, which is proved in [13], in the descriptionof F p ( E , a ) for the Krichever-Novikov equation (17) we see an infinite-dimensional Lie algebra R e ,e ,e ,which consists of certain so ( C )-valued functions on an elliptic curve. The curve and the algebra R e ,e ,e are defined in Remark 9. As discussed in Remark 9, the curve and the algebra R e ,e ,e were studiedpreviously by other authors in a different context. Remark 6.
In [13] we show that the algebras F p ( E , a ) help to obtain necessary conditions for existenceof a B¨acklund transformation (BT) between two given evolution equations. This allows us to prove anumber of non-existence results for BTs. For instance, a result of this kind is presented in Proposition 1,which is proved in [13].For any e , e , e ∈ C , we have the Krichever-Novikov equation KN( e , e , e ) given by (17). Consideralso the algebraic curve(18) C( e , e , e ) = n ( z, y ) ∈ C (cid:12)(cid:12)(cid:12) y = ( z − e )( z − e )( z − e ) o . Proposition 1 ([13]) . Let e , e , e , e ′ , e ′ , e ′ ∈ C such that e = e = e = e and e ′ = e ′ = e ′ = e ′ .If the curve C( e , e , e ) is not birationally equivalent to the curve C( e ′ , e ′ , e ′ ) , then the equation KN( e , e , e ) is not connected with the equation KN( e ′ , e ′ , e ′ ) by any B¨acklund transformation.Also, if e = e = e = e , then KN( e , e , e ) is not connected with the KdV equation by any BT. BTs of Miura type (differential substitutions) for (17) were studied in [24, 38]. According to [24, 38],the equation KN( e , e , e ) is connected with the KdV equation by a BT of Miura type iff e i = e j forsome i = j .Proposition 1 considers the most general class of BTs, which is much larger than the class of BTsof Miura type studied in [24, 38]. The definition of BTs is given in [13], using a geometric approachfrom [19].If e = e = e = e and e ′ = e ′ = e ′ = e ′ , the curves C( e , e , e ) and C( e ′ , e ′ , e ′ ) are elliptic.The theory of elliptic curves allows one to determine when C( e , e , e ) is not birationally equivalent toC( e ′ , e ′ , e ′ ). One gets a certain algebraic condition on the numbers e , e , e , e ′ , e ′ , e ′ , which allows usto formulate the result of Proposition 1 more explicitly. See [13] for details. Remark 7.
It is possible to introduce an analog of F p ( E , a ) for multicomponent evolution PDEs(19) ∂u i ∂t = F i ( x, t, u , . . . , u m , u , . . . , u m , . . . , u d , . . . , u md ) , u i = u i ( x, t ) , u ik = ∂ k u i ∂x k , i = 1 , . . . , m. In this paper we study only the scalar case m = 1. The case m > m > F p ( E , a ) for a number of multicom-ponent PDEs of Landau-Lifshitz and nonlinear Schr¨odinger types) are sketched in the preprints [10, 14]. Remark 8.
It is well known that equation (4) implies D x ( ˜ B ) − D t ( ˜ A ) + [ ˜ A, ˜ B ] = 0 for ˜ A , ˜ B givenby (7). Indeed, formulas (7) yield D x + ˜ A = G ( D x + A ) G − and D x + ˜ B = G ( D t + B ) G − . Therefore, D x ( ˜ B ) − D t ( ˜ A ) + [ ˜ A, ˜ B ] = [ D x + ˜ A, D t + ˜ B ] = [ G ( D x + A ) G − , G ( D t + B ) G − ] == G [ D x + A, D t + B ] G − = G ( D x ( B ) − D t ( A ) + [ A, B ]) G − . Hence the equation D x ( B ) − D t ( A ) + [ A, B ] = 0 implies D x ( ˜ B ) − D t ( ˜ A ) + [ ˜ A, ˜ B ] = 0. N LIE ALGEBRAS RESPONSIBLE FOR INTEGRABILITY 8
Remark 9.
In this remark we assume K = C . For any constants e , e , e ∈ C , consider the Krichever-Novikov equation (17). To study the algebras F p ( E , a ) for this equation, we need some auxiliary con-structions.Let C [ v , v , v ] be the algebra of polynomials in the variables v , v , v . Let e , e , e ∈ C such that e = e = e = e . Consider the ideal I e ,e ,e ⊂ C [ v , v , v ] generated by the polynomials(20) v i − v j + e i − e j , i, j = 1 , , . Set E e ,e ,e = C [ v , v , v ] / I e ,e ,e . In other words, E e ,e ,e is the commutative associative algebra ofpolynomial functions on the algebraic curve in C defined by the polynomials (20). (This curve is givenby the equations v i − v j + e i − e j = 0, i, j = 1 , ,
3, in the space C with coordinates v , v , v .)Since we assume e = e = e = e , this curve is nonsingular, irreducible and is of genus 1, so this is anelliptic curve. It is known that the Landau-Lifshitz equation and the Krichever-Novikov equation possess so ( C )-valued ZCRs parametrized by points of this curve [34, 5, 26, 27]. (For the Krichever-Novikovequation, the paper [26] presents a ZCR with values in the Lie algebra sl ( C ) ∼ = so ( C ).)We have the natural surjective homomorphism ρ : C [ v , v , v ] → C [ v , v , v ] / I e ,e ,e = E e ,e ,e . Setˆ v i = ρ ( v i ) ∈ E e ,e ,e for i = 1 , , α , α , α of the Lie algebra so ( C ) such that [ α , α ] = α , [ α , α ] = α ,[ α , α ] = α .Denote by R e ,e ,e the Lie subalgebra of so ( C ) ⊗ C E e ,e ,e generated by the elements α i ⊗ ˆ v i , i = 1 , , R e ,e ,e ⊂ so ( C ) ⊗ C E e ,e ,e , we can view elements of R e ,e ,e as so ( C )-valued functions on theelliptic curve in C determined by the polynomials (20).Set z = ˆ v + e . As ˆ v + e = ˆ v + e = ˆ v + e in E e ,e ,e , we have z = ˆ v + e = ˆ v + e = ˆ v + e .It is easily seen (and is shown in [27]) that the following elements form a basis for R e ,e ,e (21) α i ⊗ ˆ v i z l , α i ⊗ ˆ v j ˆ v k z l , i, j, k ∈ { , , } , j < k, j = i = k, l ∈ Z ≥ . Since the basis (21) is infinite, the Lie algebra R e ,e ,e is infinite-dimensional. It is known that the stan-dard ZCR with elliptic parameter for the (fully anisotropic) Landau-Lifshitz equation can be interpretedas a ZCR with values in this algebra [34, 7, 27].It is shown in [27] that the Wahlquist-Estabrook prolongation algebra of the (fully anisotropic)Landau-Lifshitz equation is isomorphic to the direct sum of R e ,e ,e and a 2-dimensional abelian Liealgebra.According to Proposition 2 below, the algebra R e ,e ,e shows up also in the structure of F p ( E , a ) forthe Krichever-Novikov equation. A proof of Proposition 2 is given in [13]. This proof uses some resultsof [15, 26, 27]. Proposition 2 ([13]) . For any e , e , e ∈ C , consider the Krichever-Novikov equation KN( e , e , e ) given by (17) . Let E be the infinite prolongation of this equation. Let a ∈ E . Then • the algebra F ( E , a ) is zero, • for each p ≥ , the algebra F p ( E , a ) is obtained from F p − ( E , a ) by central extension, and thekernel of the surjective homomorphism F p ( E , a ) → F ( E , a ) from (58) is nilpotent, • if e = e = e = e , then F ( E , a ) ∼ = R e ,e ,e and for each p ≥ the algebra F p ( E , a ) is obtainedfrom R e ,e ,e by applying several times the operation of central extension. ZCRs, gauge transformations, and the algebras F p ( E , a )In this section we recall some notions and results from [12], adding some clarifications.As said in Section 2, we suppose that x , t , u k take values in K , where K is either C or R . Let K ∞ bethe infinite-dimensional space with the coordinates x , t , u k for k ∈ Z ≥ . The topology on K ∞ is definedas follows.For each l ∈ Z ≥ , consider the space K l +3 with the coordinates x , t , u k for k ≤ l . One has the naturalprojection π l : K ∞ → K l +3 that “forgets” the coordinates u k ′ for k ′ > l . N LIE ALGEBRAS RESPONSIBLE FOR INTEGRABILITY 9
Since K l +3 is a finite-dimensional vector space, we have the standard topology on K l +3 . For any l ∈ Z ≥ and any open subset V ⊂ K l +3 , the subset π − l ( V ) ⊂ K ∞ is, by definition, open in K ∞ . Suchsubsets form a base of the topology on K ∞ . In other words, we consider the smallest topology on K ∞ such that the maps π l , l ∈ Z ≥ , are continuous.For a connected open subset W ⊂ K ∞ , a function f : W → K is said to be analytic if f dependsanalytically on a finite number of the coordinates x , t , u k , where k ∈ Z ≥ . (That is, f is an analyticfunction of the form f = f ( x, t, u , . . . , u m ) for some m ∈ Z ≥ .) For an arbitrary open subset ˜ W ⊂ K ∞ ,a function g : ˜ W → K is called analytic if g is analytic on each connected component of ˜ W .Since we have the topology on K ∞ and the notion of analytic functions on open subsets of K ∞ , wecan say that K ∞ is an analytic manifold. Definition 3.
Let U ⊂ K d +3 be an open subset such that the function F = F ( x, t, u , u , . . . , u d )from (1) is defined on U . (For instance, if the function F is meromorphic on K d +3 then one can take U ⊂ K d +3 to be the maximal open subset such that F is analytic on U .)The infinite prolongation E of equation (1) is defined as follows: E = π − d ( U ) ⊂ K ∞ . So E is an open subset of the space K ∞ with the coordinates x , t , u k for k ∈ Z ≥ . The topology on E isinduced by the embedding E ⊂ K ∞ .As said above, we view the space K ∞ as an analytic manifold. Since E is an open subset of K ∞ , theset E is an analytic manifold as well. Example 4.
