Three computational approaches to weakly nonlocal Poisson brackets

TL;DR

This paper compares three computational methods—distributions, pseudodifferential operators, and Poisson vertex algebras—for verifying the Jacobi identity in weakly nonlocal Poisson brackets, demonstrating their equivalence.

Contribution

It provides a unified comparison of three approaches to checking the Jacobi identity, highlighting their similarities and confirming their consistency.

Findings

All three methods yield similar computations.

The approaches are equivalent in verifying the Jacobi identity.

The methods are consistent across different mathematical frameworks.

Abstract

We compare three different ways of checking the Jacobi identity for weakly nonlocal Poisson brackets using the theory of distributions, of pseudodifferential operators and of Poisson vertex algebras, respectively. We show that the three approaches lead to similar computations and same results.

Equations

$$F=u_t^i-f^i(u^j,u_{\sigma }^j)=0$$

$$u_t^i=f^i(u^j,u_{\sigma }^j)=A^{ij}\frac{\delta H}{\delta u^i},$$

$$\{F,G{\}}_P=\int P^{ij\sigma }\frac{\delta f}{\delta u^i}{\partial }_{\sigma }\frac{\delta g}{\delta u^j}dx$$

$$P^{ij}=g^{ij}{\partial }_x+{\Gamma }_k^{ij}{u}_x^k+c^{\alpha }{w}_{\alpha k}^i{u}_x^k{\partial }_x^{-1}{w}_{\alpha h}^j{u}_x^h,$$

$$P^{ij}=B^{ij\sigma }{\partial }_{\sigma }+e^{\alpha }{w}_{\alpha }^i{\partial }_x^{-1}{w}_{\alpha }^j,$$

$${\partial }_x^{-1}=\frac{1}{2}{\int }_{-\infty }^{x}dx-\frac{1}{2}{\int }_x^{+\infty }dx$$

$$u_t^i=V_j^i(u^k)u_x^j$$

$$g^{ij}=g^{ji},$$

$$g_{,k}^{ij}={\Gamma }_k^{ij}+{\Gamma }_k^{ji},$$

$$g^{is}{\Gamma }_s^{jk}=g^{js}{\Gamma }_s^{ik},$$

$$g^{is}{w}_s^j=g^{js}{w}_s^i$$

$${\nabla }_i{w}_k^j={\nabla }_k{w}_i^j,$$

$$R_{kh}^{ij}=w_k^i{w}_h^j-w_k^j{w}_h^i.$$

$$\begin{gathered} \{u^i(x),u^j(y){\}}_P=\sum _{k\ge 0}{B}_k^{ij}(u^h,u_{\sigma }^h){\delta }^{(k)}(x-y) \\ +e^{\alpha }{w}_{\alpha }^i(u^k,u_{\sigma }^k)\nu (x-y)w_{\alpha }^j(u^k,u_{\sigma }^k) \end{gathered}$$

$$\begin{gathered} \{u^i(x),u^j(y){\}}_P,u^k(z){\}}_P+\{u^k(z),u^i(x){\}}_P,u^j(y){\}}_P \\ +\{u^j(y),u^k(z){\}}_P,u^i(x){\}}_P=0 \end{gathered}$$

$$\begin{gathered} J_{xyz}^{ijk}=\frac{\partial P_{x,y}^{ij}}{\partial u_{\sigma }^l(x)}{\partial }_x^{\sigma }{P}_{x,z}^{lk}+\frac{\partial P_{x,y}^{ij}}{\partial u_{\sigma }^l(y)}{\partial }_y^{\sigma }{P}_{y,z}^{lk}+\frac{\partial P_{z,x}^{ki}}{\partial u_{\sigma }^l(z)}{\partial }_z^{\sigma }{P}_{z,y}^{lj}+ \\ \frac{\partial P_{z,x}^{ki}}{\partial u_{\sigma }^l(x)}{\partial }_x^{\sigma }{P}_{x,y}^{lj}+\frac{\partial P_{y,z}^{jk}}{\partial u_{\sigma }^l(y)}{\partial }_y^{\sigma }{P}_{y,x}^{li}+\frac{\partial P_{y,z}^{jk}}{\partial u_{\sigma }^l(z)}{\partial }_z^{\sigma }{P}_{z,x}^{li}=0, \end{gathered}$$

$$\iiint J_{xyz}^{ijk}{p}_i(x)q_j(y)r_k(z)dxdydz$$

$$\nu (z-y)\delta (z-x)=\nu (x-y)\delta (x-z)$$

$$f(z){\delta }^{(n)}(x-z)=\sum _{k=0}^n\left(\genfrac{}{}{0ex}{}{n}{k}\right)f^{(n-k)}(x){\delta }^{(n-k)}(x-z),$$

$$\begin{gathered} a_1(x,y,z)\nu (x-y)\nu (x-z)+\text{cyclic}(x,y,z) \\ +\sum _{n\ge 0}{b}_n(x,y)\nu (x-y){\delta }^{(n)}(x-z)+\text{cyclic}(x,y,z). \end{gathered}$$

$$\sum _{m,n}{e}_{mn}(x){\delta }^{(m)}(x-y){\delta }^{(n)}(x-z)$$

$$\delta (z-x)\delta (z-y)=\delta (y-x)\delta (y-z)=\delta (x-y)\delta (x-z)$$

$$P^{\ast }(\psi {)}^j=(-1{)}^{|\sigma |}{\partial }_{\sigma }(B^{ij\sigma }{\psi }_i)-e^{\alpha }{w}_{\alpha }^j{\partial }_x^{-1}(w_{\alpha }^i{\psi }_i).$$

$$\begin{gathered} {\ell }_{P,\psi }(\phi {)}^i=\frac{\partial B^{ij\sigma }}{\partial u_{\tau }^k}{\partial }_{\sigma }{\psi }_j^1{\partial }_{\tau }{\phi }^k+e^{\alpha }\frac{\partial w_{\alpha }^i}{\partial u_{\tau }^k}{\partial }_{\tau }{\phi }^k{\partial }_x^{-1}(w_{\alpha }^j{\psi }_j) \\ +e^{\alpha }{w}_{\alpha }^i{\partial }_x^{-1}\left(\frac{\partial w_{\alpha }^j}{\partial u_{\tau }^k}{\partial }_{\tau }{\phi }^k{\psi }_j\right), \end{gathered}$$

$$[P,P]({\psi }^1,{\psi }^2,{\psi }^3)=2[{\ell }_{P,{\psi }^1}(P({\psi }^2))({\psi }^3)+\text{cyclic}({\psi }^1,{\psi }^2,{\psi }^3)],$$

$$P=$$

$$[P,P]({\psi }^1,{\psi }^2,{\psi }^3)=T({\psi }^1,{\psi }^2,{\psi }^3)=\int T^{i_1{i}_2{\sigma }_2{i}_3{\sigma }_3}{\psi }_{i_1}^1{\partial }_{{\sigma }_2}{\psi }_{i_2}^2{\partial }_{{\sigma }_3}{\psi }_{i_1}^{3}dx$$

$${\tilde{\psi }}_{\alpha }^a={\partial }_x^{-1}(w_{\alpha }^i{\psi }_i^a),$$

$$\begin{gathered} e^{\alpha }{w}_{\alpha }^i{\partial }_x^{-1}\left(\frac{\partial w_{\alpha }^j}{\partial u_{\tau }^k}{\partial }_{\tau }(B^{kp\sigma }{\partial }_{\sigma }{\psi }_p^b+e^{\alpha }{w}_{\alpha }^k{\tilde{\psi }}_{\alpha }^b{)}^k{\psi }_j^c\right){\psi }_i^a= \\ -e^{\alpha }{\tilde{\psi }}_{\alpha }^a\left(\frac{\partial w_{\alpha }^j}{\partial u_{\tau }^k}{\partial }_{\tau }(B^{kp\sigma }{\partial }_{\sigma }{\psi }_p^b+e^{\alpha }{w}_{\alpha }^k{\tilde{\psi }}_{\alpha }^b{)}^k{\psi }_j^c\right) \end{gathered}$$

$$C^{\alpha \beta k}{\tilde{\psi }}_{\alpha }^a{\tilde{\psi }}_{\beta }^b{\psi }_k^c,$$

$$C^{\alpha kj\sigma }{\tilde{\psi }}_{\alpha }^a{\partial }_{\sigma }({\psi }_j^b){\psi }_k^c,$$

$$C^{kj\sigma i\tau }{\partial }_{\tau }({\psi }_i^a){\partial }_{\sigma }({\psi }_j^b){\psi }_k^c,$$

$$\{f_{\lambda }g\}=\sum _{s\le S}{C}_s(f,g){\lambda }^s,$$

Full text

1School of Mathematics, Statistics and Actuarial Science

University of Kent, Canterbury, UK

email: M.Casati@kent.ac.uk

2Dipartimento di Matematica

Università di Milano-Bicocca, Milano, Italy

email: paolo.lorenzoni@unimib.it

3Dipartimento di Matematica e Fisica “E. De Giorgi”,

Università del Salento and Sezione INFN di Lecce

via per Arnesano, 73100 Lecce, Italy

email: raffaele.vitolo@unisalento.it

1. 1 Introduction

2. 2 Jacobi identity and distributions

3. 2.1 The Jacobi identity

4. 2.2 The algorithm

5. 3 Jacobi identity and differential operators

6. 3.1 The Jacobi identity

7. 3.2 Dictionary: distributions and differential operators

8. 3.3 The algorithm

9. 4 Jacobi identity and Poisson Vertex Algebras

10. 4.1 The Jacobi identity

11. 4.2 Dictionary: Poisson Vertex Algebras and differential operators

12. 4.3 The algorithm

13. 5 Weakly nonlocal PBHT and distributions

14. 5.1 Calculation of the Jacobi identity

15. 5.2 Calculation of the reduced form

16. 5.3 The conditions

17. 6 Weakly nonlocal PBHT and differential operators

18. 6.1 Calculation of the Jacobi identity

19. 6.2 Calculation of the reduced form

20. 6.3 The conditions

21. 7 Weakly nonlocal PBHT and Poisson Vertex Algebras

22. 7.1 Computations with the master formula

23. 7.2 Projection onto the basis

24. 7.3 The conditions

25. 8 Concluding remarks

26. 9 Appendix: three different recipes for the Jacobi identity

1 Introduction

Poisson brackets play a fundamental role in Mathematical Physics. Many partial differential equations (PDEs) can be endowed by Poisson brackets by means of a Hamiltonian operator; the prototype of such equations is the Korteweg–de Vries (KdV) equation, that was shown to be a completely integrable Hamiltonian system in [28, 17]. More precisely, an autonomous system of PDEs in evolutionary form

