Miura-reciprocal transformations and localizable Poisson pencils

We show that the equivalence classes of deformations of localizable semisimple Poisson pencils of hydrodynamic type with respect to the action of the Miura-reciprocal group contain a local representative and are in one-to-one correspondence with the equivalence classes of deformations of local semisimple Poisson pencils of hydrodynamic type with respect to the action of the Miura group.

In 2001, Dubrovin and Zhang initiated a classification programme of bi-Hamiltonian integrable PDEs in two independent variables [DZ01].The group action that they considered was that of Miura transformations, i.e., transformations depending on the field variables and, polynomially, by their derivatives of higher order through a perturbative series.
Among the questions that the above approach raises there is the issue of extending the group action to include (possibly non-local) changes of variables in one of the independent variables.Indeed, an important class of such transformations is that of reciprocal transformations, which play an important role in Mathematical Physics (see e.g.[Rog69; Rog68; Fer89; Fer91; FP03; XZ06; AG07; Abe09; BS09; LZ11; AL13]).
This paper is concerned with the action of the group of Miura-reciprocal transformations, that is a natural group of simultaneous transformations of the independent and the dependent variables of a (bi-)Hamiltonian system through a perturbative series of derivatives of the field variables.Among other things, we consider (1) the Miura-reciprocal transformations of the 1st kind and rederive from the scratch the Ferapontov-Pavlov formula for the transformation of a hydrodynamic bivector; (2) Miura-reciprocal transformations of the 2nd kind (close to identity) and classify the orbits of their action on Poisson pencils of weakly non-local bi-vectors of localizable shape with localizable semi-simple hydrodynamic leading term; (3) a smaller group of projective-reciprocal transformations and prove that they preserve the Doyle-Potëmin canonical form of the bi-vectors.
A detailed comparison between previous results and our results can be read in the following Subsections; we just stress that our result on the classification of bi-Hamiltonian integrable structures provides a natural extension for the classification program in [DZ01] (that also incorporates and explains some of the results in [LZ11]).To the best of our knowledge it is the first result in the literature that provides a systematic classification of orbits of the action of the group of Miura-reciprocal transformations in the bi-Hamiltonian context (for a single Poisson structure this type of result is established in [FP03;LZ11]).
1.1.A variety of jet space transformation groups.We consider a jet space J r (1, N), r ≥ 0, with independent variable x and dependent variables u i , i = 1, . . ., N, considered as coordinates on some open domain U ⊂ R N .Let u i,σ denote the x-derivative of u i taken σ times.
Consider the transformations (i.e.diffeomorphisms) of the jet space J r (1, n).We begin from the most general type of transformation: a reciprocal transformation coupled with a differential substitution.Reciprocal transformations in a modern setting were introduced in [Rog69;Rog68] in the study of gas dynamics, and later analyzed under a geometric viewpoint in [Fer89;Fer91] and many other authors (see e. g. [FP03; XZ06; AG07; Abe09; BS09; LZ11; AL13] and references therein).The class of differential substitutions was introduced in [Ibr85], although many particular differential substitutions were already present in the literature (in particular, the Miura transformations).
Definition 1.1.A reciprocal differential substitution is a nonlocal transformation of the independent variable x into the independent variable y of the type (1) dy = Bdx, B = B(x, u i , u i,σ ) coupled with a differential substitution of the dependent variables of the form (2) w i = Q i (x, u j , u j,σ ).
By the fact that dx(∂ x ) = 1 = dy(∂ y ) we obtain that total derivatives are related by the formula ∂ x = B∂ y .Note that, in general, the inversion of a differential substitution is a nonlocal operation.We will soon focus on a more restrictive class of transformations.
Reciprocal differential substitutions admit several interesting subclasses: • A reciprocal transformation is a nonlocal transformation of the independent variables x into the independent variable y of the type (3) dy = Bdx, B = B(x, u i , u i,σ ) coupled with the identical transformation of the dependent variables.In practical applications the functions u i depend also on an additional parameter that plays the role of "time" of the system of evolutionary PDEs u i t = F i (x, u i , u i,σ ), i = 1, ..., n governing their evolution.Taking into account this additional variable reciprocal transformations are often defined as where the function A, B are submitted to the closure condition B t = A x , that is, dy is a conservation low for the equation.Note that the coefficient A doesn't enter the transformation law for ∂ x , and thus can be disregarded throughout this paper.• A reciprocal differential substitution is said to be holonomic if there exists a differential function P such that B = ∂ x P .• A general differential substitution of (x, u i ) into (y, w j ): (4) y = P (x, u j , u j,σ ), yields a holonomic reciprocal differential substitution dy = ∂ x P dx, w i = Q i by differentiation (in this sense the two classes of transformations coincide).The above two categories of transformations, basically local and nonlocal differential substitutions, are, on the one hand, too wide to be used in the context of the classification programs for evolutionary PDEs and related geometric structures as, for instance, the one initiated by Dubrovin and Zhang in [DZ01], and on the other hand too restrictive since we are limited by fixing the parameter r ≥ 0 that controls the maximal order of jets.
