The averaging of non-local Hamiltonian structures in Whitham's method

We consider the $m$-phase Whitham's averaging method and propose a procedure of"averaging"of non-local Hamiltonian structures. The procedure is based on the existence of a sufficient number of local commuting integrals of a system and gives a Poisson bracket of Ferapontov type for the Whitham's system. The method can be considered as a generalization of the Dubrovin-Novikov procedure for the local field-theoretical brackets.


Introduction.
We consider the averaging of non-local Hamiltonian structures in Whitham's averaging method. As it is well known, the Whitham's method permits to obtain equations on the "slow" modulated parameters of exact periodic or quasi-periodic solutions of systems of partial differential equations and it was pointed out by Whitham ([1]) that these equations can be written in the Lagrangian form if the initial system possesses a local Lagrangian structure. The Lagrangian formalism for the Whitham's system is given in this case by the "averaging" of a local Lagrangian function, defined for the initial system, on the corresponding space of (quasi)-periodic solutions. Some basic questions concerning Whitham's method can be found in [1,2,3,4,5,6,7,8,9,10,11,12,13,14]. B.A. Dubrovin and S.P. Novikov investigated also the question of the conservation of local fieldtheoretical Hamiltonian structures in Whitham's method and suggested a procedure of "averaging" of a local field-theoretical Poisson bracket, giving a Poisson bracket of Hydrodynamic type for the Whitham system ( [4,7,9], see also [17]).
The Jacobi identity for the averaged bracket and the invariance of the Dubrovin-Novikov procedure was proved by the author in [18] (see also [19]) using the Dirac restriction procedure of the initial bracket on the subspace of quasi-periodic "m-phase" solutions of the initial system. The connection between the procedure of Dubrovin and Novikov and the procedure of averaging of the Lagrangian function in the case when the initial local Hamiltonian structure just follows from the local Lagrangian one was also studied in [20]. Some extension of the averaging of "local" Hamiltonian structures for the case of discrete systems is also presented in [21].
Let us also point out here that the brackets (0.1) usually appear in the theory the so-called "integrable" hierarchies (see [22,23,24]), connected with the method of the inverse scattering problem.
The most general form of the non-local Hamiltonian operators (0.1) containing only δ ′ (X − Y ) and δ(X − Y ) in the local part and the quasi-linear fluxes S ν (k)λ (U) U λ X of "hydrodynamic" type in the non-local one 1 Some general properties of the non-local brackets.
Let us consider a non-local 1-dimensional Hamiltonian structure of the type: where we have finite numbers of terms in both sums depending on a finite number of derivatives of ϕ with respect to x. We will call a local translationally invariant Hamiltonian function a functional of the form: Here ν(x − y) is the skew-symmetric function and δ (k) (x − y) is the k-th derivative of the delta-function with respect to x. We assume here that the bracket (1.1) is written in the "irreducible" form, which means that the number of terms in the second sum is the minimal possible and the sets {S (k) } and {T (k) } represent two linearly independent sets of vector-functions of the variables (ϕ, ϕ x , . . . ). From the skew-symmetry of the bracket (1.1) it follows then that the sets ofS (k) andT (k) define actually the same linear space in the space of functions and it can be easily seen that the bracket (1.1) can be represented in the "canonical" form Indeed, since the sets {S (k) } and {T (k) } span the same linear space we have just one finitedimensional space, generated by fluxes (vector fields) . . ) and a symmetric (view the skew-symmetry of the bracket and the function ν(x−y)) finite-dimensional constant 2-form, which describes their couplings in the non-local part of (1.1). So, we can write it in the canonical form according to its signature after some linear transformation of the flowsS (k) and T (k) with constant coefficients.
We should also define in every case the functional space where we consider the action of the Hamiltonian operator (1.4) and this can depend on a concrete situation. The most natural thing is to consider the functional space ϕ(x) and the algebra of functionals I[ϕ], such that their variational derivatives, multiplied by the flows S (k) (ϕ, ϕ x , . . . ), give us rapidly decreasing functions as |x| → ∞. Below we will use the functionals of the type n p=1 ϕ p (x) q p (x) dx , where q p (x) are arbitrary smooth functions with compact supports, to examine the properties of the bracket (1.4). For all the other functionals used in the considerations we will assume that they have a compatible with the bracket (1.4) form in the sense discussed above.
We will assume here for simplicity that the functions B ij k and S i (k) represent analytic functions of their arguments (maybe in some open region of the values of (ϕ, ϕ x , . . . )).
We will construct here a procedure, which gives us a bracket of Ferapontov type ( [27]- [30]) from the initial bracket (1.4) after the averaging on an appropriate family of exact m-phase solutions of a local system, which is supposed to be Hamiltonian with respect to the bracket (1.4) with a local Hamiltonian function H. So, we consider here the Whitham's method for the local fluxes (if they exist) (we assume summation over the repeated indices). For the Hamiltonian flow ξ i (x) we should have: whereĴ is the Hamiltonian operator (1.4) and L ξ is the Lie-derivative, given by the expression: Let us now consider the relation (1.10) for x and y larger than any z from the supports of q p (z). Then we will have . . ) are the derivatives of these functions with respect to the flow Here we also used that x, y > Supp q p when omitted the variational derivatives with respect to ϕ s (x) and ϕ s (y) of the non-local expressions containing the convolutions with q p (w) (the 4-th and the 5-th terms).
