DIFFERENTIAL GEOMETRY AND APPLICATIONS Proc. Conf., Aug. 28 – Sept. 1, 1995, Brno, Czech Republic Masaryk University, Brno 1996, 393–402

ANOTHER LOOK ON RECURSION OPERATORS

MICHAL MARVAN

Abstract. Recursion operators of partial differential equations are identified with Bäcklund auto-transformations of linearized diffieties. Relations to the classical concept and its recent Guthrie’s generalization are discussed.

Traditionally, a recursion operator of a PDE is a linear operator L acting on symmetries: if f is a symmetry then so is Lf. This provides a convenient way to generate infinite families of symmetries (Olver [13]). Standard reference is [14].

The presence of a recursion operator has been soon recognized as one of the attributes of the complete integrability. An exact and rich theory has been developed within the class of evolution equations (see Fokas [2] and references therein).

Recursion operators are, as a rule, nonlocal and so may turn out to be the symmetries they generate (giving a powerful source of nonlocal symmetries [10, 5]). A remarkable problem of inverting a recursion operator motivated Guthrie [3] to a generalization, with consequences for generation of nonlocal symmetries [4]. His nonlocalities are no longer limited to mere inverses D^{−1} of total derivatives.

It seems to be an interesting problem to give an interpretation of recursion operators in terms of the Vinogradov [17] category of diffieties. Recently, Krasilshchik and Kersten [6, 8] established a fundamental relation between recursion operators and deformations of diffiety structures.

In this paper (Section 3) we propose another answer, according to which recursion operators are Bäcklund auto-transformations of so called linearized diffiety (Section 2). Classical recursion operators are discussed in Section 4. The Guthrie operators turn out to be equivalent to what we call finitary linear operators, in the case of two independent variables (Section 5). To cover known examples in multidimension (e.g., the KP equation in [2]) finitary constructions are manifestly insufficient, but a theory of infinitary coverings still awaits a necessary development. In Section 6 we discuss the inversion of a recursion operator, within what we call a local setting. Finally, in Section 7 we conclude that a direct computational procedure for finding recursion operators exists, based on the mere definition.

1991 Mathematics Subject Classification. 58G37; 35A30, 35Q53.

Key words and phrases. Recursion operator, Bäcklund transformation, diffiety.

This paper is final form and no version of it will be submitted for publication elsewhere.

1. Preliminaries

We work in the category of diffieties introduced by A.M. Vinogradov [17, 9]. The objects, diffieties, are infinite-dimensional smooth manifolds equipped with an involutive distribution D of dimension m < ∞. For simplicity we assume here that our diffieties are fibred manifolds E → M over a fixed base manifold M of the same dimension m, and that the projection of the distribution on M is of maximal rank, so that D is a horizontal distribution of a flat connection in the fibration E → M (see [7]). The lift of a vector field X from M to E will be denoted by D_{X}.

A map h : E → E′ is a morphism of diffieties if h commutes with the projections on M, and D_{X}(f ∘ h) = (D′_{X}f) ∘ h for any f ∈ C^{∞}E′ and any vector field X on M. So arising category is denoted by DE_{M} here. For a different description see [11].

The passage from a system of n partial differential equations

F^{l} = 0, l = 1, …, n (1)

to a diffiety is as follows. Consider a fibred manifold Y → M. Its fibred jet manifold j^{∞}Y → M becomes a diffiety when endowed with a connection D such that D_{X} on j^{∞}Y satisfies D_{X}f ∘ j^{∞}γ = X(f ∘ j^{∞}γ) for every section γ : M → Y. Within any jet coordinate system x^{i} on M, and u_{I}^{k} in the fibres of j^{∞}Y → M, where I runs over symmetric multiindices over the i’s, we have vector fields D_{i} := D_{∂/∂x^{i}} = ∂/∂x^{i} + ∑_{I} u_{Ii}^{k} ∂/∂u_{I}^{k}. Operators D_{i}, called total derivatives, commute. Suppose now that functions F^{l} from (1) are defined on j^{∞}Y. If conditions F^{l} = 0 together with all their “differential consequences” D_{i_{1}} … D_{i_{k}}F^{l} = 0 determine a submanifold in j^{∞}Y then it admits a restriction of the total derivatives D_{i} (which we denote by the same symbol D_{i}) and, consequently, of the whole Cartan connection D. This is the diffiety naturally representing the system (1).

