Louise Gassot

Zero-dispersion limit for the Benjamin-Ono equation on the torus with single well initial data.

The Benjamin-Ono equation [START_REF] Benjamin | Internal waves of permanent form in uids of great depth[END_REF][START_REF] Ono | Algebraic solitary waves in stratied uids[END_REF] describes a certain regime of long internal waves in a two-layer uid of great depth. The parameter ε describes the balance between the dispersive term and the nonlinear term. For ε = 0, this equation becomes the inviscid Burgers equation, which causes the formation of shocks in nite time. When ε > 0, the dispersive term prevents the formation of a dispersive shock, replacing it by a train of waves (see for instance [START_REF] Miller | On the zero-dispersion limit of the Benjamin-Ono Cauchy problem for positive initial data[END_REF] for numerical simulations on the real line). It is expected that right after the breaking time, the amplitude of shock waves does not decrease with ε, while the wavelength is proportional to ε. As a consequence, the limit can exist in the weak sense in the shock region.

Zero-dispersion limit

In this paper, we describe the weak zero-dispersion limit for the Benjamin-Ono equation at all times given a single well initial data u 0 ∈ C 3 (T).

Denition 1.1 (Single well potential). We say that u 0 ∈ C 3 (T) is a single well potential if the following holds:

1. u 0 is real valued with zero mean; 2. u 0 (0) = min x∈T u(x); 3. there exists x max ∈ (0, 2π) such that u 0 > 0 on (0, x max ) and u 0 < 0 on (x max , 2π); 4. there are exactly two inection points ξ -∈ (0, x max ) and ξ + ∈ (x max , 2π) such that u 0 (ξ ± ) = 0, and the inection points are simple u 0 (ξ ± ) = 0.

The latter condition only aims at simplifying the study of the Burgers equation with initial data u 0 in the proof of Theorem 3.13, and could be removed. In this case, for every η ∈ (min u 0 , max u 0 ), η has exactly two antecedents by u 0 . We denote

x -(η) = inf{x ∈ [0, 2π] | u 0 (x) = η}, x + (η) = sup{x ∈ [0, 2π] | u ( 0 x) = η}.
According to the work of Miller and Xu on the real line [START_REF] Miller | On the zero-dispersion limit of the Benjamin-Ono Cauchy problem for positive initial data[END_REF], the relevant zero-dispersion limit for the Benjamin-Ono equation is the multivalued solution to the Burgers equation obtained by the method of characteristics. More precisely, every point u B is an image of this multivalued solution at time t with abscissa x as soon as it solves the implicit equation u B = u 0 (x -2u B t).

Given t and x, there may be several solutions u B that are denoted u B 0 (t, x) < • • • < u B 2P (t,x) (t, x), see Figure 1. We dene the signed sum of branches as u B alt (t, x) := 2P (t,x) n=0 (-1) n u B n (t, x).

(

) 1
Our main result is as follows.

Theorem 1.2 (Zero-dispersion limit for the Benjamin-Ono equation). Let u 0 ∈ C 3 (T) be a single well initial data. Let u B alt be the signed sum of branches for the multivalued solution to the inviscid Burgers equation with initial data u 0 . Then there exists a family (u ε 0 ) ε>0 ⊂ L 2 (T) of initial data such that u ε 0 → u 0 in L 2 (T) and the following holds. Uniformly on nite time intervals, the solution u ε to the Benjamin-Ono equation (BO-ε) with parameter ε and initial data u ε 0 converges weakly to u B alt in

L 2
r,0 (T) as ε → 0:

u ε u B alt .
If u 0 (x) = -β cos(x) for some β > 0, one can choose u ε 0 = u 0 for every ε > 0. Remark 1.3 (Strong (resp. weak) convergence before (reps. after) the shock time). As long as u B is a well-dened function, one knows thanks to the conservation of the L 2 norm that the convergence in Theorem 1.2 is strong. However, for instance when u 0 (x) = -β cos(x), the convergence cannot be strong right after the breaking time T for the Burgers equation. Indeed, for (t -T ) positive and small enough, there holds (see Lemma 3.17)

u ε (t) L 2 (T) = u 0 L 2 (T) > u B alt L 2 (T) .
Remark 1.4 (Convergence for small times). For C ∞ initial data, a WKB approximation of the form

u ε (t, x) = +∞ j=0
a j (t, x)ε j would enable us to get an asymptotic expansion for the zero dispersion limit up to the shock time, by transforming the problem into the Burgers equation for a 0 and transport equations for the higher order terms. However, this approach would not give access to information on the solution after the shock formation.

Zero-dispersion limit for the Benjamin-Ono equation on the torus To the best of our knowledge, not much seems to be known concerning the zero-dispersion limit for the Benjamin-Ono equation on the torus. A rst approach using Whitham modulation approximation can be found in the work of Matsuno [START_REF] Matsuno | Nonlinear modulation of periodic waves in the small dispersion limit of the Benjamin-Ono equation[END_REF]. More recently, Moll [START_REF] Moll | Finite gap conditions and small dispersion asymptotics for the classical periodic Benjamin-Ono equation[END_REF] gives a convergence result of the Lax eigenvalues in the zero-dispersion limit. However, the type of convergence is not sucient to establish that the corresponding approximate solution is a good approximation in the classical space L 2 (T). In this direction, one can also mention a similar approach in [START_REF] Moll | Exact Bohr-Sommerfeld conditions for the quantum periodic Benjamin-Ono equation[END_REF] for the quantum periodic Benjamin-Ono equation.

Our strategy relies on the recent work of Gérard, Kappeler and Topalov who constructed Birkho coordinates for the Benjamin-Ono equation in [START_REF] Gérard | On the integrability of the Benjamin-Ono equation on the torus[END_REF], which adapt to equation (BO-ε) through a rescaling.

A further study of this transformation appears in [1417] (see also [START_REF] Gérard | A nonlinear Fourier transform for the BenjaminOno equation on the torus and applications[END_REF] for a survey on the topic).

Zero-dispersion limit for the Benjamin-Ono equation on the real line Theorem 1.2 is similar to Benjamin-Ono equation the real line studied by Miller and Xu [START_REF] Miller | On the zero-dispersion limit of the Benjamin-Ono Cauchy problem for positive initial data[END_REF], where the authors prove the following result. Let u 0 be an initial data satisfying some admissibility conditions, let u B 0 (t, x) < • • • < u B 2P (t,x) (t, x) be the branches of the multivalued solution to the inviscid Burgers equation obtained by the method of characteristics, and let u B alt be the signed sum of branches as in [START_REF] Benjamin | Internal waves of permanent form in uids of great depth[END_REF]. Then there exists a sequence of initial data u ε 0 such that uniformly on compact time intervals, the solution u ε is weakly convergent in L 2 (R) to u B alt . This result also implies strong convergence for all 0 ≤ t < T , where T is the breaking time for the Burgers equation. However, the approach for the zero-dispersion limit for the Benjamin-Ono on the line needs to be restricted to positive initial data with prescribed tail behavior

|x| q+1 ∂ x u 0 x→±∞ -→ C ±
for some q > 1, and a generalization to more general potentials as in Theorem 1.2 seems still unknown.