So we have where L kĴ ij (x, y) represent the Lie derivatives ofĴ with respect to the flows (1.7) . . ) Let us use again our condition x, y > Supp q p and rewrite the above identity in the form Using the standard expression for the variational derivative and the integration by parts we obtain that this identity can be written also in the form ] is the commutator of the flows (1.7), or for any q p (z), such that x, y > Supp q p (z) .
As can be easily seen, the last term in (1.12) represents the non-local part of L ξĴ ij (x, y), which does not contain the function ν(x − y). The first term in (1.12) also contains a non-local part, however, this part contains the function ν(x − y). It is not difficult to see that this non-local part can in general be written in the "canonical" form . . ) represent some linearly independent set of analytic vector-functions of (ϕ, ϕ x , . . . ). Let us prove now, that from the identity (1.12) it follows actually that both the expressions (1.13) and should be identically equal to zero. Let us fix some value of x and consider the interval I = [x − ∆, x + ∆], such that x − ∆, x + ∆ > Supp q p (z). It is not difficult to see that for a linearly independent set of analytic functions A (s) (ϕ, ϕ y , . . . ) we can find an everywhere dense set S of analytic on the interval y ∈ [x − ∆, x + ∆] functions ϕ(y) (and infinitely smooth on the whole numerical axis), such that the functions A (s) (ϕ, ϕ y , . . . ) give a linearly independent set of analytic functions of y on the interval I for any ϕ(y) ∈ S . It is easy to see also, that for any ϕ(y) ∈ S we can find a set of analytic functions κ i (y) on the interval I , such that the functions still give a set of linearly independent analytic functions on I . According to Peano ([41]), we can claim that there exists a point y 0 ∈ I such that the Wronskian is different from 0 at the point y 0 . It is not difficult to see also, that we can assume actually that y 0 = x , so we put W (x) = 0 . Let us introduce now infinitely smooth functions ζ 0 (y), . . . , ζ Q−1 (y) having the following properties: 1) All ζ l (y) are identically equal to zero outside the interval I ; 2) All ζ l (y) and all their derivatives ζ l sy (y) , s ≥ 0 , are equal to zero at the point y = Let us say again, that the functions ζ l (y) can be easily constructed and it is most convenient to represent them in the form shown at Fig. 1.
Let us consider now the convolutions (in y) of the full expression for L ξĴ ij (x, y) with the infinitely smooth functions κ j (y) C lζ l (x + C(y − x)) , l = 0 , . . . , Q − 1 and put C → ∞ .
Easy to see that the local part of L ξĴ ij (x, y) will give us identical zero in such convolutions due to the property (2) of the functions ζ l (y) . In the same way, we will get zero in the limit C → ∞ in the non-local part (1.14) of L ξĴ ij (x, y) according to the property (4) of the functionsζ l (y) . At the same time, the non-local part (1.13) will give us the values in the limit C → ∞ according to the property (3) of the functions ζ l (y) .
Coming back now to the property W (x) = 0 and assuming that in general for the linearly independent set {S (k) } , we can claim now that the vanishing of the expression L ξĴ ij (x, y) implies in fact the relations A i (s) (ϕ, ϕ x , . . . ) = 0 for our chosen function ϕ(y) ∈ S . Using now the properties of the set S and the translational invariance of our Hamiltonian operator we conclude now that A i (s) (ϕ, ϕ x , . . . ) ≡ 0 on the full set of functions which we consider. As a result, we can claim now that the non-local part (1.13) of the expression L ξĴ ij (x, y) is in fact identically equal to zero. As a consequence, we can claim also the the non-local part (1.14) should be also identical zero on the full set of functions ϕ(x) .
Looking now at the form of the term (1.14) we can see that it represents a sum of linearly independent tensor functions of (x, y) , so we get that every coefficient, given by the integral should be in fact identically equal to zero. In view of the arbitrariness of the functions q p (z) we obtain then S (k) , S (k ′ ) ≡ 0 From (1.12) we then have also for a linearly independent set of S (k) and different q p (w) that So we obtain the statements of the theorem. Theorem 1.1 is proved.
It is also obvious that the statements of the theorem are valid for all the brackets (1.1) written in the "irreducible" form, since allS (k) andT (k) in this case are just linear combinations of the flows S (k) .

Remark.
Let us point here that the first statement of the Theorem for the non-local brackets (1.5) of Ferapontov type was proved previously by E.V. Ferapontov in [27] using differential-geometrical considerations. In [27]- [30] also the full classification of the brackets (1.5) from the differential geometrical point of view can be found.
It is easy to see now that the local functional of type (1.2) generates a local flow in the Hamiltonian structure (1.1) if and only if the derivative of its density P(ϕ, ϕ x , . . . ) with respect to any of the flows (1.7) represents total derivative with respect to x, i.e. there exist such Q (k) (ϕ, ϕ x , . . . ) that As was also pointed out by E.V. Ferapontov ([27]), this means that the integral I represents a conservation law for any of the systems (1.7).
From the Theorem 1.1 we obtain now that the flows (1.7) commute in fact with all the local Hamiltonian fluxes, generated by local functionals (1.2), since they conserve in this case both the Hamiltonian structure and the corresponding Hamiltonian functions.