For this reason, we introduce the space of differential polynomials A, and the following group of transformations, which is a subclass of the reciprocal differential substitutions.Consider a jet space J ∞ (1, N) (considered as an inductive limit of the jet spaces J r (1, N), r → ∞) with independent variable x and dependent variables u i , i = 1, . . ., N. Denote u i,σ the x-derivative of u i taken σ times.We associate with this space the algebra of functions where C ∞ (U) is the space of smooth functions on a domain U ⊂ R N in the coordinates u 1 , . . ., u N .There is a natural gradation on the algebra of densities A given by deg ∂x u i,σ .Let A d denote the deg ∂x -degree d part of A, which is a finite dimensional module over C ∞ (U).
Definition 1.2.A Miura-type reciprocal differential substitution, or Miura-reciprocal transformation for short, is a transformation of the type The formal dispersive parameter ǫ that we introduce here to control the deg ∂x -degree is, in principle, not strictly necessary but it is very convenient in particular computations and applications.
The set of all Miura-reciprocal transformations is denoted by R. It is a group with respect to the composition, and it has some distinguished subgroups: • the subgroup R DS of Miura differential substitutions, that are Miura-type reciprocal differential substitutions which are also holonomic differential substitutions of the following type: There exists P = x + P 0 , with • the subgroup of Miura transformations characterized by H 0 = 1 and H k = 0 for all k ≥ 1.This subgroup is called the Miura group G ⊂ R [DZ01] and bears his name from the transformation relating KdV and modified KdV equations introduced by Miura [Miu68].Note that the Miura group is also a subgroup of the group of Miura differential substitutions: G ⊂ R DS .
Definition 1.3.By analogy with the way the standard Miura group is typically presented, we introduce the following two subgroups.
• We define Miura-reciprocal transformations of the 1st kind to be the Miura-reciprocal transformations of the form dy = H 0 (u j )dx, (8) The group of all Miura RDS of the first type is denoted by R I .This group contains as a subgroup the group of Miura transformations of the 1st kind, G I ⊂ R I , characterized by H 0 = 1.• We define Miura-reciprocal transformations of the 2nd kind to be the Miura-reciprocal transformations of the form The group of all Miura-reciprocal transformations of the second type is denoted by R II .It contains as a subgroup the group of Miura transformations of the 2nd kind, Definition 1.4.A distinguished subgroup of R I is the group of projective reciprocal transformations P. Such transformations are characterized by the requirements that K i in Equation ( 8) is a projective transformation (in an affine chart) and H 0 is the common denominator of the projective transformation.More explicitly, dy = (a 0 j u j + a 0 0 )dx, (10) The goal of this paper is to discuss some aspects of the actions of these groups on the natural suitable geometric structures that emerge in the study of integrable systems of evolutionary PDEs.In order to describe our results we have to recall some of these structures, which we do in the rest of the introduction.1.2.Action of the transformation groups.The above group of Miura reciprocal differential substitutions act on spaces of geometric entities that play important roles in the geometric theory of integrability.In particular, it acts on: • densities, that have the form • variational vector fields, that include symmetries of partial differential equations, and have the form • covector-valued densities, that include characteristic vectors of conserved quantities of differential equations, and have the form • the Euler-Lagrange operator, which sends densities into covector-valued densities, ( 14) • variational multivectors of degree p, that include Hamiltonian operators of partial differential equations as particular bivectors.They can be regarded as maps from (p − 1)covector-valued densities to variational vector fields.
In Section 2 we will prove our change of coordinate formulae for reciprocal differential substitutions for these geometric objects.As an example, we re-derive in Section 2.1 the Ferapontov-Pavlov formula for the reciprocal transformation of a Poisson bi-vector of the differential order 1 [FP03], and this brings us to the realm of weakly non-local Poisson structures of localizable shape.
1.3.Weakly non-local Poisson bi-vectors of localizable shape.Let dependent variables u i also dependent on one external parameter, denoted by t.The most studied structures in geometric theory of integrability are the local Poisson structures needed for representation of equations of the form in Hamiltonian form (note that we don't allow possible explicit dependence of f i 's on x), that is, in the form where H = h(u k , u k,σ )dx is the Hamiltonian functional and P = d s=0 P ij s ∂ s , P ij s ∈ A defines a bi-vector which in the language of densities can be written as (in this paper bi-vectors and, more generally, multivectors are assumed to be skew-symmetric by default).
In addition to the language of densities, there is a very convenient formalism, the so-called θ-formalism, to encode the variational multivectors [Get02], see also [IVV02].Namely, extend the space A to a space Â := A[[θ σ i , i = 1, . . ., N, σ ≥ 0]], where θ σ i are formal odd variables.We often denote θ 0 i by θ i , and we extend the ∂ x operator to Â as and there is a natural θ-degree given by deg θ u i,σ = 0 and deg θ θ σ i = 1.Let Âp denote the subspace Â of θ-degree p.We can consider it as a space of densities of variational p-vectors.Let Âp d := Âd ∩ Âp .The space F := Â/∂ x Â can be considered as the space of variational multivectors.It inherits under the projection : Â → F both gradations, deg ∂x and deg θ , and F p d denotes its subspace of p-vectors of differential degree deg ∂x = d.The Schouten bracket is defined as Various cohomological computations in terms of this space and related formalism allow to efficiently control the deformation theory of Poisson bi-vectors and their pencils, see e. g. [LZ05; LZ11; DLZ06; LZ13a; CPS18; CKS18; CPS16a; CPS16b; CCS17; CCS18].