A covering (Krasilshchik and Vinogradov [10]) of a diffiety E is defined as another diffiety Ẽ together with a projection (surjective submersion and DE_{M}- morphism) p : Ẽ → E. This concept is crucial for nonlocal aspects of the PDE’s.

Any covering p : Ẽ → E is locally determined by a set of p-vertical vector fields X_{(i)} on Ẽ, i = 1, …, dim M, such that vector fields D̃_{i} = D_{i} + X_{(i)} commute, i.e.,

D_{i}X_{(j)} − D_{j}X_{(i)} + [X_{(i)}, X_{(j)}] = 0. (2)

If the covering is finite-dimensional (fibres of p have a finite dimension) and we denote by w^{k} its coordinates along the fibres then we can write X_{(i)} = X_{(i)}^{k} ∂/∂w^{k}, with X_{(i)}^{k} ∈ C^{∞}Ẽ. The diffiety Ẽ then corresponds to a system of equations

w_{x^{i}}^{k} = X_{(i)}^{k} (3)

(additionally to (1)) of which (2) is the integrability condition.

A finitary covering is said to be linear if equations (3) are linear. Linear isomorphism classes of linear one-dimensional coverings correspond to one-dimensional horizontal cohomology classes, cf. [5]. Let us recall the essentials from [18].

Let Λ^{∗}E denote the noncommutative C^{∞}E-algebra of antisymmetric forms on E, with respect to the wedge product ∧, let CΛ^{∗}E denote the ideal of contact forms, i.e., forms vanishing on the Cartan distribution. Then Λ̄^{∗}E = Λ^{∗}E/CΛ^{∗}E is a C^{∞}E-algebra, naturally identified with the algebra of horizontal forms (forms vanishing whenever at least one of the arguments is a vertical vector field). With respect to the differential d : Λ^{k}E → Λ^{k+1}E we have d(CΛ^{k}E) ⊆ CΛ^{k+1}E, so that a well-defined differential d̄: Λ̄^{k}E → Λ̄^{k+1}E is induced, giving rise to a cohomology theory. This cohomology, called horizontal, is computable by means of the Cspectral sequence [18], based on the filtration of Λ^{k}E by powers of the ideal CΛ^{k}E.

By an abelian covering [5] we mean a covering which is a composition (tower) of linear one-dimensional coverings. As a consequence of [18, Two-line theorem] no nontrivial abelian coverings exist for a non-overdetermined system if dim M > 2 (more than two independent variables). When, on the other side, dim M = 2, the one-dimensional horizontal cohomology group coincides with the group of conservation laws, and nontrivial abelian coverings may occur.

Bäcklund transformations (see [15]) admit a very transparent description in the above terms: A Bäcklund transformation from a diffiety E_{1} to a diffiety E_{2} is simply a pair of coverings Ẽ → E_{1}, Ẽ → E_{2} with a common total space Ẽ (see [10]).

2. Linearization

Denote by V E the vector bundle of vertical tangent vectors to a diffiety E over M and by τ_{E} the projection V E → E. Then V E is a diffiety in a natural way, τ_{E} is a covering, and V is a functor DE_{M} → DE_{M} (see [11] and also [10, 7, 16]). We give a self-contained exposition below.

We start with two canonical ways of lifting functions from E to V E. The first is f ↦ f ∘ τ_{E}, the second is f ↦ ḟ, where ḟ ∈ C^{∞}V E satisfies

ḟ ∘ ξ = ξf (4)

(composition of maps on the left side, action of fields on functions on the right side) for any vertical vector field ξ : E → V E. We call ḟ a linearization of the function f.

If q^{κ} are local coordinates along the fibres of E → M then functions q̇^{κ} may serve as coordinates along the fibres of V E → E. Such coordinate systems on V E will be called standard. For any function f ∈ C^{∞}E we then have

ḟ = ∑_{κ} q̇^{κ} \frac{∂f}{∂q^{κ}} . (5)

Similarly, there are two canonical ways of lifting k-forms from E to V E. The first is the plain pull-back along τ_{E}, ω ↦ τ_{E}^{∗}ω, and the second is as follows.