2 The Whitham method and the "regularity" conditions. Now we come to Whitham's averaging procedure (see [1]- [10]). Let us remind that in the m-phase Whitham's method for systems (1.6) we make a rescaling transformation X = ǫx, T = ǫt to obtain the system Then we try to find functions S(X, T ) = (S 1 (X, T ), . . . , S m (X, T )) and 2π-periodic with respect to each θ α (θ = (θ 1 , . . . , θ m )) functions such that the functions satisfy system (2.1) at any θ in any order of ǫ. It follows then that Φ i (0) (θ, X, T ) at any X and T defines an exact m-phase solution of (1.6), depending on some parameters U = (U 1 , . . . , U N ) and initial phases θ 0 = (θ 1 0 , . . . , θ m 0 ) and, besides that, we have the relations where ω α (U) and k α (U) are respectively the frequencies and the wave numbers of the corresponding m-phase solution of (1.6). The compatibility conditions of system (2.1) in the first order of ǫ together with the relations give us Whitham's system of equations on the parameters U(X, T ), which represents a quasi-linear system of hydrodynamic type Let us note here, that the representation of the modulated solutions of system (1.6) in the form (2.2) is in fact usually possible just in the one-phase situation. In the multi-phase case we can usually write down just the first term in the expansion (2.2), while the higher order corrections have in general more complicated form (see e.g. [11,12,13]). Let us say, however, that the Whitham system, defined as above, still plays the central role in description of the modulated solutions both in the one-phase and the multi-phase case.
The first procedure of averaging of local field-theoretical Poisson brackets was proposed in [4]- [9] by B.A. Dubrovin and S.P. Novikov. This procedure permits to obtain local Poisson brackets of Hydrodynamic type: for Whitham's system (2.3) from a local Hamiltonian structure for the initial system (1.6). The method of Dubrovin and Novikov is based on the presence of N (equal to the number of parameters U ν of the family of m-phase solutions of (1.6)) local integrals commuting with the Hamiltonian function (1.2) and with each other and can be described in the following way: We calculate the pairwise Poisson brackets of the densities P ν in the form according to (2.6). Then the Dubrovin-Novikov bracket on the space of functions U(X) can be written in the form where . . . means the averaging on the family of m-phase solutions of (1.6) given by the formula: 1 Here we choose the parameters U ν such that they coincide with the values of I ν on the corresponding solutions The Jacobi identity for the averaged bracket (2.7) in the general case was proved in [18] (for systems having also local Lagrangian formalism there was a proof in [20]).
Let us note here also that the procedure described above gives a Poisson bracket only if we average the initial Hamiltonian structure on a "full regular family" of m-phase solutions (see [17,18,42]). We will formulate actually more precise requirements when describe the averaging procedure in the non-local case.
Brackets (2.4) can be described from the differential-geometrical point of view. Thus, for a nondegenerated tensor g νµ we have in fact that it should represent a flat contravariant metric and the values Γ ν µγ = −g µλ b λν γ should give the Levi-Civita connection for the metric g νµ (with lower indices). The brackets (2.4) with a degenerated tensor g νµ are more complicated but also have a nice geometrical structure (see [16]). The non-local Poisson brackets (1.5) give a generalization of local Poisson brackets of Dubrovin and Novikov and are closely connected with the integrability of systems of hydrodynamic type, reducible to the diagonal form ( [25]). Namely, any system reducible to the diagonal form and Hamiltonian with respect to the bracket (1.5) satisfies in fact (see [27]- [30]) the so-called "semi-Hamiltonian" property, introduced by S.P. Tsarev ([25]), and can be integrated by Tsarev's "generalized hodograph method". In [33] the investigation of possible equivalence of the "semi-Hamiltonian" properties introduced by Tsarev and the Hamiltonian properties with respect to the bracket (1.5) can be also found.
Let us also point out here that the questions of integrability of Hamiltonian systems, which can not be written in the diagonal form, were studied in [34]- [37].
The procedure of averaging of the non-local Poisson brackets in the Whitham method and the proof of the Jacobi identity for the averaged non-local bracket resemble the same things for the local brackets. However the formulas of averaging and the proof contain in fact some essential differences, so, we have to represent here special consideration for the non-local case.
The m-phase solutions of (1.6) where ω = (ω 1 , . . . , ω m ), k = (k 1 , . . . , k m ) , are defined by 2π-periodic solutions of the system depending on ω and k as on parameters. So, we assume that for generic ω and k we obtain from (2.9) a finite-dimensional submanifold M ω,k (in the space of 2π-periodic with respect to each θ α functions), parameterized by the initial phase shifts θ α 0 and maybe also by some additional parameters r 1 , . . . , r h .

2
Combining all such M ω,k at different ω and k we obtain that the m-phase solutions of the system (1.6) can be parameterized by N = 2m + h parameters U 1 , . . . , U N , invariant with respect to the initial shifts of θ α , and the initial phase shifts θ α 0 after the choice of some "initial" functions Φ i (in) (θ, U), corresponding to the zero initial phases. The joint of the submanifolds M ω,k at all ω and k gives us a submanifold M in the space of 2π-periodic with respect to each θ α functions, which corresponds to the full family of m-phase solutions of (1.6).
For the Whitham procedure we should now require some "regularity" properties of the system of constraints (2.9). Namely (I) We require that the linearized system (2.9) has for generic ω and k exactly h + m = N − m solutions ("right eigen vectors") ξ (q)ω,k (θ, r) at the corresponding "points" of M ω,k , given by the vectors tangential to M ω,k , i.e. the functions Φ θ α (θ, r, ω, k) and Φ r q (θ, r, ω, k) (at the fixed values of ω and k).