However, studying the action of the group of Miura-reciprocal transformations we can not restrict ourselves to the local Poisson bi-vectors.As we have seen, the reciprocal transformations generate non-locality of some very particular shape, and in terms of the operator P we have to extend its possible shape to Hamiltonian operators of the form above with d = 1, x were studied by Ferapontov in [Fer95a].They belong to the larger class of weakly non-local operators, that was introduced in [MN01].Like in the local case the coefficients g ij define a metric and the coefficients Γ j lk the Christoffel symbols of the associated Levi-Civita connection but unlike in the local case the metric is no longer flat.It turns out that the Riemann tensor R and the tensor field V defining the non-local tail satisfy the conditions These are a particular instance of Ferapontov's conditions for weakly non-local Hamiltonian operators of hydrodynamic type [Fer95b].An algorithm to compute such conditions for general weakly non-local Hamiltonian operators has been developed in [CLV20] and implemented in three different computer algebra systems in [Cas+22].
A natural question here is how to extend the θ-formalism briefly recalled above to accommodate this type of non-locality.There are two recipes in the literature given in [LZ11] (specific for this case) and [LV20] (suitable for general weakly non-local operators).The identification of the two approaches should indirectly follow from the uniqueness arguments in [LZ11], but we wanted to establish an explicit identification.We do it by an explicit computation in Section 3.
Remark 1.5.It is important to comment on the action of the operator ∂ −1 x .It can be defined on ∂ x (g), for instance, as an element of a localization of A given by A(( 1 u 1,1 )), that is, we can perturbatively represent it as a series ), here C is also a constant.
Both approaches that we compare assert that for the analysis of the weakly non-local Poisson bi-vectors of localizable shape it is sufficient to formally apply ∂ −1 x to just one element −u i,1 θ i ∈ Â and denote the result by ζ, which has different meaning in these two approaches.The subsequent usage of ζ in computations implies that the extra constant that might occur by inverting ∂ x is uniformly set to C = 0. 1.4.Localizability.Consider a dispersive weakly non-local Poisson structure of localizable shape given by The leading term (d = 1) of this structure is a Poisson structure of hydrodynamic type and thus the full Poisson structure can be thought as a deformation of a Poisson structure of hydrodynamic type.If det P ij 1,0 = 0, Liu and Zhang prove in [LZ11] that there is always an element in R that turns P into a constant local Poisson structure η ij ∂ x .In the case of a purely local structure the same results is established under the action of group G in [Get02] (see also [DMS05] and [DZ01]), and in the case ǫ = 0 (that is, a purely degree 1 case) it is established under the action of the group R I in [LZ11] for N = 1, 2 and in [FP03] for N ≥ 3. Now consider a pencil P − λQ of dispersive weakly non-local Poisson structure of localizable shape.Let us fix the leading term (P − λQ)| ǫ=0 of the pencil and assume it is semi-simple.In the purely local case (that is, under the additional assumption that both P and Q are purely local), it was suggested in [LZ05; DLZ06] (see also [Lor02] for the scalar case) and proved in [LZ13a] (N = 1 case) and [CPS18; CKS18] (any N ≥ 1) that the space of orbits of the action of G II on such pencils is naturally parametrized by N smooth functions of one variable, called the central invariants.
In Section 4 we generalize these results in the following way.Let us fix the leading term (P − λQ)| ǫ=0 of the pencil and assume that P | ǫ=0 and Q| ǫ=0 are simultaneously localizable under the action of the group R I .We also still assume that (P − λQ)| ǫ=0 is semi-simple.In this case, we prove that the set of orbits of the action of R II on such pencils is also naturally parametrized by N smooth functions of one variable.Note that while the statement is literally the same as in the purely local case, it is quite different as both the group and the space of structures on which the group acts is much bigger.We show that it is still possible to read the central invariants from the symbol of the pencil.
This result is proved by a direct application of various techniques and results proposed in [LZ11; LZ13a; CPS18; CKS18].From the comparison with the computations in the local case, we obtain the following extra result: under the assumptions above, each orbit of R II contains a purely local representative.In other words, we prove that if the leading term of a semi-simple pencil P − λQ of dispersive weakly non-local Poisson structure of localizable shape is localizable by the action of the group R I , then the whole pencil is localizable by the action of the group R.