For any constants e , e , e ∈ K , we write the Krichever-Novikov equation (17) as follows u t = F ( u , u , u , u ) , (22) F ( u , u , u , u ) = u −
32 ( u ) u + ( u − e )( u − e )( u − e ) u . (23)Since the right-hand side of (22) depends on u k for k ≤
3, we have here d = 3.Let K be the space with the coordinates x , t , u , u , u , u . According to (23), the function F isdefined on the open subset U ⊂ K determined by the condition u = 0.Recall that K ∞ is the space with the coordinates x , t , u k for k ∈ Z ≥ . We have the map π : K ∞ → K that “forgets” the coordinates u k ′ for k ′ >
3. The infinite prolongation E of equation (22) is the followingopen subset of K ∞ E = π − ( U ) = (cid:8) ( x, t, u , u , u , . . . ) ∈ K ∞ (cid:12)(cid:12) u = 0 (cid:9) . Consider again an arbitrary scalar evolution equation (1). Let E be the infinite prolongation of (1).Since E is an open subset of the space K ∞ with the coordinates x , t , u k for k ∈ Z ≥ , a point a ∈ E isdetermined by the values of x , t , u k at a . Let(24) a = ( x = x a , t = t a , u k = a k ) ∈ E , x a , t a , a k ∈ K , k ∈ Z ≥ , be a point of E . The constants x a , t a , a k are the coordinates of the point a ∈ E in the coordinate system x , t , u k .We continue to use the notations introduced in Section 1. In particular, g ⊂ gl N is a matrix Liealgebra, and G ⊂ GL N is the connected matrix Lie group corresponding to g , where N ∈ Z > .According to Definition 2, a gauge transformation is a matrix-function G = G ( x, t, u , u , . . . , u l ) withvalues in G , where l ∈ Z ≥ . See also Remark 3 about gauge transformations with values in other matrixLie groups.In this section, when we speak about ZCRs, we always mean ZCRs of equation (1). For each i = 1 , A i = A i ( x, t, u , u , . . . ) , B i = B i ( x, t, u , u , . . . ) , D x ( B i ) − D t ( A i ) + [ A i , B i ] = 0 N LIE ALGEBRAS RESPONSIBLE FOR INTEGRABILITY 10 be a g -valued ZCR. The ZCR A , B is said to be gauge equivalent to the ZCR A , B if there is a gaugetransformation G = G ( x, t, u , . . . , u l ) such that A = GA G − − D x ( G ) · G − , B = GB G − − D t ( G ) · G − . Let s ∈ Z ≥ . For a function M = M ( x, t, u , u , u , . . . ), the notation M (cid:12)(cid:12)(cid:12) u k = a k , k ≥ s means that wesubstitute u k = a k for all k ≥ s in the function M . Also, sometimes we substitute x = x a or t = t a insuch functions. For example, if M = M ( x, t, u , u , u , u ), then M (cid:12)(cid:12)(cid:12) x = x a , u k = a k , k ≥ = M ( x a , t, u , u , a , a ) . The following result is obtained in [12].
Theorem 1 ([12]) . Let N ∈ Z > and p ∈ Z ≥ . Let g ⊂ gl N be a matrix Lie algebra and G ⊂ GL N bethe connected matrix Lie group corresponding to g ⊂ gl N .Consider a ZCR of order ≤ p given by (25) A = A ( x, t, u , . . . , u p ) , B = B ( x, t, u , . . . , u p + d − ) , D x ( B ) − D t ( A ) + [ A, B ] = 0 such that the functions A , B are analytic on a neighborhood of a ∈ E and take values in g .Then on a neighborhood of a ∈ E there is a unique gauge transformation G = G ( x, t, u , . . . , u l ) suchthat G ( a ) = Id and the functions (26) ˜ A = GAG − − D x ( G ) · G − , ˜ B = GBG − − D t ( G ) · G − satisfy ∂ ˜ A∂u s (cid:12)(cid:12)(cid:12)(cid:12) u k = a k , k ≥ s = 0 ∀ s ≥ , (27) ˜ A (cid:12)(cid:12)(cid:12) u k = a k , k ≥ = 0 , (28) ˜ B (cid:12)(cid:12)(cid:12) x = x a , u k = a k , k ≥ = 0 . (29) Furthermore, one has the following. • The function G depends only on x , t , u , . . . , u p − . ( In particular, if p = 0 then G depends onlyon x , t . ) • The function G is analytic on a neighborhood of a ∈ E . • The functions (26) take values in g and satisfy ˜ A = ˜ A ( x, t, u , . . . , u p ) , ˜ B = ˜ B ( x, t, u , . . . , u p + d − ) , (30) D x ( ˜ B ) − D t ( ˜ A ) + [ ˜ A, ˜ B ] = 0 . (31) So the functions (26) form a g -valued ZCR of order ≤ p .Note that, according to our definition of gauge transformations, G takes values in G . The property G ( a ) = Id means that G ( x a , t a , a , . . . , a p − ) = Id . Definition 4.
Fix a point a ∈ E given by (24), which is determined by constants x a , t a , a k . A ZCR(32) A = A ( x, t, u , u , . . . ) , B = B ( x, t, u , u , . . . ) , D x ( B ) − D t ( A ) + [ A , B ] = 0is said to be a -normal if A , B satisfy the following equations ∂ A ∂u s (cid:12)(cid:12)(cid:12)(cid:12) u k = a k , k ≥ s = 0 ∀ s ≥ , (33) A (cid:12)(cid:12)(cid:12) u k = a k , k ≥ = 0 , (34) B (cid:12)(cid:12)(cid:12) x = x a , u k = a k , k ≥ = 0 . (35) N LIE ALGEBRAS RESPONSIBLE FOR INTEGRABILITY 11
Remark 10.
For example, the ZCR ˜ A, ˜ B described in Theorem 1 is a -normal, because ˜ A , ˜ B obey (27),(28), (29). Theorem 1 implies that any ZCR on a neighborhood of a ∈ E is gauge equivalent to an a -normal ZCR. Therefore, following [12], we can say that properties (33), (34), (35) determine a normalform for ZCRs with respect to the action of the group of gauge transformations on a neighborhood of a ∈ E . Remark 11.
The functions A , B , G considered in Theorem 1 are analytic on a neighborhood of a ∈ E .Therefore, the g -valued functions ˜ A , ˜ B given by (26) are analytic as well.Since ˜ A , ˜ B are analytic and are of the form (30), these functions are represented as absolutelyconvergent power series˜ A = X l ,l ,i ,...,i p ≥ ( x − x a ) l ( t − t a ) l ( u − a ) i . . . ( u p − a p ) i p · ˜ A l ,l i ...i p , (36) ˜ B = X l ,l ,j ,...,j p + d − ≥ ( x − x a ) l ( t − t a ) l ( u − a ) j . . . ( u p + d − − a p + d − ) j p + d − · ˜ B l ,l j ...j p + d − , (37) ˜ A l ,l i ...i p , ˜ B l ,l j ...j p + d − ∈ g . For each k ∈ Z > , we set(38) V k = n ( i , . . . , i k ) ∈ Z k +1 ≥ (cid:12)(cid:12)(cid:12) ∃ r ∈ { , . . . , k } such that i r = 1 , i q = 0 ∀ q > r o . In other words, for k ∈ Z > and i , . . . , i k ∈ Z ≥ , one has ( i , . . . , i k ) ∈ V k iff there is r ∈ { , . . . , k } suchthat ( i , . . . , i r − , i r , i r +1 , . . . , i k ) = ( i , . . . , i r − , , , . . . , . Set also V = ∅ . So the set V is empty.Using formulas (36), (37), we see that properties (27), (28), (29) are equivalent to(39) ˜ A l ,l ... = ˜ B ,l ... = 0 , ˜ A l ,l i ...i p = 0 , ( i , . . . , i p ) ∈ V p , l , l ∈ Z ≥ . Remark 12.
Let L be a Lie algebra and m ∈ Z ≥ . Consider a formal power series of the form C = X l ,l ,i ,...,i m ≥ ( x − x a ) l ( t − t a ) l ( u − a ) i . . . ( u m − a m ) i m · C l ,l i ...i m , C l ,l i ...i m ∈ L . Set D x ( C ) = X l ,l ,i ,...,i m D x (cid:0) ( x − x a ) l ( t − t a ) l ( u − a ) i . . . ( u m − a m ) i m (cid:1) · C l ,l i ...i m , (40) D t ( C ) = X l ,l ,i ,...,i m D t (cid:0) ( x − x a ) l ( t − t a ) l ( u − a ) i . . . ( u m − a m ) i m (cid:1) · C l ,l i ...i m . (41)The expressions(42) D x (cid:0) ( x − x a ) l ( t − t a ) l ( u − a ) i . . . ( u m − a m ) i m (cid:1) ,D t (cid:0) ( x − x a ) l ( t − t a ) l ( u − a ) i . . . ( u m − a m ) i m (cid:1) are functions of the variables x , t , u k . Taking the corresponding Taylor series at the point (24), weview (42) as power series. Then (40), (41) become formal power series with coefficients in L .According to (5), one has D t = ∂∂t + P k ≥ D kx ( F ) ∂∂u k , where F = F ( x, t, u , . . . , u d ) is given in (1).When we apply D t in (41), we view F as a power series, using the Taylor series of the function F .Let n ∈ Z ≥ and consider another formal power series R = X q ,q ,j ,...,j n ≥ ( x − x a ) q ( t − t a ) q ( u − a ) j . . . ( u n − a n ) j n · R q ,q j ...j n , R q ,q j ...j n ∈ L . Then the Lie bracket [
C, R ] is defined in the obvious way and is a formal power series with coefficientsin L . N LIE ALGEBRAS RESPONSIBLE FOR INTEGRABILITY 12
According to the described procedure, the expression D x ( R ) − D t ( C ) + [ C, R ] is well defined and is aformal power series with coefficients in L . Remark 13.
The main idea of the definition of the Lie algebra F p ( E , a ) can be informally outlined asfollows. According to Theorem 1 and Remark 11, any ZCR (25) of order ≤ p is gauge equivalent to aZCR given by functions ˜ A , ˜ B that are of the form (36), (37) and satisfy (31), (39).To define F p ( E , a ) in terms of generators and relations, one can regard ˜ A l ,l i ...i p , ˜ B l ,l j ...j p + d − from (36), (37)as abstract symbols. Then one can say that the Lie algebra F p ( E , a ) is generated by the symbols ˜ A l ,l i ...i p ,˜ B l ,l j ...j p + d − for l , l , i , . . . , i p , j , . . . , j p + d − ∈ Z ≥ so that relations for these generators are provided byequations (31), (39).The details of this construction are presented below. To avoid confusion in notation, we introducenew symbols A l ,l i ...i p , B l ,l j ...j p + d − , which will be generators of the algebra F p ( E , a ).Fix p ∈ Z ≥ and consider formal power series A = X l ,l ,i ,...,i p ≥ ( x − x a ) l ( t − t a ) l ( u − a ) i . . . ( u p − a p ) i p · A l ,l i ...i p , (43) B = X l ,l ,j ,...,j p + d − ≥ ( x − x a ) l ( t − t a ) l ( u − a ) j . . . ( u p + d − − a p + d − ) j p + d − · B l ,l j ...j p + d − , (44)where(45) A l ,l i ...i p , B l ,l j ...j p + d − , l , l , i , . . . , i p , j , . . . , j p + d − ∈ Z ≥ , are generators of a Lie algebra, which is described below.We impose the equation(46) D x ( B ) − D t ( A ) + [ A , B ] = 0 , which is equivalent to some Lie algebraic relations for the generators (45). The left-hand side of (46) isdefined by the procedure described in Remark 12. Also, we impose the following condition(47) A l ,l ... = B ,l ... = 0 , A l ,l i ...i p = 0 , ( i , . . . , i p ) ∈ V p , l , l ∈ Z ≥ . Definition 5.