(II) We also require that the number of linearly independent "left eigen vectors" κ (q)ω,k (θ, r), orthogonal to the image of the introduced linear operator, is exactly the same (N − m) as the number of the "right eigen vectors" ξ (q)ω,k (θ, r) for generic ω and k. In addition, we will assume that the corresponding κ (q)ω,k (θ, r) also depend continuously on the parameters U ν on M.
The requirements (I) and (II) are actually closely connected with the Whitham procedure and the asymptotic solutions (2.2). Indeed, it is not difficult to see that every k-th term in the expansion (2.2) is determined by the defined above linear system with a nontrivial right-hand part, depending on the previous terms of (2.2). For resolvability of these systems we have in any case to require the orthogonality of the right-hand part to all the "regular left eigen vectors" κ (q)ω,k (θ, r), corresponding to the zero eigen values. The corresponding orthogonality conditions in the first order of ǫ together with the relations k T = ω X give a system of (N − m) + m = N equations, which coincides (by definition) with the Whitham's system of equations (2.3).
Let us now discuss the requirements (I) and (II) from the Hamiltonian point of view.
First of all, like in the procedure of Dubrovin and Novikov, for the procedure of averaging of the bracket (1.4) we need a set of integrals I ν , ν = 1, . . . , N, satisfying the following requirements: which generates a local flow with respect to the bracket (1.4).
As was pointed above we should require then that the local flows (1.7), defined by the bracket (1.4) in the "canonical" (or "irreducible") form, conserve all the I ν , i.e. the time derivatives of the corresponding P ν (ϕ, ϕ x , . . . ) with respect to each of the flows (1.7) represent total derivatives with respect to x for some functions F ν (k) (ϕ, ϕ x , . . . ). (B) All I ν commute with each other and with the Hamiltonian function (1.2) can be regarded as independent coordinates U 1 , . . . , U N on the family of m-phase solutions of (1.6). 3 From the requirements above we immediately obtain that the flows (2.11) commute with our initial system (1.6) and with each other.
From Theorem 1.1 we obtain also that the commutative flows (1.7), defined by the Poisson bracket, also commute with (1.6) and (2.11) since they conserve the corresponding Hamiltonian functions and the Hamiltonian structure (1.4).

Now we can consider the functionals
on the space of the quasiperiodic functions (with m wave numbers).
It is easy to see now that the local fluxes (1.6), (1.7) and (2.11), being considered on the space of the quasiperiodic functions, also conserve the values ofĪ ν andH and commute with each other, since these properties can be expressed just as local relations containing ϕ, ϕ x , . . . and the time derivatives of the densities P ν (ϕ, ϕ x , . . . ), P H (ϕ, ϕ x , . . . ) at the same point x.
Now we can conclude that all the fluxes (1.7) and (2.11) leave invariant the family of m-phase solutions, given by (2.9), and can generate on it only linear shifts of the initial phases θ α 0 , which follows from the commutativity of the flows and with the flows ϕ i t α = ϕ i θ α and on the space of 2π-periodic with respect to each θ α functions and the conservation of the functionals I ν (i.e. U ν on M) by the flows (2.17) and (2.18). (Here k α are m wave numbers of the function ϕ(x).) So, we obtain that our family of m-phase solutions of (1.6) represents also a family of m-phase solutions for systems (1.7) and (2.11), and we can consider also the Whitham equations for these systems, based on the family M.
We can also conclude that in our situation the variational derivatives of the functionals (2.15) and (2.16) with respect to ϕ(θ) at the points of the submanifold M represent some linear combinations of the corresponding "left eigen vectors" κ (q) (θ + θ 0 , U) (see [7]- [10] and references therein). Indeed, from the conservation of the functionals (2.15) and (2.16) by the flows ϕ i t α = ϕ i θ α and we can conclude that the convolution of their derivatives (with respect to ϕ i (θ)) with the system of constraints (2.9) is identically zero for all the periodic functions with respect to all θ α and for any k 1 , . . . , k m and ω 1 , . . . , ω m . So we can take the variational derivative of the corresponding expression with respect to ϕ j (θ ′ ) and then omit the second variational derivative ofĪ ν andH according to the conditions (2.9). After that we obtain that the variational derivatives ofĪ ν andH are also orthogonal to the image of the linearized operator (2.9) at the points of M and so represent some linear combinations of κ (q) (θ + θ 0 , U) on M. β has the full rank and is reversible. So, we get the differentials dk β as some linear combinations of differentials N ν=1 λ α ν (U) dU ν , corresponding to the functionals with zero derivatives on M So Lemma 2.1 now follows from (2.20).

Remark 1.
As can be seen from the proof of Lemma 2.1, the variational derivatives ofĪ ν on M should span the full (N − m)-dimensional linear space, generated by all κ (q) (θ + θ 0 , U), if we want to take P ν as a set of independent coordinates on M. It is essential also that we consider the full family of m-phase solutions, given by (2.9) at different ω and k, (but not its "submanifold") and have m independent relations (2.22) on N differentials dU ν from m relations (2.20).

Remark 2.
Let us note here that the equations (2.21) were introduced at first by S.P. Novikov in [15] as the definition of the m-phase solutions for the KdV equation.
Let us now prove a technical lemma which we will need later.