It is worth to mention that this result also generalizes and put in the right context a theorem of Liu and Zhang [LZ11, Theorem 1.3] that states that if two local Poisson pencils with the leading semi-simple hydrodynamic term are related by a reciprocal transformation, then their central invariants are the same.
can always be transformed by the action of the group G I to an operator of the shape Mokhov made the following interesting conjecture: The form ( 22) is called the Doyle-Potëmin form of a local homogeneous bi-vector of differential degree deg ∂x ≥ 2.
It was recently proved that in the cases of homogeneous local Poisson structures of degree d = 2 [VV] and d = 3 [FPV14] the form ( 22) is preserved by the action of the group P. In Section 5, thanks to our change of coordinates formulae from Subsection 1.2, we generalize the above results to local homogeneous bi-vectors (i.e., not necessarily Poisson structures) of degree d ≥ 2 and prove that the group P preserves the set of local bi-vectors of Doyle-Potëmin form.A nice example of application to a Hamiltonian operator for the Dubrovin-Zhang hierarchy is pointed out.1.6.Acknowledgments.S. S. and R. V. were supported by the Netherlands Organization for Scientific Research.P. L. and R. V. are supported by funds of INFN (Istituto Nazionale di Fisica Nucleare) by IS-CSN4 Mathematical Methods of Nonlinear Physics.P. L. is supported by funds of H2020-MSCA-RISE-2017 Project No. 778010 IPaDEGAN.P. L. and R. V. are also thankful to GNFM (Gruppo Nazionale di Fisica Matematica) for supporting activities that contributed to the research reported in this paper.

Formulae for the action
The goal of this Section is to compute from the scratch the effect of general reciprocal differential substitutions on variational (multi)vector fields and related geometric objects.It is clear that a reciprocal differential substitution given by dy = Bdx (or y = P in the holonomic case), w i = Q i , also yields a coordinate change of the y-derivative variables: It is convenient to introduce the Fréchet derivative1 of a differential function F ∈ A, as where X = X i δ u i is a variational vector field.The formal adjoint of the above operator is acting on covector-valued densities.
A change of coordinates formula for Hamiltonian operators under the action of differential substitutions was already given in [Mok87;Olv88].We rephrase the arguments of the proof in [Olv88] and obtain change of coordinates formulae for the geometric objects that we listed in Subsection 1.2 which turn out to be valid in the more general case of reciprocal differential substitutions.
We observe that also in [LZ11] there are formulae for coordinate change, but their validity is limited to the action of Miura reciprocal transformations on operators of localizable shape, while we do not have this limitation.
First of all, we provide a formula for the coordinate change of an variational vector field under a differential substitution.
Proposition 2.1.Let X i δ u i = Y i δ w i be a variational vector field in the coordinate systems (x, u i,σ ) and (y, w i,σ ), respectively, where the latter coordinates systems are related by a holonomic reciprocal differential substitution y = P , w i = Q i .Then the following change of coordinate formula holds: The proof uses arguments that provide a change of coordinates formula for Euler-Lagrange operators in [Olv93], Theorem 4.8 and Exercise 5.49.Let (28) be functions that are put in correspondence by a transformation.We can consider a oneparameter family of such functions defined by the variation field where The transformed variation field is given by Y i (y) = ∂ ǫ g i (y, ǫ) ǫ=0 .As variation fields do not depend on ǫ we have (32) We have: In the non-holonomic case, we have to regard the differential function P as the primitive of a differential function B, P = ∂ −1 x B, and we obtain the following Corollary.Corollary 2.2.In the non-holonomic case of the reciprocal differential substitution dy = Bdx, w i = Q i the following change of coordinate formula holds for an variational vector field where (36) which is very useful in computations.Dualizing the computation above, we obtain the formulae for the change of coordinates formula for the Euler-Lagrange operator.
Corollary 2.3.Let the coordinate systems (x, u i,σ ) and (y, w i,σ ), respectively, where the latter coordinates systems are related by a reciprocal differential distribution dy = Bdx, w i = Q i , and let E x i , E y i be the Euler-Lagrange operator with respect to the coordinates (x, u i σ ), (y, w i,σ ).Then the change of coordinate formula is , with D given by Equation (36).
In the holonomic case, the formula reduces to the known formula in [Olv93, Exercise 5.49] (with D given by Equation ( 27)).In the particular case of a differential substitution of the dependent variable only we have ℓ B = 0 and the above formula reduces to the well-known formula Corollary 2.4.Consider a covector-valued density Ξ i du i ⊗ dx = Ψ i dw i ⊗ dy in the coordinate systems (x, u i,σ ) and (y, w i,σ ) related by dy = Bdx, w i = Q i .Then we have the following change of coordinates formula: , where D is as in Equation (36) (or as in Equation (27) in the holonomic case y = P ).
Finally, we obtain the following: Proposition 2.5.Consider a reciprocal differential substitution dy = Bdx, w i = Q i and let P ij x , P ij y be its coordinate expressions of a (possibly non-local or non-Poisson) bi-vector with respect to the coordinates (x, u i σ ) and (y, w i σ ).Then we have the change of coordinate formula (39) where D is as in Equation (36).