Recall that the manifold E is the infinite prolongation of equation (1), and a ∈ E is givenby (24), where the constants x a , t a , a k are the coordinates of the point a in the coordinate system x , t , u k . For each p ∈ Z ≥ , the Lie algebra F p ( E , a ) is defined in terms of generators and relations as follows.The algebra F p ( E , a ) is given by the generators (45), relations (47), and the relations arising from (46)when we substitute (43), (44) in (46).This description of F p ( E , a ) is sufficient for the present paper. A more detailed definition of F p ( E , a )is given in [12]. Note that condition (47) is equivalent to the following equations ∂ A ∂u s (cid:12)(cid:12)(cid:12)(cid:12) u k = a k , k ≥ s = 0 ∀ s ≥ , (48) A (cid:12)(cid:12)(cid:12) u k = a k , k ≥ = 0 , (49) B (cid:12)(cid:12)(cid:12) x = x a , u k = a k , k ≥ = 0 . (50)So the algebra F p ( E , a ) is generated by the elements (45). Theorem 2 below, which is proved in [12],says that the elements (51) generate the algebra F p ( E , a ) as well. This fact is very useful in computationsof F p ( E , a ) for concrete equations, because the set of the elements (51) is much smaller than that of (45).We will use Theorem 2 in Section 4. N LIE ALGEBRAS RESPONSIBLE FOR INTEGRABILITY 13
Theorem 2 ([12]) . The elements (51) A l , i ...i p , l , i , . . . , i p ∈ Z ≥ , generate the algebra F p ( E , a ) . Remark 14.
Let g be a finite-dimensional matrix Lie algebra. By Theorem 1, for any g -valued ZCR (25)of order ≤ p on a neighborhood of a ∈ E , there is a unique gauge transformation G such that G ( a ) = Idand the functions (26) obey (27), (28), (29). Furthermore, Theorem 1 says that the functions (26) takevalues in g and satisfy (30), (31).Consider the Taylor series (36), (37) of the functions (26). Properties (27), (28), (29) are equivalentto (39). Properties (31), (39) imply that the following homomorphism(52) µ : F p ( E , a ) → g , µ (cid:0) A l ,l i ...i p (cid:1) = ˜ A l ,l i ...i p , µ (cid:0) B l ,l j ...j p + d − (cid:1) = ˜ B l ,l j ...j p + d − , is well defined. Here ˜ A l ,l i ...i p , ˜ B l ,l j ...j p + d − ∈ g are the coefficients of the power series (36), (37). The defini-tion (52) of µ implies that the ZCR given by (36), (37) takes values in the Lie subalgebra µ (cid:0) F p ( E , a ) (cid:1) ⊂ g .It is shown in [12] that the ZCR (25) is uniquely determined (up to gauge equivalence) by thecorresponding homomorphism µ : F p ( E , a ) → g .On the other hand, consider an arbitrary homomorphism ˜ µ : F p ( E , a ) → g . Applying ˜ µ to the coeffi-cients of the power series (43), (44), we get the following power series with coefficients in g A = X l ,l ,i ,...,i p ( x − x a ) l ( t − t a ) l ( u − a ) i . . . ( u p − a p ) i p · ˜ µ (cid:0) A l ,l i ...i p (cid:1) , (53) B = X l ,l ,j ,...,j p + d − ( x − x a ) l ( t − t a ) l ( u − a ) j . . . ( u p + d − − a p + d − ) j p + d − · ˜ µ (cid:0) B l ,l j ...j p + d − (cid:1) . (54)Since (43), (44) obey (46), the power series (53), (54) satisfy D x ( B ) − D t ( A ) + [ A , B ] = 0. UsingDefinition 6, we can say that the formal power series (53), (54) constitute a formal ZCR of order ≤ p with coefficients in g . If the power series (53), (54) converge to analytic functions, then they constitutea g -valued ZCR of order ≤ p .The described correspondence between g -valued ZCRs and homomorphisms µ : F p ( E , a ) → g allowsone to say that the algebra F p ( E , a ) is responsible for ZCRs of order ≤ p .Suppose that p ≥
1. According to Definition 5, to define the algebra F p ( E , a ), we take formal powerseries (43), (44) and impose conditions (46), (47). The Lie algebra F p ( E , a ) is given by the generators A l ,l i ...i p , B l ,l j ...j p + d − and the relations arising from (46), (47). Similarly, to define the algebra F p − ( E , a ),we take formal power seriesˆ A = X l ,l ,i ,...,i p − ( x − x a ) l ( t − t a ) l ( u − a ) i . . . ( u p − − a p − ) i p − · ˆ A l ,l i ...i p − , ˆ B = X l ,l ,j ,...,j p + d − ( x − x a ) l ( t − t a ) l ( u − a ) j . . . ( u p + d − − a p + d − ) j p + d − · ˆ B l ,l j ...j p + d − and impose the following conditions D x (cid:0) ˆ B (cid:1) − D t (cid:0) ˆ A (cid:1) + (cid:2) ˆ A , ˆ B (cid:3) = 0 , (55) ˆ A l ,l ... = ˆ B ,l ... = 0 , ˆ A l ,l i ...i p − = 0 , ( i , . . . , i p − ) ∈ V p − , l , l ∈ Z ≥ . (56)The Lie algebra F p − ( E , a ) is given by the generators ˆ A l ,l i ...i p − , ˆ B l ,l j ...j p + d − and the relations arisingfrom (55), (56).This implies that the map(57) A l ,l i ...i p − i p δ ,i p · ˆ A l ,l i ...i p − , B l ,l j ...j p + d − j p + d − δ ,j p + d − · ˆ B l ,l j ...j p + d − N LIE ALGEBRAS RESPONSIBLE FOR INTEGRABILITY 14 determines a surjective homomorphism F p ( E , a ) → F p − ( E , a ). Here δ ,i p and δ ,j p + d − are the Kroneckerdeltas. We denote this homomorphism by ϕ p : F p ( E , a ) → F p − ( E , a ).According to Remark 14, the algebra F p ( E , a ) is responsible for ZCRs of order ≤ p , and thealgebra F p − ( E , a ) is responsible for ZCRs of order ≤ p −
1. The constructed homomorphism ϕ p : F p ( E , a ) → F p − ( E , a ) reflects the fact that any ZCR of order ≤ p − ≤ p . Thus we obtain the following sequence of surjective homomorphisms of Lie algebras(58) . . . ϕ p +1 −−−→ F p ( E , a ) ϕ p −→ F p − ( E , a ) ϕ p − −−−→ . . . ϕ −→ F ( E , a ) ϕ −→ F ( E , a ) . Some results on F p ( E , a ) for equations (9)In this section we study the algebras (58) for equations of the form (9), where f = f ( x, t, u , . . . , u q − )is an arbitrary function and q ∈ { , , } .Let E be the infinite prolongation of equation (9). According to Definition 3, E is an open subset of thespace K ∞ with the coordinates x , t , u k for k ∈ Z ≥ . For equation (9), the total derivative operators (5)are(59) D x = ∂∂x + X k ≥ u k +1 ∂∂u k , D t = ∂∂t + X k ≥ D kx (cid:0) u q +1 + f ( x, t, u , . . . , u q − ) (cid:1) ∂∂u k . Consider an arbitrary point a ∈ E given by (24), where the constants x a , t a , a k are the coordinates of a in the coordinate system x , t , u k .Let p ∈ Z > such that p ≥ q + δ q, , where δ q, is the Kronecker delta. According to Definition 5, thealgebra F p ( E , a ) can be described as follows. Consider formal power series A = X l ,l ,i ,...,i p ≥ ( x − x a ) l ( t − t a ) l ( u − a ) i . . . ( u p − a p ) i p · A l ,l i ...i p , (60) B = X l ,l ,j ,...,j p +2 q ≥ ( x − x a ) l ( t − t a ) l ( u − a ) j . . . ( u p +2 q − a p +2 q ) j p +2 q · B l ,l j ...j p +2 q (61)satisfying A l ,l i ...i p = 0 if ∃ r ∈ { , . . . , p } such that i r = 1 , i m = 0 ∀ m > r, (62) A l ,l ... = 0 ∀ l , l ∈ Z ≥ , (63) B ,l ... = 0 ∀ l ∈ Z ≥ . (64)Then A l ,l i ...i p , B l ,l j ...j p +2 q are generators of the Lie algebra F p ( E , a ), and the equation(65) D x ( B ) − D t ( A ) + [ A , B ] = 0provides relations for these generators (in addition to relations (62), (63), (64)).Condition (62) is equivalent to(66) ∂∂u s ( A ) (cid:12)(cid:12)(cid:12)(cid:12) u k = a k , k ≥ s = 0 ∀ s ≥ . Using (59), one can rewrite equation (65) as(67) ∂∂x ( B ) + p +2 q X k =0 u k +1 ∂∂u k ( B ) + [ A , B ] = ∂∂t ( A ) + p X k =0 (cid:16) u k +2 q +1 + D kx (cid:0) f ( x, t, u , . . . , u q − ) (cid:1)(cid:17) ∂∂u k ( A ) . Here we view f ( x, t, u , . . . , u q − ) as a power series, using the Taylor series of the function f at thepoint (24). Differentiating (67) with respect to u p +2 q +1 , we obtain(68) ∂∂u p +2 q ( B ) = ∂∂u p ( A ) . N LIE ALGEBRAS RESPONSIBLE FOR INTEGRABILITY 15
Since q ∈ { , , } , from (68) it follows that B is of the form(69) B = u p +2 q ∂∂u p ( A ) + B ( x, t, u , . . . , u p +2 q − ) , where B ( x, t, u , . . . , u p +2 q − ) is a power series in the variables x − x a , t − t a , u − a , . . . , u p +2 q − − a p +2 q − . Differentiating (67) with respect to u p +2 q , u p + i for i = 1 , . . . , q − ∂ ∂u p ∂u p ( A ) + ∂ ∂u p +1 ∂u p +2 q − ( B ) = 0 , ∂ ∂u p + s ∂u p +2 q − ( B ) = 0 , ≤ s ≤ q − . Therefore, B = B ( x, t, u , . . . , u p +2 q − ) is of the form(70) B = u p +1 u p +2 q − (cid:16) δ q, − (cid:17) ∂ ∂u p ∂u p ( A ) + u p +2 q − B ( x, t, u , . . . , u p ) + B ( x, t, u , . . . , u p +2 q − ) . Here B ( x, t, u , . . . , u p ) is a power series in the variables x − x a , t − t a , u − a , . . . , u p − a p , and B ( x, t, u , . . . , u p +2 q − ) is a power series in the variables x − x a , t − t a , u − a , . . . , u p +2 q − − a p +2 q − . Lemma 1.