Lemma 2.2
Let us introduce the additional densities at any U and θ 0 .
Proof. According to Lemma 2.1 we should not take into account variations of the form of Φ (in) (θ + θ 0 , U) when we consider infinitesimal changes of the values of the functionals k α (Ī) on M. So, the only source for a change of these functionals on M is the dependence on the wave numbers k in the expressionsĪ where the values of ∂Ī ν [ϕ]/∂k β on M are given by the integral expressions from (2.24). Since the values of the functionals k α (Ī) on M coincide by definition with the wave numbers k α , we obtain the relation (2.24). Lemma 2.2 is proved.
For the evolution of the densities P ν (ϕ, ϕ x , . . . ) according to our system (1.6) we can also write the relations d dt P ν (ϕ, ϕ x , . . . ) ≡ ∂ x Q νH (ϕ, ϕ x , . . . ) (2.25) and the Whitham's system (2.3) can be also written in the following "conservative" form for the parameters U ν = P ν , which gives an equivalent form of the Whitham equations.
The conservative form (2.26) of the Whitham's system will be very convenient in our considerations of the averaging of Hamiltonian structures.
Let us now put some "regularity" requirements about the joint M of the submanifolds M ω,k for all ω and k, corresponding to the full set of the m-phase quasiperiodic solutions of the system (1.6).
(III) We require that M represents an (N + m)-dimensional submanifold in the space of the 2π-periodic with respect to each θ α functions.
The property (III) means nothing but the fact that the shapes of the solutions of (2.9) are all different at different ω and k in the space of the 2π-periodic vector-functions of θ so that ω and k can be reconstructed from them. It is easy to see that this requirement corresponds to the generic situation. We will use here the property (III) in our procedure of averaging of bracket (1.1).
We will work with the full family of 2π-periodic solutions of (2.9) which will also depend on the "slow" variables X and T . To define the corresponding submanifold in the space of functions ϕ(θ, X, T ) we should extend the coordinates U ν as functionals of ϕ(θ) in the vicinity of our submanifold M. This can be easily done (see [18]) in the following way: Let introduce N different functionals such that their valuesĀ ν are functionally independent on the functions from the submanifold M. Then we can express U ν = U ν (Ā) in terms ofĀ ν on M and after that extend them as the functionals U ν (A) on the space of 2π-periodic with respect to each θ α functions.
We can also expand the coordinates θ α 0 (see [18]) in the vicinity of M by introduction of, say, functionals Now we consider the system where θ α 0 [ϕ] and U ν [ϕ] are the functionals in the vicinity of M, as a system of constraints, which defines M in the space of 2π-periodic with respect to each θ α functions.
From the invariance of the submanifold M with respect to the flows (1.7) and (2.11) we can also write here the relations for any i, k and ν at any point (U, θ 0 ) of M, where k α = k α [Φ] can be considered now as the values of the corresponding functionals on M.
We now introduce the space of functions ϕ(θ, X, T ), depending on "slow" parameters X and T and 2π-periodic with respect to each θ α . Systems (2.27), considered independently at different X, give us a system of constraints defining the submanifold M ′ in the space of functions ϕ(θ, X), corresponding to m-phase solutions of (1.6) depending on the additional parameters X and T .
It will be actually convenient to introduce also the "modified" constraints (2.27) and take U ν (X), θ α 0 (X) and G i [U,θ 0 ] (θ, X), such that as coordinates in the vicinity of M ′ instead of the ϕ i (θ, X). It is easy to see also that we can find uniquely ϕ i (θ, X) from the relations and the values of U ν (X) and θ α 0 (X) under the conditions (2.31). 4

Remark.
Certainly we have here some freedom in the choice of the constraints G i (θ, X). For example we can take also the expressions (2.27) as a system of constraints defining M ′ . We prefer here to take the constraints in the form (2.30) just to fix the uniform orthogonality conditions (2.31) in the vicinity of M ′ .
We will need also another coordinate system in the vicinity of M ′ , which differs from the described above by a transformation, depending on the small parameter ǫ and singular at ǫ → 0. Namely, we recall our integrals (2.5) make a transformation X = ǫx and define the functionals on the space of 2π-periodic with respect to each θ α functions ϕ(θ, X). Let us also introduce the functionals for some fixed point X 0 . We have identically θ * α 0 (X 0 ) ≡ 0 (2.34) As was shown in [18], we can also obtain the values of U ν (X) and θ α 0 (X) from J ν (X), θ * α 0 (X) and θ α 0 (X 0 ) on M ′ as formal series in powers of ǫ and we will have for these series The form of the relation (2.35) will be important in our considerations, so we reproduce here the calculations from [18].
After the integration with respect to θ, which removes the singular at ǫ → 0 phase shift θ 0 in the argument of Φ (in) , we obtain on M ′ : The sum in (2.37) contains a finite number of terms. The functions ζ ν (k) and ζ ν are the integrated with respect to θ functions P ν (k) and P ν respectively. So, since is satisfied by the solution J ν (X) ≡ U ν (X) according to the definition of the parameters U ν . Since we suppose that system (2.38) has a generic form we will assume that (locally) this is the only solution and put J ν (X) = U ν (X) in the zero order of ǫ.