[TABLE]

in two independent variables $t$, $x$ and in $n$ dependent variables $u^{1}$, …, $u^{n}$, where $u^{j}_{\sigma}$ are $\sigma$-times $x$-derivatives of $u^{j}$, is said to be Hamiltonian if it can be written as

[TABLE]

where $\mathchoice{\frac{\delta}{\delta u^{i}}}{\delta/\delta u^{i}}{\delta/\delta u^{i}}{\delta/\delta u^{i}}$ are the variational derivatives, $H=\int h(u^{j},u^{j}_{\sigma})\,dx$ is the Hamiltonian functional and $A^{ij}=A^{ij\sigma}\partial_{\sigma}$ is a matrix partial differential operator in total derivatives $\partial_{\sigma}=\partial_{x}\circ\cdots\circ\partial_{x}$ ($\sigma$-times). The operator $A$ is required to define a Poisson bracket on the space of local functionals as follows. Given two local functionals $F=\int f\,dx$ and $G=\int g\,dx$ where $f=f(u^{j},u^{j}_{\sigma})$ and similarly for $g$, the operation

[TABLE]

is required to be a Poisson bracket. This requirement implies strong differential constraints on the coefficients $P^{ij\sigma}=P^{ij\sigma}(u^{k},u^{k}_{\sigma})$ coming from the skew-symmetry and the Jacobi identity for $\{\,,\,\}_{P}$, as we will see later on.

It was soon realized that Poisson brackets for integrable systems should be extended to include nonlocal terms (or pseudo-differential operators) in the definition: one of the first examples is the Poisson bracket defined by the operator $u_{x}\partial_{x}^{-1}u_{x}$ for the Krichever–Novikov equation appeared in [26]. A general class of nonlocal Poisson brackets was introduced and studied by Ferapontov and Mokhov in [11] and further generalized in [12, 13]. This class is made of operators $P$ of the form

[TABLE]

where all coefficients are functions of the field variables $(u^{i})$ and $g$ is assumed to be non degenerate. The operator $P$ is homogeneous with respect to $x$-derivation in the sense of Dubrovin and Novikov [8], and generalizes the local homogeneous Poisson brackets defined therein. Purely nonlocal Hamiltonian operators generalizing the nonlocal structure of Krichever-Novikov equation have been studied in [22, 14].

The above class was further enlarged in [20] to Poisson brackets defined by operators of the form111We will use the form of the nonlocal part that is called “reduced” in [20].

[TABLE]

where the coefficients $B^{ij\sigma}$ and $w^{i}_{\alpha}$, $w^{j}_{\beta}$ can depend on all field variables $u^{k}$ and a finite number of derivatives $u^{k}_{\sigma}$, and $e^{\alpha}$ are constants. The operator $\partial_{x}^{-1}$ is defined to be

[TABLE]

acting on a suitable space of rapidly decreasing vector functions; the operator is formally skew-adjoint. Poisson brackets defined by operators of the type of $J$ are said to be weakly non-local [20]; Poisson brackets of the subclass of $P$ in (4) are said to be weakly non-local Poisson brackets of hydrodynamic type (PBHT).

Weakly non-local Poisson brackets play an important role in the theory of integrable systems. It was proved in [20] that the higher Hamiltonian operators for the KdV equation generated by composition with the Recursion operator (known since the late ’70) are all weakly non-local. It was conjectured that every hydrodynamic system of PDEs

[TABLE]

is Hamiltonian with respect to a suitable weakly nonlocal Poisson bracket of hydrodynamic type. As the dispersionless limit of a large class of evolutionary systems of PDEs consists in a system of the type of (7), it is natural to expect that weakly nonlocal Poisson brackets, and in particular those that can be obtained as a deformation of a weakly nonlocal Poisson bracket of hydrodynamic type, are of utmost interest.

Despite their importance, the use of nonlocal Poisson brackets is quite limited, especially if compared to local homogeneous Poisson brackets of hydrodynamic type. The most important reason for that is related to the much higher computational difficulties with respect to the local case. Indeed, the very first moment when such difficulties are met is the verification of the Jacobi identity for the coefficients of a candidate differential operator $P$.

More in detail, the problem is to reach a divergence-free form of the Jacobi expression $\{\{F,G\}_{P},H\}_{P}+\text{cyclic}$ when there are nonlocal terms, in order to check (or impose) the vanishing of a necessary and sufficient set of differential expressions of the coefficients of $P$.

In the literature we can find (at least) three approaches to the Hamiltonian formalism for PDEs:

the approach with distributions, quite spread in Theoretical Physics [8, 9, 10]; 2. 2.

the approach with differential operators, more common between pure mathematicians [2, 5, 23]; 3. 3.

a new algebraic approach based on Poisson Vertex Algebras, introduced in [1] for local Poisson brackets and later extended to nonlocal Poisson brackets in [4].

The third approach relies on an algebraic formulation in order to give a mathematically well-grounded basis to the theory of nonlocal Poisson brackets. In particular, a canonical form of pseudo-differential operators is achieved by the systematic use of formal series (see [23]), which, on the other hand, lay outside the more traditional differential-geometric picture (see the Conclusions for further remarks).

The aim of this work is to illustrate an algorithmic procedure to compute the Jacobi identity for weakly nonlocal Poisson brackets in the three formalisms above.

In the case of distributions, the algorithm has been introduced in [19] in order to study the bi-Hamiltonian structure of the Liouville and sine-Gordon PDEs. In the case of differential operators, the algorithm is shown here for the first time thanks to the explicit correspondence between the languages of distributions and differential operators. In the case of Poisson Vertex Algebra, the algorithm has been described in [4] (a nontrivial example can be found in [3]).

In all cases, the algorithm consists in the reduction of the Jacobi identity to a reduced form, in such a way to make the identity effectively computable. This is practically achieved by means of identities between distributions, or integration by parts, or algebraic manipulations. In all cases the interesting fact is that the computations lead to the same conditions in the end, with a precise correspondence that is explicitly written as a ‘dictionary’ between the three formalisms.

In order to illustrate the algorithm and the correspondence between the different formalisms we will consider the case of weakly nonlocal Poisson brackets of hydrodynamic type. We will obtain the following conditions on the coefficients in $P^{ij}$ (assuming $\det g^{ij}\neq 0$), namely

[TABLE]

where $\nabla$ is the linear connection with Christoffel symbols $\Gamma^{k}_{ij}=-g_{il}\Gamma^{lk}_{j}$ and $R^{ij}_{kh}=g^{is}R^{j}_{skh}$ is the Riemannian curvature. The above conditions, first obtained in [12] (for details of computations see [24]), admit the following interpretation: the first three equations tell us that the functions $g^{ij}$ can be interpreted as the contravariant components of a (pseudo)-euclidean metric and $\Gamma^{ij}_{k}$ as Christoffel symbols of the corresponding Levi-Civita connection, while the remaining equations coincide with the classical Gauss-Peterson-Mainardi-Codazzi equations for a hypersurface in a (pseudo)-Euclidean space. The (pseudo)-metric $g$ and the affinor $w$ can be identified with the induced metric and the Weingarten operator respectively.

The paper is organized as follows. In Sections $2,3$ and $4$ we explain the algorithm to check Jacobi identity in the three formalisms and we write a sort of dictionary between the three approaches. The remaining sections are devoted to illustrate the algorithm in the case of weakly nonlocal Poisson brackets of hydrodynamic type. In order to simplify the computations we consider the subcase where the nonlocal tail contains a single term but the computations can be performed in the same way in the general case.

Acknowledgments.

We thank E. Ferapontov, J. Krasil’shchik, M. Pavlov, D. Valeri, A. Verbovetsky, Y. Zhang for useful discussions. M. C. is supported by EPSRC grant EP/P012698/1. P.L. is supported by MIUR - FFABR funds 2017 and by research funds of the University of Milano - Bicocca. P.L is grateful to the Department of Mathematics and Physics “E. De Giorgi” of the Università del Salento for the kind hospitality and for supporting his visit. R.V. acknowledges the support of the Department of Mathematics and Physics “E. De Giorgi” of the Università del Salento and of the Istituto Nazionale di Fisica Nucleare, Sezione di Lecce, IS-CSN4 Mathematical Methods of Nonlinear Physics. Finally, we thank GNFM for supporting activities that contributed to the research reported in this paper.

2 Jacobi identity and distributions

Here we briefly introduce weakly nonlocal Poisson brackets as distributions and describe the algorithm for bringing the Jacobi identity to a reduced canonical form.

2.1 The Jacobi identity

Following [20], we consider weakly nonlocal Poisson brackets of the form

[TABLE]

where $\nu(x-y)=\frac{1}{2}\operatorname{sgn}(x-y)$.

The Jacobi identity

[TABLE]

can be written as [10]

[TABLE]

where $P^{ij}_{x,y}=\{u^{i}(x),u^{j}(y)\}_{P}$. The vanishing of the distribution $J^{ijk}_{xyz}$ means that for any choice of the test functions $p_{i}(x),q_{j}(y),r_{k}(z)$ the triple integral

[TABLE]

should vanish.

2.2 The algorithm

Following [19], we present a procedure to collect together all terms which are related by a distributional identity. We call the result of this procedure the reduced form of the Jacobi identity.

Using the identity

[TABLE]

and its two obvious analogues obtained by a cyclic permutation of the variables, together with their differential consequences, we can eliminate all terms containing $\nu(z-y)\delta^{(n)}(z-x)$, $\nu(y-x)\delta^{(n)}(y-z)$, $\nu(x-z)\delta^{(n)}(x-y)$ producing nonlocal terms containing $\nu(x-y)\delta^{(n)}(x-z)$, $\nu(z-x)\delta^{(n)}(z-y)$, $\nu(y-z)\delta^{(n)}(y-x)$ and additional local terms. 2. 2.