Recall that q ∈ { , , } and p ≥ q + δ q, . We have (71) D x (cid:16) ∂ ∂u p ∂u p ( A ) (cid:17) + h A , ∂ ∂u p ∂u p ( A ) i = 0 . Proof.
Since D x = ∂∂x + P k ≥ u k +1 ∂∂u k , one has(72) ∂∂u n (cid:0) D x ( Q ) (cid:1) = D x (cid:16) ∂∂u n ( Q ) (cid:17) + ∂∂u n − ( Q ) ∀ n ∈ Z > for any Q = Q ( x, t, u , u , . . . , u l ). Here Q is either a function or a power series.In what follows we sometimes use the notation B u n = ∂∂u n ( B ) , B u m u n = ∂ ∂u m ∂u n ( B ) , m, n ∈ Z ≥ . Using (72), one gets(73) ∂ ∂u m ∂u n (cid:0) D x ( B ) (cid:1) = ∂∂u m (cid:16) D x (cid:16) ∂∂u n ( B ) (cid:17) + ∂∂u n − ( B ) (cid:17) = D x ( B u m u n ) + B u m − u n + B u m u n − ∀ m, n ∈ Z > . We will need also the formula(74) D t ( A ) = ∂∂t ( A ) + p X k =0 (cid:16) u k +2 q +1 + D kx (cid:0) f ( x, t, u , . . . , u q − ) (cid:1)(cid:17) ∂∂u k ( A ) , which follows from (59).Consider the case q = 1. Then, by our assumption, p ≥
1. Equation (9) reads u t = u + f ( x, t, u , u ).According to (69), (70), for q = 1 one has(75) B = u p +2 ∂∂u p ( A ) −
12 ( u p +1 ) ∂ ∂u p ∂u p ( A ) + u p +1 B ( x, t, u , . . . , u p ) + B ( x, t, u , . . . , u p ) . Since we assume q = 1 and p ≥
1, formula (74) implies(76) ∂ ∂u p ∂u p +2 (cid:0) D t ( A ) (cid:1) = ∂ ∂u p ∂u p − ( A ) , ∂ ∂u p +1 ∂u p +1 (cid:0) D t ( A ) (cid:1) = 0 . N LIE ALGEBRAS RESPONSIBLE FOR INTEGRABILITY 16
Using (73), (75), (76), one obtains(77) (cid:16) ∂ ∂u p ∂u p +2 − ∂ ∂u p +1 ∂u p +1 (cid:17)(cid:16) D x ( B ) − D t ( A ) + [ A , B ] (cid:17) == D x (cid:16) B u p u p +2 − B u p +1 u p +1 (cid:17) + B u p − u p +2 − ∂ ∂u p ∂u p − ( A ) + h A , B u p u p +2 − B u p +1 u p +1 i == 32 (cid:18) D x (cid:16) ∂ ∂u p ∂u p ( A ) (cid:17) + h A , ∂ ∂u p ∂u p ( A ) i(cid:19) . Since D x ( B ) − D t ( A ) + [ A , B ] = 0 by (65), equation (77) implies (71) in the case q = 1.Now let q = 2. Then p ≥
2. Equation (9) reads u t = u + f ( x, t, u , u , u , u ). Using (69), (70), for q = 2 one obtains(78) B = u p +4 ∂∂u p ( A ) − u p +1 u p +3 ∂ ∂u p ∂u p ( A ) + u p +3 B ( x, t, u , . . . , u p ) + B ( x, t, u , . . . , u p +2 ) . Applying the operator ∂ ∂u p +2 ∂u p +3 to equation (67) and using (78), we get(79) ∂ ∂u p +2 ∂u p +2 ( B ) − ∂ ∂u p ∂u p ( A ) = 0 . Since we assume q = 2 and p ≥
2, formula (74) implies ∂ ∂u p ∂u p +4 (cid:0) D t ( A ) (cid:1) = ∂ ∂u p ∂u p − ( A ) , (80) (cid:16) − ∂ ∂u p +1 ∂u p +3 + 12 ∂ ∂u p +2 ∂u p +2 (cid:17)(cid:0) D t ( A ) (cid:1) = 0 . (81)Using (73), (78), (79), (80), (81), one obtains(82) (cid:16) ∂ ∂u p ∂u p +4 − ∂ ∂u p +1 ∂u p +3 + 12 ∂ ∂u p +2 ∂u p +2 (cid:17)(cid:16) D x ( B ) − D t ( A ) + [ A , B ] (cid:17) == D x (cid:16) B u p u p +4 − B u p +1 u p +3 + 12 B u p +2 u p +2 (cid:17) + B u p − u p +4 − ∂ ∂u p ∂u p − ( A )++ h A , B u p u p +4 − B u p +1 u p +3 + 12 B u p +2 u p +2 i == 52 (cid:18) D x (cid:16) ∂ ∂u p ∂u p ( A ) (cid:17) + h A , ∂ ∂u p ∂u p ( A ) i(cid:19) . As D x ( B ) − D t ( A ) + [ A , B ] = 0, equation (82) yields (71) in the case q = 2.Finally, consider the case q = 3. Then p ≥ u t = u + f ( x, t, u , u , u , u , u , u ). According to (69), (70), for q = 3 we have(83) B = u p +6 ∂∂u p ( A ) − u p +1 u p +5 ∂ ∂u p ∂u p ( A ) + u p +5 B ( x, t, u , . . . , u p ) + B ( x, t, u , . . . , u p +4 ) . Differentiating (67) with respect to u p +5 , u p + i for i = 2 , , − ∂ ∂u p ∂u p ( A ) + ∂ ∂u p +2 ∂u p +4 ( B ) = 0 , ∂ ∂u p +3 ∂u p +4 ( B ) = ∂ ∂u p +4 ∂u p +4 ( B ) = 0 . Therefore, B = B ( x, t, u , . . . , u p +4 ) is of the form(84) B = u p +2 u p +4 ∂ ∂u p ∂u p ( A ) + u p +4 B ( x, t, u , . . . , u p +1 ) + B ( x, t, u , . . . , u p +3 ) N LIE ALGEBRAS RESPONSIBLE FOR INTEGRABILITY 17 for some B ( x, t, u , . . . , u p +1 ) and B ( x, t, u , . . . , u p +3 ).Applying the operator ∂ ∂u p +3 ∂u p +4 to equation (67) and using (83), (84), we obtain(85) ∂ ∂u p +3 ∂u p +3 ( B ) + ∂ ∂u p ∂u p ( A ) = 0 . Since we consider the case when q = 3 and p ≥
4, formula (74) implies ∂ ∂u p ∂u p +6 (cid:0) D t ( A ) (cid:1) = ∂ ∂u p ∂u p − ( A ) , (86) (cid:16) − ∂ ∂u p +1 ∂u p +5 + ∂ ∂u p +2 ∂u p +4 − ∂ ∂u p +3 ∂u p +3 (cid:17)(cid:0) D t ( A ) (cid:1) = 0 . (87)Using (73), (83), (84), (85), (86), (87) one gets(88) (cid:16) ∂ ∂u p ∂u p +6 − ∂ ∂u p +1 ∂u p +5 + ∂ ∂u p +2 ∂u p +4 − ∂ ∂u p +3 ∂u p +3 (cid:17)(cid:16) D x ( B ) − D t ( A ) + [ A , B ] (cid:17) == D x (cid:16) B u p u p +6 − B u p +1 u p +5 + B u p +2 u p +4 − B u p +3 u p +3 (cid:17) + B u p − u p +6 − ∂ ∂u p ∂u p − ( A )++ h A , B u p u p +6 − B u p +1 u p +5 + B u p +2 u p +4 − B u p +3 u p +3 i == 72 (cid:18) D x (cid:16) ∂ ∂u p ∂u p ( A ) (cid:17) + h A , ∂ ∂u p ∂u p ( A ) i(cid:19) . Since D x ( B ) − D t ( A ) + [ A , B ] = 0, equation (88) implies (71) in the case q = 3. (cid:3) Lemma 2.
One has (89) ∂ ∂u k ∂u p ∂u p ( A ) = 0 ∀ k ∈ Z ≥ . Proof.
Suppose that (89) does not hold. Let k be the maximal integer such that ∂ ∂u k ∂u p ∂u p ( A ) = 0.Equation (66) for s = k + 1 says(90) ∂∂u k +1 ( A ) (cid:12)(cid:12)(cid:12)(cid:12) u k = a k , k ≥ k +1 = 0 . Differentiating (71) with respect to u k +1 , we obtain(91) ∂ ∂u k ∂u p ∂u p ( A ) + h ∂∂u k +1 ( A ) , ∂ ∂u p ∂u p ( A ) i = 0 . Substituting u k = a k in (91) for all k ≥ k + 1 and using (90), one gets ∂ ∂u k ∂u p ∂u p ( A ) = 0, whichcontradicts our assumption. (cid:3) Using equation (66) for s = p and equation (89) for all k ∈ Z ≥ , we see that A is of the form(92) A = ( u p − a p ) A ( x, t ) + A ( x, t, u , . . . , u p − ) , where A ( x, t ) is a power series in the variables x − x a , t − t a and A ( x, t, u , . . . , u p − ) is a power seriesin the variables x − x a , t − t a , u − a , . . . , u p − − a p − .From (89), (92) it follows that equation (71) reads(93) 2 ∂∂x ( A ) + 2[ A , A ] = 0 . N LIE ALGEBRAS RESPONSIBLE FOR INTEGRABILITY 18
Note that condition (63) implies(94) A (cid:12)(cid:12)(cid:12) u k = a k , k ≥ = 0 . Substituting u k = a k in (93) for all k ≥ ∂∂x ( A ) = 0 . Combining (95) with (93), one obtains(96) [ A , A ] = 0 . In view of (60), (92), we have(97) A = X l ,l ,i ,...,i p − ≥ ( x − x a ) l ( t − t a ) l ( u − a ) i . . . ( u p − − a p − ) i p − · A l ,l i ...i p − According to (60), (92), (95), one has(98) A = X l ≥ ( t − t a ) l · ˜ A l , ˜ A l = A ,l ... ∈ F p ( E , a ) . Combining (92), (97), (98) with Theorem 2, we see that the elements(99) ˜ A , A l , i ...i p − , l , i , . . . , i p − ∈ Z ≥ , generate the algebra F p ( E , a ). Substituting t = t a in (96) and using (97), (98), one gets(100) (cid:2) ˜ A , A l , i ...i p − (cid:3) = 0 ∀ l , i , . . . , i p − ∈ Z ≥ . Since the elements (99) generate the algebra F p ( E , a ), equation (100) yields(101) (cid:2) ˜ A , F p ( E , a ) (cid:3) = 0 . Lemma 3.
