A Nekhoroshev theorem for some perturbations of the Benjamin-Ono equation with initial data close to finite gap tori

We consider a perturbation of the Benjamin Ono equation with periodic boundary conditions on a segment. We consider the case where the perturbation is Hamiltonian and the corresponding Hamiltonian vector field is analytic as a map form energy space to itself. Let $\epsilon$ be the size of the perturbation. We prove that for initial data close in energy norm to an $N$-gap state of the unperturbed equation all the actions of the Benjamin Ono equation remain $\cO(\epsilon^{\frac{1}{2(N+1)}})$ close to their initial value for times exponentially long with $\epsilon^{-\frac{1}{2(N+1)}}$.


Introduction
In this paper we prove a Nekhoroshev type theorem for perturbations of the Benjamin-Ono (BO) equation on the torus.It is well known that the BO equation is a Hamiltonian integrable system, but only recently it has been understood how to introduce Birkhoff coordinates [GK21], namely some regularized action angle variables, for such a system with periodic boundary conditions.We will call finite gap states, functions u which, when written in Birkhoff coordinates, have only a finite number of coordinates different from zero -see [GK21] for a justification of this denomination.Here we consider a small Hamiltonian perturbation of the BO equation and prove that, for initial data close to finite gap initial states, all the actions of the BO equation remain close to their initial value for times which are exponentially long with a power of ǫ −1 , ǫ being the size of the perturbation.The main point is that we are not confined neither to small amplitude initial data nor to neighbourhood of nonresonant tori: as far as we know this is the first result of this kind for PDEs.Our result however has a strong limitation: we deal only with perturbations whose Hamiltonian vector field is an analytic map from the phase space to itself, and in view of the fact that the Poisson structure of the BO equation is the Gardner's one, namely ∂ x , this means that the Hamiltonian must depend on some antiderivative of the state u and typically the vector field is non local.The main example of Hamiltonian perturbation that we can treat is given by ǫP (u), with (1.1) where F (x, y) is analytic in the second variable and continuous in the first one.We also cover Hamiltonian versions of perturbations dealt with by [Gas22].We will comment further on this point after the statement of the main theorem.
Notwithstanding this defect, we think that the fact that exponentially long stability estimates hold for some perturbations of the BO equation, close to finite dimensional tori could spread some light on the behaviour of perturbed integrable Hamiltonian systems.In particular this has to be confronted with the paper [BK05] in which a mechanism of possible instability of infinite dimensional tori was exhibited.
Finally we recall that all known Nekhoroshev type results for PDEs can be split into two types: results dealing with neighbourhoods of nonresonant tori (see [Bou00, BFG20, BG21, KM21]) -which are essentially stability results for such tori -, and results dealing with small amplitude solutions, mainly in perturbations of nonresonant systems (see e.g.[Bou96, Bam03, BG06, BDGS07, BFM22] and references therein).We also mention the paper [Bam99] in which a Nekhoroshev type result controlling also neighbourhood of resonant tori has been proved, for small amplitude solutions in perturbations of the integrable NLS.
The proof of the main result of the present paper is a variant of the proof of [Bam99] (see also [BG93]) which is based on Lochak's proof of Nekhoroshev's theorem [Loc92,LN92].We emphasize that this is made possible by the fact that, as shown by the works [GK21,GKT23], the Hamiltonian of the BO equation, written in Birkhoff coordinates, has a very simple and explicit structure and this allows us to verify all the geometric properties needed in order to apply Lochak's technique.
Acknowledgements The present research was founded by the PRIN project 2020XB3EFL Hamiltonian and dispersive PDEs.It was also supported by GNFM.The second author is supported by the project ANR-23-CE40-0015-01.