Using the identity

[TABLE]

(and its cyclic permutations) we can eliminate the dependence on $z$ in the coefficients of the terms containing $\nu(x-y)\delta^{(n)}(x-z)$, the dependence on $y$ in the coefficients of the terms containing $\nu(z-x)\delta^{(n)}(z-y)$ and the dependence on $x$ in the coefficients of the terms containing $\nu(y-z)\delta^{(n)}(y-x)$. After the first two steps the nonlocal part of $J^{ijk}_{xyz}$ has the form

[TABLE] 3. 3.

The local part of $J^{ijk}_{xyz}$ (which contain also some additional terms coming from the nonlocal part) can be treated as usual and reduced to the form

[TABLE]

using the identities (and their differential consequences)

[TABLE]

and the identities (14).

It is easy to check that no further simplifications are possible. We will see later that the fulfillment of the Jacobi identity turns out to be equivalent to the vanishing of each coefficient in the reduced form.

3 Jacobi identity and differential operators

3.1 The Jacobi identity

The conditions under which the bracket (3) is a Poisson bracket can be written as requirements on the differential operator $P$ (5). We recall that the operator $P$ is a variational bivector [2, 5, 23], hence it is defined up to total divergencies. We consider Poisson brackets defined by differential operators of the form $P^{ij}=B^{ij\sigma}\partial_{\sigma}+e^{\alpha}w^{i}_{\alpha}\partial_{x}^{-1}w^{j}_{\alpha}$ (5). Then, it is well-known that

• •

the skew-symmetry of $\{,\}_{J}$ is equivalent to the formal skew-adjointness of $P$, $P^{*}=-P$;

• •

the Jacobi identity for $\{,\}_{P}$ is equivalent to the vanishing of the Schouten bracket $[P,P]=0$.

Note that the Schouten bracket of two variational bivectors is a variational three-vector, i.e., it is a skew-symmetric differential operator with three arguments whose value is defined up to total divergencies.

In coordinates, the formal adjoint $P^{*}$ of the operator $P$ is

[TABLE]

We stress that the non-local summand of weakly nonlocal operators is skew-adjoint by construction: we have $(\partial_{x}^{-1})^{*}=-\partial_{x}^{-1}$.

Let us denote by $\ell_{P,\psi}(\varphi)$ the linearization of the (coefficients of the) operator $P$. We have the following coordinate expressions:

[TABLE]

where we used (105) and the fact that $\partial_{x}^{-1}$ commutes with linearization. Then, we have the following expression for the Schouten bracket:

[TABLE]

where square brackets denote the fact that the expression is calculated up to total divergencies. We observe that the expression of the Schouten bracket of two operators can be written in different ways, which differ up to total divergencies. In the Appendix we wrote two more expressions that are more commonly used in the formalism of differential operators, together with a proof of their equivalence.

3.2 Dictionary: distributions and differential operators

Here we present a dictionary between the language of operators and the language of distributions for the reader’s convenience. The calculus with distributions is defined in [10, Subsect. 2.3].

The following notation for a local multivector coincide:

[TABLE]

In particular the value of the multivector in the distributional notation is obtained by evaluating it on test vector functions of the arguments $x^{1}$, …, $x^{k}$. The above correspondence can easily be extended between the nonlocal multivectors (9) and (5). Then, it is clear that the expressions (11) and (20) coincide up to the evaluation on test vector functions.

3.3 The algorithm

The result of the Schouten bracket $[P,P]$ (20) is a three-vector and has the following coordinate expression:

[TABLE]

$T^{i_{1}i_{2}\sigma_{2}i_{3}\sigma_{3}}$ is defined up to total divergencies: this means that three-vectors of the type $\partial_{x}\left(T^{i_{1}\sigma_{1}i_{2}\sigma_{2}i_{3}\sigma_{3}}\partial_{\sigma_{1}}\psi^{1}_{i_{1}}\partial_{\sigma_{2}}\psi^{2}_{i_{2}}\partial_{\sigma_{3}}\psi^{3}_{i_{1}}\right)$ are zero. It immediately follows that a local three-vector which is of order zero in one of its arguments is zero if and only if its coefficients are zero.

The algorithm in Section 2.2 translates into the language of differential operators as follows. Let us introduce the notation

[TABLE]

where $a$ refers to the particular argument of the operator. Then, the vector functions $\psi^{1}$, $\psi^{2}$, $\psi^{3}$ play the role of test vector functions of the variables $x$, $y$, $z$ in the language of distributions.

The first step in Section 2.2 is not needed in the differential operator formalism, as it boils down to a change in the variable of integration (and its differential consequences). 2. 2.

The second step aims at bringing the nonlocal part of the three-vector in the reduced form (15). To this aim, we remark that the reduced form of the distributions implies that there is no distribution of the type $\nu(x-y)$ acting on two vector test functions. This means effecting the following substitution (up to total divergencies)

[TABLE]

After such a substitution, we observe that the generic summands of (19) are of three types:

[TABLE]

where $C$’s are functions of $(u^{i},u^{i}_{\sigma})$. The reduced form of the three-vector in the formalism of differential operators amounts at bringing the operator to a canonical form where the arguments $\psi^{a}$, $\psi^{b}$, $\psi^{c}$ are a fixed sequence of integers (say, $1$, $2$, $3$) or its cyclic permutations (in the previous example, $3$, $1$, $2$ and $2$, $3$, $1$). This task can always be achieved by integration by parts that will produce the required terms plus extra terms. 3. 3.

The third step of the algorithm amounts at bringing the local part into a reduced form. This is achieved with the usual procedure of integrating by parts the three-vector with respect to one distinguished argument (say $\psi^{3}$) in such a way that the result will be of order zero in that argument.

4 Jacobi identity and Poisson Vertex Algebras

4.1 The Jacobi identity

Definition 1**.**

A (nonlocal) Poisson vertex algebra (PVA) is a differential algebra $(\mathcal{A},\partial)$ endowed with a derivation $\partial$ and a bilinear operation $\{\cdot_{\lambda}\cdot\}\colon\mathcal{A}\otimes\mathcal{A}\to\mathbb{R}((\lambda^{-1}))\otimes\mathcal{A}$ called a (nonlocal) $\lambda$-bracket, satisfying the following set of properties:

$\{\partial f_{\lambda}g\}=-\lambda\{f_{\lambda}g\}$ (left sesquilinearity), 2. 2.

$\{f_{\lambda}\partial g\}=(\lambda+\partial)\{f_{\lambda}g\}$ (right sesquilinearity), 3. 3.

$\{f_{\lambda}gh\}=\{f_{\lambda}g\}h+\{f_{\lambda}h\}g$ (left Leibnitz property), 4. 4.

$\{fg_{\lambda}h\}=\{f_{\lambda+\partial}h\}g+\{g_{\lambda+\partial}h\}f$ (right Leibnitz property), 5. 5.

$\{g_{\lambda}f\}=-{}_{\to}\{f_{-\lambda-g}g\}$ (PVA skew-symmetry), 6. 6.

$\{f_{\lambda}\{g_{\mu}h\}\}-\{g_{\mu}\{f_{\lambda}h\}\}=\{\{f_{\lambda}g\}_{\lambda+\mu}h\}$ (PVA-Jacobi identity).

In the notation for the bracket, the symbol separating the two arguments is the formal parameter of the expansion. We denote

[TABLE]

with $C_{s}(f,g)\in\mathcal{A}$; the argument signals that each of the coefficients of the expansion depends on the two elements $f$ and $g$ in $\mathcal{A}$. Such an expansion is bounded by $0\leq s\leq S$ for local PVAs and is not bounded from below for nonlocal PVAs.

The special notation used on the RHS of Property 4 is to be understood as

[TABLE]

Similarly, the RHS of Property 5 (the skewsymmetry) reads

[TABLE]

For a nonlocal $\lambda$ bracket, the three terms of PVA-Jacobi identity do not necessarily belong to the same space, because of the double infinite expansion of the brackets (in terms of $(\lambda,\mu)$, $(\mu,\lambda)$ and $(\lambda,\lambda+\mu)$, respectively). A bracket is said to be admissible if all the three terms can be (not uniquely) expanded as

[TABLE]

Only admissible brackets can define a nonlocal PVA. We denote the space where the PVA-Jacobi identity of admissible brackets takes values by $V_{\lambda,\mu}$. This space can be decomposed by the total degree $d$ in $(\lambda,\mu,\lambda+\mu)$; elements of each homogeneous component $V^{(d)}_{\lambda,\mu}$ can be uniquely expressed in the basis [6]

[TABLE]

This filtration in the total degree $d$ and the subsequent choice of a basis plays a crucial role in obtaining the normal form for the PVA-Jacobi identity.

The main result used to perform most of the computations is the so called master formula. Under the hypothesis that the differential algebra $\mathcal{A}$ is generated by the elements $(u^{i})$, the $\lambda$-bracket between any two elements of $\mathcal{A}$ is explicitly given by

[TABLE]

Thus, the structure of a PVA is defined by the matrix of the $\lambda$ brackets between the generators $\{u^{i}_{\lambda}u^{j}\}=P^{ji}(\lambda)$.

In the nonlocal case, expressions such as $(\lambda+\partial)^{p}$ for $p<0$ arise in $P^{ji}(\lambda+\partial)$ from the master formula (29). In such cases, the rigorous approach – working for any kind of nonlocality – is to expand the negative powers of $(\lambda+\partial)$ as

[TABLE]

In the weakly nonlocal case this can be avoided, relying only on Properties (1)-(4) of the lambda bracket. More details on this will be provided in Section 7.2.

4.2 Dictionary: Poisson Vertex Algebras and differential

operators

The connection between the theory of PVA and Hamiltonian operator is given by Theorem 4.8 in [6, pag. 261]. In short, there is a 1-1 correspondence between $\lambda$-brackets of a (nonlocal) PVA and (pseudo)differential Hamiltonian operators; the entries of the matrix $P^{ji}(\lambda)$ correspond to the differential operator $P^{ij}$ (5) after the formal replacement of $\lambda$ by $\partial$.

More precisely, the equivalence between the expression of the Poisson bracket (3) and the expression of a $\lambda$-bracket according with the master formula (29) is:

[TABLE]

using (29). The PVA-Jacobi identity for a triple of generators $(u^{i},u^{j},u^{k})$ can also be expressed by means of differential operators. First of all, we compute the PVA-Jacobi identity using the master formula; we have

[TABLE]

The PVA-Jacobi identity is $J^{ijk}_{\lambda,\mu}(P,P)=$ (32) $-$ (33) $-$ (34) $=0$. We evaluate the expression on three covectors $\psi^{1}_{i}\psi^{2}_{j}\psi^{3}_{k}$, and regard each power of $\lambda$ as derivations acting on $\psi^{1}$, and each power of $\mu$ as derivations acting on $\psi^{2}$. Then, the three summands correspond to

[TABLE]

respectively, and the PVA-Jacobi identity is the vanishing of the Schouten bracket $[P,P]$ in the form of (109).