One has (102) (cid:2) ˜ A l , F p ( E , a ) (cid:3) = 0 ∀ l ∈ Z ≥ . Proof.
We prove (102) by induction on l . The property (cid:2) ˜ A , F p ( E , a ) (cid:3) = 0 has been obtained in (101).Let r ∈ Z ≥ such that (cid:2) ˜ A l , F p ( E , a ) (cid:3) = 0 for all l ≤ r . Since ∂ l ∂t l ( A ) (cid:12)(cid:12)(cid:12)(cid:12) t = t a = l ! · ˜ A l , we get(103) (cid:20) ∂ l ∂t l ( A ) (cid:12)(cid:12)(cid:12)(cid:12) t = t a , ∂ m ∂t m ( A ) (cid:12)(cid:12)(cid:12)(cid:12) t = t a (cid:21) = 0 ∀ l ≤ r, ∀ m ∈ Z ≥ . Applying the operator ∂ r +1 ∂t r +1 to equation (96), substituting t = t a , and using (103), one obtains0 = ∂ r +1 ∂t r +1 (cid:0) [ A , A ] (cid:1) (cid:12)(cid:12)(cid:12)(cid:12) t = t a = r +1 X k =0 (cid:18) r + 1 k (cid:19) · (cid:20) ∂ k ∂t k ( A ) (cid:12)(cid:12)(cid:12)(cid:12) t = t a , ∂ r +1 − k ∂t r +1 − k ( A ) (cid:12)(cid:12)(cid:12)(cid:12) t = t a (cid:21) == (cid:20) ∂ r +1 ∂t r +1 ( A ) (cid:12)(cid:12)(cid:12)(cid:12) t = t a , A (cid:12)(cid:12)(cid:12)(cid:12) t = t a (cid:21) == (cid:20) ( r + 1)! · ˜ A r +1 , X l ,i ,...,i p − ( x − x a ) l ( u − a ) i . . . ( u p − − a p − ) i p − · A l , i ...i p − (cid:21) , which implies(104) (cid:2) ˜ A r +1 , A l , i ...i p − (cid:3) = 0 ∀ l , i , . . . , i p − ∈ Z ≥ . N LIE ALGEBRAS RESPONSIBLE FOR INTEGRABILITY 19
Equation (101) yields(105) (cid:2) ˜ A , ˜ A r +1 (cid:3) = 0 . Since the elements (99) generate the algebra F p ( E , a ), from (104), (105) we get (cid:2) ˜ A r +1 , F p ( E , a ) (cid:3) = 0. (cid:3) Theorem 3.
Let E be the infinite prolongation of an equation of the form (9) with q ∈ { , , } . Let a ∈ E . For each p ∈ Z > , consider the surjective homomorphism ϕ p : F p ( E , a ) → F p − ( E , a ) from (58) .If p ≥ q + δ q, then (106) [ v , v ] = 0 ∀ v ∈ ker ϕ p , ∀ v ∈ F p ( E , a ) . In other words, if p ≥ q + δ q, then the kernel of ϕ p is contained in the center of the Lie algebra F p ( E , a ) .For each k ∈ Z > , let ψ k : F k + q − δ q, ( E , a ) → F q − δ q, ( E , a ) be the composition of the homomorphisms F k + q − δ q, ( E , a ) → F k + q − δ q, ( E , a ) → · · · → F q + δ q, ( E , a ) → F q − δ q, ( E , a ) from (58) . Then (107) [ h , [ h , . . . , [ h k − , [ h k , h k +1 ]] . . . ]] = 0 ∀ h , . . . , h k +1 ∈ ker ψ k . In particular, the kernel of ψ k is nilpotent.Proof. Let p ≥ q + δ q, . Combining (92), (69), (98) with the definition of the homomorphism ϕ p : F p ( E , a ) → F p − ( E , a ) , we see that ker ϕ p is equal to the ideal generated by the elements ˜ A l , l ∈ Z ≥ . Then (106) followsfrom (102).So we have proved that the kernel of the homomorphism ϕ p : F p ( E , a ) → F p − ( E , a ) is contained inthe center of the Lie algebra F p ( E , a ) for any p ≥ q + δ q, .Let us prove (107) by induction on k . Since ψ = ϕ q + δ q, , for k = 1 property (107) follows from (106).Let r ∈ Z > such that (107) is valid for k = r . Then for any h ′ , h ′ , . . . , h ′ r +2 ∈ ker ψ r +1 we have(108) (cid:2) ϕ r + q + δ q, ( h ′ ) , (cid:2) ϕ r + q + δ q, ( h ′ ) , . . . , (cid:2) ϕ r + q + δ q, ( h ′ r ) , (cid:2) ϕ r + q + δ q, ( h ′ r +1 ) , ϕ r + q + δ q, ( h ′ r +2 ) (cid:3)(cid:3) . . . (cid:3)(cid:3) = 0 , because ϕ r + q + δ q, ( h ′ i ) ∈ ker ψ r for i = 2 , , . . . , r + 2. Equation (108) says that(109) (cid:2) h ′ , (cid:2) h ′ , . . . , (cid:2) h ′ r , (cid:2) h ′ r +1 , h ′ r +2 (cid:3)(cid:3) . . . (cid:3)(cid:3) ∈ ker ϕ r + q + δ q, . Since ker ϕ r + q + δ q, is contained in the center of F r + q + δ q, ( E , a ), property (109) yields (cid:2) h ′ , (cid:2) h ′ , (cid:2) h ′ , . . . , (cid:2) h ′ r , (cid:2) h ′ r +1 , h ′ r +2 (cid:3)(cid:3) . . . (cid:3)(cid:3)(cid:3) = 0 . So we have proved (107) for k = r + 1. Clearly, property (107) implies that ker ψ k is nilpotent. (cid:3) Now we prove a result which is used in Example 7.
Theorem 4.
Let E be the infinite prolongation of the equation (110) u t = u + f ( x, t, u , u , u , u ) for some function f = f ( x, t, u , u , u , u ) such that ∂ f∂u ∂u ∂u = 0 . ( More precisely, we assume that thefunction ∂ f∂u ∂u ∂u is not identically zero on any connected component of the manifold E . Usually, themanifold E is connected, and then our assumption means that ∂ f∂u ∂u ∂u is not identically zero on E . ) Then F ( E , a ) = F ( E , a ) = 0 and F p ( E , a ) is nilpotent for all a ∈ E , p > .Proof. Consider an arbitrary point a ∈ E given by (24). According to Definition 5, the algebra F ( E , a )for equation (110) can be described as follows. Consider formal power series A = X l ,l ,i ,i ≥ ( x − x a ) l ( t − t a ) l ( u − a ) i ( u − a ) i · A l ,l i ,i , (111) N LIE ALGEBRAS RESPONSIBLE FOR INTEGRABILITY 20 B = X l ,l ,j ,...,j ≥ ( x − x a ) l ( t − t a ) l ( u − a ) j . . . ( u − a ) j · B l ,l j ...j (112)satisfying(113) A l ,l i , = A l ,l , = B ,l ... = 0 , l , l , i ∈ Z ≥ . Then A l ,l i ,i , B l ,l j ...j are generators of the algebra F ( E , a ), and the equation(114) D x ( B ) − D t ( A ) + [ A , B ] = 0provides relations for these generators (in addition to relations (113)). Note that here D t ( A ) is given byformula (74) for q = 2 and p = 1, so we have(115) D t ( A ) = ∂∂t ( A ) + (cid:0) u + f ( x, t, u , u , u , u ) (cid:1) ∂∂u ( A ) + (cid:0) u + D x (cid:0) f ( x, t, u , u , u , u ) (cid:1)(cid:1) ∂∂u ( A ) . Similarly to (78), from (114) we deduce that B is of the form(116) B = u ∂∂u ( A ) − u u ∂ ∂u ∂u ( A ) + u B ( x, t, u , u ) + B ( x, t, u , u , u , u ) , where B ( x, t, u , u ) is a power series in the variables x − x a , t − t a , u − a , u − a and B ( x, t, u , u , u , u ) is a power series in the variables x − x a , t − t a , u − a , u − a , u − a , u − a .Differentiating (114) with respect to u , u and using (116), we get(117) ∂ ∂u ∂u ( B ) = ∂ ∂u ∂u ( A ) + ∂ f∂u ∂u · ∂∂u ( A ) . Using (72), (115), (116), (117), one can verify that(118) (cid:16) − ∂ ∂u ∂u ∂u + 12 ∂ ∂u ∂u ∂u (cid:17)(cid:16) D x ( B ) − D t ( A ) + [ A , B ] (cid:17) == 12 ∂ f∂u ∂u ∂u · (cid:18) D x (cid:16) ∂∂u ( A ) (cid:17) + h A , ∂∂u ( A ) i − ∂∂u ( A ) (cid:19) . Since D x ( B ) − D t ( A ) + [ A , B ] = 0 by (114), equation (118) implies(119) ∂ f∂u ∂u ∂u · (cid:18) D x (cid:16) ∂∂u ( A ) (cid:17) + h A , ∂∂u ( A ) i − ∂∂u ( A ) (cid:19) = 0 . As the analytic function ∂ f∂u ∂u ∂u is not identically zero on any connected component of the manifold E ,equation (119) yields(120) D x (cid:16) ∂∂u ( A ) (cid:17) + h A , ∂∂u ( A ) i − ∂∂u ( A ) = 0 . Since ∂∂u ( A ) = 0, differentiating (120) with respect to u , we obtain ∂ ∂u ∂u ( A ) = 0.Recall that A is of the form (111). As A l ,l i , = 0 for all l , l , i ∈ Z ≥ by (113), equation ∂ ∂u ∂u ( A ) = 0yields ∂∂u ( A ) = 0. Combining the equation ∂∂u ( A ) = 0 with (120), one gets ∂∂u ( A ) = 0.Combining the equations ∂∂u ( A ) = ∂∂u ( A ) = 0 with (113), we get A l ,l i ,i = 0 for all l , l , i , i ∈ Z ≥ .Since, by Theorem 2, the algebra F ( E , a ) is generated by the elements A l , i ,i for l , i , i ∈ Z ≥ , weobtain F ( E , a ) = 0. As one has the surjective homomorphism F ( E , a ) → F ( E , a ) in (58), one gets F ( E , a ) = 0.According to Theorem 3 for q = 2, for any k ∈ Z > the kernel of the homomorphism ψ k : F k +1 ( E , a ) → F ( E , a )is nilpotent. Since F ( E , a ) = 0, this implies that F p ( E , a ) is nilpotent for all p > (cid:3) N LIE ALGEBRAS RESPONSIBLE FOR INTEGRABILITY 21 ZCRs with values in infinite-dimensional Lie algebras
According to Definition 1 and Remark 1, for a finite-dimensional Lie algebra g , a ZCR with valuesin g is given by analytic functions A ( x, t, u , u , . . . ), B ( x, t, u , u , . . . ) with values in g satisfying (4).Sometimes one needs to consider ZCRs with values in infinite-dimensional Lie algebras. An exampleof such a ZCR is studied in Section 6.3.For an arbitrary infinite-dimensional Lie algebra L , the notion of analytic functions with values in L is not defined. Because of this, a theory for ZCRs with values in infinite-dimensional Lie algebras isdeveloped below by using formal power series instead of analytic functions.Consider an arbitrary scalar evolution equation (1). Let E be the infinite prolongation of (1). Fix apoint a ∈ E given by (24), which is determined by constants x a , t a , a k . Definition 6.
