Vey theorem in infinite dimensions and its application to KdV

We consider an integrable infinite-dimensional Hamiltonian system in a Hilbert space $H=\{u=(u_1^+,u_1^-; u_2^+,u_2^-;....)\}$ with integrals $I_1, I_2,...$ which can be written as $I_j={1/2}|F_j|^2$, where $F_j:H\to \R^2$, $F_j(0)=0$ for $j=1,2,...$ . We assume that the maps $F_j$ define a germ of an analytic diffeomorphism $F=(F_1,F_2,...):H\to H$, such that dF(0)=id$, $(F-id)$ is a $\kappa$-smoothing map ($\kappa\geq 0$) and some other mild restrictions on $F$ hold. Under these assumptions we show that the maps $F_j$ may be modified to maps $F_j^\prime$ such that $F_j-F_j^\prime=O(|u|^2)$ and each $\frac12|F'_j|^2$ still is an integral of motion. Moreover, these maps jointly define a germ of an analytic symplectomorphism $F^\prime: H\to H$, the germ $(F^\prime-id)$ is $\kappa$-smoothing, and each $I_j$ is an analytic function of the vector $(\frac12|F'_j|^2,j\ge1)$. Next we show that the theorem with $\kappa=1$ applies to the KdV equation. It implies that in the vicinity of the origin in a functional space KdV admits the Birkhoff normal form and the integrating transformation has the form `identity plus a 1-smoothing analytic map'.


Introduction
In his celebrated paper [Vey78] J. Vey proved a local version of the Liouville-Arnold theorem which we now state for the case of an elliptic singular point. 1 Consider the standard symplectic linear space (R 2n x , ω 0 ), ω 0 = n j=1 dx j ∧ dx n+j . Let H(x) = O(|x| 2 ) be a germ of an analytic function 2 and V H be the corresponding Hamiltonian vector field. It has a singularity at zero and we assume that in a suitable neighbourhood O of the origin, H has n commuting analytic integrals H 1 = H, H 2 , . . . , H n such that H j (x) = O(|x| 2 ) for each j, the quadratic forms d 2 H j (0), 1 ≤ j ≤ n, are linearly independent and for all sufficiently small numbers δ 1 , . . . , δ n we have {x : H j (x) = δ j ∀ j} ⋐ O. Then in the vicinity of the origin exist symplectic analytic coordinates {y 1 , . . . , y 2n } (i.e. n j=1 dy j ∧dy n+j = ω 0 ) such that each hamiltonian H r (x) may be written as H r (x) =Ĥ r (I 1 , . . . , I n ), I j = 1 2 (y 2 j + y 2 n+j ), whereĤ 1 , . . . ,Ĥ n are germs of analytic functions on R n .
Vey's proof relies on the Artin theorem on a system of analytic equations, so it applies only to analytic finite-dimensional Hamiltonian systems. The theorem was developed and generalised in [Eli84,Ito89,Eli90,Zun05]. In [Eli84,Eli90] Eliasson suggested a constructive proof of the theorem, which applies both to smooth and analytic hamiltonians and may be generalised to infinite-dimensional systems. In this work we use Eliasson's arguments to get an infinite-dimensional version of Vey's theorem, applicable to integrable Hamiltonian PDE. Namely, we consider the l 2 -space h 0 , formed by sequences u = (u + 1 , u − 1 , u + 2 , u − 2 , . . . ), provide it with the symplectic form ω 0 = ∞ j=1 du + j ∧ du − j , and include h 0 in a scale {h j , j ∈ R} of weighted l 2 -spaces. Let us take any space h m , m ≥ 0, and in a neighbourhood O of the origin in h m consider commuting analytic hamiltonians I 1 , I 2 , . . . . We assume that I j = O( u 2 m ) ≥ 0 ∀ j and that this system of functions is regular in the following sense: There are analytic maps F j : O → R 2 , j ≥ 1, such that I j = 1 2 |F j | 2 and i) the map F = (F 1 , F 2 , . . . ) : O → h m is an analytic diffeomorphism on its image, ii) dF (0) = id and the mapping F − id analytically maps O → h m+κ for some κ ≥ 0 (i.e., F − id is κ-smoothing). Moreover, for any u ∈ O the linear operator dF (u) * − id continuously maps h m to h m+κ .