4.3 The algorithm

For the local case, the expression of the PVA-Jacobi identity is a polynomial in $\lambda$ and $\mu$, and the vanishing of the coefficients of $\lambda^{p}\mu^{q}$ corresponds to the vanishing of the coefficients for $\partial^{p}(\psi^{1}_{i})\partial^{q}(\psi^{2}_{j})\psi^{3}_{k}$.

In the nonlocal case, the PVA-Jacobi identity is a Laurent series in $\lambda^{-1}$, $\mu^{-1}$ and $(\lambda+\mu)^{-1}$ living in the space $V_{\lambda,\mu}$ defined in Section 4.1: in the weakly nonlocal case, these coefficients come, respectively, from the expansion of $(\lambda+\partial)^{-1}$, $(\mu+\partial)^{-1}$, $(\lambda+\mu+\partial)^{-1}$.

From the computation of the PVA-Jacobi identity we obtain seven types of terms including one or two nonlocal factors, together with the pure local terms; each of them corresponds to the types of summands in the three-vector of the Schouten identity in (23), as detailed in (26) and following. They are

$A^{ijk}\lambda^{p}\mu^{q}$ with $p,q\geq 0$, corresponding to $\partial^{p}(\psi_{i}^{1})\partial^{q}(\psi_{j}^{2})\psi^{3}_{k}$; 2. 2.

$w^{k}(\lambda+\mu+\partial)^{-1}A^{ij}\lambda^{p}$ with $p\geq 0$, corresponding to $\partial^{p}(\psi_{i}^{1})\psi_{j}^{2}\tilde{\psi}^{3}$; 3. 3.

$w^{k}(\lambda+\mu+\partial)^{-1}A^{ij}\mu^{p}$ with $p>0$, corresponding to $\psi_{i}^{1}\partial^{p}(\psi_{j}^{2})\tilde{\psi}^{3}$; 4. 4.

$[(\lambda+\partial)^{-1}w^{i}]A^{jk}\mu^{p}$ with $p\geq 0$, corresponding to $\tilde{\psi}^{1}\partial^{p}(\psi_{j}^{2})\psi^{3}_{k}$; 5. 5.

$[(\mu+\partial)^{-1}w^{j}]A^{ki}\lambda^{p}$ with $p\geq 0$, corresponding to $\partial^{p}(\psi_{i}^{1})\tilde{\psi}^{2}\psi^{3}_{k}$; 6. 6.

$w^{k}(\lambda+\mu+\partial)^{-1}A^{j}(\lambda+\partial)^{-1}w^{i}$, corresponding to $\tilde{\psi}^{1}\psi_{j}^{2}\tilde{\psi}^{3}$; 7. 7.

$w^{k}(\lambda+\mu+\partial)^{-1}A^{i}(\mu+\partial)^{-1}w^{j}$, corresponding to $\psi_{i}^{1}\tilde{\psi}^{2}\tilde{\psi}^{3}$; 8. 8.

$[(\lambda+\partial)^{-1}w^{i}]A^{k}[(\mu+\partial)^{-1}w^{j}]$, corresponding to $\tilde{\psi}^{1}\tilde{\psi}^{2}\psi^{3}_{k}$.

The square brackets denote that the differential operators obtained by the expansion of the pseudodifferential operator do not act outside them.

Note that the expansion of the terms 3 and 7 is not expressed in the basis for $V_{\lambda,\mu}$ we have chosen; on the other hand, terms 3 and 5 do not correspond to the choice of coefficients for the normalization algorithm of the previous Sections (when one takes the cyclic ordering $\tilde{\psi}^{a}\partial^{p}(\psi^{b})\psi^{c}$).

We give a different treatment of the terms including at most one nonlocal expression and of the ones with two: in the first case, we bring them to a form whose expansion is automatically expressed in our chosen basis for $V_{\lambda,\mu}$; in the second case, we show that the vanishing of the term 7, together with the other ones, is equivalent to the vanishing of the corresponding terms in the expansion on the basis.

Finally, we comment on the equivalence between the vanishing of the PVA-Jacobi identity on our chosen basis and as a result of the normalization algorithm of the previous Sections.

Proposition 2**.**

The terms of type $w^{k}(\lambda+\mu+\partial)^{-1}A^{ij}\mu^{p}$ can be brought to a combination of terms of type $A^{ijk}\lambda^{p}\mu^{q}$ with $p,q\geq 0$ and $w^{k}(\lambda+\mu+\partial)^{-1}A^{ij}\lambda^{p}$, reducing the PVA-Jacobi identity to the expansion of seven terms.

Proof.

From the expansion $(\lambda+\mu+\partial)^{p}=\sum_{l=0}^{p}\binom{p}{l}\mu^{p-l}(\lambda+\partial)^{l}$ we can rewrite a term of the form $w^{k}(\lambda+\mu+\partial)^{-1}A^{ij}\mu^{p}$ as

[TABLE]

which gives

[TABLE]

The expression in the square bracket has top degree $p-1$ in $\mu$. Repeating the operation we obtain only local terms or terms of the type 2. ∎

Theorem 3**.**

The PVA-Jacobi identity, expressed using terms of the type $1$, $2$, $4$ – $8$ as above, can always be expressed in the space $V^{(d)}_{\lambda,\mu}$. This latter expression vanishes if and only if the former does.

Proof.

Expressing the PVA-Jacobi identity in the space $V^{(d)}_{\lambda\,\mu}$, for all for $d\leq D$, means expanding it on the basis $\lambda^{p}\mu^{d-p}$, $p\in\mathbb{Z}$ and $(\lambda+\mu)^{-p}\lambda^{d+p}$, $p>0$. Terms of type 1 do not need to be expanded, as they are already expressed in the basis for $V^{(d)}_{\lambda,\mu}$, $d\geq 0$.

For the types with one nonlocal term, namely $2$, $4$ and $5$ in the previous list, the expansions of the pseudodifferential operators give the series

[TABLE]

which are in our chosen basis of $V^{(d)}_{\lambda,\mu}$, for $d\leq p-1$. The vanishing of the $t=0$ term in the expansion is a sufficient and necessary condition for the vanishing of the whole series: all the subsequent terms in the expansion vanish if the first one does, and it must vanish because that is the only one in $V^{(p-1)}_{\lambda,\mu}$ containing the factor $(\lambda+\mu)^{-1}$ (resp. $\lambda^{-1}$ and $\mu^{-1}$). It is hence enough to check (or impose) the vanishing of the coefficients $A$ or $w$. However, the vanishing of the $w$ terms coincides with the dropping of the nonlocal part of the $\lambda$ bracket, so the condition is only on $A$’s.

A similar point can be made for the types 6 and 8 with the double nonlocality: their expansion is expressed in our chosen basis and starts, respectively, with $(\lambda+\mu)^{-1}\lambda^{-1}$ and $\lambda^{-1}\mu^{-1}$ in $V^{(-2)}_{\lambda,\mu}$. The expansion of the term 7 starts with $A^{i}w^{k}w^{j}(\lambda+\mu)^{-1}\mu^{-1}$, which is not an element in the basis of $V^{(-2)}_{\lambda,\mu}$. However, this is a term we can rearrange as an infinite series

[TABLE]

for some fixed constants $c_{m}$.

Note that elements in $V^{(-2)}_{\lambda,\mu}$ could be obtained by the expansions (for $t=1$) of the previous terms with only one nonlocality. However, the vanishing of the elements in $V^{(-1)}_{\lambda,\mu}$ implies their vanishing, too, and hence we can focus on the terms arising from the expansion of double nonlocalities only.

It is straightforward to see that we get only one expression in front of $(\lambda+\mu)^{-1}\lambda^{-1}$ (from type 6) and $(\lambda+\mu)^{-2}$ (from our rearrangement of type 8); on the other hand, there could be two sources of terms of the form $\lambda^{-1}\mu^{-1}$. The vanishing of either $A$ or $w$ for all $i,j,k$ in the first two cases is a necessary and sufficient condition; once that this has been imposed or checked, the only surviving class of terms of the form $\lambda^{-1}\mu^{-1}$ comes from the expansion of 7.

Since the vanishing of $w$ is equivalent to simply dropping the nonlocal term of the operator, the condition we need to consider is only the vanishing of the expressions $A$’s. ∎

Remark 4**.**

The above theorem has two important consequences.

This algorithm always yields a divergence-free form of the Jacobi identity; this means that the Jacobi identity holds if and only if the coefficient of the Laurent series in the spaces $V^{(d)}_{\lambda,\mu}$ vanish. 2. 2.

There is no need to expand in Laurent series: indeed, the expansion is always ruled by the zeroth-order coefficients, which are just the coefficients of the terms $2$, $4$ – $8$.

Remark 5**.**

Writing the PVA-Jacobi identity on our chosen basis of $V^{(d)}_{\lambda,\mu}$ for $d\geq-2$ yields a different result than the one obtained with the algorithm described in Section 3.3. For the terms with one nonlocality, indeed, the PVA-Jacobi identity produces the coefficients corresponding to $\partial^{p}(\psi_{i}^{1})\psi_{j}^{2}\tilde{\psi}^{3}$, $\partial^{p}(\psi_{j}^{2})\psi_{k}^{3}\tilde{\psi}^{1}$ and $\partial^{p}(\psi_{i}^{1})\psi_{j}^{3}\tilde{\psi}^{2}$, while the latter is replaced by $\partial^{p}(\psi_{i}^{3})\psi_{j}^{1}\tilde{\psi}^{2}$ in Section 3.3.

Nevertheless, the sets of condition given by the vanishing of the coefficients in front of the terms obtained with the two different algorithms are equivalent. Let us demonstrate it assuming that the terms of type 5 are

[TABLE]

corresponding to

[TABLE]

This latest expression is equivalent, up to total derivatives, to

[TABLE]

The vanishing of expression (43) at top degree implies the vanishing of the lower degree coefficients, being hence equivalent to the vanishing of (41).