Let L be a (possibly infinite-dimensional) Lie algebra. A formal ZCR of order ≤ p withcoefficients in L is given by formal power series A = X l ,l ,i ,...,i p ≥ ( x − x a ) l ( t − t a ) l ( u − a ) i . . . ( u p − a p ) i p · A l ,l i ...i p , (121) B = X l ,l ,j ,...,j p + d − ≥ ( x − x a ) l ( t − t a ) l ( u − a ) j . . . ( u p + d − − a p + d − ) j p + d − · B l ,l j ...j p + d − (122)such that A l ,l i ...i p , B l ,l j ...j p + d − ∈ L , (123) D x ( B ) − D t ( A ) + [ A , B ] = 0 . (124)If the power series (121), (122) satisfy (33), (34), (35) then this formal ZCR is said to be a -normal . Example 5.
Since (43), (44) obey (46), (48), (49), (50) and A l ,l i ...i p , B l ,l j ...j p + d − ∈ F p ( E , a ), the powerseries (43), (44) constitute an a -normal formal ZCR of order ≤ p with coefficients in F p ( E , a ). Example 6.
Consider a ZCR of order ≤ p with values in a finite-dimensional Lie algebra g given by g -valued functions A = A ( x, t, u , . . . , u p ), B = B ( x, t, u , . . . , u p + d − ) satisfying (4). If the functions A , B are analytic on a neighborhood of the point a ∈ E , then the Taylor series of these functions constitutea formal ZCR of order ≤ p with coefficients in g .For any vector space V , we denote by gl ( V ) the vector space of linear maps V → V . The space gl ( V )is an associative algebra with respect to the composition of such maps and is a Lie algebra with respectto the commutator. We denote by Id V ∈ gl ( V ) the identity map Id V : V → V .Let m, n ∈ Z ≥ . Consider power series P = X l ,l ,i ,...,i m ≥ ( x − x a ) l ( t − t a ) l ( u − a ) i . . . ( u m − a m ) i m · P l ,l i ...i m , (125) Q = X l ,l ,i ,...,i n ≥ ( x − x a ) l ( t − t a ) l ( u − a ) i . . . ( u n − a n ) i n · Q l ,l i ...i n with coefficients P l ,l i ...i m , Q l ,l i ...i n ∈ gl ( V ).The product P Q is defined in the standard way, using the associative multiplication of the coefficients.The power series D x ( P ), D t ( P ), [ P, Q ] are defined as described in Remark 12. Thus
P Q , D x ( P ), D t ( P ),[ P, Q ] are power series in the variables x − x a , t − t a , u k − a k with coefficients in gl ( V ).If the coefficient P , ... ∈ gl ( V ) in (125) is invertible (i.e., the linear map P , ... : V → V is invertible),then we can consider the power series P − such that P P − = P − P = Id V .For any Lie algebra L , there is a (possibly infinite-dimensional) vector space V such that L is iso-morphic to a Lie subalgebra of gl ( V ). For example, one can use the following well-known construction.Denote by U( L ) the universal enveloping algebra of L . Using the canonical embedding L ⊂ U( L ), we N LIE ALGEBRAS RESPONSIBLE FOR INTEGRABILITY 22 get the injective homomorphism of Lie algebras ξ : L ֒ → gl (U( L )) , ξ ( v )( w ) = vw, v ∈ L ⊂ U( L ) , w ∈ U( L ) , vw ∈ U( L ) . So one can set V = U( L ).As said above, Theorem 1 about analytic ZCRs is proved in [12]. Similarly, one can prove the followinganalog of Theorem 1 for formal ZCRs. Theorem 5.
Let p ∈ Z ≥ . Consider a vector space V and a Lie subalgebra L ⊂ gl ( V ) . Note that V and L can be infinite-dimensional. Consider a formal ZCR of order ≤ p with coefficients in L given bypower series A , B satisfying (121) , (122) , (123) , (124) .Then there is a unique power series of the form (126) G = Id V + X l ,l ,i ,...,i m ≥ l + l + i + ··· + i m > ( x − x a ) l ( t − t a ) l ( u − a ) i . . . ( u m − a m ) i m · G l ,l i ...i m , G l ,l i ...i m ∈ gl ( V ) , such that the power series (127) ˜ A = GAG − − D x ( G ) · G − , ˜ B = GBG − − D t ( G ) · G − satisfy ∂ ˜ A ∂u s (cid:12)(cid:12)(cid:12)(cid:12) u k = a k , k ≥ s = 0 ∀ s ≥ , (128) ˜ A (cid:12)(cid:12)(cid:12) u k = a k , k ≥ = 0 , (129) ˜ B (cid:12)(cid:12)(cid:12) x = x a , u k = a k , k ≥ = 0 . (130) Furthermore, one has the following. • The power series (126) depends only on the variables x − x a , t − t a , u k − a k for k = 0 , , . . . , p − .That is, one can write m = p − in (126) . ( In particular, if p = 0 then (126) depends only on x − x a , t − t a . ) • The power series (127) are of the form ˜ A = X l ,l ,i ,...,i p ≥ ( x − x a ) l ( t − t a ) l ( u − a ) i . . . ( u p − a p ) i p · ˜ A l ,l i ...i p , (131) ˜ B = X l ,l ,j ,...,j p + d − ≥ ( x − x a ) l ( t − t a ) l ( u − a ) j . . . ( u p + d − − a p + d − ) j p + d − · ˜ B l ,l j ...j p + d − (132) for some ˜ A l ,l i ...i p , ˜ B l ,l j ...j p + d − ∈ L and obey (133) D x (˜ B ) − D t (˜ A ) + [˜ A , ˜ B ] = 0 . That is, ˜ A , ˜ B constitute a formal ZCR of order ≤ p with coefficients in L . Equations (128) , (129) , (130) say that this ZCR is a -normal. • The power series (126) satisfies the following. (134)
The coefficients of the power series ∂∂x ( G ) · G − , ∂∂t ( G ) · G − , ∂∂u k ( G ) · G − , k ∈ Z ≥ , belong to L . Fix a vector space V and a Lie subalgebra L ⊂ gl ( V ). A formal power series of the form (126)satisfying (134) is called a formal gauge transformation . It is easily seen that formal gauge transforma-tions constitute a group with respect to the associative multiplication of power series with coefficientsin gl ( V ). Formulas (127) determine an action of the group of formal gauge transformations on the setof formal ZCRs with coefficients in L .The formal ZCR given by (127) is gauge equivalent to the formal ZCR given by A , B satisfying (121),(122), (123), (124). N LIE ALGEBRAS RESPONSIBLE FOR INTEGRABILITY 23
Remark 15.
Equations (128), (129), (130), (133) imply that the following homomorphism µ : F p ( E , a ) → L , µ (cid:0) A l ,l i ...i p (cid:1) = ˜ A l ,l i ...i p , µ (cid:0) B l ,l j ...j p + d − (cid:1) = ˜ B l ,l j ...j p + d − , is well defined, where ˜ A l ,l i ...i p , ˜ B l ,l j ...j p + d − ∈ L are the coefficients of the power series (131), (132). Theo-rem 5 implies that any formal ZCR of order ≤ p with coefficients in L is gauge equivalent to an a -normalformal ZCR corresponding to a homomorphism µ : F p ( E , a ) → L .We will use this in Remark 18, in order to get some information about the algebra F ( E , a ) forequation (142).6. Integrability conditions and examples of proving non-integrability
Necessary conditions for integrability.
As said in Section 1, in this paper, integrability ofPDEs is understood in the sense of soliton theory and the inverse scattering method, relying on the useof ZCRs.For each scalar evolution equation (1), in Section 3 we have defined the family of Lie algebras F p ( E , a ),where E is the infinite prolongation of (1), a is a point of the manifold E , and p ∈ Z ≥ . In this subsectionand in Subsection 6.2 we show that, using the algebras F p ( E , a ), one obtains some necessary conditionsfor integrability of equations (1). Examples of the use of these conditions in proving non-integrabilityfor some equations are given in Example 7 and in Subsection 6.2.In this subsection, g is a finite-dimensional matrix Lie algebra, and G is the connected matrix Lie groupcorresponding to g . (The precise definition of G is given in Definition 2.) A gauge transformation is amatrix-function G = G ( x, t, u , . . . , u l ) with values in G , where l ∈ Z ≥ . ZCRs and gauge transformationsare supposed to be defined on a neighborhood of a point a ∈ E .As said in Section 1, all algebras are supposed to be over the field K , where K is either C or R , andthe variables x , t , u k take values in K . Definition 7. A g -valued ZCR A = A ( x, t, u , u , . . . ) , B = B ( x, t, u , u , . . . ) , D x ( B ) − D t ( A ) + [ A, B ] = 0is called gauge-nilpotent if there is a gauge transformation G = G ( x, t, u , . . . , u l ) such that the functions˜ A = GAG − − D x ( G ) · G − , ˜ B = GBG − − D t ( G ) · G − take values in a nilpotent Lie subalgebra of g . In other words, a g -valued ZCR is gauge-nilpotent iff itis gauge equivalent to a ZCR with values in a nilpotent Lie subalgebra of g .It is known that a ZCR with values in a nilpotent Lie algebra cannot establish integrability of a givenequation (1). Therefore, a gauge-nilpotent ZCR cannot establish integrability of (1) either, because agauge-nilpotent ZCR is equivalent to a ZCR with values in a nilpotent Lie algebra.Hence the property(135) “there is g such that equation (1) possesses a g -valued ZCR which is not gauge-nilpotent”can be regarded as a necessary condition for integrability of equation (1).It is shown in [12] that, for any g -valued ZCR of order ≤ p , there is a homomorphism µ : F p ( E , a ) → g such that this ZCR is gauge equivalent to a ZCR with values in the Lie subalgebra µ (cid:0) F p ( E , a ) (cid:1) ⊂ g .The construction of µ is described in Remark 14.Therefore, if for each p ∈ Z ≥ and each a ∈ E the Lie algebra F p ( E , a ) is nilpotent then any ZCRof (1) is gauge-nilpotent, which implies that equation (1) is not integrable. This yields the followingresult. Theorem 6.