Proof.The proposition has already been proved in [Mok87;Olv88] for the particular case of Hamiltonian operators and differential substitutions.In our case, the proof follows from the fact that P ij maps covector-valued densities into variational vector fields.So, we can use the change of coordinates formulae for these two geometric objects (independently of the Hamiltonian property) and find the above result, that holds also in the case of (nonlocal) reciprocal differential substitutions.
Remark 2.6.The same argument can be applied to multivector fields considered as maps from multicovector-valued densities to variational vector fields.For instance, in the same set-up as Theorem 2.5 let T ijk and T ijk be the coordinate expressions of a trivector.Then (40) Consider the change of x given by as an element of R I , that is, we assume that B = B(u j ).Let a local Poisson bracket of differential degree 1 be given by the operator Convention 2.7.Throughout the computations in this Section it is important for us to distinguish between ∂ x u k and ∂ y u k , so we use the notation u k x and u k y rather than u k,1 .Proposition 2.8.The action of the reciprocal transformation (41) on the operator (42) produces a weakly non-local operator of localizable shape, whose local part is given explicitly as and the non-local part is equal to is exactly the covariant Christoffel symbol for the metric g ij B 2 , so we indeed reproduce the Ferapontov-Pavlov formula in [FP03].
Proof of Proposition 2.8.By Theorem 2.5 the bi-vector P ij is transformed under the substitution (41) to Expanding the brackets, we have the following four summands (we intentionally keep derivatives in x instead of y as long as possible): Thus the non-local term is given by Equation (44), and the local term is given by where the latter expression is equal to (43).

2.2.
Weakly non-local bi-vectors of localizable shape.The goal of this Section is to prove that the space of weakly non-local bi-vectors of localizable shape is closed under the action of reciprocal differential substitutions.We narrow the scope to the Miura-type substitutions R as in Definition 1.2.Let us consider the effect of a reciprocal transformation of the form (41) on a general weakly nonlocal bi-vector of localizable shape: Note that both P ij loc and P ij nonloc define skew-symmetric bi-vectors, that is (P * loc ) ij = −P ji loc and (P * nonloc ) ij = −P ji nonloc .Proposition 2.10.Consider a Miura-reciprocal transformation in R given by dy = Bdx, w i = Q i .Under this transformation any weakly non-local bi-vector P ij of localizable shape (51) is transformed into a weakly non-local bi-vector of localizable shape.
Remark 2.11.In principle, this proposition follows from the arguments of [LZ11] and [FP03].However, it can also be directly obtained using Theorem 2.5.
Proof.Repeating mutatis mutandis the proof of Proposition 2.8 one can check that the local part P ij loc produces a weakly non-local operator of localizable shape (the only thing that matters for that computation is skew-symmetry of the bi-vector defined by P ij loc ).So let us focus on the non-local part and we use Convention 2.7 here and below in computations.We have: (we omit the summation over s and t for brevity).We compute (52) as follows.First, note that Here loc are the terms where we collect some purely local operators.Furthermore, Here O j BuV Q , O j BV uQ , O i QuV B , and O i QV uB are some scalar local operators, whose main property is that (O j BuV Q ) * = −O j QV uB and (O j BV uQ ) * = −O j QuV B .We omit their explicit formulas.Finally, where O BuV B and O BV uB are scalar local operators such that O * BuV B = −O BV uB .We omit their explicit formulas, but we use below that Now we collect the terms together.Firstly, we list all terms with B x that emerged in (54) and (55): x B Note some cancellations: the non-local terms in (53) cancel with the corresponding summands in the first and the second line of (56), two non-local terms in the second and third line of (54) cancel with the two terms in the third and forth line of (56), are there are two terms in the latter lines that cancel each other.So, modulo the purely local terms, (52) is equal to the sum of the following four expressions: (1)∂ −1 y w j y + loc, which is manifestly a weakly non-local operator of localizable shape.

Schouten bracket for weakly non-local operators of localizable shape
The goal of this Section is to compare two ways to encode weakly non-local Poisson structures of localizable shape: the one given in [LZ11] (by design only working for the localizable shape case) and [LV20] (it is working for general weakly non-local case, but we specialize it for the localizable shape).In principle, the identification of these two approaches follows from the uniqueness property of the bracket, c.f. [LZ11, Theorem 2.4.1], but we want to present an explicit computation for this identification.
3.1.The two approaches.In both approaches the weakly non-local p-vectors of localizable shape are encoded as where P L ∈ Âp , P N ∈ Âp−1 , and (59) The difference in two approaches is the meaning of ζ.In the approach of and the space of weakly non-local multivectors of localizable shape is defined as E := S/∂ x S.
In the approach of [LV20], ζ is not a new dependent variable, but rather an expression in the existing dependent variables (still of differential degree deg ∂x ζ = 0 and multivector degree deg θ ζ = 1), such that Equation ( 59) is satisfied for the standard operator For instance, one can find such a function in Â(( 1 u 1,1 )), cf.[DLZ06].To this end, one looks for a unique solution ∂x Once the objects are defined, we have two different formulae for the Schouten bracket in these two approaches: • The formula in the approach of [LV20] is Recall that ζ is regarded as a function of (u i σ , θ σ i ) in the variational derivatives (which are denoted by δu i and δθ i for that reason).