The same result can be obtained in the framework of Poisson vertex algebras introducing the symbol $\nu=-\lambda-\mu-\partial$, representing derivations acting on $\psi^{3}$ [7, Section 4.1].

5 Weakly nonlocal PBHT and distributions

5.1 Calculation of the Jacobi identity

In this section we will consider, as an example, weakly nonlocal Poisson bracket of hydrodynamic type, of the form (4). In the language of distribution it has the form

[TABLE]

(we will use only one ‘tail summand’ to make calculations simpler) where $\delta_{xy}=\delta(x-y)$ e $\nu_{xy}=\nu(x-y)$. We assume $g$ to be non degenerate. In what follows, an index after a comma denotes a partial derivative with respect to the corresponding field variable, e.g. $g^{ij}_{,k}=\partial g^{ij}/\partial u^{k}$.

From the skew-symmetry the two conditions (8a), (8b) follow, namely: $g^{ij}=g^{ji}$ and $g^{ij}_{,k}=\Gamma^{ij}_{k}+\Gamma^{ji}_{k}$. We apply now the reducing procedure explained in Section 2.2. Since $P^{ij}_{xy}$ depend only on $u(x)$ and $u_{x}$ each sum in (11) contains only two terms. The Jacobi identity can be rewritten as

[TABLE]

5.2 Calculation of the reduced form

The first summand in (45) is

[TABLE]

The coefficients of the reduced form are listed below.

• •

The coefficient of $\delta^{\prime}_{xy}\delta^{\prime}_{xz}$ is $g^{lk}g^{ij}_{,l}$.

• •

The coefficient of $\nu_{xy}\nu_{xz}$ is $w^{i}_{s,l}w^{l}_{m}u^{m}_{x}u^{s}_{x}w^{j}_{t}u^{t}_{y}w^{k}_{n}u^{n}_{z}$.

• •

The coefficient of $\delta^{\prime}_{xy}\delta_{xz}$ is $g^{ij}_{,l}\Gamma^{lk}_{t}u^{t}_{x}$.

• •

The coefficient of $\delta_{xy}\delta^{\prime}_{xz}$ is $g^{lk}\Gamma^{ij}_{s,l}u^{s}_{x}$.

• •

The coefficient of $\delta_{xy}\delta_{xz}$ is $\Gamma^{ij}_{s,l}\Gamma^{lk}_{t}u^{t}_{x}u^{s}_{x}-g^{ij}_{,l}w^{l}_{s}u^{s}_{x}w^{k}_{t}u^{t}_{x}$.

• •

The coefficient of $\delta_{yx}\nu_{yz}$ is $-\partial_{y}(g^{ij}_{,l}w^{l}_{s}u^{s}_{y})w^{k}_{t}u^{t}_{z}+\Gamma^{ij}_{s,l}w^{l}_{r}u^{r}_{x}u^{s}_{x}w^{k}_{t}u^{t}_{z}$.

• •

The coefficient of $\delta_{xz}\nu_{xy}$ is $w^{i}_{s,l}\Gamma^{lk}_{r}u^{r}_{x}u^{s}_{x}w^{j}_{t}u^{t}_{y}$.

• •

The coefficient of $\delta^{\prime}_{yx}\nu_{yz}$ is $-g^{ij}_{,l}w^{l}_{s}u^{s}_{y}w^{k}_{t}u^{t}_{z}$.

• •

The coefficient of $\delta^{\prime}_{xz}\nu_{xy}$ is $g^{lk}w^{i}_{s,l}u^{s}_{x}w^{j}_{t}u^{t}_{y}$.

The second summand in (45) is

[TABLE]

The coefficients of the reduced form are listed below:

• •

The coefficient of $\delta_{xy}\delta_{xz}$ is ${w^{i}_{s}u^{s}_{x}g^{lk}w^{j}_{t,l}u^{t}_{x}}$.

• •

The coefficient of $\nu_{xz}\delta^{\prime}_{zy}$ is $-w^{i}_{s}u^{s}_{x}g^{lk}w^{j}_{t,l}u^{t}_{z}$.

• •

The coefficient of $\nu_{xz}\delta_{zy}$ is ${-w^{i}_{s}u^{s}_{x}\partial_{z}\left(g^{lk}w^{j}_{t,l}u^{t}_{z}\right)+w^{i}_{s}u^{s}_{x}w^{j}_{t,l}\Gamma^{lk}_{m}u^{m}_{y}u^{t}_{y}}$.

• •

The coefficient of $\nu_{xy}\nu_{yz}$ is ${w^{i}_{s}u^{s}_{x}w^{j}_{t,l}w^{l}_{m}u^{m}_{y}u^{t}_{y}w^{k}_{n}u^{n}_{z}}$.

The third summand in (45) is

[TABLE]

The coefficients of the reduced form are listed below.

• •

The coefficient of $\delta^{\prime\prime}_{xy}\delta_{xz}$ is $-g^{lj}g^{ki}_{,l}$.

• •

The coefficient of $\delta^{\prime}_{xy}\delta^{\prime}_{xz}$ is $-g^{lj}g^{ki}_{,l}$.

• •

The coefficient of $\delta^{\prime}_{xy}\delta_{xz}$ is $-\partial_{x}\left(g^{lj}g^{ki}_{,l}\right)-g^{ki}_{,l}\Gamma^{lj}_{t}u^{t}_{x}+g^{lj}\Gamma^{ki}_{s,l}u^{s}_{x}$.

• •

The coefficient of $\delta_{xy}\delta^{\prime}_{xz}$ is $-g^{ki}_{,l}\Gamma^{lj}_{t}u^{t}_{x}$.

• •

The coefficient of $\delta_{xy}\delta_{xz}$ is $-\partial_{x}\left(g^{ki}_{,l}\Gamma^{lj}_{t}u^{t}_{x}\right)+\Gamma^{ki}_{s,l}\Gamma^{lj}_{t}u^{t}_{x}u^{s}_{x}-g^{ki}_{,l}w^{l}_{s}u^{s}_{x}w^{j}_{t}u^{t}_{x}$.

• •

The coefficient of $\delta_{xz}\nu_{xy}$ is $-\partial_{x}\left(g^{ki}_{,l}w^{l}_{s}u^{s}_{x}\right)w^{j}_{t}u^{t}_{y}+\Gamma^{ki}_{s,l}w^{l}_{r}u^{r}_{x}u^{s}_{x}w^{j}_{t}u^{t}_{y}$.

• •

The coefficient of $\delta_{zy}\nu_{zx}$ is $w^{k}_{s,l}\Gamma^{lj}_{t}u^{t}_{z}u^{s}_{z}w^{i}_{t}u^{t}_{x}$.

• •

The coefficient of $\delta^{\prime}_{zy}\nu_{zx}$ is $g^{lj}w^{k}_{s,l}u^{s}_{z}w^{i}_{t}u^{t}_{x}$.

• •

The coefficient of $\delta^{\prime}_{xz}\nu_{xy}$ is $-g^{ki}_{,l}w^{l}_{s}u^{s}_{x}w^{j}_{t}u^{t}_{y}$.

• •

The coefficient of $\nu_{zx}\nu_{zy}$ is $w^{k}_{s,l}w^{l}_{m}u^{m}_{z}u^{s}_{z}w^{i}_{t}u^{t}_{x}w^{j}_{n}u^{n}_{y}$.

The fourth summand in (45) is

[TABLE]

The coefficients of the reduced form are listed below:

• •

The coefficient of $\delta_{xy}\delta_{xz}$ is $w^{k}_{s}u^{s}_{x}g^{lj}w^{i}_{t,l}u^{t}_{x}$.

• •

The coefficient of $\nu_{zy}\delta^{\prime}_{yx}$ is $-w^{k}_{s}u^{s}_{z}g^{lj}w^{i}_{t,l}u^{t}_{y}$.

• •

The coefficient of $\nu_{zy}\delta_{xy}$ is $-w^{k}_{s}u^{s}_{z}\partial_{y}\left(g^{lj}w^{i}_{t,l}u^{t}_{y}\right)-w^{k}_{s}u^{s}_{z}w^{i}_{t,l}\Gamma^{lj}_{m}u^{m}_{x}u^{t}_{x}$.

• •

The coefficient of $\nu_{zx}\nu_{xy}$ is $w^{k}_{s}u^{s}_{z}w^{i}_{t,l}w^{l}_{m}u^{m}_{x}u^{t}_{x}w^{j}_{n}u^{n}_{y}$.

The fifth summand in (45) is

[TABLE]

The coefficients of the reduced form are

• •

The coefficient of $\delta^{\prime}_{xy}\delta^{\prime}_{xz}$ is $-g^{li}g^{jk}_{,l}$.

• •

The coefficient of $\delta_{xy}\delta^{\prime\prime}_{xz}$ is $-g^{li}g^{jk}_{,l}$.

• •

The coefficient of $\delta_{xy}\delta^{\prime}_{xz}$ is $-\partial_{x}\left(g^{li}g^{jk}_{,l}\right)+g^{jk}_{,l}\Gamma^{li}_{t}u^{t}_{y}-g^{li}\Gamma^{jk}_{s,l}u^{s}_{x}$.

• •

The coefficient of $\delta^{\prime}_{xy}\delta_{xz}$ is $-g^{li}\Gamma^{jk}_{s,l}u^{s}_{x}$.

• •

The coefficient of $\delta_{xy}\delta_{xz}$ is $-\partial_{x}\left(g^{li}\Gamma^{jk}_{s,l}u^{s}_{x}\right)+\Gamma^{jk}_{s,l}\Gamma^{li}_{t}u^{t}_{x}u^{s}_{x}-g^{jk}_{,l}w^{l}_{s}u^{s}_{x}w^{i}_{t}u^{t}_{x}$.

• •

The coefficient of $\delta_{zy}\nu_{zx}$ is $-\partial_{z}\left(g^{jk}_{,l}w^{l}_{s}u^{s}_{z}\right)w^{i}_{t}u^{t}_{x}+\Gamma^{jk}_{s,l}w^{l}_{s}u^{s}_{y}u^{s}_{y}w^{i}_{t}u^{t}_{x}$.

• •

The coefficient of $\delta^{\prime}_{zy}\nu_{zx}$ is $-g^{jk}_{,l}w^{l}_{s}u^{s}_{z}w^{i}_{t}u^{t}_{x}$.