Main result
Consider the Benjamin-Ono (BO) equation We also need the notation H s 0 (T, C) for H s functions with zero mean value.
is a diffeomorphism; Φ and its inverse Φ −1 , map bounded sets to bounded sets. 2 .The map is symplectic, in the sense that in terms of the variables ξ n the Hamilton equations (2.3) take the form 3 .In terms of the variables ξ the BO Hamiltonian takes the form In [GKT22a,GKT21] it was also proved that the map Φ is real analytic, in the sense that, for any s > −1/2, Φ extends to a holomorphic map from a complex neighbourhood of H s r,0 in H s 0 (T, C) to h s+1/2 := h s+1/2 + ⊕ h s+1/2 + .A similar result holds for Φ −1 .This is crucial for our application.
From now on we will work in the energy space H 1/2 r,0 .To state our main result, for any integer N ≥ 1, consider the set of N gap states, namely (2.8) The positive quantities γ n (u) := |ξ n (u)| 2 , n = 1, . . ., N, are called the gaps of the state u ∈ G N .In section 7, remark 7.2(i) of [GK21], it is proved that G N is a dense open subset of the symplectic 2N-dimensional manifold where P r denotes the usual Poisson kernel Theorem 2.2.Consider a Hamiltonian system with Hamiltonian function (2.9) H := H BO + ǫP , and assume that there exists N ≥ 1 such that P extends to a real analytic function on a neighbourhood of G N , which is bounded on bounded sets.Also assume that its Hamiltonian vector field X P extends to a real analytic function from an open neighbourhood of r,0 and it is is bounded on bounded sets.Fix two positive parameters 0 < E m < E M , then the following holds true: there exist constants ǫ * , C 1 , ..., C 5 , independent of ǫ s.t.if |ǫ| < ǫ * and the initial datum u 0 fulfills then along the flow of the Hamiltonian system (2.9) one has for all times t fulfilling A family of examples of perturbations P fulfilling our assumptions is (2.15) with F continuous in the first variable and globally analytic in the second variable.This gives rise to the perturbed Benjamin-Ono equation , where Π 0 is the projector on states of zero mean: and f (x, y) := ∂ y F (x, y).
A second example is the Hamiltonian variant of the damping used in [Gas22] and is given by which gives rise to the perturbed BO equation (2.17) ∂ t u = H∂ 2 x u − ∂ x (u 2 ) + ǫ ( u, sin(.)cos x − u, cos(.)sin x) .The fact that this perturbation is Hamiltonian is what guarantees the exponentially long times of stability of the actions.

Proof of Theorem 2.2.
3.1.Scheme of the proof.We recall that, since we are studying large amplitude solutions, the frequencies and the resonance relations that they fulfill depend on the initial datum, so one has to develop also the so called geometric part of the proof of Nekhoroshev's theorem.Here we use Lochak's approach to the geometric part.
We recall that Lochak's proof is based on the idea of first proving long time stability of resonant tori and then showing that any initial datum falls in the domain of stability of some resonant torus, thus deducing long time stability of every torus.
We now describe more in detail the two steps and illustrate the way they are developed here for a perturbation of the BO equation.
First of all we fix a resonant finite gap torus, with gaps γ * 1 , ..., γ * N , and expand the Hamiltonian of the BO equation close to it.This is done explicitely in Eq. (3.7).One has with y * n a linear expression in γ * n (given explicitely by (3.2) and (3.3)) and H 4 given by (2.6).Neglecting the term H 4 one gets the expression of the linearized Hamiltonian at the torus, in particular one gets that the frequency of motion of such a linearized Hamiltonian are given by If one chooses the gaps γ * in such a way that y n (γ * ) = kn q with k 1 , ..., k N , q ∈ Z then the linearized dynamics turn out to be periodic with period q.
The first important remark by Lochak is that it is particularly simple and effective to make averages when the unperturbed dynamics is periodic.So the next step consists in averaging the complete non integrable system close to a fixed resonant torus.However in order to be able to perform the subsequent steps of the proof one has to be very quantitative and to keep into account the size of the neighbourhood of the torus in which the normal form is valid and the dependence of all the constants on the period of the unperturbed dynamics.This is done in Theorem 3.6, which in turn is obtained by applying the main normal form theorem of [Bam99] which is recalled in the appendix (see Theorem A.1). Theorem 3.6 requires as input a precise estimate of the size of the different parts of the Hamiltonian and of their vector fields in a complex neighbourhood of the resonant torus.This is obtained in Lemma 3.4 and Lemma 3.5.