After that we can resolve system (2.37) by iterations, taking on the initial step U ν (X) = J ν (X). The substitution of (2.35) into (2.37) under the condition of the non-singularity of matrix ∂ζ ν (J,U ) ∂U µ | U =J will sequentially define the functions u ν (k) . So we obtain the relations (2.35) and (2.36). Now we can take also the values of J ν (X), θ * α 0 (X), θ α 0 (X 0 ) and G i [U [ϕ],θ 0 [ϕ]] (θ, X) with the restrictions (2.34) and also We define now a Poisson bracket on the space of functions ϕ(θ, X) by the formula which is just a rescaling of the bracket (1.4), multiplied by δ(θ−θ ′ ). We normalize here the δ-function δ(θ − θ ′ ) by (2π) m . The pairwise Poisson brackets of the constraints G i [U,θ 0 ] (θ, X) on M ′ can be written in the form (we can omit the Poisson brackets of L i k and L j s on M ′ and also the brackets of the functionals θ α 0 [ϕ] and U ν [ϕ] from Φ (in) in (2.30) since they are multiplied by the convolutions of the corresponding L-operators with the "right eigen vectors" Φ (in)θ α and Φ (in)U ν , which are zero on M ′ ).
Brackets (2.41) evidently satisfy the orthogonality conditions: for q = 1, . . . , N + m in the coordinates J(X), θ * 0 (X) and θ 0 (X 0 ) on the submanifold M ′ . We note now that every derivative with respect to X or Y appears in the bracket (2.40) with the multiplier ǫ but, being applied to the functions on M ′ , contains the nonzero at ǫ → 0 term k α (J) ∂/∂θ α . Now we formulate the statement about the structure of the bracket (2.40) on M ′ in the coordinates J(X), θ * 0 (X) and θ 0 (X 0 ).

Lemma 2.3
The pairwise Poisson brackets of constraints G i [U,θ 0 ] (θ, X) on M ′ have no singular terms at ǫ → 0 and no non-local terms in the zero order of ǫ (ǫ 0 ) at any fixed coordinates J ν (X), θ * α 0 (X) and θ α 0 Proof.
The first statement is evident for the local part of bracket (2.40), since any differentiation with respect to X in it appears with the multiplier ǫ and has the regular at ǫ → 0 form k α (J(X))∂/∂θ α + O(ǫ), being applied to the functions of the form (2.44). So, we should check only the non-local part of (2.40), which contains the multiplier ǫ −1 in it. But, according to the relation (2.28) and also (2.35), we have that the terms arising on the both sides of ν(X − Y ) (the convolutions ofL with S (k) (Φ, k α Φ θ α , . . . )) are of order of ǫ on M ′ in the coordinates J ν (X) and θ * α 0 (X). So, we obtain that all the non-local part of (2.41) is of the order of ǫ on M ′ at any fixed coordinates J ν (X), θ * α 0 (X) and θ α 0 (X 0 ).

Lemma 2.3 is proved.
Let us formulate now the last "regularity" property of the submanifold M ′ with respect to the Poisson structure (2.40).
We consider in the coordinates J ν (X), θ * α 0 (X) and θ α 0 (X 0 ) on M ′ a linear non-homogeneous system on the functions having the form After all differentiations with respect to X we can omit the term which appears in all functions depending on θ and X in (2.45), and then consider the system (2.45) at the zero order of ǫ. From Lemma 2.3 we have that in the zero order of ǫ the brackets {G i (θ, X), G j (θ ′ , Y )} on M ′ do not include non-local terms, containing ν(X − Y ). For the derivatives with respect to X, which arise with the multiplier ǫ from the local terms of {ϕ k (τ, X), ϕ s (σ, Y )}| M ′ , we should take in the zero order of ǫ only the main part k α (J) ∂/∂θ α . So, in the zero order of ǫ we obtain from (2.45) just linear systems of integro-differential equations with respect to θ and θ ′ on the functions f j (θ ′ , X), independent at different X. We have also that the right-hand side of (2.45) satisfies at any X and ǫ the compatibility conditions (2.42) (let us remind that U ν [J, θ * 0 ] are the asymptotic series at ǫ → 0).
(IV) We require that the system (2.45) is resolvable on M ′ for any F [ϕ](ǫ) in the class of 2πperiodic with respect to all θ α functions and its solutions can be represented in the form of regular at ǫ → 0 asymptotic series The condition (IV) is responsible for the Dirac restriction of the bracket (2.40) on the submanifold M ′ . Now we prove a statement which will be very important for our averaging procedure.
First we note that the Poisson brackets of ϕ i (θ, X) with the functionals J ν (Y ) can be written in the form . . ) (we have integrated with respect to θ ′ ). So, the flow generated by the functional q(Y )J ν (Y )dY (where q(Y ) has a compact support) can be written as As can be easily seen, the local terms of (2.47) have the form fulfilled in the general case. It can be actually shown that this requirement can be significantly weakened and replaced by resolvability of system (2.45) just for everywhere dense set of parameters U [ϕ] on M ′ , using the approach represented in [42,43]. We will, however, use here the assumption, formulated above, since the methods used in [42,43] require in fact noticeably longer considerations.
where the term in the brackets is just the flow, generated by the functional In the non-local part of (2.47) (the last expression) we have the convolution of the "slow" functions q Y (Y ) with the rapidly oscillating F ν (k) (ϕ, ǫϕ Y , . . . ), where ϕ i (θ, Y ) has the form (2.44). So, in the leading order of ǫ we can neglect the dependence on θ of the last integral of (2.47) and take the averaged with respect to θ values F ν (k) on M ′ instead of the exact F ν (k) (ϕ, ǫϕ Y , . . . ) in the integral expression in (2.47).