• •

The coefficient of $\delta^{\prime}_{yx}\nu_{yz}$ is $g^{li}w^{j}_{s,l}u^{s}_{y}w^{k}_{t}u^{t}_{z}$.

• •

The coefficient of $\nu_{yz}\nu_{yx}$ is $w^{j}_{s,l}w^{l}_{m}u^{m}_{y}u^{s}_{y}w^{k}_{t}u^{t}_{z}w^{i}_{n}u^{n}_{x}$.

The sixth summand in (45) is

[TABLE]

The coefficients of the reduced form are listed below

• •

The coefficient of $\delta_{xy}\delta_{xz}$ is $w^{j}_{s}u^{s}_{x}g^{li}w^{k}_{t,l}u^{t}_{x}$.

• •

The coefficient of $\nu_{xy}\delta^{\prime}_{xz}$ is $w^{j}_{s}u^{s}_{y}g^{li}w^{k}_{t,l}u^{t}_{x}$.

• •

The coefficient of $\nu_{xy}\delta_{xz}$ is $+w^{j}_{s}u^{s}_{y}\partial_{x}\left(g^{li}w^{k}_{t,l}u^{t}_{x}\right)-w^{j}_{s}u^{s}_{y}w^{k}_{t,l}\Gamma^{li}_{m}u^{m}_{x}u^{t}_{x}$.

• •

The coefficient of $\nu_{yz}\nu_{zx}$ is $+w^{j}_{s}u^{s}_{y}w^{k}_{t,l}w^{l}_{m}u^{m}_{z}u^{t}_{z}w^{i}_{n}u^{n}_{x}$.

The seventh summand in (45) is

[TABLE]

The coefficients of the reduced form are listed below:

• •

The coefficient of $\delta_{xy}\delta^{\prime\prime}_{xz}$ is $g^{lk}\Gamma^{ij}_{l}$.

• •

The coefficient of $\delta^{\prime\prime}_{xz}\nu_{xy}$ is $w^{i}_{l}g^{lk}w^{j}_{r}\,u^{r}_{y}$.

• •

The coefficient of $\delta_{xy}\delta^{\prime}_{xz}$ is $\Gamma^{ij}_{l}\Gamma^{lk}_{t}u^{t}_{x}+\Gamma^{ij}_{l}g^{lk}_{,s}u^{s}_{x}$.

• •

The coefficient of $\delta_{xy}\delta_{xz}$ is $\Gamma^{ij}_{l}\Gamma^{lk}_{t,s}u^{s}_{x}u^{t}_{x}+\Gamma^{ij}_{l}\Gamma^{lk}_{t}u^{t}_{xx}+\Gamma^{ij}_{l}w^{l}_{s}u^{s}_{x}w^{k}_{t}u^{t}_{x}$.

• •

The coefficient of $\delta_{yx}\nu_{yz}$ is $\Gamma^{ij}_{l}w^{l}_{s,m}u^{m}_{x}u^{s}_{x}w^{k}_{t}u^{t}_{z}+\Gamma^{ij}_{l}w^{l}_{s}u^{s}_{xx}w^{k}_{t}u^{t}_{z}$.

• •

The coefficient of $\delta_{xz}\nu_{xy}$ is

[TABLE]

• •

The coefficient of $\delta^{\prime}_{xz}\nu_{xy}$ is $w^{i}_{l}g^{lk}_{,s}u^{s}_{x}w^{j}_{r}\,u^{r}_{y}+w^{i}_{l}\Gamma^{lk}_{t}u^{t}_{x}w^{j}_{r}\,u^{r}_{y}$.

• •

The coefficient of $\nu_{xz}\nu_{xy}$ is $w^{i}_{l}w^{l}_{s,m}u^{m}_{x}u^{s}_{x}w^{k}_{t}u^{t}_{z}w^{j}_{r}\,u^{r}_{y}+w^{i}_{l}w^{l}_{s}u^{s}_{xx}w^{k}_{t}u^{t}_{z}w^{j}_{r}\,u^{r}_{y}$.

The eighth summand in (45) is

[TABLE]

The coefficients of the reduced form are listed below:

• •

The coefficient of $\delta_{xy}\delta_{xz}$ is $w^{i}_{s}u^{s}_{x}w^{j}_{l}g^{lk}_{,m}u^{m}_{x}-w^{i}_{s}u^{s}_{x}\partial_{x}(w^{j}_{l}g^{lk})+w^{i}_{s}u^{s}_{x}w^{j}_{l}\Gamma^{lk}_{t}u^{t}_{x}$.

• •

The coefficient of $\nu_{xz}\delta^{\prime}_{zy}$ is

[TABLE]

• •

The coefficient of $\nu_{xz}\delta^{\prime\prime}_{zy}$ is $w^{i}_{s}u^{s}_{x}w^{j}_{l}g^{lk}$.

• •

The coefficient of $\nu_{xz}\delta_{zy}$ is

[TABLE]

• •

The coefficient of $\nu_{xy}\nu_{yz}$ is $w^{i}_{s}u^{s}_{x}w^{j}_{l}w^{l}_{r,m}u^{m}_{y}u^{r}_{y}w^{k}_{t}u^{t}_{z}+w^{i}_{s}u^{s}_{x}w^{j}_{l}w^{l}_{r}u^{r}_{yy}w^{k}_{t}u^{t}_{z}$.

• •

The coefficient of $\delta_{xy}\delta^{\prime}_{xz}$ is $w^{i}_{s}u^{s}_{x}w^{j}_{l}g^{lk}$.

• •

The coefficient of $\delta^{\prime}_{xy}\delta_{xz}$ is $-w^{i}_{s}u^{s}_{x}w^{j}_{l}g^{lk}$.

The ninth summand in (45) is

[TABLE]

The coefficients of the reduced form are listed below

• •

The coefficient of $\delta^{\prime\prime}_{xy}\delta_{xz}$ is $g^{lj}\Gamma^{ki}_{l}$.

• •

The coefficient of $\delta^{\prime}_{xy}\delta_{xz}$ is $\Gamma^{ki}_{l}g^{lj}_{,s}u^{s}_{x}+\Gamma^{ki}_{l}\Gamma^{lj}_{t}u^{t}_{x}$.

• •

The coefficient of $\delta_{xy}\delta_{xz}$ is $\Gamma^{ki}_{l}\Gamma^{lj}_{t,s}u^{s}_{x}u^{t}_{x}+\Gamma^{ki}_{l}\Gamma^{lj}_{t}u^{t}_{xx}+\Gamma^{ki}_{l}w^{l}_{s}u^{s}_{x}w^{j}_{t}u^{t}_{x}$.

• •

The coefficient of $\delta_{xz}\nu_{xy}$ is $\left(\Gamma^{ki}_{l}w^{l}_{s,m}u^{m}_{x}u^{s}_{x}+\Gamma^{ki}_{l}w^{l}_{s}u^{s}_{xx}\right)w^{j}_{t}u^{t}_{y}$.

• •

The coefficient of $\delta_{zy}\nu_{zx}$ is \big{(}w^{k}_{l}\Gamma^{lj}_{t}u^{t}_{zz}+w^{k}_{l}\Gamma^{lj}_{t,s}u^{s}_{z}u^{t}_{z}+w^{k}_{l}w^{l}_{s}u^{s}_{z}w^{j}_{t}u^{t}_{z}\big{)}w^{j}_{r}\,u^{r}_{x}.

• •

The coefficient of $\delta^{\prime}_{zy}\nu_{zx}$ is $w^{k}_{l}g^{lj}_{,s}u^{s}_{z}w^{i}_{r}\,u^{r}_{x}+w^{k}_{l}\Gamma^{lj}_{t}u^{t}_{z}w^{i}_{r}\,u^{r}_{x}$.

• •

The coefficient of $\nu_{zy}\nu_{zx}$ is $w^{k}_{l}w^{l}_{s,m}u^{m}_{z}u^{s}_{z}w^{j}_{t}u^{t}_{y}w^{i}_{r}\,u^{r}_{x}+w^{k}_{l}w^{l}_{s}u^{s}_{zz}w^{j}_{t}u^{t}_{y}w^{i}_{r}\,u^{r}_{x}$.

• •

The coefficient of $\delta^{\prime\prime}_{zy}\nu_{zx}$ is $w^{k}_{l}g^{lj}w^{i}_{r}\,u^{r}_{x}$.

The tenth summand in (45) is

[TABLE]

The coefficients of the reduced form are listed below:

• •

The coefficient of $\delta_{xy}\delta_{xz}$ is $w^{k}_{s}u^{s}_{x}w^{i}_{l}g^{lj}_{,m}u^{m}_{x}+w^{k}_{s}u^{s}_{x}w^{i}_{l}\Gamma^{lj}_{t}u^{t}_{x}+\partial_{x}(w^{k}_{s}u^{s}_{x})w^{i}_{l}g^{lj}$.

• •

The coefficient of $\nu_{yz}\delta^{\prime}_{yx}$ is $w^{k}_{s}u^{s}_{z}w^{i}_{l}g^{lj}_{,m}u^{m}_{y}+w^{k}_{s}u^{s}_{z}w^{i}_{l}\Gamma^{lj}_{t}u^{t}_{y}-2w^{k}_{s}u^{s}_{z}\partial_{y}(w^{i}_{l}g^{lj})$.

• •

The coefficient of $\nu_{zy}\delta_{yx}$ is

[TABLE]

• •

The coefficient of $\nu_{zx}\nu_{xy}$ is $w^{k}_{s}u^{s}_{z}w^{i}_{l}w^{l}_{r,m}u^{m}_{x}u^{r}_{x}w^{j}_{t}u^{t}_{y}+w^{k}_{s}u^{s}_{z}w^{i}_{l}w^{l}_{r}u^{r}_{xx}w^{j}_{t}u^{t}_{y}$.

• •

The coefficient of $\delta^{\prime}_{xy}\delta_{xz}$ is $2w^{k}_{s}u^{s}_{x}w^{i}_{l}g^{lj}$.

• •

The coefficient of $\delta_{xy}\delta^{\prime}_{xz}$ is $w^{k}_{s}u^{s}_{x}w^{i}_{l}g^{lj}$.

• •

The coefficient of $\nu_{zy}\delta^{\prime\prime}_{yx}$ is $w^{k}_{s}u^{s}_{z}w^{i}_{l}g^{lj}$.