Finally, Miller and Xu establish a similar zero-dispersion convergence result for the Benjamin-Ono hierarchy in [START_REF] Miller | The Benjamin-Ono hierarchy with asymptotically reectionless initial data in the zero-dispersion limit[END_REF], and it would be interesting to compare their result to the Benjamin-Ono hierarchy on the torus, see Remark 3.16.

The approach of Miller and Xu is based on inverse scattering transform techniques, rst formally derived by Fokas and Ablowitz [START_REF] Fokas | The inverse scattering transform for the Benjamin-Ono equationa pivot to multidimensional problems[END_REF], and then rigorously written by Coifman and Wickerhauser [START_REF] Coifman | The scattering transform for the Benjamin-Ono equation[END_REF] for small and decaying data, see also [START_REF] Kaup | The inverse scattering transform for the BenjaminOno equation[END_REF]. The strategy is as follows. The initial data is rst approximated by a rational potential of Klaus-Shaw type u ε 0 , that is, a rational potential with only one bump. The authors rst guess the right approximate eigenvalues λ n (ε) and phase constants of u ε (0) in order for the scattering problem to approximate well the solution. Then they prove that for every time t (in particular for t = 0), there holds weak convergence of u ε (t) to u B alt (t) as ε → 0. This approximation is necessary in order to have an explicit inversion formula for the scattering data. A possible progress might come from the recent work on the direct scattering problem for the Benjamin-Ono equation [START_REF] Wu | Simplicity and Finiteness of Discrete Spectrum of the BenjaminOno Scattering Operator[END_REF][START_REF] Wu | Jost Solutions and the Direct Scattering Problem of the BenjaminOno Equation[END_REF], and from the construction of a Birkho map started in the paper of Sun [START_REF] Sun | Complete integrability of the BenjaminOno equation on the multi-soliton manifolds[END_REF].

In [START_REF] Miller | Direct Scattering for the BenjaminOno Equation with Rational Initial Data[END_REF], Miller and Wetzel establish exact formulae for positive rational initial conditions with simple poles. Using these formulae, the authors are able to derive a precise asymptotic expansion for the scattering data in the zero dispersion limit [START_REF] Miller | The scattering transform for the BenjaminOno equation in the small-dispersion limit[END_REF]. In this special case, the asymptotics enable to choose the initial data u 0 itself instead of an ε-dependent initial data u ε (0) in the zero-dispersion limit problem. As we will see in part 2.2, however, this approach seems uncertain for rational potentials on the torus. On the torus, we are still able to provide a precise asymptotic expansion for the eigenvalues when u 0 (x) = -β cos(x) is not a nite gap potential in Theorem 1.6 below, and we hope to extend this asymptotic expansion to more general initial data in a subsequent work.

Zero-dispersion limit for the KdV equation The zero-dispersion limit problem was rst investigated for the Korteveg-de Vries equation on the real line by Lax and Levermore [START_REF] Lax | The small dispersion limit of the Korteweg-de Vries equation[END_REF] ∂

t u -3∂ x (u 2 ) + ε 2 ∂ xxx u = 0,
when the initial data is negative or zero and decays suciently fast at innity. The authors construct approximate scattering data, or approximate initial data u ε 0 , such that the solutions u ε (t) are convergent to some limit in the weak sense when ε → 0, uniformly on compact time intervals. The limit is dierent from the Benjamin-Ono equation, and can only be expressed implicitly as the solution of some variational problem. This approach was adapted to positive initial data in [START_REF] Venakides | The zero dispersion of the Korteweg-de Vries equation for initial potentials with non-trivial reection coecient[END_REF]. The authors use WKB methods in order to approximate the scattering data associated to the KdV equation, and their analysis is based on the inverse scattering transform. Venakides then describes the nature of oscillations that appear after the dispersive shock time in [START_REF] Venakides | The Korteweg-de Vries equation with small dispersion: higher order Lax-Levermore theory[END_REF]. The theory was developed in [START_REF] Deift | New results in small dispersion KdV by an extension of the steepest descent method for Riemann-Hilbert problems[END_REF] using the steepest-descent method in order to get strong convergence results. A further renement of these asymptotics can be found in the work of Claeys and Grava [START_REF] Miller | The scattering transform for the BenjaminOno equation in the small-dispersion limit[END_REF], who exhibit in particular a universal wave prole starting from ε-independent initial data.

On the torus, Venakides [START_REF] Venakides | The zero dispersion limit of the Korteweg-de Vries equation with periodic initial data[END_REF] computes the weak zero dispersion limit for periodic initial data.

The author proves that a shock appears for small dispersion parameter at the breaking time of the Burgers equation, causing the emergence of rapid oscillations with wavenumbers and frequencies of order O(1/ε). In this purpose, he establishes asymptotics on the exact solution instead of relying on an approximation. More recently, an asymptotic expansion of the spectral parameters has then been established in [START_REF] Deng | Small dispersion limit of the Kortewegde Vries equation with periodic initial conditions and analytical description of the ZabuskyKruskal experiment[END_REF] with the cosine initial data in order to justify the Zabusky-Kruskal experiment [START_REF] Zabusky | Interaction of solitons in a collisionless plasma and the recurrence of initial states[END_REF].

Distribution of the Lax eigenvalues

Our strategy of proof relies on the complete integrability of the Benjamin-Ono equation in the sense that it admits Birkho coordinates, constructed in in [START_REF] Gérard | On the integrability of the Benjamin-Ono equation on the torus[END_REF]. This transformation enables us to consider general initial data in L 2 r,0 (T), that is, in L 2 (T), real-valued and with mean zero. The construction of Birkho coordinates relies on the eigenvalues λ n (u 0 ; ε) of the Lax operator L u 0 (ε), and some phase constants θ n (u 0 ; ε) depending on the eigenfunction of L u 0 (ε) (see part 2.1 for more details). A careful study of the spectral parameters λ n (u 0 ; ε) and θ n (u 0 ; ε) for n ≥ 1 leads us to introduce the following asymptotic approximation for the Lax eigenvalues and phase constants.