After that we obtain, that the non-local term of (2.47) gives us in the zero order of ǫ a linear combination of the flows S (k) (ϕ, ǫϕ X , . . . ), considered on the functions at any fixed point X.
From the invariance of the submanifold M with respect to the flows (2.17) and (2.18) we can conclude now that the flow (2.47), being considered at the points of M ′ with fixed coordinates J(X), θ * 0 (X), θ 0 (X 0 ) in the zero order of ǫ, leaves M ′ invariant and generates on it a linear evolution of the initial phases with some frequencies Ω αν [q] (X) . Here we use the formula (2.35) for U[J, θ * 0 ] and we can claim now that the Poisson brackets of the functionals θ α 0 (X) with q(Y )J ν (Y )dY at the points of M ′ with fixed coordinates J ν (X), θ * α 0 (X) and θ 0 (X 0 ) have the form Let us now prove the relation at the points of M ′ with fixed values of J ν (X), θ * α 0 (X) and θ α 0 (X 0 ). Using again the relation (2.35) we can write for (2.47) at the points of M ′ where [J, θ * 0 ] means a regular at ǫ → 0 dependence on J, J X , θ * 0X , . . . .
We are interested in the evolution of the functionals where the densities Π µ i(k) were introduced in (2.23). It is easy to see that (2.50) does not change J µ (X) at the zero order of ǫ and we can also state that the terms of the order of ǫ in (2.50) (i.e. ǫη i (θ + . . . , X)) are unessential for the evolution of k(J(X)) on M ′ at the order of ǫ. Indeed, their contribution to the evolution of J µ (X) in the order of ǫ is where we should take only the main term k γ (J(X)) ∂/∂θ γ for the derivatives ǫ ∂/∂X in the formula (2.51). After the integration by parts we have for this contribution But after the substitution of the main part of ϕ i (θ, X) (according to (2.35)) into the densities Π µ i(k) (ϕ, ǫϕ X , . . . ) we obtain in the leading order of ǫ the convolution of η(θ, X) with the variational derivative of the functionalĪ µ , introduced in (2.15), with respect to ϕ(θ, X). Our statement follows now from Lemma 2.1, which claims that the variational derivatives of the functionals k α (Ī[ϕ]) are identically equal to zero on the space of m-phase solutions of (1.6).
Consider now the first term of (2.50). We have that the evolution of J µ (X), which is responsible for the evolution of k(J), is given by the expression where s(X, ǫ) ≡ θ * 0 (X) + θ 0 (X 0 ) + 1 ǫ X X 0 k(J(X ′ ))dX ′ . The first term here after the substitution of exact ϕ i in the form on M ′ , as can be easily seen, is just while the second term on M ′ in the leading order of ǫ is equal to i.e. the relation (2.49). Now, using (2.48) and (2.49), we can write that for any q(Y ) in our coordinates on M ′ . So, we have {θ * α 0 (X), J ν (Y )}| M ′ = O(ǫ) at any fixed coordinates J ν (X), θ * α 0 (X) and θ α 0 (X 0 ). Lemma 2.4 is proved.
Let us now describe the averaging procedure of the Poisson bracket (1.4) on the family of m-phase solutions of (1.6) under the conditions of "regularity" formulated above.
which generate local flows according to the Poisson bracket (1.4) and the averaged densities of which can be taken as parameters U ν on the space of m-phase solutions of (1.6) (the conditions (A)-(C)). Then under the "regularity" conditions (I)-(IV) for the space of m-phase solutions of (1.6) we can construct a Poisson bracket of Ferapontov type (1.5) for the "slow" parameters U ν (X) by the following procedure: We calculate the pairwise Poisson brackets of P ν (ϕ, ϕ x , . . . ) in the form where is a finite number of terms in the both sums). Here we have the total derivatives of the functions F ν (k) and F µ (k) with respect to x and y as a corollary of the fact that both I ν and I µ generate local flows according to the Poisson bracket (1.4). From the commutativity of the set {I ν } we have also for some functions Q νµ (ϕ, ϕ x , . . . ). Then for the "slow" coordinates U ν (X) = P ν (X) we can define a Poisson bracket by the formula where the averaged values are the functions of U(X) and U(Y ) at the corresponding points X and Y . Bracket (3.2) satisfies the Jacobi identity and is invariant with respect to the choice of the set {I 1 , . . . , I N }, satisfying (A)-(C), if the choice of these integrals is not unique, i.e.
The most difficult part is to prove the Jacobi identity for the bracket (3.2). For this we use the Dirac restriction of the Poisson bracket (2.40) on the submanifold M ′ with the coordinates J ν (X), θ * α 0 (X) and θ α 0 (X 0 ) on it. According to the Dirac restriction procedure we should find for J ν (X), θ * α 0 (X) and θ α 0 (X 0 ) the corrections of the form such that the fluxes, generated in the Hamiltonian structure (2.40) by the"functionals" J ν (X) + V ν (X), θ * α 0 (X) + W α (X) and θ α 0 After that we put for the Dirac restriction on M ′ and, in the same way, (and so on).