The eleventh summand in (45) is

[TABLE]

The coefficients of the reduced form are listed below

• •

The coefficient of $\delta_{xy}\delta^{\prime}_{xz}$ is $-\Gamma^{jk}_{l}g^{li}_{,s}u^{s}_{x}+2\partial_{x}(g^{li}\Gamma^{jk}_{l})-\Gamma^{jk}_{l}\Gamma^{li}_{t}u^{t}_{x}$.

• •

The coefficient of $\delta^{\prime}_{xy}\delta_{xz}$ is $2\partial_{x}(g^{li}\Gamma^{jk}_{l})-\Gamma^{jk}_{l}g^{li}_{,s}u^{s}_{x}-\Gamma^{jk}_{l}\Gamma^{li}_{t}u^{t}_{x}$.

• •

The coefficient of $\delta_{xy}\delta_{xz}$ is

[TABLE]

• •

The coefficient of $\delta_{xy}\delta^{\prime\prime}_{xz}$ is $g^{li}\Gamma^{jk}_{l}$.

• •

The coefficient of $\delta^{\prime}_{xy}\delta^{\prime}_{xz}$ is $2g^{li}\Gamma^{jk}_{l}$.

• •

The coefficient of $\delta^{\prime\prime}_{xy}\delta_{xz}$ is $g^{li}\Gamma^{jk}_{l}$.

• •

The coefficient of $\delta_{yz}\nu_{zx}$ is $(\Gamma^{jk}_{l}w^{l}_{s,m}u^{m}_{z}u^{s}_{z}+\Gamma^{jk}_{l}w^{l}_{s}u^{s}_{zz})w^{i}_{t}u^{t}_{x}$.

• •

The coefficient of $\delta_{yx}\nu_{yz}$ is $(w^{j}_{l}\Gamma^{li}_{t,s}u^{s}_{y}u^{t}_{y}+w^{j}_{l}\Gamma^{li}_{t}u^{t}_{yy}+w^{j}_{l}w^{l}_{s}u^{s}_{y}w^{i}_{t}u^{t}_{y})w^{k}_{r}\,u^{r}_{z}$.

• •

The coefficient of $\delta^{\prime}_{yx}\nu_{yz}$ is $w^{j}_{l}g^{li}_{,s}u^{s}_{y}w^{k}_{r}\,u^{r}_{z}+w^{j}_{l}\Gamma^{li}_{t}u^{t}_{y}w^{k}_{r}\,u^{r}_{z}$.

• •

The coefficient of $\nu_{yx}\nu_{yz}$ is $w^{j}_{l}w^{l}_{s,m}u^{m}_{y}u^{s}_{y}w^{i}_{t}u^{t}_{x}w^{k}_{r}\,u^{r}_{z}+w^{j}_{l}w^{l}_{s}u^{s}_{yy}w^{i}_{t}u^{t}_{x}w^{k}_{r}\,u^{r}_{z}$.

• •

The coefficient of $\delta^{\prime\prime}_{yx}\nu_{yz}$ is $w^{j}_{l}g^{li}w^{k}_{r}\,u^{r}_{z}$.

The twelfth (and last) summand in (45) is

[TABLE]

The coefficients of the reduced form are listed below

• •

The coefficient of $\delta_{xy}\delta_{xz}$ is

[TABLE]

• •

The coefficient of $\nu_{yx}\delta^{\prime\prime}_{xz}$ is $w^{j}_{s}u^{s}_{y}w^{k}_{l}g^{li}$.

• •

The coefficient of $\nu_{yx}\delta^{\prime}_{xz}$ is $2w^{j}_{s}u^{s}_{y}\partial_{x}(w^{k}_{l}g^{li})-w^{j}_{s}u^{s}_{y}w^{k}_{l}\Gamma^{li}_{t}u^{t}_{x}-w^{j}_{s}u^{s}_{y}w^{k}_{l}g^{li}_{,m}u^{m}_{x}$.

• •

The coefficient of $\delta_{xy}\delta^{\prime}_{xz}$ is $-2w^{j}_{s}u^{s}_{x}w^{k}_{l}g^{li}$.

• •

The coefficient of $\delta^{\prime}_{xy}\delta_{xz}$ is $-w^{j}_{s}u^{s}_{x}w^{k}_{l}g^{li}$.

• •

The coefficient of $\nu_{yx}\delta_{xz}$ is

[TABLE]

• •

The coefficient $\nu_{yz}\nu_{zx}$ is $w^{j}_{s}u^{s}_{y}w^{k}_{l}w^{l}_{r,m}u^{m}_{z}u^{r}_{z}w^{i}_{t}u^{t}_{x}+w^{j}_{s}u^{s}_{y}w^{k}_{l}w^{l}_{r}u^{r}_{zz}w^{i}_{t}u^{t}_{x}$.

5.3 The conditions

Collecting all similar terms we get the following conditions

• •

The coefficients of $\delta^{\prime\prime}_{xy}\delta_{xz}$, $\delta^{\prime}_{xy}\delta^{\prime}_{xz}$ and $\delta_{xy}\delta^{\prime\prime}_{xz}$ vanish iff (8c) holds, namely $g^{li}\Gamma^{jk}_{l}=g^{lj}\Gamma^{ik}_{l}$. Combining this condition with skew-symmetry of the bracket (8b) we obtain that $\Gamma^{i}_{jk}=-g_{jl}\Gamma^{li}_{k}$ are the Christoffel symbols of the Levi-Civita connection of $g$.

• •

The coefficients of products of step functions vanish.

• •

The coefficients of $\delta^{\prime\prime}_{xz}\nu_{xy}$, $\delta^{\prime\prime}_{zy}\nu_{zx}$ and $\delta^{\prime\prime}_{yx}\nu_{yz}$ vanish iff (8d) holds, namely $g_{ik}w^{k}_{j}=g_{jk}w^{k}_{i}$.

• •

Using the above conditions the coefficient of $u^{s}_{xx}\delta_{xy}\delta_{xz}$ can be written as

[TABLE]

where $R^{ijk}_{s}$ is Riemann tensor (in upper indices). This yields the condition (8f).

• •

The coefficient of $u^{t}_{xx}\nu_{xy}\delta_{xz}$ (up to a common factor) is

[TABLE]

which yields the condition (8e).

• •

The coefficient of $\delta^{\prime}_{xy}\delta_{xz}$ is a linear combination of the coefficient (57) repeated two times.

• •

The coefficients of $u^{r}_{x}u^{s}_{x}\delta_{xy}\delta_{xz}$ vanish using the $x$-derivative of the condition (57). Indeed, the coefficient reduces to

[TABLE]

Using the condition (8e) we obtain

[TABLE]

Using again (57) we obtain

[TABLE]

• •

The coefficient of $u^{t}_{y}u^{r}_{x}u^{s}_{x}\nu_{xy}\delta_{xz}$ vanishes due to the previous conditions. Indeed

[TABLE]

Similar computations hold for the coefficients of $\nu_{yz}\delta_{yx}$ and $\nu_{xz}\delta_{zy}$.

• •

The coefficient of $u^{s}_{y}u^{t}_{x}\delta^{\prime}_{xz}\nu_{xy}$ is

[TABLE]

which vanishes upon (8e).

6 Weakly nonlocal PBHT and differential operators

Here we will just show the main steps of the algorithm in Section 3.3.

We assume that

[TABLE]

where $\det(g^{ij})\neq 0$ and $\epsilon_{\alpha}\in\mathbb{R}$.

We will compute the conditions of Hamiltonianity of the operator $P$ of the type (59) using formula (83) and the Algorithm 3.3. Let us set:

[TABLE]

The conditions of skew-adjointness are obvious.

6.1 Calculation of the Jacobi identity

From now on we will assume $L$ to be skew-adjoint ($N$ is skew-adjoint by construction).

Lemma 6**.**

We have

[TABLE]

We begin by computing the linearization of $L$ and $N$. Let us introduce the new non-local scalar functions

[TABLE]

Lemma 7**.**

The linearization of $L$ and $N$ have the following expressions:

[TABLE]

We compute the first summand of (61):

[TABLE]

We compute the second summand of (61):

[TABLE]

We compute the third summand of (61). Here we integrate non-local expressions by parts in order to concentrate integrals in expressions of the form (62):

[TABLE]

We compute the fourth summand of (61). Here we integrate non-local expressions by parts in order to concentrate integrals in expressions of the form (62):

[TABLE]

The three-vector (61) can be written, after adding the cyclically permuted summands, as the sum $T_{l}+T_{n}$, where $T_{l}$ is the local part and $T_{n}$ is the non-local part.

6.2 Calculation of the reduced form

Now, we fix the indices $1$, $2$, $3$ and we bring the nonlocal part to the normal form with respect to the three ordered indices. This means that the terms which are quadratic in the nonlocal expressions (62) shall be preserved, while terms which are linear in the nonlocal expressions should be brought to one of the following forms by integrating by parts:

[TABLE]

For example, $g^{ij}_{,k}w^{k}_{m}u^{m}_{x}\tilde{\psi}^{2}\psi^{3}_{i}\partial_{x}\psi^{1}_{j}$ must be replaced by -\partial_{x}\big{(}g^{ij}_{,k}w^{k}_{m}u^{m}_{x}\tilde{\psi}^{2}\psi^{3}_{i}\big{)}\psi^{1}_{j} (of course, up to a total divergence). After the above computational step we can write the final form of the nonlocal part of the three-vector. Note that $T_{n}$ acquired some local terms at the end of the first step of the algorithm. We introduce the notation $T_{n}=T_{N}+T_{nL}$, where $T_{N}$ is the non-local part and $T_{nL}$ is the local part of $T_{n}$ after the first step of the algorithm. We have, after collecting like terms:

[TABLE]

plus a cyclic permutation of the above terms. Moreover:

[TABLE]

plus a cyclic permutation of the last summand.

Let us introduce the notation $T_{L}=T_{l}+T_{nL}$. We shall bring each summand of $T_{L}$ to the canonical form

[TABLE]

where $c^{jpi}$ are coefficient functions, using integration by parts on summands that contain $\partial_{x}^{l}\psi^{3}_{i}$ with $l>0$. We have:

[TABLE]

6.3 The conditions

The vanishing of coefficients of the $3$-vector $T$ yields the conditions on $P$ to be Hamiltonian. Below we list the basic elements of $T$ and the conditions that arise from their coefficients. We assume the condition of skew-adjointness of $P$.