We also make some mild assumptions concerning Cauchy majorants for the maps F − id and dF (u) * − id, see in Section 1. The main result of this work is the following theorem: See Section 1 for a more detailed statement of the result and see Section 4 for its proof. In Section 3 we develop some infinite-dimensional techniques, needed for our arguments.
Theorem 0.1 applies to study an integrable Hamiltonian PDE in the vicinity of an equilibrium. In Section 2 we apply it to the KdV equation under zero-meanvalue periodic boundary conditionṡ u(t, x) = 1 4 u xxx + 6uu x , x ∈ S 1 = R/2πZ, To apply Theorem 0.1 we first normalise the symplectic form ν to the canonical form ω 0 . To do this we write any u(x) ∈ H m 0 as Fourier series, , and consider the map Then T : H m 0 → h m+1/2 is an isomorphism for any m, and T * ω 0 = ν. The Lax operator for the KdV hierarchy is the Sturm-Liouville operator L u = −∂ 2 /∂x 2 − u(x). Let γ 1 , γ 2 , . . . be the lengths of its spectral gaps. Then γ 2 j (u), j ≥ 1, are commuting analytic functionals which are integrals of motion for all equations from the hierarchy. In [Kap91] T. Kappeler suggested a way to use the spectral theory of the operator L u to construct germs of analytic maps . In Sections 5 we show that the map Ψ = (Ψ 1 , Ψ 2 , . . . ) meets assumptions i), ii) with κ = 1 (see Theorem 2.1). So the system of integrals I j (v) = 1 2 |Ψ j (v)| 2 , j ≥ 1, is regular. Accordingly, Theorem 0.1 implies the following result (see Section 2): Theorem 0.2. For any m ≥ 0 there exists a germ of an analytic symplectomorphism Ψ : (H m 0 , ν) → (h m+1/2 , ω 0 ), dΨ(0) = T , such that a) the germ Ψ − T defines a germ of an analytic mapping H m 0 → h m+3/2 ; b) each γ 2 j , j ≥ 1, is an analytic function of the vectorĪ = ( 1 2 |Ψ j (u)| 2 , j ≥ 1). Similar, a hamiltonian of any equation from the KdV hierarchy is an analytic function ofĪ (provided that m is so big that this hamiltonian is analytic on the space H m 0 ); c) the maps Ψ, corresponding to different m, agree. That is, if Ψ m j corresponds to m = m j , j = 1, 2, then Ψ m 1 = Ψ m 2 on h max(m 1 ,m 2 ) .
Moreover, Remark 4) to Theorem 1.1 with k = 2 and Remark at the end of Section 5 jointly imply that the mapΨ equals Ψ • T up to O(u 3 ): Assertion b) of the theorem means that the map Ψ puts KdV (and other equations from the KdV hierarchy) to the Birkhoff normal form.
In a number of publications, starting with [Kap91], T. Kappeler with collaborators established existence of a global analytic symplectomorphism which satisfies assertion b) of Theorem 0.2, see in [KP03]. Our work shows that a local version of Kappeler's result follows from Vey's theorem. What is more important, it specifies the result by stating that a local transformation which integrates the KdV hierarchy may be chosen '1-smoother than its linear part'. This specification is crucial to study qualitative properties of perturbed KdV equations, e.g. see [KP09]. A global symplectomorphism Ψ as above integrates the KdV equation, i.e. puts it to the Birkhoff normal form. Similar, the linearised KdV equatioṅ u = u xxx may be integrated by the (weighted) Fourier transformation T . An integrating transformation Ψ is not unique . 3 For the linearised KdV we do not see this ambiguity since T is the only linear integrating symplectomorphism. In the KdV case the best transformation Ψ is the one which is the most close to the linear map T = dΨ(0) in the sense that the map Ψ − T is the most smoothing. Motivated by Theorem 0.2 and some other arguments (see in [KP09]), we are certain that there exists a (global) integrating symplectomorphism Ψ such that Ψ − T is 1-smoother than T . 4 In Proposition 2.2 we show that if a germ of an integrating analytic transformation Ψ is such that Ψ − T is κ-smoothing, then κ ≤ 3/2. We conjecture that the 1-smoothing is optimal. Acknowledgment. We thank H. Eliasson for discussion of the Vey theorem.