2.1. Proposition. There exists a unique map Λ^{k}E → Λ^{k}V E, ω ↦ ω̇ such that 1^{∘}. ξ^{∗}ω̇ = L_{ξ}ω (the Lie derivative) for any vertical vector field ξ : E → V E. 2^{∘}. (ω + σ). = ω̇ + σ̇. 3^{∘}. (ω ∧ σ). = ω̇ ∧ τ_{E}^{∗}σ + τ_{E}^{∗}ω ∧ σ̇. 4^{∘}. (dω). = dω̇.

Proof. For k = 0 we have the above linearization of functions. Conditions 2^{∘}–4^{∘} then determine ω̇ uniquely for any k (force a coordinate formula). Finally, 1^{∘} holds for a sum, a wedge product and a differential if it holds for the arguments.

The form ω̇ ∈ Λ^{k}V E is said to be a linearization of the k-form ω.

For a vector field Ξ on E we define a V -lift Ξ^{V} as a vector field on V E satisfying

Ξ^{V} (q ∘ τ_{E}) = Ξq ∘ τ_{E}, Ξ^{V} (q̇) = (Ξq). (6)

for any function q ∈ C^{∞}E. Obviously, these formulas determine the field Ξ^{V} uniquely within one coordinate chart, hence its global existence follows.

Now, let E be a diffiety. For any vector field X on M the field D_{X}^{V}, the V -lift of the field D_{X}, depends on the fields X C^{∞}M-linearly, i.e., induces a connection. It is easily verified that the connection is flat, giving rise to a diffiety (V E, D^{V}). We call it a linearization of the diffiety E.

Let us show that this diffiety coincides with that introduced in [11]. Indeed, the Cartan connection on E can be equivalently described as a section e_{D} of the projection j^{1}E → E. This section induces a section e_{D}^{V} : V E ⟶^{V e_{D}} V j^{1}E ≅ j^{1}V E of the projection j^{1}V E → V E, i.e., a connection on V E. In standard coordinates e_{D} is described by q_{i}^{k} = D_{i}q^{k}, so that e_{D}^{V} is given by q_{i}^{k} = D_{i}q^{k}, (q̇^{k})_{i} = (q_{i}^{k}). = (D_{i}q^{k}).. The verification is finished by comparison with formulas (6).

Obviously, if the diffiety E is determined by equations (1) then the linearization V E is determined by equations (1) together with their linearizations (5):

∑_{k,I} u̇_{I}^{k} \frac{∂F^{l}}{∂u_{I}^{k}} = 0. (7)

In accordance with [9, Section 7.4] we define a symmetry of a diffiety E ∈ DE_{M} as a vertical vector field ξ satisfying [ξ, D_{X}] = 0 for any vector field X on M.

2.2. Proposition. A section ξ : E → V E of the projection τ_{E} is a symmetry of E if and only if it is a DE_{M}-morphism E → V E.

Proof. A map ξ : E → V E is a DE_{M}-morphism if and only if it preserves the Cartan connection, i.e., if D_{X}(f ∘ ξ) = (D_{X}^{V}f) ∘ ξ for any X and any function f on V E. It is sufficient to check this condition for functions of the form q ∘ τ_{E} and q̇, for q ∈ C^{∞}E. Obviously, D_{X}(q ∘τ_{E} ∘ξ) = D_{X}q = D_{X}q ∘τ_{E} ∘ξ = D_{X}^{V}(q ∘τ_{E})∘ξ always holds, while D_{X}(q̇ ∘ ξ) − (D_{X}^{V} q̇) ∘ ξ = D_{X}(ξq) − (D_{X}q). ∘ ξ = D_{X}(ξq) − ξ(D_{X}q) = [D_{X}, ξ]q is zero for any q ∈ C^{∞}E if and only if [D_{X}, ξ] = 0, which finishes the proof.

2.3. Proposition. For any form ω ∈ Λ^{k}E and any vector fields Ξ_{1}, …, Ξ_{k} on E it holds ω̇(Ξ_{1}^{V} , …, Ξ_{k}^{V}) = ω(Ξ_{1}, …, Ξ_{k})..