$\psi^{1}_{j}\partial_{x}^{2}\psi^{2}_{p}\psi^{3}_{i}$:

the coefficient is

[TABLE]

and corresponds to the coefficient of $\delta^{\prime\prime}_{xy}\delta_{xz}$ and similar terms in Section 5.3. Its vanishing is equivalent to the condition $\Gamma^{jp}_{h}g^{hi}=\Gamma^{ip}_{h}g^{hj}$.

$\partial_{x}\psi^{1}_{j}\partial_{x}\psi^{2}_{p}\psi^{3}_{i}$:

the coefficient vanish on account of the above condition.

$\tilde{\psi}^{1}\partial_{x}^{2}\psi^{2}_{p}\psi^{3}_{i}$:

the coefficient is

[TABLE]

and corresponds to the coefficients of $\delta^{\prime\prime}_{xz}\nu_{xy}$ and similar terms in Section 5.3.

$\psi^{1}_{j}\psi^{2}_{p}\psi^{3}_{i}$:

This coefficient is a differential polynomial; the coefficient of $u^{k}_{xx}$ reduces to

[TABLE]

using (77). This corresponds to the coefficient of $u^{k}_{xx}\delta_{xy}\delta_{xz}$ in Section 5.3.

$\tilde{\psi}^{2}\psi^{3}_{i}\psi^{1}_{j}$:

This coefficient is a differential polynomial; the coefficient of $u^{m}_{xx}$ reduces to

[TABLE]

and corresponds to the coefficient of $u^{m}_{xx}\nu_{xy}\delta_{xz}$ in Section 5.3. The coefficient is equal to $g^{kj}(\nabla_{k}w^{i}_{m}-\nabla_{m}w^{i}_{k})$.

The correspondence between the coefficients of the three-vector in the language of operators and of distributions extends to all remaining terms; there is no need to repeat the computation that shows that all other coefficients vanish on account of the above conditions.

7 Weakly nonlocal PBHT and Poisson Vertex Algebras

In this section we will use the master formula to compute the skewsymmetry condition and the PVA-Jacobi identity for the $\lambda$ bracket

[TABLE]

corresponding to the weakly non-local Hamiltonian operator (60). As before, we split the operator in the local and nonlocal parts

[TABLE]

Enforcing Property 5 of Section 4.1 on the two $\lambda$ brackets (81) and (82) gives the conditions for the corresponding operator to be skewsymmetric.

Indeed, for the local part, we have

[TABLE]

which implies the conditions (8a), (8b), and it is easy to prove that the nonlocal part is skewsymmetric by construction.

7.1 Computations with the master formula

For convenience, we split the PVA-Jacobi identity on the generators – defined in Section 4.2 – in the four parts

[TABLE]

where the last two terms correspond to the Schouten bracket $[L,N]$.

The purely local part $J^{ijk}_{\lambda,\mu}(L,L)$ is a straightforward application of the master formula:

[TABLE]

[TABLE]

[TABLE]

where all the monomials are of the form $\lambda^{p}\mu^{q}$ with $p,q\geq 0$.

Computing the expressions with nonlocal terms is more complicated. However, it is possible to rely on Leibniz’s and sesquilinearity properties of the $\lambda$ brackets to split the problem into smaller chunks. The basic observation, that can be proved by expanding the $(\lambda+\partial)^{-1}$ expression, is that

[TABLE]

Let us start with $J^{ijk}_{\lambda,\mu}(N,L)$. Combining this with the left and right Leibnitz properties we get

[TABLE]

The notation \big{(}i\leftrightarrow j,\lambda\leftrightarrow\mu\big{)} we used in Equation (7.1) means that all the terms are to be replaced with the ones obtained by switching the corresponding indices and parameters. The expressions of the form $[(\lambda+\partial)^{-1}A]$ enclosed within square brackets denote terms on which derivation operators “from outside” do not act and containing derivations which do not act “on the outside”.

The computation of the terms of $J^{ijk}_{\lambda,\mu}(L,N)$ is straightforward for the first two addends

[TABLE]

[TABLE]

In the computation of the third one, we exploit the identity $(A+\partial)^{-1}B(A+\partial)C=(A+\partial)^{-1}(A+\partial)BC-(A+\partial)^{-1}[\partial B]C$

[TABLE]

We compute now the expression for $J^{ijk}_{\lambda,\mu}(N,N)$. We have

[TABLE]

[TABLE]

[TABLE]

Note that in the last passage we have used the same identity as in Equation (7.1) to simplify the terms of the form $(\lambda+\mu+\partial)^{-1}A(\lambda+\mu+\partial)B$.

7.2 Projection onto the basis

The Jacobi identity lives in the previously defined space $V_{\lambda,\mu}$. The partial results of our computation are not all expressed in such a form that, after the expansion of the nonlocal terms, will produce elements on the basis $\lambda^{p}\mu^{d-p}$ and $(\lambda+\mu)^{-q}\lambda^{d+q}$ for $p\in\mathbb{Z}$, $q>0$ for all $d\in\mathbb{Z}$.

All the double nonlocal terms in $J^{ijk}_{\lambda,\mu}(N,N)$ cancel out, leaving with a simplified expression

[TABLE]

There are terms which would not expand in the chosen basis in the last line of Equation (7.1). We have

[TABLE]

and

[TABLE]

The full PVA-Jacobi identity can be then be obtained in terms of the previously computed expressions, provided that we replace the last line in Equation (7.1) with the expression (7.2) $+$ (7.2). The full form of $J^{ijk}_{\lambda,\mu}$ is then

[TABLE]

7.3 The conditions

Assuming the skewsymmetry of the bracket $P$, the PVA-Jacobi equation is symmetric for cyclic permutations of $(i,\lambda),(j,\mu),(k,\nu=-\lambda-\mu-\partial)$, and it is fulfilled if and only if all the coefficients in the basis of $V_{\lambda,\mu}$ we have chosen vanish. We report here the coefficients corresponding to the elements of Section 6.3, under the condition of skewsymmetry for the bracket.

• •

The coefficient of $\lambda^{2}$ is

[TABLE]

whose vanishing, given the skewsymmetry of the bracket, is equivalent to $g^{il}\Gamma^{kj}_{l}=g^{kl}\Gamma^{ij}_{l}$.

• •

The coefficient of $\lambda\mu$ would be

[TABLE]

which vanishes on account of the previous condition.

• •

The coefficient of $\left((\mu+\partial)^{-1}w^{j}_{n}u^{n}_{x}\right)\lambda^{2}$ is

[TABLE]

• •

The expression that is obtained when $\lambda^{0}\mu^{0}=1$ is a differential polynomial. The coefficient multiplying $u^{s}_{xx}$ is

[TABLE]

where the two summands with $ww$ come from (7.1).

• •

The expression $\left((\mu+\partial)^{-1}w^{j}_{n}u^{n}_{x}\right)$ is a differential polynomial. The coefficient multiplying $u^{s}_{xx}$ is

[TABLE]

which is equal to Equation (79) (after the exchange of the free indices $(j,m)\leftrightarrow(k,s)$).

8 Concluding remarks

In this paper we have considered three different approaches to the problem of verifying Jacobi identity for weakly non local Poisson brackets of hydrodynamic type, and we have showed their equivalence. While the equivalence between the approach based on distributions and the approach based on (pseudo)-differential operators is quite straightforward, the equivalence between the first two approaches and the approach based on Poisson vertex algebras is more subtle and requires additional work (Proposition 2 and Theorem 3). The final result is an algorithmic procedure adapted to the three different formalisms. It is also clear that the procedure can be used to check the compatibility of two Poisson brackets, or that the Schouten bracket between two distinct Hamiltonian operators vanish.

One of the most important consequences of the existence of a well-defined procedure for the computation of the Jacobi identity is that the procedure can be easily programmed in a computer algebra system. We plan to do this in a future work.

The extension of this work to weakly nonlocal symplectic operators [21] seems possible along the lines presented in this paper, and might prove to be useful. More generally, a differential-geometric theory of nonlocal integrability operators (i.e., Hamiltonian operators, symplectic operators and recursion operators for symmetries and conserved quantities) for partial differential equations, including weakly nonlocal operators, already exist [16], but it does not include Schouten brackets, the formulation of the symplectic property and the formulation of the hereditary property for recursion operators (the variational Nijenhuis bracket). However, this seems to be a possible goal and at the moment it is in development, see [15] for latest advances.

9 Appendix: three different recipes for the Jacobi

identity

In this section we will show that the expression of the Jacobi identity can be written in three different ways up to total divergencies.

We recall that the formal adjoint is defined by the equality

[TABLE]

where $\langle,\rangle$ is the pairing between vectors and covectors and square brackets mean that the result is an equivalence class up to total divergencies. In what follows we will need the standard facts [2, 18]:

[TABLE]

and

[TABLE]

where $\mathcal{E}$ is the Euler–Lagrange operator. If $P^{*}=-P$, then it is easy to prove [18] that

[TABLE]

Theorem 8**.**

Let $P$, $Q$ be skew-adjoint variational bivectors. Then, the following formulae for the Schouten bracket coincide up to total divergencies:

[TABLE]

Proof.

The equivalence between (108) and (109) is given by the following formulae (all equalities are up to total divergencies!):

[TABLE]

The equivalence between (108) and (110) is given by the following formulae:

[TABLE]

In the cyclic sum in (110) all terms of the form $\langle Q(\psi^{i}),\ell_{\psi^{j}}(P(\psi^{k}))$ cancel if the computation is restricted to covector-valued densities $\psi$ that lie in the image of the Euler–Lagrange operator $\mathcal{E}$: $\psi=\mathcal{E}(F)$, where $F=\int f\,dx$. In that case, we have $\ell_{\psi}=\ell^{*}_{\psi}$. The proof is completed by the remark that multivector identities hold true in general even if they are proved on the image of $\mathcal{E}$ only [18]. ∎

Remark 9**.**

The expression (110) was used in [13] in order to check the Jacobi identity. The expressions (108) and (109) are more commonly used (see e.g. [2, 5, 23]). In particular, the expression (108) is the formula that we use throughout this paper. The three expressions do not exhaust all possibilities; see the above references for more exotic expressions of the Jacobi identity.