1 The main theorem.
Consider a scale of Hilbert spaces {h m , m ∈ R}. A space h m is formed by complex sequences u = (u j ∈ C, j ≥ 1) and is regarded as a real Hilbert space with the Hilbert norm (1.1) 3 in difference with the mapping to the action variables u → I • Ψ(u), which is unique. 4 We are cautious not to claim that the symplectomorphism Ψ, constructed in [KP03], possesses this extra smoothness since it is normalised by the condition It is not obvious that an optimal global symplectomorphism satisfies this condition, and we do not know if the local symplectomorphism Ψ from Theorem 0.2 meets it.
We will denote by ·, · the scalar product in h 0 : u, v = u j · v j = Re u jvj . For any linear operator A : h m → h n we will denote by A * : h −n → h −m the operator, conjugated to A with respect to this scalar product.
Definition 1. An analytic germ F as above is called normally analytic (n.a.) if F defines a germ of a real analytic map h m R → h n R , where the space h m R is formed by real sequences (u j ), given the norm (1.1). That is, each (1.3) Here the linear map Φ(|u|) = Φ F (|u|) ∈ L(h m R , h m+κ R ) has non-negative matrix elements and defines an analytic germ |u| → The notion of a n.a. germ formalizes the method of Cauchy majorants in a way, convenient for our purposes. We study the class of n.a. germs and its subclass A m,κ in Section 3.
We will write elements of the spaces h m as u = (u k ∈ C, k ≥ 1), u k = u + k + iu − k , u ± k ∈ R, and provide h m , m ≥ 0, with a symplectic structure by means of the two-form This form may be written as ω 0 = idu ∧ du. Here and below for any antisymmetric (in h 0 ) operator J we denote by Jdu ∧ du the 2-form (Jdu ∧ du)(ξ, η) =< Jξ, η > . (1.4) The form ω 0 is exact, ω 0 = dα 0 , where For a map f : h m → h −m , f (u)du stands for the one-form Re f j (u)ξ j .
Then there exists a germ Ψ + : h m → h m which satisfies i), ii) with the same κ, and such that a) foliation of the vicinity of the origin in h m by the sets is the same as by the sets { Ψ + j 2 = const j , ∀j}.
2) By the item a) of the theorem each I j (Ψ(u)) is a function of the vector I + = {I +j = 1 2 |Ψ +j | 2 , j ≥ 1}. In fact, I j is an analytic function of I + with respect to the norm I + = |I +j |j 2m . E.g., see the proof of Lemma 3.1 in [Kuk00].
3) The map Ψ + is obtained from Ψ in a constructive way, independent from m.

5) The theorem above is an infinite-dimensional version of Theorem C in [Eli90] which is the second step in Eliasson's proof of the Vey theorem.
At the first step he proves that any n commuting integrals H 1 , . . . , H n as in Introduction can be written in the form ii). In difference with his work we have to assume that the integrals are of the form ii), where the maps Ψ 1 , Ψ 2 , . . . have additional properties, specified in i). Fortunately, we can check i) and ii) for some important infinite-dimensional systems.

Application to the KdV equation
To apply Theorem 1.1 we need a way to construct germs of analytic maps Ψ : h m → h m which satisfy i) and ii). Examples of such maps may be obtained from Lax-integrable Hamiltonian PDEṡ (2.1) (We normalized original Hamiltonian PDEs and wrote them as the Hamiltonian systems (2.1) in the symplectic space as in Section 1). The Lax operator L u , corresponding to equation (2.1), is such that its spectrum σ(L u ) is an integral of motion for (2.1). Spectral characteristics of L u may be used to construct (real)-analytic germs Ψ j : h m → R 2 ≃ C such that the functions 1 2 |Ψ j | 2 , j ≥ 1, are functionally independent integrals of motion. For some integrable equations these germs jointly define a germ of an analytic diffeomorphism u → Ψ = (Ψ 1 , Ψ 2 , . . . ), satisfying i) and ii). Below we show that this is the case for the KdV equation. Our construction is general and directly applies to some other integrable equations (e.g. to the defocusing Schrödinger equation). Consider the KdV equation (0.1). This is a Hamiltonian equation in any Sobolev space H m 0 , m ≥ 1, given symplectic structure by the form ν, see Introduction. It is Lax-integrable with the Lax operator L u = −∂ 2 /∂x 2 − u(x). Let γ 1 (u), γ 2 (u), . . . be the sizes of spectral gaps of L u (e.g., see in [Kuk00,KP03]). It is well known that γ 2 1 (u), γ 2 2 (u), . . . are commuting analytic integrals of motion for (0.1), as well as for other equations from the KdV hierarchy, see in [KP03].