Then, in order to describe the result of the normal form theorem and the way the normal form is used in order to prove stability of the resonant tori we have to introduce some notations.We denote by h ω the Hamiltonian generating the linearized dynamics at the torus: and remark that, since I n are the difference of the actions and the gaps of the resonant torus γ * n , they are not necessarily positive, so that h ω is not positive definite.The normal form Theorem 3.6 ensures the existence of a canonical transformation, defined in a neighbourhood of size R of the resonant torus, conjugating the Hamiltonian of the perturbed BO equation to a Hamiltonian of the form , where Z has the property that {h ω , Z} = 0 and has the same size as the original perturbation, namely ǫ, while R is a remainder which is exponentially small in the inverse of Remark that the small parameter µ contains the size ǫ of the perturbation, the size R of the neighbourhood and the period q of the linearized dynamics at the torus.
We describe now how to use such a normal form to show that the resonant torus is stable over long times.To this end remark that if in H ′ one neglects the exponentially small remainder R, then one has two integrals of motion, namely H ′ and h ω .If one considers also the remainder, then h ω is no more an integral of motion, but it moves by a small quantity over very long times (while H ′ remains an integral of motion).We aim now at exploiting the convexity of H 4 in order to control the motion of each one of the actions.To this end write the conservation of energy for H ′ : evaluating at t = 0 or at a general instant of time, one has reorganizing the terms and using the triangular inequality, one gets Now, all the terms in the right hand side are small over exponentially long times, at least if H 4 (0) is small, i.e., if the initial datum is close to the resonant torus.Then the explicit form of H 4 allows us to ensure that all the quantities s l are small at time t, and therefore that the solution is close to the resonant torus at time t.However the situation is slightly subtler since one needs closeness in some H s topology, namely a topology in which one can ensure the perturbation to be small.However the topology in which H 4 is convex is weaker than an H s topology.So, in order to conclude the proof one also uses the almost conservation of h ω : the fact that h ω (t) is small provides some additional information and allows us control the distance of the solution from the resonant torus in the energy norm.This is obtained in Subsection 3.3.
After establishing the long time stability of resonant tori one wants to extend stability to general finite gap tori.
Let us fix a torus whose stability has to be studied.This is done by fixing its gaps, say γ 0 1 , ..., γ 0 N , correspondingly one gets a sequence of frequency modulations y 0 1 , ..., y 0 N , and one looks for a resonant sequence y * 1 , ..., y * N close to it.This is done by using Dirichlet theorem, according to which, for any Q > 1 there exist k 1 , ..., k N ∈ Z and Z + ∋ q ≤ Q such that By the fact that the map γ → y is invertible, one then identifies a torus with gaps γ * , at a distance R ≃ O from the original one, on which the linearized dynamics is periodic with period q.Thus, the actions remain close to the actions of the resonant torus for a time exponentially long with the inverse of , therefore they also remain close to their initial value.Choosing Q in such a way that the two terms of the small parameter are equal, one gets the wanted exponential estimate valid for all initial data.
We emphasize that a few technical points have also to be taken into account: they are related to the fact that the above conservation arguments are presented in the coordinates introduced by the normalizing change of coordinate, which is different for different resonant tori.The other point is that one has to show that the domain in which the normal form is performed is not left by the initial data one is studying.This is the heart of the proof of Theorem 3.8.

3.2.
Normal form close to resonant tori.From now on, in order to take advantage of the analyticity assumption of the Hamiltonian function, we work in the complexification of the phase space, which means that the variables ξ n have to be considered as independent of ξn , so we will consider as phase space h 1 := h 1 + ⊕ h 1 + and a point of h 1 will be denoted (ξ, η) ≡ ((ξ n ) n≥1 , (η n ) n≥1 ).The real subspace then corresponds to η n = ξn .