Let us denote for η ∈ R

F (η) := 1 2π Leb(x ∈ [0, 2π] | u 0 (x) ≥ η).
The eigenvalues are expected to follow some quantization rule depending on the distribution function F as follows (see also Figure 2).

Denition 1.5 (Admissible approximate initial data). Let u 0 be a single well initial data. We say that the family of approximate initial data

(u ε 0 ) ε>0 is admissible if u ε 0 L 2 (T) → u 0 L 2 (T)
as ε → 0 and the following holds. For every δ > 0, there exist C(δ), K(δ) > 0 such that for every ε > 0, the eigenvalues λ n = λ n (u ε 0 ; ε) and phase constants θ n = θ n (u ε 0 ; ε) satisfy the following properties.

(Small eigenvalues) If

λ n + ε, λ p + ε ∈ Λ -(δ) = [-max(u 0 ) + δ, -min(u 0 ) -δ], then for ε ≤ ε 0 (δ)
small enough,

-λn -λp F (η) dη -(p -n)ε ≤ C(δ)ε √ ε.

(Large eigenvalues) We denote

Λ + (δ) = [-min(u 0 ) + δ, ∞). If λ n + ε, λ p + ε ∈ Λ + (δ) = [-min(u 0 ) + δ, K(δ)], then |λ p -λ n -(p -n)ε| ≤ C(δ)ε √ ε, whereas if λ n + ε, λ p + ε ∈ [K(δ), ∞), then |λ p -λ n -(p -n)ε| ≤ δ.

(Other eigenvalues

) There are at least δ Cε and at most Cδ ε eigenvalues λ n satisfying λ n + ε ∈ [-min(u 0 ) -δ, -min(u 0 ) + δ], and at least 1 C(δ)ε and at most Cδ ε eigenvalues in the region

λ n + ε ∈ [-max(u 0 ), -max(u 0 ) + δ].

(Phase constants)

θ n+1 -θ n -π - x + (-λ n ) + x -(-λ n ) 2 ≤ C(δ)ε √ ε. (2)
By applying the inverse Birkho transformation (Φ ε ) -1 , the choice of a family of eigenvalues and phase factors denes an approximate initial data u ε 0 . The set of admissible initial data is never empty, see Lemma 3.15. This distribution of the spectral parameters is completely justied in the case of the cosine initial data thanks to the following Theorem, which is our second main result. Theorem 1.6 (Distribution of the Lax eigenvalues for cosine initial data). Assume that u 0 (x) = -β cos(x) for some β > 0. Then the family (u 0 ) ε>0 of ε-independent initial data is admissible. Remark 1.7 (Other eigenvalues). Note that we need to remove two small regions λ+ε ∈ [-β, -β +δ), λ + ε ∈ (β -δ, β + δ) and one large region λ + ε ∈ (K(δ), ∞) in our analysis, for which a uniform asymptotic expansions of the eigenvalues is not known. The reason is that we need uniform bounds in the method of stationary phase and in the Laplace method, but the stationary point goes to the limits of integration at -β, β and ∞. To prove Theorem 1.6, we will see that the eigenfunctions f n (u 0 ; ε) ∈ L 2 + (T) of the Lax operator must have a holomorphic extension to the open unit disk in the complex plane. Therefore, the pole at 0 has to satisfy some constraints in order to meet the holomorphy property, leading to an asymptotic expansion of the eigenvalues λ n (u 0 ; ε) as ε → 0.

As a consequence, we will see that the initial data u 0 (x) = -β cos(x) is well-approximated by a solution for which γ n (u

0 ; ε) = |ζ n (u 0 ; ε)| 2 = 0 when n ≥ β+O(1)
ε , which is actually to a N ε -soliton with N ε ≈ β ε . This insight leads us to dene approximate Lax eigenvalues for more general single well initial data from the distribution function F in Denition 1.5. In order to derive the entire Birkho coordinates which completely characterize the approximate initial data u ε 0 , we needed to choose the phase constants θ n (u ε 0 ; ε) for n ≥ 1, but we dot not have a justication for our choice yet. When u 0 (x) = -β cos(x), we take advantage of the fact that all the phase constants vanish.

Strategy of proof

Let u ε be the solution to (BO-ε) with initial data u ε 0 satisfying the admissibility conditions from Denition 1.5. In this part, we explain how to obtain weak convergence of u ε to u B alt when ε → 0. We rst make a link between the matrix M (u 0 ; ε) = (M n,p (u 0 ; ε)) n,p≥0 of the shift operator S :

h ∈ L 2 + (T) → e ix h ∈ L 2 + (T) (dened later in (7)
) as a function of the spectral parameters, and the Fourier coecients of u 0 ∈ L 2 r,0 (T) (see Proposition 3.1)

u 0 (k) = ε Tr(M (u 0 ; ε) k ).
Then, it is quite direct that for a single well potential there also holds (see Proposition 3.3)

u 0 (k) = -i 2kπ max u 0 min u 0 (e -ikx + (η) -e -ikx -(η) ) dη,
where x ± (λ) are the antecedents of λ by u 0 dened below Denition 1.1. One can make a parallel between those two formulas and [START_REF] Miller | On the zero-dispersion limit of the Benjamin-Ono Cauchy problem for positive initial data[END_REF], where Miller and Xu make a link between the k-th moment Tr(M (u 0 ; ε) k ) and some integral depending on (x + 2λt -x ± (λ)) k+1 (see their Proposition 4.1).

Using the asymptotics of the eigenvalues, we justify an approximation of ε Tr(M (u

ε 0 ; ε) k ) under the form ε Tr(M (u ε 0 ; ε) k ) ≈ -i 2kπ max u 0 min u 0 (e -ikx + (η) -e -ikx -(η) ) dη, (3)
where the right-hand side does not depend on ε but only on u 0 . In this approximation, we neglect the terms M n,p (u ε 0 ; ε) which are o-diagonal |n -p| ≥ ε -r for some ε > 0, and the terms where the index n corresponds to an eigenvalue outside the small-eigenvalue region λ n + ε ∈ Λ -(δ). Our approximation method then enables us to deduce an approximation of ε Tr(M (u ε (t); ε) k ):

ε Tr(M (u ε (t); ε) k ) ≈ -i 2kπ max u 0 min u 0 (e -ik(x + (η)+2ηt
) -e -ik(x -(η)+2ηt) ) dη.

For small times, we recognize the right hand side as the k-th Fourier coecient for the time evolution of u 0 under the Burgers equation. Indeed, for the Burgers solution u B , one would have x B ± (η, t) = x ± (η) + 2ηt, so that we have actually proven

u ε (t)(k) ≈ u B (t)(k).
After the breaking time for the Burgers equation, the right hand side becomes the alternate sum of Fourier coecients for the branches of the multivalued solution of the Burgers equation obtained with the method of characteristics, denoted u B alt .