In Section 5 we show that the spectral theory of L u may be used to construct an analytic germ Ψ : h 1/2 → h 1/2 , Ψ = (Ψ 1 , Ψ 2 , . . . ), Ψ j ∈ R 2 , with the following properties: Applying Theorems 2.1 and 1.1 to the KdV equation, written in the variables v = T (u) ∈ h m ′ , we get Theorem 0.2, stated in the Introduction. Indeed, assertions a) and b) follow from the two theorems and Remark 2 to Theorem 1.1 since the hamiltonian of any n-th KdV is a function of the lengths of spectral gaps. Assertion c) follows from Remark 3.
Towards the optimality of Theorems 0.2 and 2.1 we have the following partial results.
Proof. We may assume that κ ≥ 1. Denote by G the germ It defines a germ of analytic mapping H 1 0 → R and can be written as an absolutely and uniformly convergent series It follows from assumption b) that K 2l+1 , l = 1, 2, . . . , vanish identically. In particular, K 3 (u) ≡ 0. Together with (2.2) this leads to the relations valid for each u ∈ H 1 0 . If κ ≥ 2 we have an obvious contradiction. It remains to consider the case when 1 ≤ κ < 2. Now u H 2−κ 0 ≤ u 2−κ H 1 0 |u| κ−1 L 2 and the inequality above implies that Proof. Assume that κ > 2. Keeping the notations above we still have The first term in the r.h.s. clearly belongs to . So the second term in the r.h.s. also belong to H m ′ +κ−2 . Since κ − 2 > 0, then the sum of the first two terms cannot cancel identically the third, belonging to H m ′ 0 . Contradiction.
3 Properties of normally analytic germs Substituting series in series, collecting similar terms and replacing u j andū j by |u j | we get G • F (|u|). Next consider G • F (|u|). This series is obtained by the same procedure as G • F (|u|), but instead of calculating the modulus of an algebraical sum of similar terms we take the sum of their moduli. As |a + b| ≤ |a| + |b|, we get G • F ≤ G • F . Since both series have non-negative coefficients and G • F defines an analytic germ h n 1 R → h n 3 R , the assertion follows.
Lemma 3.2. If F : h m → h m is a n.a. germ such that F 1 = dF (0) = id, then the germ G = F −1 exists and is n.a.
Here and below we freely identify n-homogeneous maps with the corresponding n-linear symmetric forms. Since F (G(v)) = v, we have the recursive relations For the same reasons as in the proof of Lemma 3.1 we have

These recursive formulas define a germ of an analytic map
For a n.a. germ F : h m → h n consider its differential, which we regard as a germ Lemma 3.3. Germ (3.1) is n.a. and dF (|u|)|v| ≤ dF (|u|)|v|.
iii) If F ∈ A m,κ , then the map u → dF (u)u also belongs to A m,κ .
Proof. i) If F, G ∈ A m,κ , then F •G : h m → h m+κ is n. a. by Lemma 3.1. It remains to verify that it satisfies (1.3). We have d(F •G(u)) * = dG(u) * dF (G(u)) * . Arguing as when proving Lemma 3.1 we get iii) We skip an easy proof (cf. arguments in the proof of Lemma 3.6).
Proof. Denote a solution for (3.2) as u = u(t; v), and decompose u(t; v) in series in v: . . , u kr (s))ds.
Arguing by induction we see that the sum ∞ k=1 u k (t, v) defines a n.a. germ. This is the germ of the map ϕ t (v).