Proof. By induction on k. For k = 1 and a generator ω = f dg, where f, g ∈ C^{∞}E, we have ω̇(Ξ^{V}) = ḟ dg(Ξ^{V})+f dġ(Ξ^{V}) = ḟΞ^{V} g+fΞ^{V} ġ = ḟΞg+f(Ξg). = (fΞg). = ω(Ξ).. The induction step follows by formulas 2^{∘}, 3^{∘} of Proposition 2.1.

2.4. Corollary. For any contact form ω ∈ CΛ^{k}E the form ω̇ is contact again: ω̇ ∈ CΛ^{k}V E. Consequently, the map ω ↦ ω̇ maps horizontal forms on E to horizontal forms on V E and commutes with the horizontal differential.

This corollary opens the way for lifting the whole C-spectral sequence from E to V E, different from simple pulling-back by τ_{E}. However, we shall not need this here.

3. Recursion operators

3.1. Definition. A recursion operator for a diffiety E is a pair of coverings K, L : R → V E over the linearization V E of E such that the diagram

[figure: \begin{aligned}& R \\ V E & & V E \\ & E, \end{aligned}; K; L; τ_{E}; τ_{E}] (8)

is commutative. A recursion operator is said to be linear whenever 1^{∘} The covering R is linear over E. 2^{∘} The morphisms K, L are linear over E.

A recursion operator is said to be finitary whenever 3^{∘} Both coverings K, L are finite-dimensional over V E.

Thus, a recursion operator is a special case of a Bäcklund auto-transformation of the linearized diffiety. Note that coverings K, L may have different dimensions.

3.2. Example (KdV). The KdV equation, which we write as u_{t} = u_{xxx} + uu_{x}, is known to admit a recursion operator (Lenard operator) L : v ↦ D_{xx}v + \frac{2}{3}uv + \frac{1}{3}u_{x}D_{x}^{−1}v. The corresponding commutative diagram (8) is easily seen to be

[figure: \begin{matrix}w_{x} = v \\ w_{t} = v_{xx} + uv \\ v_{t} = v_{xxx} + uv_{x} + u_{x}v \\ u_{t} = u_{xxx} + uu_{x} \end{matrix} ; \begin{matrix}v_{t} = v_{xxx} + uv_{x} + u_{x}v \\ u_{t} = u_{xxx} + uu_{x} \end{matrix}  \begin{matrix}v_{t} = v_{xxx} + uv_{x} + u_{x}v \\ u_{t} = u_{xxx} + uu_{x} \end{matrix} ; u_{t} = u_{xxx} + uu_{x}.; K : v=w_{x}; L: v=v_{xx}+ \frac{2}{3} uv+ \frac{1}{3} u_{x}w]

Here the linearized equation v_{t} = v_{xxx}+uv_{x}+u_{x}v comes out by formula (7) with u̇ = v. The variables v, v_{x}, v_{xx}, etc. are coordinates in the fibres of the projection τ_{KdV} : V (KdV) → KdV. The covering K is one-dimensional. Its unique nonlocal coordinate w satisfies w_{x} = v, w_{t} = v_{xx} + uv, i.e., corresponds to the traditional D_{x}^{−1}v. It can be shown that the covering L is of dimension three.

3.3. Action on symmetries. If f is a symmetry of an equation E possessing a recursion operator L then Lf may turn out to be a symmetry, or merely a shadow of a symmetry (see Krasilshchik and Vinogradov [10]). Let us look at this point here.

Let s be a symmetry of the diffiety E, i.e., a morphism s : E → V E which is a section of the projection τ_{E} : V E → E. Consider a diagram

[figure: \begin{matrix}Ẽ & R & V E \\ E & V E & E, \end{matrix}; P′; P; L; K; τ_{E}; s; id; τ_{E}]

where the left-hand square is a pullback, i.e., P : Ẽ → E is a covering over E induced by s. The whole diagram is commutative and hence the morphism s̃ := L ∘ P′ : Ẽ → V E satisfies τ_{E} ∘ s̃ = P, i.e., is a morphism of coverings over E.

When the covering P is trivial (Ẽ ≅ E × ℝ^{n}) then s̃ is simply an n-parametric family of symmetries of E, otherwise s̃ is the shadow of a nonlocal symmetry [10] (shadows may be defined as morphisms Ẽ → V E over E). The process of reconstruction of the true nonlocal symmetry out of its shadow (detailed in loc. cit.) is left out in this paper.