• The formula in the approach of [LZ11] is Here ζ is regarded as an extra dependent variable, and the operator Ê is defined as 3.2.Identification of the two approaches.We prove the following: where we used that Using these formulas, we obtain If we want to treat ζ as a new dependent variable, we have Ê(P )∂ ζ Q = Ê(P )Q N (and similarly for the other summand in the formula), so we have to prove that (−1) Let us use the following property of the operator Ê: Substituting Equations ( 73) and ( 74) into (71), we see that the statement of the theorem reduces to the following equality: (−1) In order to prove this equality, our strategy is move x (−u i,1 θ i ) to the other factor (P N or Q N ) using integration by parts.We have: Substituting the above expressions into the equality (75) that we shall prove, we are led to the simplified equality: Integrating by parts the summands containing ∂ x δ θ i P , ∂ x δ θ i Q we obtain the further simplification of the equality (75) (note that deg θ P N = deg θ P − 1): (−1) Expanding the total derivatives on the right-hand side we easily see that the above equality is an identity.This completes the proof of the theorem.

Pencils of weakly non-local bi-vectors of localizable shape
In this Section we compute the bi-Hamiltonian cohomology for a semi-simple pencil of weakly non-local Poisson bi-vectors of localizable shape of differential order deg ∂x = 1 satisfying the extra condition: the pencil of these bi-vectors should be localizable (or, equivalently, they should be simultaneously localizable) with respect to the Miura-reciprocal group.As a result of this computation and some further arguments we prove the following theorem: Theorem 4.1.Let P 1 and P 2 be two of commuting non-local Poisson bi-vectors of localizable shape.We assume that P 1 and P 2 have dispersive expansion given by P a = ∞ i=1 ǫ i−1 P a,i , deg ∂x P a,i = i, a = 1, 2, i = 1, 2, . . . .If the leading terms of degree deg ∂x = 1, P 1,1 and P 2,1 , are simultaneously localizable under the action of the Miura-reciprocal group and form a semi-simple Poisson pencil, then the full dispersive brackets P 1 and P 2 are simultaneously localizable under the action of the Miurareciprocal group.
In order to prove this theorem, we have to make a few preliminary computations with bi-Hamiltonian cohomology, following the ideas in [LZ11] , and E := S/∂ x S. Let P 1 , P 2 ∈ S 2 1 such that P 1 and P 2 form a pencil of Poisson structures (possibly nonlocal, but then they are automatically weakly non-local of localizable shape, since it is the only type of non-locality accommodated in the space E), that is, we assume that Recall that there is a group R I of the Miura-reciprocal transformations of the 1st kind acting on them, see Equation (8).We assume that the pencil P 2 −λP 1 is localizable under the action of R I .We also assume that the pencil formed by P 1 and P 2 is semi-simple, which together with the assumption of localizability implies that the we can choose the coordinates x, u 1 , . . ., u N such that the densities P 1 and P 2 of the bivectors P 1 and P 2 take the form We are interested to classify the equivalence classes of the higher order dispersive deformations of the Poisson pencil P 2 − λP 1 in E with respect to the Miura-reciprocal transformations of the 2nd kind, R II .Let d i := ad P i : E → E, i = 1, 2. Then the deformation problem is controlled by the bi-Hamiltonian cohomology BH p d (E, d 1 , d 2 ) of cohomological degree p = 2 and p = 3 and of differential degrees d ≥ 2 and d ≥ 4, respectively.It is a rather standard argument, see e. g. [LZ11, Proposition 3.3.5].The only extra bit that one needs in our case, that is, the space E and the group R II of Miura-reciprocal transformations of the 2nd kind, in comparison with the usual local case, that is, the space F and the group G II of Miura transformations of the 2nd kind, is the identification of the action of the Lie algebra of R II on weakly non-local bi-vectors (or, more generally, multivectors) of localizable shape with the adjoint action of E 1 on E 2 (resp., E).This is established in [LZ11, Theorems 2.5.7 and 2.6.5]4.1.2.Bi-Hamiltonian cohomology computation.We prove the following Theorem 4.2.We have: Proof.For the proof we use that for d ≥ 2 we have [LZ13b, Lemma 4.4]: In order to compute H p d (E[λ], d 2 −λd 1 ), we recall the definition of D i := D P i : S → S from [LZ11]: Note that [∂ x , D i ] = 0 (by direct computation).We prove that it is a homological vector field (which is not true in general, for a non-local bi-vector P i ): Lemma 4.3.For a purely local bivector P the operator D P does not depend on the choice of a purely local density P .Moreover, for purely local densities of the bivectors P, Q ∈ Â2 and for any T ∈ S we have: and where In particular, for a purely local density P of a Poisson bivector P we have D 2 P = 0 on S. Remark 4.4.The statements of Lemma 4.3 do not hold for not purely local densities.