For any vector ξ, dϕ t (v)ξ = w(t) is a solution of the linearized equatioṅ So dϕ t (v)ξ = U(t)ξ, where the linear operator U(t) may be calculated as follows This series converges if u(0) = v m ≪ 1. Taking the adjoint to the integral above we see that dϕ t (u) − id satisfies (1.3) and the corresponding operator Φ t (|v|) meets the estimate Replacing |u(t n )| by ϕ tn (|v|) we see that the operator Φ t defines an analytic Let G 0 , F 0 ∈ A m,κ . Denote F (u) = u+F 0 (u). The arguments in Section 4 use the map B(u) = dG 0 (u) * (iF (u)).

Proof of the main theorem
In this section we prove Theorem 1.1, following the scheme, suggested in Section VI of [Eli90]. To overcome corresponding infinite-dimensional difficulties we check recursively that all involved germs Ψ of transformations of the phase-space h m are of the form id +Ψ 0 , where Ψ 0 ∈ A m,κ .
Step 1. At this step we will achieve that the average in angles of the form ω 1 equal to ω 0 .
To prove this we first note that Using the Cartan formula (e.g., see Lemma 1.2 in [Kuk00]) we have The 1-form in the r.h.s. equals and the assertion follows.
is antisymmetric and can be written as To prove the lemma it suffices to show that since then the assertion would follow by the arguments, used to establish (4.3). Moreover, by continuity it suffices to verify the relation at a point v = (v 1 , v 2 , . . . ) such that v j = 0 for all j. Due to ii), {I j (v), I k (v)} ω 1 = 0 for any j and k. That is Consider the space Σ v = span{v j 1 j , j ≥ 1} (as before 1 r = (0, . . . , 1, . . . ), where 1 is on the r-th place). Its orthogonal complement in h 0 is iΣ v = span{iv j 1 j , j ≥ 1}. Relations (4.5) imply that J 1 (v)ξ, η = 0 for any ξ, η ∈ Σ v . Hence, , then for small v this linear operator is an isomorphism. As So ω 1 (χ i , χ j ) = J 1 J 1 ξ i , J 1 ξ j = − ξ i , J 1 ξ j = 0 and the lemma is proved.
By (4.3) and the lemma above, relation (4.3) also holds with α 0 replaced by α △ . That is, Also note that by (4.1) For any function g(v) and for j = 1, 2, . . . denote Due to (4.6), (4.7) the system of equations (4.2) is solvable and its solution is given by an explicit formula due to J. Moser (see in [Eli90]): Consider the germ f of a function in h m : If the series converges in C 0 (h m ), as well as the series for χ j (f ), j ≥ 1, then f is a solution of (4.2).

Proof of Theorem 2.1
The construction of Ψ that we present below follows the ideas of [Kap91] (also see [Kuk00], pp. 42-44). It relies on the spectral theory of the corresponding Lax operator L u = −∂ 2 x − u. It will be convenient for us to allow for complex-valued potentials u: We write u(x) as Fourier series u(x) = 1 is an isomorphism. Here and below we use the notations A sequence w called real if F −1 (w) is a real-valued function. That is, if w j = w −j for each j. We view L u as an operator on L 2 (R/4πZ) with the domain D(L u ) = H 2 (R/4πZ). The spectrum of L u is discrete and for u real is of the form x , that is, to the set σ(L 0 ) = {j 2 /4, j ≥ 0}. More precisely, one has For j ≥ 1 we will denote by E j (u) the invariant two-dimensional subspace of L u , corresponding to the eigenvalues λ 2j−1 (u), λ 2j (u), and by P j (u) the spectral projection on E j (u): Here γ j is a contour in the complex plane which isolates λ 2j−1 and λ 2j from other eigenvalues of L u . For the computations that will be performed below we fix the contours as γ j = {λ ∈ C, |λ − j 2 /4| = δ 0 j}, δ 0 > 0 small. Clearly, u → P j (u), j ≥ 1, are analytic 6 maps from V δ = {u ∈ L 0 2 (S 1 , C), u L 2 ≤ δ} to L(L 2 , H 2 ), L 2 = L 2 (R/4πZ), H 2 = H 2 (R/4πZ), provided δ is sufficiently small. Furthermore, it is not difficult to check that for j ≥ 1 and u ∈ V δ . Here P j0 is the spectral projection of L 0 , corresponding to a double eigenvalue j 2 /4: Following [Kat66], see also [Kap91], we introduce the transformation operators U j (u), j ≥ 1: It follows from (5.1) that the maps u → U j (u) are well defined and analytic on V δ . It turns out (see [Kat66]) that the image of U j (u) is E j (u) and for u real one has For j ∈ Z 0 let us set Here (·, ·) stands for the standard scalar product in L 2 ([0, 4π], C): (f, g) = fḡ dx.