Plan of the paper

In section 2, we introduce the Birkho coordinates and spectral parameters associated to (BO-ε). In section 3, we use the approximate spectral data in order to approximate the k-th Fourier coecient of the solution at time t by the k-th Fourier coecient of the relevant solution to the Burgers equation. In 2 Birkho coordinates in the zero dispersion limit In all that follows, the constants C may change from line to line, and are denoted by C(δ) if they depend on a parameter δ which is not xed. Some parameters 0 < c < r < 1 are also xed all throughout this paper.

Lax eigenvalues for the Benjamin-Ono equation

Our main tool is the description of complete integrability for the Benjamin-Ono equation on the torus from [START_REF] Gérard | On the integrability of the Benjamin-Ono equation on the torus[END_REF], in which Birkho coordinates are constructed in the case ε = 1. Let us adapt the setting to equation (BO-ε) with arbitrary parameter ε.

For u ∈ L 2 r,0 (T), we denote L u (ε) the Lax operator for the Benjamin-Ono equation with parameter ε L u (ε) = εD -T u .

We have written D = -i∂ x , moreover, T u : h ∈ L 2 + (T) → Π(uh) ∈ L 2 + (T) is a Toeplitz operator, where Π is the Szeg® projector onto the subspace L 2 + (T) of L 2 (T) of functions with positive Fourier frequencies.

Let (λ n (u; ε)) n≥0 be the eigenvalues of L u (ε) in increasing order. Let (f n (u; ε)) n≥0 be the corresponding eigenfunctions dened through the additional conditions 1|f 0 > 0 and f n |e ix f n-1 > 0, n ≥ 1.

Then according to [START_REF] Gérard | On the integrability of the Benjamin-Ono equation on the torus[END_REF], the gaps lengths γ n (u; ε) = λ n (u; ε) -λ n-1 (u; ε) -ε are nonnegative, and for any n ≥ 1, there holds

λ n (u; ε) = nε + λ 0 (u; ε) + n k=1 γ k (u; ε) = nε - +∞ k=n+1 γ k (u; ε). Finally, dening h 1/2 = {(ζ n ) n≥1 ∈ C N | n≥1 n|ζ n | 2 < ∞}, the Birkho transform is written Φ ε : u ∈ L 2 r,0 (T) → (ζ n (u; ε)) n≥1 ∈ h 1/2 .
In the construction of this transformation, one can see that |ζ n (u; ε)| 2 = γ n (u; ε). Moreover, the phase constants are dened by θ n (u;

ε) = arg(ζ n (u; ε)) for ζ n (u; ε) = 0.
The following Parseval formula holds true

u 2 L 2 (T) = 2ε n≥1 nγ n (u; ε). (4)
In order to check this identity for general parameter ε, it might be useful to note that L u (ε) = εL u/ε (1), so that f n (u; ε) = f n (u/ε; 1), then ζ n (u; ε) = √ εζ n (u/ε; 1), and nally γ k (u; ε) = εγ k (u/ε; 1). As a consequence, we can use the Parseval formula with parameter ε = 1 from [START_REF] Gérard | On the integrability of the Benjamin-Ono equation on the torus[END_REF].

Finally, the eigenvalue λ n (u; ε) satises the min-max formula

λ n (u; ε) = max dim F =n min{ L u (ε)h|h | h ∈ H 1 + ∩ F ⊥ , h L 2 = 1}.
Therefore, the smallest eigenvalue

λ 0 (u; ε) = min{ L u (ε)h|h | h ∈ H 1 + , h L 2 = 1} is bounded below by λ 0 (u; 0) = -max{ u||h| 2 | h ∈ H 1 + , h L 2 = 1} ≥ -max x∈T u(x).
(5)

Inversion formula

Given u ∈ L 2 r,0 (T), one has the inversion formula [START_REF] Gérard | On the integrability of the Benjamin-Ono equation on the torus[END_REF] Πu(z) = (Id -zM (u; ε)) -1 X(u; ε)|Y (u; ε)

with

X(u; ε) = (-λ p (u; ε) 1|f p (u; ε) ) p≥0 , Y (u; ε) = ( 1|f n (u; ε) ) n≥0 ,
and M (u; ε) is the matrix of the shift operator S :

u ∈ L 2 + → e ix u ∈ L 2 + with coecients M n,p (u; ε) = f p (u; ε)|Sf n (u; ε) , M n,p (u; ε) =    µ n+1 (u; ε) √ κp(u;ε) √ κ n+1 (u;ε) ζ p (u; ε)ζ n+1 (u; ε) 1 λp(u;ε)-λn(u;ε)-ε if p = n + 1 µ n+1 (u; ε) if p = n + 1. (7)
In this denition, µ n and κ n are functions of the eigenvalues (λ n (u; ε)) n dened in [START_REF] Fokas | The inverse scattering transform for the Benjamin-Ono equationa pivot to multidimensional problems[END_REF] and in [START_REF] Gassot | The third order Benjamin-Ono equation on the torus: well-posedness, traveling waves and stability[END_REF].

In order to get an asymptotic expansion of the eigenvalues, the strategy used in Miller, Wetzel [START_REF] Miller | Direct Scattering for the BenjaminOno Equation with Rational Initial Data[END_REF][START_REF] Miller | The scattering transform for the BenjaminOno equation in the small-dispersion limit[END_REF] is restricted to N -gap solutions. On the torus, when u satises ζ n (u; ε) = 0 for every n ≥ N (u is a N -gap for the parameter ε), the inversion formula becomes Πu(z) = -2εz∂ z log det (Id -zM N -1 (u; ε)) .

Recall that a function in the Hardy space f ∈ L 2 + (T) admits an holomorphic expansion to the open unit disk D in C as follows. Expand f in Fourier modes f (x) = n≥0 c n e inx , then the holomorphic expansion of f is written f

(z) = n≥0 c n z n . Conversely, let z ∈ D ⊂ C and Q(z) = N j=1 (1 -q j z) with q j ∈ C and 0 < |q j | < 1, then Πu(z) = ε N j=1 q j z 1 -q j z denes a N -gap for the parameter ε and Q(z) = det(Id -zM N -1 (u; ε)). Such a N -gap u has a meromorphic extension on C u(z) = ε N j=1 q j z 1 -q j z + ε N j=1 q j z -q j ,
with poles q j inside the unit disk and 1/q j outside of the unit disk. One diculty of this approach on the torus is the following observation. If we replace ε by ε/2, then

Πu(z) = - ε 2 2N -1 j=0
q j z 1 -q j z with q j+N = q j , and u becomes a 2N -gap for the parameter ε/2. As a consequence, we expect that a N -gap for the parameter ε becomes a N/ε-gap for the parameter ε. The number of poles and zeroes of the meromorphic extension of u being increasing with ε, we do not expect to get a uniform approximation of the eigenvalues when using the steepest descent method as ε → 0.