Proof of Lemma 4.3.Firstly, we check the D P does not depend on the choice of a local density P .To this end, we remind the definitions and basic properties of Ê and ∂ x .We have: With the last three equations we immediately see that for any local X ∈ Â Adding to the latter expression the same one with interchanged P and Q and using that for purely local densities we obtain that the coefficient of ∂ ζ on the left hand side of Equation ( 92) is equal to which is the coefficient of ∂ ζ on the right hand side of Equation (92).The coefficients of all other components of the vector fields on the left hand side of Equation ( 92) are computed in a very similar way.Lemma 4.3 implies that D 2 −λD 1 is a differential on S[λ], and we have a short exact sequence and it implies a long exact sequence in the cohomology which reads Lemma 4.5.We have Also, H 3 2 (S[λ], D 2 − λD 1 ) ∼ = 0. Proof.This lemma can be derived from [CKS18, Theorems 2.12 and 2.13].Indeed, Lemma 4.3 in particular implies that we have a bicomplex (S[λ], D loc , D ζ ) with the differentials given by D ζ := ( Ê(P 2 ) − λ Ê(P 1 ))∂ ζ and D loc := D 2 − λD 1 − D ζ .We start a spectral sequence associated with this bicomplex.Obviously, it converges on the second page.The computation of the first page splits as which implies all desired vanishings (for p ≤ d the only non-trivial cohomology groups are Theorems 2.12 and 2.13]).
Since both H i i (S[λ], D loc ) = 0 for i = 1, 2, 4, and the induced differential on the first page has the (p, d)-degree (1, 1), we conclude that which gives the answer for (p, d) = (2, 3).Now, the special cases of these computations for p = 2, d ≥ 2 and p = 3, d ≥ 4 imply all statements of Theorem 4.2.
An immediate corollary of Theorem 4.2 is the following: Corollary 4.7.Let P 2 − λP 1 be a semi-simple pencil of local Poisson bivectors of differential order 1.We consider the higher order dispersive extensions of P 2 − λP 1 in the realm of weakly non-local Poisson pencils of localizable shape, that is, we consider Poisson pencils The space of orbits of the action of the group R II (the group of Miura-reciprocal transformation of the 2nd kind) onto the set of these dispersive extensions is isomorphic to the space This result is strikingly similar to the corresponding statement in the local case, cf.[CPS18, Theorem 1], see also [LZ05;LZ13a;DLZ06].However, in the local case both the space F where the deformations of P 2 − λP 1 are allowed as well as the group G II acting on them are much smaller than in Corollary 4.7.Our next goal is to compare these two situations.Corollary 4.7 implies that there is a R II invariant map C : We have the following Proposition 4.9.The composition c • ι : Proof.Basically, we want to show that any cohomology class in H 2 3 (E) has a representative with a purely local density.Let a or, in other words, that there exist b Thus, the cocycle a 2 + a 1 ζ is cohomologous to a 2 + ( Ê(P 2 ) − λ Ê(P 1 ))(e 0 ) + u i,1 θ i f 1 , which gives a pure local deformation for P 2 − λP 1 .
Taking into account that that c is a bijection, an immediate corollary of Proposition 4.9 is the following: Corollary 4.10.The map ι is surjective (and hence a bijection).In particular, every orbit of the action of R II on T nl contains a purely local representative.
It is just a different way to state Theorem 4.1, so this corollary also completes the proof of Theorem 4.1 4.3.Roots of the characteristic polynomial of the symbol.In the purely local case the central invariants, besides a purely cohomological definition, can be computed directly from a representative of a deformation (see [DLZ06] for details).More precisely, one has to compute the eigenvalues of the symbol of a representative of a deformation, which behave as scalars with respect to the Miura group action.In this section we extend this viewpoint to the invariants of the Miura-reciprocal group.
First, we recall the construction from [DLZ06].Let Consider the symbol of the densities of the bi-vector The construction of the Miura group invariants from the eigenvalues of the symbol is based on the following lemma: which are the formal power series in ǫ with the coefficients given by smooth functions in u 1 , . . ., u N , with the leading terms in ǫ given by m i = u i + O(ǫ).These eigenvalues are further used to derive the closed formulas for the central invariants of a pencil ∞ d=1 ǫ d−1 (P 2,d − λP 1,d ) ∈ T loc .
In the weakly non-local case of localizable shape, the densities of This means that λ i → r i + (Bǫ) 2 λ i 2 + (Bǫ) 4 λ i 4 + • • • or, equivalently, that λ i 2k → B 2k λ i 2k .The result then follows from the transformation rule for the contravariant metric (see (43)): f i → B 2 f i .In the case of reciprocal transformation of 2nd kind we observe that they do not affect the symbol of the pencil.Indeed a bivector transforms according to the following rule ǫ k H k (u j , u j x , . . ., u j σ ), H k ∈ A k .