3.2.1.Preliminary estimates.Following [GK21] we will denote by γ n := ξ n η n the gaps.We now recall some formulae from [GKT22b].In terms of the Birkhoff coordinates the BO equation takes the form ξn By these formulae it is easy to see that the nonlinear correction to the action to frequency map, namely the map γ → y, is invertible and its inverse is with the convention y 0 := 0. We remark that ω n (γ) are the frequency of motion on the torus corresponding to the state with actions γ.
In particular it follows that for N-gap states one has y n = y N ∀n ≥ N + 1, and viceversa, given any N dimensional vector (y 1 , ..., y N ) with the open condition y N > y N −1 and 2y n > y n+1 + y n−1 , n = 1, . . ., N −1, one gets a corresponding N gap state.
Fix now a reference N dimensional invariant torus with gaps γ * ≡ (γ * 1 , ..., γ * N ) that we will also assume to fulfill (3.5) E m < γ * n < E M , ∀n = 1, ..., N ; correspondingly we consider the "frequency vector" y * ≡ (y * 1 , ..., y * N ) defined by (3.2), (3.3).In particular we thus have We expand the BO Hamiltonian at γ * .Writing I n := γ n − γ * n , n = 1, . . ., N, and I n := γ n , n ≥ N + 1, one gets, having set γ * n := 0 for n ≥ N + 1, (3.7) We now introduce action angle variables for the first N modes, namely we introduce variables (I n , φ n ) N n=1 by (3.8) The variables in the phase space will now be ) .and the BO-Hamiltonian turns out to be (3.7) with namely the gaps of index larger than N must be considered as functions of the Birkhoff coordinates.
We will call C the map that to the variables (I, φ, (ξ n ) n≥N +1 , (η n ) n≥N +1 ) associates the Birkhoff variables, namely with (3.10) Since we work in the complex extension of the phase space the variables I, φ will be assumed to be complex.Definition 3.1.A state will be said to be real if one has We now define the norm in the phase space.It will depend on a parameter R which will be eventually linked to ǫ.So we define (3.13) From now on the balls will be intended with respect to this norm.The complex ball with center u and radius R 1 in this norm will be denoted by B R 1 (u).
A function which will play a fundamental role in the rest of the paper is We now define the domain G which will then be extended to the complex domain in order to apply Theorem A.1.We fix a parameter ǫ 1 and define (3.16) Most of the time, the balls will be taken of radius R with R the same parameter in the definition of the norm (3.13).
From now on we will use the notation a b in order to mean "there exists a constant C independent of ǫ, ǫ 1 , R such that a ≤ Cb ".
A first property of the states in G is given by the next lemma Lemma 3.2.For u ≡ (I, φ, ξ, η) ∈ G one has Proof.By the reality of the state one gets s n ∈ R, thus (2.6) implies We come to (3.19)(3.20).First we fix L such that (L + 1) 2 − 2y * N ≥ (L + 1) 2 /2.Notice that the size of L is controlled by E M .Then we have where we have used h ω ≤ ǫ 1 on G.This implies also n≥N +1 n 2 γ n ǫ 1 .Now, one has, for l ≥ N + 1, which clearly holds also for n ≤ N.This gives Then we remark that, by Lemma 3.2 one has that N , thus, by Lemma 3.3, provided ǫ 1 and R are small enough, one has that and furthermore the set in the left hand side is bounded.Furthermore, if R and ǫ 1 are small enough, then C(G R ) is in the domain of analyticity of Φ and furthermore Φ(C(G R )) is contained in the domain of analyticity and boundedness of P and of its vector field.It follows that since the supremum is taken over a domain smaller than the domain of boundedness of P .Furthermore, since one also has We still have to introduce the action angle variables.To this end we use We compute now dC.To this end remark first that this linear map is the identity on the modes with index larger than N, so consider the smaller indexes: denoting (ξ ′ , η ′ ) T = dC(u)(h I , h φ ) T , one has, for n ≤ N, whose inverse linear map is To evaluate the norm of this map, we compute , which is ensured by R 2 < E m /8, the second term in the right hand side is bounded uniformly in R. The first term is bounded by a constant times R −1 , so one gets (3.31).

Normal form.