Instead of using this approach, we rather consider the trigonometric polynomial -β cos, which meromorphic extension to the complex plane

z+z -1 2
has a pole at the origin z = 0 but nowhere else.

This pole could have a higher order if one considers more general trigonometric polynomials, but this order does not depend on ε, which makes us hope to be able to extend our result to trigonometric polynomials in the future.

Approximate Birkho coordinates

In this part, we justify that if Theorem 1.6 is true for u 0 (x) = -β cos(x), then (u 0 ) ε>0 denes an admissible family of initial data in the sense of Denition 1.5. Then we state some consequences of Denition 1.5.

Concerning the cosine function, it is enough to establish that all phase constants are equal to 0.

Proposition 2.1 (Phase constants for the cosine function). Fix a parameter β > 0 and consider the initial data u(x) = -β cos(x). For every n ≥ 1, there holds

ζ n (u; ε) > 0, hence θ n (u; ε) = 0.
Proof. It is enough to tackle the case ε = 1. From [START_REF] Gérard | On the integrability of the Benjamin-Ono equation on the torus[END_REF], equation (2.7), one has for every n ≥ 0

L u Sf n = SL u f n + Sf n -uSf n |1 1. For the potential u(x) = -β cos(x), since Sf n ∈ L 2 + (T), there holds u|Sf n = - β 2 e ix |Sf n = - β 2 1|f n , so that L u Sf n = (λ n + 1)Sf n + β 2 f n |1 1. Note that γ n = κ n | 1|f n | 2
, where κ n is nonzero and dened in [START_REF] Fokas | The inverse scattering transform for the Benjamin-Ono equationa pivot to multidimensional problems[END_REF]. If γ n = 0 for some n ≥ 1, then L u Sf n = (λ n + 1)Sf n , which implies that f n+1 = Sf n , λ n+1 = λ n + 1 and γ n+1 = 0. Conversely, if γ n+1 = 0, then Lemma 2.5 from [START_REF] Gérard | On the integrability of the Benjamin-Ono equation on the torus[END_REF] implies that L u Sf n = λ n+1 Sf n = (λ n + 1)Sf n , consequently, 1|f n = 0 and γ n = 0. We conclude that either all the Birkho coordinates of u are zero, which is impossible, either all the Birkho coordinates of u are nonzero.

Finally, from [START_REF] Gérard | On the integrability of the Benjamin-Ono equation on the torus[END_REF], Lemma 2.6, one has for every n, p ≥ 0

(λ p -λ n -1) f p |Sf n = -f p |1 u|Sf n . When p = n + 1, we get γ n+1 f n+1 |Sf n = β 2 f n+1 |1 1|f n .
Since γ n+1 > 0 and f n+1 |Sf n > 0 by denition of the eigenfunctions f n , we deduce that for every n ≥ 0, there holds

ζ n+1 ζ n > 0.
Using that ζ 0 = 1, we conclude that the Birkho coordinates of u are all real and positive.

The following result summarizes the properties of the approximate Birkho coordinates both in the case u 0 (x) = -β cos(x) and in the case of a general single well potential.

Corollary 2.2 (Consequences of Denition 1.5). Let u 0 be a single well initial data. We choose an admissible family (u ε 0 ) ε>0 as in Denition 1.5. We consider the approximate Lax eigenvalues

λ n = λ n (u ε 0 ; ε). Let δ > 0, then for ε ≤ ε 0 (δ)
small enough the following holds. 1. For every n ≥ 0, we have

max(u 0 ) -λn F (η) dη -nε ≤ Cδ + C(δ)ε √ ε. (8) 2. (Large eigenvalues) If -λ n -ε ∈ Λ -(δ) = [-max(u 0 ) + δ, -min(u 0 ) -δ],
then there holds

∞ k=n+1 γ k ≤ Cδ + C(δ)ε √ ε.

(Two-region eigenvalues) If

-λ n -ε ∈ Λ -(δ) and -λ p -ε ∈ Λ + (δ) = [-min(u 0 ) + δ, ∞), then |p -n| ≥ δ Cε .
We also state the bounds that we will use on the distribution function.

Lemma 2.3 (Lipschitz properties of the distribution function). Fix a general single well initial data. Let δ > 0. There exists

C(δ) > 0 such that F , x + and x -are C(δ)-Lipschitz on [min(u 0 )+δ, max(u 0 )- δ]. Moreover, F (η) ≥ 1/C(δ) for η ≤ max(u 0 ) -δ.
3 Asymptotic expansion of the Fourier coecients for single well initial data

In this section, we establish Theorem 1.2 by proving the convergence of every Fourier coecient of u ε (t) to the Fourier coecient of u B alt (t).

Fourier coecients as a trace

Proposition 3.1 (Fourier coecients and trace of the shift matrix). For any u ∈ L 2 r,0 (T) and k ≥ 1, there holds

u(k) = ε Tr(M (u; ε) k ). Proof. Let H 1 (u) = 1 2π 2π 0 u 2 (x)
dx be the mass of the solution. We use the dierentiation formula

dH 1 (u). cos(kx) -i dH 1 (u). sin(kx) = 2 u| cos(kx) -2i u| sin(kx) = 2 u(k).
Since the Parseval formula gives

H 1 (u) = 2ε n≥1 nγ n (u; ε), taking the dierential leads to dH 1 (u).h = 2ε n≥1 n dγ n (u; ε).h. Given that γ n (u; ε) = εγ n (u/ε; 1) and f n (u; ε) = f n (u/ε; 1), one has dγ n (u; ε).h = -|f n (u; ε)| 2 -|f n-1 (u; ε)| 2 |h . We get that dH 1 (u).h = -2ε n≥1 n |f n (u; ε)| 2 -|f n-1 (u; ε)| 2 |h .
This leads to the telescopic sum

dH 1 (u).h = 2ε n≥1 |f n (u; ε)| 2 |h . But we note that Tr(M (u; ε) k ) = n f n (u; ε)|S k f n (u; ε) = n f n (u; ε)| cos(kx)f n (u; ε) -i n f n (u; ε)| sin(kx)f n (u; ε) ,
which leads to the identity with h = cos(kx) and h = sin(kx).

Remark 3.2. When ε is xed, it is possible to prove that the following expanded formula for the trace

u(k) = ε Tr(M k ) = ε n 1 ,...,n k+1 ≥1 n 1 =n k+1 k i=1 M n i ,n i+1 .
is absolutely convergent, but we will see in Proposition 3.8 that one can bound the sum of absolute values of the terms by some constant C(δ) independent of ε.