3.4. Example (KdV). We start with the KdV symmetry s : v = 1 + tu_{x}. From the defining equations of R we easily see that w, now as the nonlocal variable of the induced covering Ẽ, satisfies w_{x} = 1+tu_{x}, w_{t} = u+tu_{t}. These equations are readily integrated as w = tu + x + const (w represents a trivial conservation law) so that the induced covering is trivial via a map E × ℝ → Ẽ, c ↦ (w = tu + x + c). Hence, v′ = v_{xx} + \frac{2}{3}uv + \frac{1}{3}u_{x}w = tu_{t} + \frac{1}{3}xu_{x} + \frac{2}{3}u + \frac{1}{3}cu_{x} is a one-parameter family of symmetries (cf. a similar discussion of mKdV in [3]).

Continuing with s = tu_{t} + \frac{1}{3}xu_{x} + \frac{2}{3}u, we see that the defining equations of Ẽ, w_{x} = tu_{xxx} + tuu_{x} + \frac{1}{3}xu_{x} + \frac{2}{3}u, w_{t} = tu_{txx} + tuu_{t} + \frac{1}{3}xu_{t} + \frac{4}{3}u_{xx} + \frac{2}{3}u^{2} represent a nontrivial conservation law (reducible to z_{x} = u, z_{t} = u_{xx} + \frac{1}{2}u^{2} by w = tu_{xx} + \frac{1}{3}xu + \frac{1}{2}tu^{2} + \frac{1}{3}z). Thus, the covering Ẽ is nontrivial and we obtain a shadow of a true nonlocal symmetry in this case.

4. Classical recursion operators

4.1. Definition. A linear finitary recursion operator (8) is said to be classical if

4^{∘} The covering K : R → V E is abelian (see Section 2 for the definition of abelian coverings).

For finding classical recursion operators it is important to know the horizontal cohomology of linearized diffieties V E.

4.2. Definition. A diffiety E is said to be V -simple if there exist no nontrivial horizontal one-dimensional cohomology classes of the linearized diffiety V E other than [τ_{E}^{∗}σ] and [σ̇] for horizontal forms σ ∈ Λ̄^{1}E (cf. Corollary 2.4).

Concrete but unpublished computations support a hypothesis that equations of mathematical physics are, as a rule, V -simple. Below we sketch two examples to support another hypothesis, that the abelian coverings for classical recursion operators are due to cohomology classes of the form [σ̇], and not [τ_{E}^{∗}σ].

4.3. Example (KdV). In Example 3.2 the underlying exact 1-form of the covering R over V E was just v dx+(v_{xx} +uv) dt, a linearization of the KdV conservation law u dx + (u_{xx} + \frac{1}{2}u^{2}) dt.

4.4. Example (sin-Gordon equation). We write the sin-Gordon (Bonnet) equation B as u_{xy} = sin u. Olver [13] found its recursion operator to be v ↦ v′ = D_{xx}v + u_{x}^{2}v − u_{x}D_{x}^{−1}(u_{xx}v). The linearization V B is given by a pair u_{xy} = sin u, v_{xy} = v cos u, while the covering R over V B is determined by equations w_{x} = u_{xx}v, w_{y} = u_{x}v_{y}. This covering is abelian, due to an exact horizontal 1-form σ = u_{xx}v dx + u_{x}v_{y} dy, which is not a linearization of any exact 1-form on B.

However, σ may be adjusted by adding an appropriate horizontal differential d̄f, f ∈ C^{∞}V B, since [σ + d̄f] = [σ]. Choosing f = −u_{x}v, we get σ + d̄f = −u_{x}v_{x} dx + v sin u dy, a linearization of the conservation law −\frac{1}{2}u_{x}^{2} dx + cos u dy. The recursion operator then becomes v′ = v_{xx} − u_{x}w̃ for w̃_{x} = −u_{x}v_{x}, w̃_{t} = v sin u.

5. Guthrie’s recursion operators

We assume in this section a linear finitary recursion operator K, L : R → V E to be given. We denote by v^{ι}, ι ∈ I, the nonlocal variables of the covering V E → E, and by w^{k}, k = 1, …, dim_{K} R, the nonlocal variables of the covering K.