Lemma 5.1. For u real, one has Proof. Assertion i) is obvious (see (5.3)). To check ii), consider It follows from (5.2), (5.3) that the vectors e j , e −j form a real orthonormal basis of E j (u). Let M j (u) be the matrix of the self-adjoint operator − √ π(L u − j 2 /4) E j (u) in this basis: Consider the deviators M D j , j ≥ 1, (for a 2 × 2 matrix M its deviator is the traceless matrix M − ( 1 2 tr M)I): By construction, one has |z j (u)| 2 = 4(a 2 j + b 2 j ) = π(λ 2j (u) − λ 2j−1 (u)) 2 . Functions z j (u), j ∈ Z 0 , are analytic functions of u ∈ V δ , vanishing at zero. They can be represented by absolutely and uniformly converging Taylor series that we will write in terms of the Fourier coefficients w = F (u).
where Z j n (w) are bounded n-homogeneous functionals on h 0 (Z 0 ): K j n (·) being a symmetric function on Z n 0 . Notice that As a consequence, (5.8) Substituting (5.8) into (5.5), one gets which gives that Z 1 (w) = w and In a similar way, one can show that (5.10) The structure of higher order terms Z j n (w) is described by the following lemma which is the key technical step of our analysis.
Here R is a positive constant, independent of j and n.
Postponing the proof of this lemma till the end of the section, we proceed with the construction of the map Ψ. Introduce the map F that associates to w ∈ h 0 (Z 0 ) the sequence F (w) = (z j (u), j ∈ Z 0 ), u = F −1 (w). Since Z 1 (w) = w, we write F as the sum where Z 2 (w) = (Z j 2 (w), j ∈ Z 0 ). Notice that, by the construction, As a direct consequence of Lemma 5.2 (i), (ii) one gets is analytic and normally analytic.
, is real analytic.
Proof. We have Here B j n and B j n are the n-linear forms with the kernels B j n and B j n respectively: To prove the lemma it is sufficient to show that for n ≥ 3 the poly-linear map B n = (B j n , j ∈ Z 0 ) : h m R × · · · × h m R → h m+2 R is bounded and verifies: It follows from Lemma 5.2 (i) that Combining this inequality with (ii) of Lemma 5.2 and using once more (5.11) we get (5.12) for any n ≥ 3.
We next denote byD the operator of multiplication by the diagonal matrix diag(|j| 1/2 , j ∈ Z 0 ). It defines isomorphismsD : h r → h r−1/2 , r ∈ R. Let us set m ′ = m + 1 2 ≥ 1 2 . For any analytic germ H : h m → h m+a we will denote by HD the germ Notice that the operations F → FD and F → F commute.
Proof. As in the proof of Lemma 5.4, it is sufficient to prove the statement, corresponding to dFD 2 (v) t . We write , is real analytic. Therefore it is also real analytic as a map from ). Next consider ZD 2 =D −1 Z 2D . Note that Z 2 (w) = constD −2 w * D −2 w. Accordingly, dZ 2 (w)(f ) = constD −2 w * D −2 f , and where m ′ + 1/2 ≥ 1, and since the convolution defines a continuous bilinear map h r × h r → h r if r > 1/2, then we have is bounded, which concludes the proof of Lemma 5.5.
This concludes the proof of Theorem 2.1. It remains to prove Lemma 5.2. We will obtain it as a consequence of Lemma 5.7. Z j n (w) (see (5.6)) can be represented as Z j n (w) = i=(i 1 ,i 2 ,...,in)∈Z n 0K j n (i)w i 1 w i 2 . . . w in , (5.13) where for n ≥ 2K j n satisfies K j n (i) ≤ R n A j n (i), A j n (i) = δ j, (5.14) Here < j >= (1 + j 2 ) 1/2 .
Remark. The difference between representation (5.7) and (5.13) is thatK j n (i) are not required to be symmetric functions.