Let E be the infinite prolongation of an equation of the form (1) . If for each p ∈ Z ≥ andeach a ∈ E the Lie algebra F p ( E , a ) is nilpotent, then this equation is not integrable.In other words, the property (136) “there exist p ∈ Z ≥ and a ∈ E such that the Lie algebra F p ( E , a ) is not nilpotent” N LIE ALGEBRAS RESPONSIBLE FOR INTEGRABILITY 24 is a necessary condition for integrability of equation (1) . For some classes of equations (1) one can find a nonnegative integer r such that for any k > r thealgebra F k ( E , a ) is obtained from F k − ( E , a ) by central extension. This implies that for any k > r thealgebra F k ( E , a ) is obtained from F r ( E , a ) by applying several times the operation of central extension.Then condition (136) should be checked for p = r .For example, according to Theorem 3 and Remark 5, for equations of the form (9) we can take r = q − δ q, . According to Proposition 2, for the Krichever-Novikov equation (17) one can take r = 1.Let us show how this works for equations (9). Theorem 7.
Let E be the infinite prolongation of an equation of the form (9) with q ∈ { , , } . Let a ∈ E . If the Lie algebra F q − δ q, ( E , a ) is nilpotent, then F p ( E , a ) is nilpotent for all p ∈ Z ≥ .Proof. According to Theorem 3 and Remark 5, for every p ≥ q + δ q, the Lie algebra F p ( E , a ) is obtainedfrom F q − δ q, ( E , a ) by applying several times the operation of central extension.Since the homomorphisms (58) are surjective, for each ˜ p ≤ q − δ q, we have a surjective homomor-phism F q − δ q, ( E , a ) → F ˜ p ( E , a ).Clearly, these properties imply the statement of Theorem 7. (cid:3) Combining Theorem 7 with Theorem 6, we obtain the following.
Theorem 8.
Let E be the infinite prolongation of an equation of the form (9) with q ∈ { , , } .If for all a ∈ E the Lie algebra F q − δ q, ( E , a ) is nilpotent, then for each p ∈ Z ≥ any ZCR of order ≤ pA = A ( x, t, u , u , . . . , u p ) , B = B ( x, t, u , u , . . . ) , D x ( B ) − D t ( A ) + [ A, B ] = 0 is gauge-nilpotent. Hence, if F q − δ q, ( E , a ) is nilpotent for all a ∈ E , then equation (9) is not integrable.In other words, the property (137) “the Lie algebra F q − δ q, ( E , a ) is not nilpotent for some a ∈ E ”is a necessary condition for integrability of equations of the form (9) . Remark 16.
In this paper we study integrability by means of ZCRs. Another well-known approachto integrability uses symmetries and conservation laws. Many remarkable classification results for sometypes of equations (1) possessing higher-order symmetries or conservation laws are known (see, e.g., [24,25, 31] and references therein).However, it is also known that the approach of symmetries and conservation laws is not completelyuniversal for the study of integrability. For a given evolution equation, non-existence of higher-ordersymmetries and conservation laws does not guarantee non-integrability. For example, in [28] one can finda scalar evolution equation which is connected with KdV by a Miura-type transformation and is, there-fore, integrable, but does not possess higher-order symmetries and conservation laws. In Subsection 6.3we present this equation and a ZCR for it.Examples of the situation when two evolution equations are connected by a Miura-type transformationbut only one of the equations possesses higher-order symmetries can be found also in [35, 37]. (In [35, 37]Miura-type transformations are called differential substitutions.)When we speak about symmetries and conservation laws, we mean the standard notions of localsymmetries and conservation laws [24, 25, 31], which may be of arbitrarily high order with respectto the variables u k . One can try to consider also nonlocal symmetries depending on so-called nonlocalvariables (see, e.g., [18, 19, 37] and references therein), but the theory of nonlocal symmetries is much lessdeveloped than that of local symmetries. A classification result for equations of order 2 satisfying certainintegrability conditions related to existence of higher-order weakly nonlocal symmetries is presentedin [37]. Remark 17.
In this subsection we study ZCRs with values in finite-dimensional Lie algebras. Usingthe theory presented in Section 5, one can show that the results of this subsection are valid also forformal ZCRs with coefficients in arbitrary (possibly infinite-dimensional) Lie algebras.
N LIE ALGEBRAS RESPONSIBLE FOR INTEGRABILITY 25
Example 7.
Consider (9) in the case q = 2. Let E be the infinite prolongation of an equation of theform(138) u t = u + f ( x, t, u , u , u , u ) . According to Theorem 4, if ∂ f∂u ∂u ∂u = 0 then F ( E , a ) = 0 for all a ∈ E .Combining this with Theorem 8, we get the following. If ∂ f∂u ∂u ∂u = 0 then equation (138) is notintegrable. (As said above, this means that, for any p ∈ Z ≥ , any ZCR of order ≤ p for this equation isgauge-nilpotent.)6.2. An evolution equation related to the H´enon-Heiles system.
The following scalar evolutionequation was studied by A. P. Fordy [6] in connection with the H´enon-Heiles system(139) u t = u + (8 α − β ) u u + (4 α − β ) u u − αβu u , where α , β are arbitrary constants. (In [6] these constants are denoted by a , b , but we use the symbol a for a different purpose.)If α = β = 0 then (139) is the linear equation u t = u . Since we intend to study nonlinear PDEs, inwhat follows we suppose that at least one of the constants α , β is nonzero. We want to determine forwhich values of α , β equation (139) is not integrable.The following facts were noticed in [6]. • If α + β = 0 then (139) is equivalent to the Sawada-Kotera equation. (That is, if α + β = 0 then(139) can be transformed to the Sawada-Kotera equation by scaling of the variables. As saidabove, we assume that at least one of the constants α , β is nonzero.) • If 6 α + β = 0 then (139) is equivalent to the 5th-order flow in the KdV hierarchy. • If 16 α + β = 0 then (139) is equivalent to the Kaup-Kupershmidt equation.So in the cases α + β = 0, 6 α + β = 0, 16 α + β = 0 equation (139) is equivalent to a well-knownintegrable equation.Now we need to study the case(140) α + β = 0 , α + β = 0 , α + β = 0 . As discussed in Remark 16, there are several different approaches to the notion of integrability ofPDEs. According to [6] and references therein, in the case (140) equation (139) is not integrable inthe approach of symmetries and conservation laws. (This means that the equation does not possesshigher-order symmetries and conservation laws.) However, according to Remark 16, this fact does notguarantee non-integrability of (139) in some other approaches.Let us see what the structure of the algebras F p ( E , a ) can say about integrability or non-integrabilityof equation (139) in the case (140). Lemma 4 is proved in [13]. Lemma 4 ([13]) . Let E be the infinite prolongation of equation (139) . Let a ∈ E . Then • the Lie algebra F ( E , a ) is obtained from F ( E , a ) by central extension, • if (140) holds and α = 0 , the algebra F ( E , a ) is isomorphic to the direct sum of the -dimensionalLie algebra sl ( K ) and an abelian Lie algebra of dimension ≤ , • if α = 0 and β = 0 , the Lie algebra F ( E , a ) is nilpotent and is of dimension ≤ . Combining Lemma 4 with Theorem 3 and Remark 5, we get the following.
Theorem 9.
Let E be the infinite prolongation of equation (139) . Let a ∈ E . Then one has the following. • For any p ∈ Z ≥ , the kernel of the surjective homomorphism F p ( E , a ) → F ( E , a ) from (58) isnilpotent. The algebra F p ( E , a ) is obtained from the algebra F ( E , a ) by applying several timesthe operation of central extension. N LIE ALGEBRAS RESPONSIBLE FOR INTEGRABILITY 26 • If (140) holds and α = 0 , then F ( E , a ) is isomorphic to the direct sum of sl ( K ) and anabelian Lie algebra of dimension ≤ , and for each p ∈ Z ≥ there is a surjective homomor-phism F p ( E , a ) → sl ( K ) with nilpotent kernel. • If α = 0 and β = 0 , the Lie algebra F p ( E , a ) is nilpotent for all p ∈ Z ≥ . According to Theorem 9, if α = 0 and β = 0 then for any p ∈ Z ≥ the Lie algebra F p ( E , a ) for (139)is nilpotent. Then Theorem 6 implies that equation (139) is not integrable in the case when α = 0 and β = 0.Now it remains to study the case when (140) holds and α = 0. Before doing this, we need to discusssomething else. All our experience in the study of the algebras F p ( E , a ) for various evolution equationssuggests that the following conjecture is valid. Conjecture 1.
Let E be the infinite prolongation of a (1+1) -dimensional evolution equation (1) . Sup-pose that the equation is integrable. Then there exist p ∈ Z ≥ and a ∈ E such that • the Lie algebra F p ( E , a ) is infinite-dimensional, • for any nilpotent ideal I ⊂ F p ( E , a ) , the quotient Lie algebra F p ( E , a ) / I is infinite-dimensionalas well. This conjecture is supported by the following examples.
Example 8.
According to [12] for the KdV equation, the Lie algebra F ( E , a ) is isomorphic to the directsum sl ( K [ λ ]) ⊕ K , where K is a 3-dimensional abelian Lie algebra. Hence sl ( K [ λ ]) is embedded in F ( E , a ) as a subalgebra of codimension 3.Evidently, the infinite-dimensional Lie algebra sl ( K [ λ ]) does not have any nontrivial nilpotent ideals.Hence, for any nilpotent ideal I ⊂ F ( E , a ), one has I ∩ sl ( K [ λ ]) = 0 and, therefore, dim I ≤
3. Thisimplies that Conjecture 1 is valid for the KdV equation.
Example 9.