As it has been explained in Section 2, R has to be given by a set of equations (3). These equations are linear over E if and only if X_{(i)}^{k} = A_{(i)}_{l}^{k}w^{l} + B_{(i)}^{k}, where each B_{(i)}^{k} is linear in v^{ι}, i.e., B_{(i)}^{k} = B_{(i)}_{ι}^{k}v^{ι}. Symbols A_{(i)}_{l}^{k}, B_{(i)}_{ι}^{k} denote functions on E. The integrability conditions (2) are (in matrix notation)

D_{j}A_{(i)} − D_{i}A_{(j)} + A_{(i)}A_{(j)} − A_{(j)}A_{(i)} = 0, (9) D_{j}B_{(i)} − D_{i}B_{(j)} + A_{(i)}B_{(j)} − A_{(j)}B_{(i)} = 0. (10)

The linear operator L is then of the form (F_{l}^{κ}, G_{ι}^{κ} are functions on E)

v′^{κ} = F_{l}^{κ}w^{l} + G_{ι}^{κ}v^{ι}.

It is easy to see that these are exactly the formulas of Guthrie [3, (3.2–3)].

5.1. Remarks. (1) Guthrie assumes additionally some covering Ẽ over E to be given [3, (3.1)]. The difference is that A_{(i)}_{l}^{k}, B_{(i)}_{ι}^{k}, F_{l}^{κ}, and G_{ι}^{κ} are then functions on Ẽ. However, this clearly amounts to what one might call a nonlocal setting (having a recursion operator on Ẽ instead of E). In Section 6 we demonstrate that for the purpose of mere inversion the local approach is sufficient.

(2) Obviously, (9) determines a zero-curvature representation of the equation E (is a Maurer–Cartan condition). Moreover, upon introducing extended matrices

A_{(i)} := ( \begin{matrix}A_{(i)} & B_{(i)} \\ 0 & 0 \end{matrix} ) (11)

(one column (B_{(i)}^{k}) from the right and one row of zeroes from below) conditions (9, 10) reduce to a single Maurer–Cartan condition D_{j}A_{(i)} − D_{i}A_{(j)} + A_{(i)}A_{(j)} − A_{(j)}A_{(i)} = 0, and represent a zero-curvature representation for V E.

(3) While the linearity property of recursion operators seems to be universal (symmetries form a linear space), it is clear from the following proposition that general recursion operators in multidimension cannot be finitary.

5.2. Proposition. For any finitary linear recursion operator (8) the covering K : R → E is trivial whenever dim M > 2 and the equation E is not overdetermined.

Proof. From [12, Corollary 4.4] it follows that the matrices A_{(i)} can be reduced to zero matrices by an appropriate change of coordinates in the fibres of K. Then (10) simplifies to a condition that columns B_{(i)} consist of components of exact horizontal 1-forms, which are all trivial by [18, Two-line theorem], and hence they determine a trivial covering.

6. Inverting a recursion operator

An inversion of a general Bäcklund transformation amounts to mere interchanging of the two coverings, and the same applies to recursion operators. Hence, for a recursion operator given by diagram (4) the inverted operator is simply

[figure: \begin{aligned}& R \\ V E & & V E \\ & E, \end{aligned}; K′=L; L′=K; τ_{E}; τ_{E}]

Obviously, the inversion of a finitary linear recursion operator is a finitary linear recursion operator. Moreover, as we shall see, the process of inversion is basically algorithmic.

6.1. Example (mKdV). Let us recompute the example of mKdV [3] in the local setting. We write the mKdV equation E as u_{t} = u_{xxx}−u^{2}u_{x}. Then the linearization V E is a pair u_{t} = u_{xxx} − u^{2}u_{x}, v_{t} = v_{xxx} − u^{2}v_{x} − 2uu_{x}v. The mKdV recursion operator L : z = v_{xx}− \frac{2}{3}u^{2}v− \frac{2}{3}u_{x}w (where z stands for u̇ in the second copy of V E) comes out of a covering R, given by w_{x} = uv, w_{t} = uv_{xx} − u_{x}v_{x} + u_{xx}v − u^{3}v (due to a linearization of an mKdV conservation law \frac{1}{2}u^{2} dx + (uu_{xx} − \frac{1}{2}u_{x}^{2} − \frac{1}{4}u^{4}) dt).