After calculation of the brackets in (3.3)-(3.5) and the substitution of ϕ(θ, X) in the form (2.44) we obtain regular at ǫ → 0 systems for the functionsv ν j (X, θ, Z, ǫ),w α j (X, θ, Z, ǫ) andō α j (θ, Z, ǫ), which coincide with the system (2.45). From the arguments analogous to those used in Lemma 2.3 and the fact that the flows, generated by the functionals J ν (X), leave invariant the submanifold M ′ at the zero order of ǫ (at fixed coordinates J(X), θ * 0 (X), θ 0 (X 0 )) we have also that the right-hand sides of these systems are regular at ǫ → 0 in these coordinates.
So, according to (IV), we can find the functionsv ν j ,w α j andō α j in the form of regular at ǫ → 0 asymptotic series. (The functions v ν j (X, θ, Z, ǫ), w α j (X, θ, Z, ǫ) and o α j (θ, Z, ǫ) are not uniquely defined but it is easy to show that this does affect the Dirac restriction of the bracket (2.40) on M ′ according to the formulas (3.6)-(3.8)).
Besides that, as was mentioned above, the flows (2.47), generated by the functionals q(X) J µ (X) dX on the functions (2.44), leave invariant the submanifold M ′ at the zero order of ǫ and generate a linear evolution of the initial phases. So, we can conclude that the right-hand side of the linear system (3.3) contains no zero powers of ǫ and we should start the expansion for v ν i (X, θ, Z, ǫ) from the first power. Now we havev According to the relations above and Lemma 2.
for arbitrary smooth q ν (X) and q µ (Y ) with compact supports. For the Poisson brackets of the densities P ν (ϕ, ǫϕ X , . . . ) according to (2.40) we have the expression: Now we should substitute the functions ϕ i (θ, X), ϕ i (θ, Y ) on M ′ in the form and respectively. It is easy to see that the local part of (3.10) gives us the expression in the coordinates J(X), θ * 0 (X) and θ 0 (X 0 ) on M ′ , where s(X) ≡ θ * 0 (X)+θ 0 (X 0 )+ 1 ǫ X X 0 k(J(X ′ ))dX ′ . Here we used only the main part of (3.11) and its derivatives in the second term of (3.13) and replaced U ν (X) by J ν (X) according to (2.35) in the arguments of the averaged functions modulo the higher orders of ǫ.
The non-local part of (3.10) gives for (3.9) the following equalities: where we used the integration by parts for the generalized functions.
We can see now that in the first term of the expression above in the both regions X > Y and X < Y we have the convolution with respect to X and Y of the "slow" functions q ν X (X) q µ Y (Y ) with the rapidly oscillating expressions . . in the main order of ǫ. Here . . . means the averaging with respect to phases θ α . Now, since small ∆X and ∆Y lead to the changes of phase equal to 1 ǫ k α (J(X))∆X + O((∆X) 2 ) and 1 ǫ k α (J(Y ))∆Y + O((∆Y ) 2 ), it is not very difficult to see that in the sense of "generalized" limit (i.e. in the sense of the convolutions with the "slow" functions of X and Y ) we can replace these oscillating expressions in the main order of ǫ just by their mean values where . . . means the averaging on the space of m-phase solutions.
The dependence of {J ν (X), J µ (Y )} D on J(X), J X (X), θ * 0X (X), . . . is regular at ǫ → 0 and, as can be easily seen from (3.15), we have no dependence of θ * 0 in the first order of ǫ. So, it is easy to see now that the Jacobi identities for the bracket {. . . , . . . } D on M ′ with coordinates J(X), θ * 0 (X) and θ 0 (X 0 ), written for the fields J ν (X), J µ (Y ) and J λ (Z) in the order of ǫ 2 , coincide with the corresponding Jacobi identities for the bracket (3.2).
So we proved the Jacobi identity for the bracket (3.2).
The skew-symmetry of the bracket (3.2) is just a trivial corollary of the skew-symmetry of (2.40).
We now prove the invariance of the bracket (3.2) with respect to the choice of the integrals I ν . The proof is just the same as in the local case and we will just reproduce it here.
Theorem 3.1 is proved.

Remark.
From Theorem 3.1 it also follows in particular that the procedure (3.2) is insensitive to the addition of the total derivatives with respect to x to the densities P ν (ϕ, ϕ x , . . . ). This fact, however, can be also obtained from an elementary consideration of the definition of bracket (3.2).

Proof.
It is easy to check by direct substitution that any ofH µ generates the "conservative" form of the Whitham's system for the corresponding flow (2.11). It is easy to see also that this flow conserves any ofH ν and so allH ν andH µ commute view the bracket (3.2). The same property for the Hamiltonian functionH (and also for the integral of the averaged density of any local integral I, commuting with H and I ν and generating a local flow view (1.4)) can be now obtained from the invariance of (3.2) with respect to the set {I ν }, since we can use the Hamiltonian function H in the form (1.2) as one of the integrals instead of any of I ν . Theorem 3.2 is proved.
We can also see that the functionalsH ν give us conservation laws for our Whitham system.
From the Theorem 1.1 it follows also that the flows U ν T = ∂ X F ν (k) (U) commute with all the local flows, generated by local functionals h(U)dX in the Hamiltonian structure (3.2), and it can be also seen that they give us the Whitham's equations for the corresponding flows (1.7).
It can be easily seen also that the described procedure can be applied in the same way to the brackets (1.1) written also in the "irreducible" form and not only in the "canonical" one.