From now on we take (3.35) so that, on G R , one has We now fix a resonant torus, make a normal form close to it and use h ω and the energy as Lyapunov functions in order to prove long time stability of this torus.For definiteness we avoid here the use of the symbol .
Theorem 3.6.Under the assumptions of Theorem 2.2, assume that with k 1 , ..., k n , q ∈ Z + , then there exist positive constants µ * , K 1 , ..., K 5 (independento of k n and q) such that, if then there exists an analytic canonical transformation T : G 3R/4 → G R with T (G 3R/4 ) ⊃ G R/2 which is close to identity, namely and is such that

with
• Z is analytic on G 3R/4 toghether with its Hamiltonian vector field and fulfills furthermore it is in normal form, namely one has • R is analytic on G 3R/4 toghether with its Hamiltonian vector field and fulfills Proof.The theorem is a direct application of Theorem A.1 of the appendix: it is obtained by inserting the values of the constants obtained from Lemmas 3.4, 3.5.We just remark that we have Applying Remark A.2, one also gets the following result.
Remark 3.7.By the above preliminary estimates one has Theorem 3.8.There exist positive constants µ * ≪ 1 and K, K ≫ 1 with the following properties: assume and consider initial data u 0 with Then, for all times t fulfilling the solution of the perturbed BO equation (2.3), (2.9), fulfills is not bigger than 3R 2 /4.In particular h ω (t * ) ≤ 3R 2 /4.Notice that the second estimate of (3.58) is ensured by (3.52) and the first one by (3.50).We come to H 4 .Exploiting the conservation of the Hamiltonian (3.42), one gets taking also into account (3.49) one gets with ǫ 2 to be determined later.We look for a resonant torus close to it.Consider γ 0 n := |ξ 0 n | 2 , for n = 1, ..., N. We approximate the torus with gaps γ 0 n by a resonant torus.To this end define y 0 n by (3.2) so that |y 0 n | ≤ N 2 E M and use Dirichlet's theorem to approximate y 0 n by a rational vector.Indeed Dirichlet's theorem ensures that for every Q ≥ 1 there exist integers k 1 , ..., k N , q ∈ Z with 1 ≤ q ≤ Q such that We can now compute h ω (0): passing to the action variables I n , we have H 4 (0) ≤ (K + 1)ǫ 2 2 ≤ 2ǫ 2 2 .In order to ensure (3.51), we take (3.67)R 2 := 40N 3 E M qQ 1/N .We now aim at applying Theorem 3.8.Then we would like to optimize the value of Q in order to maximize the time of validity of the estimates.Inserting (3.67) in (3.38) we get (with a suitable C) (3.68) µ ≤ C 1 qQ 1/N + ǫqQ 1/N q ≤ C 1 Q 1/N + ǫQ 2+1/N .This would lead to choose Q by imposing the two terms in brackets to be equal.This would give (3.69) ) , but we also have to ensure the validity of the last inequality in (3.50), while (3.69) can only ensure R 4 ≥ (40N 3 E M ) 2 ǫ, which is not necessarily bigger than Kǫ.For this reason we take (3.70) , so that (3.71) In particular, concerning the second estimate, we remark that, by the Dirichlet theorem, q is smaller than Q, but it can be of order 1.Inserting in the different estimates one concludes the proof.
Proof.We proceed by a bootstrap argument.Assume that there exists t satisfying (3.52) and such that (3.53) does not hold.Denote by t * a time of minimal absolute value satisfying (3.52) and such that(3.54)hω(u(t* )) = R 2 or H 4 (u(t * )) = R 4 ,and let us look for a contradiction.First, in view of (3.54), we can make the canonical transformation T provided by Theorem 3.6 near the trajectory for |t| ≤ |t * |, and we set,for |t| ≤ |t * |, u = T (u ′ ) , h ω (t) := h ω (u(t)) , h ′ ω (t) := h ω (u ′ (t)) , and similarly for H 4 .From (3.51) and (3.48) we have, with a constant C that can change from line to line, and in particular, in view of (3.15) and (3.35), u(t) ∈ G.