To invert this operator all we need is to reconstruct the covering over the second copy of V E out of the above data. The formula for L gives a relation v_{xx} = z + \frac{2}{3}u^{2}v + \frac{2}{3}u_{x}w, which leads us to the choice of the nonlocal variables over V E as w, v and s = v_{x}, subject to the relations

w_{x} = uv,
v_{x} = s,
s_{x} = \frac{2}{3}u_{x}w + \frac{2}{3}u^{2}v + z,

and

w_{t} = \frac{2}{3}uu_{x}w + (u_{xx} − \frac{1}{3}u^{3})v − u_{x}s + uz,
v_{t} = \frac{2}{3}u_{xx}w − \frac{1}{3}u^{2}s + z_{x},
s_{t} = (\frac{2}{3}u_{xxx} − \frac{2}{9}u^{2}u_{x})w + (\frac{2}{3}uu_{xx} − \frac{2}{9}u^{4})v − \frac{2}{3}uu_{x}s + z_{xx} − \frac{1}{3}u^{2}z.

It is easy to verify the integrability conditions (9) and (10) and that v satisfies the linearized equation whenever z does. Thus, we have arrived at a covering L′ : R′ → V E, (w, v, s) ↦ v. Its fibre dimension is one (recall that s = v_{x} is a local variable with respect to L′), as is that of the covering K : R → V E. Hence we naturally identify R′ = R, L′ = K, K′ = L with the inverted recursion operator.

The above covering R′ can be simplified (rewriting the defining system via extended matrices of Remark 5.2(2) and applying gauge transformations is very helpful). We present only the result: Upon a coordinate transformation p^{+} = w+\sqrt{3/2}, p^{−} = w − \sqrt{3/2}, q = −\frac{2}{3}uw + s we get

p_{x}^{±} = ±\sqrt{\frac{2}{3}}up^{±} ± \sqrt{\frac{3}{2}}q,
q_{x} = z,
p_{t}^{±} = ±\sqrt{\frac{2}{3}}(u_{xx} − \frac{1}{3}u^{3})p^{±} − (u_{x} ± \frac{1}{\sqrt{6}}u^{2})q ± \sqrt{\frac{3}{2}}z_{x} + uz,
q_{t} = z_{xx} − u^{2}z,

while the operator K acquires the form v = p^{+} − p^{−}.

The covering R′ → V E is non-abelian, but still decomposes as a tower R′ ≅ P^{+} ⊕ P^{−} → Q → V E, where Q → V E is due to the linearization of an mKdV conservation law u dx + (u_{xx} − \frac{1}{3}u^{3}) dt, and P^{+} ⊕ P^{−} → Q is the Whitney sum of two coverings corresponding to the (decoupled) equations on p^{+}, p^{−}, respectively. The coverings P^{+}, P^{−} are non-isomorphic.

7. Finding recursion operators

Procedures to find recursion operators have been proposed by many authors, see, e.g., [1, 8, 14]. See also, e.g., [2] for deriving recursion operators from Lax pairs. Finding finitary linear recursion operators on the basis of the mere definition is, however, also possible. Steps to be followed are:

1. Compute a (rich enough) covering R over V E (abelian for classical operators).

2. Find all linear morphisms R → V E over E. For methods to find abelian coverings over diffieties see [18, 5]. Hints of Section 4 may be helpful. Finding coverings of Section 5 is possible by the procedure of [12].

Step 2 consists of solving a system of equations in total derivatives, which is, in fact, a part of the defining system for symmetries of R. Therefore, virtually any software to compute generating functions of symmetries is capable of finding recursion operators. Selecting surjective morphisms out of those found in Step 2 (and excluding the identity morphism) is then easy. (If we drop the linearity requirement then also all symmetries s of E, as morphisms R → E →^{s} V E, will be found.)

Concrete computations confirm feasibility, and will be published elsewhere.

Acknowledgements

This research has been supported by grant No. 201/93/2245 from the Czech Grant Agency. Symbolic computations related to Section 7 are in progress in the Supercomputing Centre of Masaryk University in Brno. Special thanks are due to I.S. Krasilshchik for his continuous attention to this work, and reading the manuscript.

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Dept. Math., Silesian University at Opava Bezručovo nám. 13, 746 01 Opava, CZECH REPUBLIC

E-mail: Michal.Marvan@fpf.slu.cz