In [START_REF] Miller | On the zero-dispersion limit of the Benjamin-Ono Cauchy problem for positive initial data[END_REF] Proposition 4.1, Miller and Xu make a link between the k-th moment Tr(M (u; ε) k ) and some integral depending on (x + 2λt -x ± (λ)) k+1 . In our setting, we can guess what is the corresponding formula on the torus by expressing the k-th Fourier coecient of u in a dierent manner. Proposition 3.3. For any single well potential u ∈ C 1 r,0 (T) (see Denition 1.1), we have

u(k) = -i 2kπ max u min u (e -ikx + (η) -e -ikx -(η) ) dη.
Proof. We integrate by parts

2π u(k) = u(x)e -ikx -ik 2π 0 - i k 2π 0 ∂ x u(x)e -ikx dx,
where the crochet vanishes by periodicity. Let x max ∈ [0, 2π] the unique point for which u(x max ) = max T u. We split the integral between the zones [0, x max ] on which u is increasing, and [x max , 2π] on which u is decreasing. This leads to

2π u(k) = - i k xmax 0 ∂ x u(x)e -ikx dx - i k 2π xmax ∂ x u(x)e -ikx dx.
Then we make the change of variable η = u(x) (or x -(η) = x) in the rst term of the right hand side, and η = u(x) (or x + (η) = u(x)) in the second term of the right hand side. Since in both cases there holds dη = ∂ x u(x) dx, we get

2π u(k) = - i k max(u) min(u) e -ikx + (η) dη - i k min(u) max(u)
e -ikx -(η) dη.

Upper bounds for the Fourier coecients

We now x some k ∈ Z and estimate the k-th Fourier coecient u ε 0 (k) of the approximate initial data u ε 0 , where the rate of convergence may depend on k.

In this part, we rst establish some upper bounds on the matrix coecients M n,p (u ε 0 ; ε). As a consequence, we justify that in the formula for ε Tr(M (u ε 0 ; ε) k ), we can neglect the terms when the indexes n, p are too o-diagonal |p -n| ≥ ε -r for some 0 < r < 1, and when the Lax eigenvalue λ n is not in the region λ n + ε ∈ Λ -(δ).

Up to replacing u ε 0 by some very close initial data v ε 0 in the proof, one can assume that for every n, there holds γ n (u ε 0 ; ε) = 0. Indeed, let T, δ 0 > 0. By continuity of the ow map for (BO-ε), there exists c 1 (ε

) such that if u ε 0 -v ε 0 L 2 ≤ c 1 (ε), then we have sup t∈[0,T ] u ε (t) -v ε (t) L 2 ≤ δ 0 . (9)
By continuity of the inverse Birkho map Φ(ε) -1 , there exists c 2 (ε) > 0 such that if

k εk|ζ k (u ε 0 ; ε) -ζ k (v ε 0 ; ε)| 2 < c 2 (ε), then u ε 0 -v ε 0 L 2 ≤ c 1 (ε). We choose v ε 0 under the form ζ k (v ε 0 ; ε) = ε k (ε) > 0 as soon as ζ k (u ε 0 ; ε) = 0, ζ k (v ε 0 ; ε) = ζ k (u ε 0 ; ε) otherwise, with ε k (ε)
small enough so as to satisfy the above inequality. Then inequality (9) holds. As a consequence, for every t ∈ [0, T ], there holds

sup t∈[0,T ] | u ε (t)(k) -v ε (t)(k)| ≤ δ 0 ,
and since δ 0 is arbitrary, the convergence of the Fourier coecients for v ε are enough to conclude the proof of Theorem 1.2 for the family (u ε 0 ) ε . In what follows, we x ε > 0 and drop the ε in the notation, for instance λ n stands for λ n (u ε 0 ; ε). Recall that when γ n+1 = 0, then [START_REF] Gérard | On the integrability of the Benjamin-Ono equation on the torus[END_REF] M n,p (u ε 0

; ε) = √ a n γ n+1 γ p 1 λ p -λ n -ε e i(θ n+1 -θp)
where

a n = µ n+1 κ p κ n+1 > 0, κ n = 1 λ n -λ 0 ∞ p=1 p =n 1 - γ p λ p -λ n , (10)
µ n+1 = 1 - γ n+1 λ n+1 -λ 0 ∞ p=1 p =n+1 1 - γp λp-λ n+1 1 - γp λp-λn-ε . ( 11
)
Lemma 3.4 (Formula for a n ). The following formula holds for every n ≥ 1

a n γ n γ n+1 ε 2 = 1 + ε λ n -λ 0 ∞ p=1 p =n 1 - ε 2 (λ p -λ n ) 2 .
Proof. We simplify the product a n = µ n+1 κn κ n+1 as

a n = λ n -λ 0 + ε λ n -λ 0     ∞ p=1 p =n,n+1 1 - γp λp-λn 1 - γp λp-λn-ε     1 -γ n+1 λ n+1 -λn 1 - γn λn-λ n+1 • 1 - γn λn-λ n+1 1 -γn -ε = λ n -λ 0 + ε λ n -λ 0     ∞ p=1 p =n,n+1 λ p-1 -λ n + ε λ p -λ n λ p -λ n -ε λ p-1 -λ n     ε 2 (ε + γ n+1 )(ε + γ n )
.

Then one can re-index the product to get

a n = λ n -λ 0 + ε λ n -λ 0     ∞ p=1 p =n 1 + ε λ p -λ n 1 - ε λ p -λ n     λ n-1 -λ n + ε λ n-1 -λ n -1 λ n+1 -λ n -ε λ n+1 -λ n -1 ε 2 (ε + γ n+1 )(ε + γ n ) so that a n = λ n -λ 0 + ε λ n -λ 0     ∞ p=1 p =n 1 + ε λ p -λ n 1 - ε λ p -λ n     ε 2 γ n+1 γ n .
Lemma 3.5 (Bounds for a n ). There exist C > 0 such that for every ε > 0 and δ > 0, the following holds. For every n ≥ 1,

a n γ n γ n+1 ε 2 + a n + a n γ n+1 ε + a n γ n ε ≤ C.
Proof. These inequalities are a direct consequence of the formula for a n . Indeed, we have

0 ≤ a n γ n γ n+1 ε 2 ≤ 1 + ε λ n -λ 0 ≤ 2.
Similarly, since

γn ε+γn = 1 + ε λ n-1 -λn and γ n+1 ε+γ n+1 = 1 - ε λ n+1 -λn , we get 0 ≤ a n ≤ 1 + ε λ n -λ 0 1 + ε λ n+1 -λ n 1 - ε λ n-1 -λ n ε 2 (ε + γ n+1 )(ε + γ n ) ≤ 8, 0 ≤ a n γ n ε ≤ 1 + ε λ n -λ 0 1 - ε λ n-1 -λ n ε ε + γ n ≤ 4, 0 ≤ a n γ n+1 ε ≤ 1 + ε λ n -λ 0 1 + ε λ n+1 -λ n ε ε + γ n+1 ≤ 4.
Now, we remove the coecients which are too far from the diagonal in the sense that |n -p| ≥ ε -r for some xed parameter 0 < r < 1. We also justify that we can neglect the coecients outside the region λ n + ε ∈ Λ -(δ).