Recall that the KdV hierarchy consists of commuting flows, which are scalar evolutionequations of orders 2 k + 1 for k ∈ Z > . The standard sl ( K )-valued ZCR for the KdV hierarchy dependspolynomially on a parameter λ . Therefore, this ZCR can be viewed as a ZCR with values in sl ( K [ λ ]).It can be shown that this gives a surjective homomorphism F ( E , a ) → sl ( K [ λ ]) for each equation inthe hierarchy. Since the Lie algebra sl ( K [ λ ]) is infinite-dimensional and does not have any nontrivialnilpotent ideals, this implies that dim F ( E , a ) = ∞ and dim F ( E , a ) / I = ∞ for any nilpotent ideal I ⊂ F ( E , a ), so Conjecture 1 holds true for each equation in the KdV hierarchy. Using similar arguments,one can show that Conjecture 1 is valid also for many other hierarchies of integrable evolution equationspossessing a ZCR with a parameter. Example 10.
According to Proposition 2, for the Krichever-Novikov equation KN( e , e , e ) in the casewhen e i = e j for all i = j , the algebra F ( E , a ) is isomorphic to the infinite-dimensional Lie algebra R e ,e ,e . Using the basis (21) of this algebra, it is easy to show that R e ,e ,e does not have any nontrivialnilpotent ideals. Therefore, F ( E , a ) is infinite-dimensional and does not have any nontrivial nilpotentideals, which implies that Conjecture 1 is valid in this case.According to [38], if e , e , e ∈ C are such that e i = e j for some i = j , then the Krichever-Novikovequation KN( e , e , e ) is connected by a Miura-type transformation with the KdV equation. Using thisfact and the fact that Conjecture 1 is valid for the KdV equation, one can show that Conjecture 1 holdstrue for the equation KN( e , e , e ) when e i = e j for some i = j . Example 11.
In this paper we study scalar evolution PDEs (1). As said in Remark 7, it is possible tointroduce an analog of F p ( E , a ) for multicomponent evolution PDEs (19). Therefore, one can try to checkConjecture 1 for multicomponent evolution PDEs. Computations in [14] show that Conjecture 1 holdstrue for the Landau-Lifshitz, nonlinear Schr¨odinger equations (which can be regarded as 2-componentevolution PDEs) and for a number of other multicomponent PDEs.Now return to the study of equation (139) in the case when (140) holds and α = 0. According toTheorem 9, for any a ∈ E and any p ∈ Z ≥ there is a surjective homomorphism ψ : F p ( E , a ) → sl ( K ) N LIE ALGEBRAS RESPONSIBLE FOR INTEGRABILITY 27 with nilpotent kernel. Let I ⊂ F p ( E , a ) be the kernel of ψ . Then I is a nilpotent ideal of F p ( E , a ), andwe have dim F p ( E , a ) / I = dim sl ( K ) = 3 . Then Conjecture 1 implies that equation (139) is not integrable in this case.6.3.
A zero-curvature representation.
Consider the KdV equation(141) u t = u xxx + uu x , u = u ( x, t ) , and the equation(142) v ˜ t = v v ˜ x ˜ x ˜ x + 3 v v ˜ x v ˜ x ˜ x − ˜ x v ˜ x + 3˜ xv, v = v (˜ x, ˜ t ) , where subscripts denote derivatives. We assume that x , t , u , ˜ x , ˜ t , v take values in K .According to [28], equation (142) is connected with KdV (141) by the following Miura-type transfor-mation(143) ˜ t = t, ˜ x = u x , v = u xx . In [28] the variables ˜ x and ˜ t are denoted by y and s .Using the methods of [8, 9, 36], it is shown in [28] that equation (142) does not possess higher-ordersymmetries and conservation laws. (As explained in Remark 16, when we speak about symmetries andconservation laws, we mean the standard notions of local symmetries and conservation laws [24, 25, 31].)We are going to present a ZCR for equation (142).Consider the infinite-dimensional Lie algebra sl ( K [ λ ]) ∼ = sl ( K ) ⊗ K K [ λ ] and the map ∂ λ : sl ( K [ λ ]) → sl ( K [ λ ]) , ∂ λ (cid:0) q ⊗ f (cid:1) = q ⊗ ∂f∂λ , q ∈ sl ( K ) , f ∈ K [ λ ] . We set K ∂ λ = (cid:8) c∂ λ (cid:12)(cid:12) c ∈ K (cid:9) . That is, K ∂ λ is the one-dimensional vector subspace spanned by themap ∂ λ in the vector space of all K -linear maps sl ( K [ λ ]) → sl ( K [ λ ]).One has the following Lie algebra structure on the vector space sl ( K [ λ ]) ⊕ K ∂ λ (cid:2) m + c ∂ λ , m + c ∂ λ (cid:3) = [ m , m ] + c ∂ λ ( m ) − c ∂ λ ( m ) , m , m ∈ sl ( K [ λ ]) , c , c ∈ K . Consider the following functions with values in sl ( K [ λ ]) ⊕ K ∂ λ A (˜ x, v ) = (cid:18) − vλ v (cid:19) + ˜ xv ∂ λ , (144) B (˜ x, v, v ˜ x , v ˜ x ˜ x ) = (cid:18) ˜ x vv ˜ x ˜ x + v x + ˜ x v + λ ( v − λvv ˜ x ˜ x − λv x ) − λ ˜ x v − λ − ˜ x (cid:19) ++ (cid:16) vv ˜ x − ˜ x ( vv ˜ x ˜ x + v x ) − ˜ x v (cid:17) ∂ λ . (145)It is straightforward to check that these functions satisfy the zero-curvature condition(146) D ˜ x ( B ) − D ˜ t ( A ) + [ A, B ] = 0 , where D ˜ x , D ˜ t are the total derivative operators corresponding to equation (142). Therefore, the func-tions (144), (145) form a ZCR for equation (142). This ZCR takes values in the infinite-dimensional Liealgebra sl ( K [ λ ]) ⊕ K ∂ λ .This ZCR for equation (142) can be obtained from the standard ZCR of the KdV equation (141) bymeans of the Miura-type transformation (143) and a linear change of variables. Remark 18.
Let E be the infinite prolongation of equation (142). According to Definition 3, E can beidentified with the space K ∞ with the coordinates˜ x, ˜ t, v, v ˜ x , v ˜ x ˜ x , v ˜ x ˜ x ˜ x , v ˜ x ˜ x ˜ x ˜ x , . . . Then A (˜ x, v ) and B (˜ x, v, v ˜ x , v ˜ x ˜ x ) given by (144), (145) are rational functions on E with values in the Liealgebra sl ( K [ λ ]) ⊕ K ∂ λ . N LIE ALGEBRAS RESPONSIBLE FOR INTEGRABILITY 28
Set V = sl ( K [ λ ]) ⊕ K ∂ λ . Consider the Lie algebra gl ( V ) which consists of K -linear maps V → V .We have the following injective homomorphism of Lie algebras ψ : sl ( K [ λ ]) ⊕ K ∂ λ ֒ → gl ( V ) , ψ ( r )( s ) = [ r, s ] , r ∈ sl ( K [ λ ]) ⊕ K ∂ λ , s ∈ V = sl ( K [ λ ]) ⊕ K ∂ λ , which is the adjoint representation of sl ( K [ λ ]) ⊕ K ∂ λ . Hence sl ( K [ λ ]) ⊕ K ∂ λ can be regarded as a Liesubalgebra of gl ( V ).Take a point a ∈ E such that v = 0 at a . Then the functions A (˜ x, v ) and B (˜ x, v, v ˜ x , v ˜ x ˜ x ) are analyticon a neighborhood of a ∈ E .Taking the Taylor series of these functions, we get power series with coefficients in sl ( K [ λ ]) ⊕ K ∂ λ .Equation (146) implies that the Taylor series of A (˜ x, v ), B (˜ x, v, v ˜ x , v ˜ x ˜ x ) constitute a formal ZCR oforder ≤ sl ( K [ λ ]) ⊕ K ∂ λ .Using Theorem 5 and Remark 15 for the Lie algebra L = sl ( K [ λ ]) ⊕ K ∂ λ ⊂ gl ( V ), we obtain thatthis formal ZCR is gauge equivalent to an a -normal formal ZCR corresponding to a homomorphism µ : F ( E , a ) → sl ( K [ λ ]) ⊕ K ∂ λ . (The homomorphism µ is uniquely determined by the ZCR A (˜ x, v ), B (˜ x, v, v ˜ x , v ˜ x ˜ x ).)Using methods of [12, 13], one can show that(147) µ (cid:0) F ( E , a ) (cid:1) contains the subalgebra sl ( K [ λ ]) ⊂ sl ( K [ λ ]) ⊕ K ∂ λ .Since the infinite-dimensional Lie algebra sl ( K [ λ ]) is of codimension 1 in sl ( K [ λ ]) ⊕ K ∂ λ and doesnot have any nontrivial nilpotent ideals, property (147) yields the following. For any nilpotent ideal I ⊂ F ( E , a ), the quotient Lie algebra F ( E , a ) / I is infinite-dimensional. Therefore, Conjecture 1 is validfor equation (142). Acknowledgements
Gianni Manno (GM) and Sergei Igonin (SI) acknowledge that the present research has been partiallysupported by the following projects and grants: • PRIN project 2017 “Real and Complex Manifolds: Topology, Geometry and holomorphic dy-namics” by the Ministry of Education, University and Research (MIUR), Italy; • “Connessioni proiettive, equazioni di Monge-Amp`ere e sistemi integrabili” by Istituto Nazionaledi Alta Matematica (INdAM), Italy; • “MIUR grant Dipartimenti di Eccellenza 2018-2022 (E11G18000350001)”, Italy; • “Finanziamento alla ricerca 2017-2018 (53 RBA17MANGIO)” by Politecnico di Torino; • “FIR-2013 Geometria delle equazioni differenziali” by INdAM.GM is a member of Gruppo Nazionale per le Strutture Algebriche, Geometriche e le loro Applicazioni(GNSAGA) of INdAM. The work of SI was carried out within the framework of the State Programmeof the Ministry of Education and Science of the Russian Federation, project number 1.13560.2019/13.1.SI would like to thank A. P. Fordy, I. S. Krasilshchik, A. V. Mikhailov, and V. V. Sokolov foruseful discussions. SI is grateful to the Max Planck Institute for Mathematics (Bonn, Germany) forits hospitality and excellent working conditions during 06.2010–09.2010, when part of this research wasdone. References [1] P. J. Caudrey, R. K. Dodd, and J. D. Gibbon. A new hierarchy of Korteweg-de Vries equations.
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