Thus we have Since the symbol of the bivector contains only the subset of the coefficients which depend only on the u's but not on their x-derivatives the second term and the third terms above cannot contribute to it.This implies that the symbol of each bivector defining the pencil is unaffected by these transformations.For Miura reciprocal transformations (5) the transformation rule for the symbol of the pencil is obtained combining the Lemma 4.11 with the above rule.It turns out that the symbol of the pencil transforms in the following way In this Section we make a first step towards the study of the projective-reciprocal group action.Consider a local operator of homogeneous differential order d + 2, d ≥ 2 of the form We call this presentation of an operator the Doyle-Potëmin form (see Subsection 1.5).
We prove the following theorem: Proof.Consider an element of the group P given by dy = A 0 dx, (123) where A i := a i j u j + a i 0 , i = 0, 1, . . ., N. Since the functions A i and A 0 do not depend on the higher jet variables, Theorem 2.5 implies that the operator P ij = ∂ x • Q ij • ∂ x in the coordinates y, w 1 , . . ., w N is represented as Now we see that

1. 5 .
Projective group and Doyle-Potëmin form.Finally, we consider the action of the group P ⊂ R I .It is a quite small group with transparent structure, and we expect that in general the orbits of its action should have a rich geometric structure.In this paper we find a surprising connection of this group to a conjecture of Mokhov on the possible form of the local Poisson structures of differential degree deg ∂x ≥ 2. It was independently proved by Doyle [Doy93] and Potëmin [Pot91; Pot97] that homogeneous local Poisson structures of differential degree d = 2, 3, i.e. of the form (21) d s=0

Conjecture 1. 6 (
See [Mok98, Proposition 2.3 and text afterwards]).Let P = d+2 e=1 P ij e ∂ d+2−e x be a local operator of homogeneous differential order d + 2 (that is, deg ∂x P ij e = e), d ≥ 0. Assume that P defines a Poisson bracket.Then there exists a local skew-symmetric operator

2. 1 .
The Ferapontov-Pavlov formula.Let us apply a special case of Theorem 2.5 to a local Poisson bi-vector of order deg ∂x = 1 and a reciprocal transformation in R that only changes the independent variable.This should reproduce the Ferapontov-Pavlov formula first derived in [FP03, Section 3] (based on[Fer95a]).
[LZ11], ζ is a new dependent variable such that deg ∂x ζ = 0 and deg θ ζ = 1.The new space of multivector densities is defined as S := Â[ζ], equipped with the operator

Theorem 3. 1 .
The identity map Â[ζ] → Â[ζ] induces the isomorphism of the Lie algebras of local multivector fields defined by the Schouten brackets in these two approaches.Proof.We represent any density P ∈ Â[ζ] as P = P L + ζP N and consider ζ to be a nonlocal function.Note that subsequent steps in [CPS18; CKS18].4.1.Bi-Hamiltonian cohomology.4.1.1.Setup for a deformation problem.Recall that following Liu and Zhang [LZ11] we denote S := Â[ζ], with ∂ x : S → S given by which implies the first assertion of the lemma.Now, Equation (91) is obvious from the definition of the Schouten bracket.So we focus on Equation (92).Let us compute the coefficient of ∂ ζ on the left hand side.Using the vanishing of ∂ ζ derivatives, we have:

4. 2 .
Comparison with the purely local deformations.Within this section it is important to have a notation that distinguishes between the operator ∂ x as given by Equation (93) on the space S = Â[ζ] and its purely local version ∂x := ∂ x + u i,1 θ i ∂ ζ defined both on S and on A. Note that on S the operator ∂x commutes with multiplication by ζ.Let T nl denote the space of dispersive weakly non-local Poisson pencils of localizable shape∞ d=1 ǫ d−1 (P 2,d − λP 1,d) ∈ E with the fixed leading term P 2,1 − λP 1,1 = P 2 − λP 1 that is purely local and semi-simple.Let T loc denote the space of dispersive local Poisson pencils∞ d=1 ǫ d−1 (P 2,d − λP 1,d ) ∈ F with the same fixed leading term P 2,1 − λP 1,1 = P 2 − λP 1 .The group R II acts on T nl and the group G II acts on T loc .Moreover, there is a natural embedding I : T loc → T nl that is G II -equivariant (G II acts on T nl as a subgroup of R II ).The map I induces a map of the sets of orbits ι :T loc /G II → T nl /R II .Proposition 4.8.The map ι is injective.Proof.This proposition immediately follows from [LZ11, Theorem 1.3] and[CPS18, Theorem  2].By the latter result in the local case, we have an isomorphism of sets c : T loc /G II → N i=1 C ∞ (R, u i ) (these are the so-called central invariants in the local case).On the other hand, [LZ11, Theorem 1.3] states that for any x, y ∈ T loc /G II such that ι(x) = ι(y) we have c(x) = c(y).Hence, x = y, and ι is surjective.

Theorem 5. 1 .
The projective group preserves the Doyle-Potëmin form of an operator.More precisely, the image of a homogeneous skew-symmetric operator of the form∂ x • Q ij • ∂ x , deg ∂x Q ij = d ≥0 under the action of an element of P is a homogeneous skew-symmetric operator of the form ∂ x • Qij • ∂ x , deg ∂x Qij = d ≥ 0