Remark 3.6 (Absolute convergence). In order to establish error bounds, we rst prove absolute convergence of the summand. Thanks to Miller and Xu [START_REF] Miller | On the zero-dispersion limit of the Benjamin-Ono Cauchy problem for positive initial data[END_REF], Lemma 4.7, we know the convergence of the series

m 1 ,...,m k ∈Z m 1 +•••+m k =0 k i=1 1 |m i | = 1 2π 2π 0 g(θ) k dθ,
where g(θ) := -log(2(1 -cos(θ)) for 0 < θ < 2π satises g ∈ L 2 r,0 (T) and g(k) = 1/|k| for k = 0. Lemma 3.7 (Bounds for λ n -λ p ). For every n, p ≥ 0 such that n = p, there holds

ε 2 (λ p -λ n ) 2 ≤ 1 |p -n| 2 .
Moreover, when p = n + 1 one has

ε |λ p -λ n -ε| ≤ 2 |p -n| ,
whereas when p = n + 1, one has

1 λ n+1 -λ n -ε = 1 γ n+1 .
Proof. The rst claim comes from the formula for p ≥ n

p k=n+1 γ k + (p -n)ε = λ p -λ n ≥ (p -n)ε.
To establish the second claim, we remark that when p > n,

λ p -λ n -ε = (p -n -1)ε + p k=n+1 γ k ≥ (p -n -1)ε ≥ 0,
and when p < n,

λ p -λ n -ε = (p -n -1)ε - n k=p+1 γ k ≤ (p -n -1)ε ≤ 0.
As a consequence, when p = n, we have proven that

|λ p -λ n -ε| ≥ |p -n -1|ε ≥ |p-n| 2 ε.
Proposition 3.8 (Restriction to small eigenvalues close to the diagonal). Let 0 < r < 1. There exists C(δ) > 0 such that for every ε > 0,

ε n 1 ,...,n k+1 ≥1 n 1 =n k+1 k i=1 √ a n i γ n i+1 γ n i +1 1 |λ n i+1 -λ n i -ε| ≤ C(δ).
Moreover, there holds

u ε 0 (k) -ε n 1 ,...,n k+1 ≥1, n 1 =n k+1 , |n i -n i+1 |≤ε -r , λn i +ε∈Λ -(δ) k i=1 √ a n i γ n i+1 γ n i +1 1 λ n i+1 -λ n i -ε e i(θ n i +1 -θn i+1 ) ≤ C(δ)ε r + Cδ.
Proof. Let us in the proof denote the sum of absolute values

S := ε n 1 ,...,n k+1 ≥1 n 1 =n k+1 k i=1 √ a n i γ n i+1 γ n i +1 1 |λ n i+1 -λ n i -ε| .
We deduce from Lemma 3.7 that

S ≤ Cε n 1 ,...,n k+1 ≥1 n 1 =n k+1   n i ≥1 √ a n i       n i ≥1 n i+1 =n i +1 √ γ n i+1 γ n i +1 ε     k i=1 2 |n i+1 -n i | . ( 12
)
Using Lemma 3.5 on the bounds of a n , one can note that every term of the form √ a n i

√ γn i γ n i +1 ε , √ a n i √ γ n i +1 √ ε , √ a n i √ γn i √ ε and √ a n i is bounded by C.
We split the upper bound in several regions |p -n| ≥ ε -r , λ n + ε ∈ Λ(δ), and λ n + ε ∈ Λ + (δ), for which we expect the sum to be small, and one region λ n + ε ∈ Λ -(δ), for which we expect the sum to be bounded.

Other eigenvalues We rst assume that λ n 1 + ε ∈ Λ(δ), and denote

S Λ c := ε n 1 ,...,n k+1 ≥1 n 1 =n k+1 λn 1 +ε ∈Λ(δ) k i=1 √ a n i γ n i+1 γ n i +1 1 |λ n i+1 -λ n i -ε| .
As for inequality [START_REF] Gérard | A nonlinear Fourier transform for the BenjaminOno equation on the torus and applications[END_REF], Lemma 3.7 gives

S Λ c ≤ Cε n 1 ,...,n k+1 ≥1 n 1 =n k+1 λn 1 +ε ∈Λ(δ) n i ≥1 √ a n i n i ≥1 n i+1 =n i +1 √ γ n i+1 γ n i +1 ε k i=1 2 |n i+1 -n i | .
We make the change of variable n = n 1 , m i = n i+1 -n i for 1 ≤ i ≤ k. Then thanks to Lemma 3.5 bounding the terms involving a n , there holds

S Λ c ≤ Cε n≥1 λn+ε ∈Λ(δ) m 1 ,...,m k ∈Z m 1 +•••+m k =0 k i=1 1 |m i | .
Using the bound from Remark 3.6, we deduce that

S Λ c ≤ Cε n≥1 λn+ε ∈Λ(δ)
1.

Finally, we note that there are at most Cδ/ε possible indexes n such that λ n + ε ∈ Λ(δ), so that

S Λ c ≤ Cδ.
The same applies if n 1 is replaced by some other index n i in the sum. In the following cases, we can therefore assume that λ n i + ε ∈ Λ(δ) for every i.

O-diagonal terms

We now consider the terms such that |n 2 -n 1 | ≥ ε -r . Then assuming that ε is small, we get from Lemma 3.7 that

|λ n 2 -λ n 1 -ε| ≥ 1 2 ε 1-r .
Let us denote

S of f := ε n 1 ,...,n k+1 ≥1 n 1 =n k+1 |n 2 -n 1 |>ε -r ∀j,λn j +ε∈Λ(δ) k i=1 √ a n i γ n i+1 γ n i +1 1 λ n i+1 -λ n i -ε .
Using Lemma 3.7 as in inequality [START_REF] Gérard | A nonlinear Fourier transform for the BenjaminOno equation on the torus and applications[END_REF] and the bound on a n from Lemma 3.5 we get the upper bound

S of f ≤ Cε r n 1 ,...,n k+1 ≥1 n 1 =n k+1 ∀j,λn j +ε∈Λ(δ) k i=1 √ γ n i+1 γ n i +1 1 n i+1 =n i +1 .
We rst remove the sum over n i+1 when the condition n i+1 = n i + 1 is met. More precisely, we dene m 1 as the smallest index n i such that n i-1 + 1 = n i (this is always possible because n 1 = n k+1 ), with the convention n 0 := n k . Then let 0 ≤ d 1 ≤ k be such that for 0 ≤ i ≤ d 1 , we have n i = n 1 + i and n 1+d 1 = n 1 + d 1 , we dene m 2 := n 1+d 1 and so on by induction. The upper bound becomes

S of f ≤ Cε r k l=1 0≤d 1 ,...,d l+1 ≤k m 1 ,...,m l+1 ≥1 m 1 =m l+1 , m j+1 =m j +1 ∀j,λm j +ε∈Λ(δ) l i=1 √ γ m i+1 γ m i +d i .
Since λ 0 = -k≥1 γ k ≥ -max(u 0 ) thanks to (5), we have

m i √ γ m i γ m i +d i ≤ C.
As a consequence,

S of f ≤ Cε r .
The same applies if the condition |n 2 -n 1 | ≥ ε -r is replaced by the condition |n i+1 -n i | ≥ ε -r for some i. In the following cases, we therefore assume that for every i, there holds |n i+1 -n i | ≤ ε -r .

In this case, if ε < ε 0 (δ), then when λ n 1 + ε ∈ Λ + (δ) (resp. Λ -(δ)), there holds λ n i + ε ∈ Λ + (δ/2) (resp. Λ -(δ/2)) for every i. Indeed, Denition 1.5 with the parameter δ/2 implies that there are at least 1 C(δ)ε ≥ ε -r eigenvalues in [-max(u 0 ) + δ/2, -max(u 0 ) + δ], and the same holds in [-min(u 0 ) + δ/2, -min(u 0 ) + δ].

As a consequence, in the remaining cases, we can assume all the eigenvalues to be large (in Λ + (δ)) at the same time, or all the eigenvalues to be small (in Λ -(δ)) at the same time.

Large eigenvalues Let

S + := ε n 1 ,...,n k+1 ≥1, n 1 =n k+1 |n i+1 -n i |≤ε -r , λn i +ε∈Λ + (δ) k i=1 √ a n i γ n i+1 γ n i +1 1 λ n i+1 -λ n i -ε .
Using Lemma 3.7 as in inequality [START_REF] Gérard | A nonlinear Fourier transform for the BenjaminOno equation on the torus and applications[END_REF], we know that

S + ≤ Cε n 1 ,...,n k+1 ≥1, n 1 =n k+1 |n i+1 -n i |≤ε -r , λn i +ε∈Λ + (δ) n i ≥1 √ a n i n i ≥1 n i+1 =n i +1 √ γ n i+1 γ n i +1 ε k i=1 2 |n i+1 -n i |
.

Since n k+1 = n 1 , there exists an index n i such that n i+1 = n i + 1. Up to multiplying the upper bound by some reordering constant C, one can assume that this index is n 1 so that the term

√ γ n 2 γ n 1 appears in the upper bound. If n i+1 = n i + 1 for every 2 ≤ i ≤ k, then n 1 = n k+1 = n 2 + k -1.
Otherwise, let i 0 be the rst index 2 ≤ i ≤ k such that n i+1 = n i + 1, we know that the term √ γ n i 0 +1 γ n i 0 +1 also appears in the upper bound, where n i 0 +1 = n 2 + i 0 -1. As a consequence, we have proven that there exists 1 ≤ j 0 = j 0 (n 1 , . . . , n k ) ≤ k such that both √ γ n 2 and √ γ n 2 +j 0 appear in the upper bound.

Then the bounds on a n from Lemma 3.5 imply

S + ≤ Cε n 1 ,...,n k+1 ≥1, n 1 =n k+1 |n i+1 -n i |≤ε -r , λn i +ε∈Λ + (δ) n 2 =n 1 +1 √ γ n 2 γ n 2 +j 0 ε k i=1 2 |n i+1 -n i | .
The bound from Remark 3.6, coupled with the change of variable n = n 2 , m i = n i+1 -n i for 2 ≤ i ≤ k + 1 (with the convention n k+2 := n 2 ), leads to

S + ≤ C k j 0 =1 n≥1 λn+ε∈Λ + (δ) √ γ n γ n+j 0 .
Finally, using Corollary 2.2, there holds

n≥1 λn∈Λ + (δ) γ n ≤ C(δ)ε √ ε + Cδ.
and we deduce

S + ≤ C(δ)ε √ ε + Cδ.
Small eigenvalues In the last scenario, let

S -:= ε n 1 ,...,n k+1 ≥1, n 1 =n k+1 |n i+1 -n i |≤ε -r , λn i +ε∈Λ -(δ) k i=1 √ a n i γ n i+1 γ n i +1 1 λ n i+1 -λ n i -ε .
We apply the argument from the former paragraph to get

S -≤ C k j 0 =1 n≥1 λn+ε∈Λ -(δ) √ γ n γ n+j 0 .
This is bounded by C thanks to the lower bound (5) on λ 0 .

Summing the upper bounds for every one of the cases, we get the Proposition.

Approximation of the Fourier coecients

In this part, we express all the terms from the approximation of u ε 0 (k) in Proposition 3.8 as a function of F (-λ n ) uniquely. Theorem 3.9 (Fourier coecients as a Riemann sum). Let 0 < c < r < 1. For every δ > 0, there exist C(δ) > 0, ε 0 (δ) > 0, and a function R of ε uniquely, tending to zero as ε → 0, such that for every

0 < ε < ε 0 (δ), u ε 0 (k) -ε n≥1 λn+ε∈Λ -(δ) sinc(kπF (-λ n ))e -ik x + (-λn)+x -(-λn) 2 ≤ C(δ)(ε r-c + ε 1-2r ) + R(ε) + Cδ. ( 13
)
The proof of Theorem 3.9 decomposes in several steps. We rst approximate λ p -λ n and a n by functions of F (-λ n ) only. Then we simplify the sum obtained by replacing the terms by their approximation.

Lemma 3.10 (Eigenvalues and distribution function). Let λ

n + ε, λ p + ε ∈ Λ -(δ) such that |p -n| ≤ ε -r , then we have F (-λ n ) 2 (p -n) 2 - ε 2 (λ p -λ n ) 2 ≤ C(δ) √ ε |p -n| 1+1/2r and F (-λ n ) p -n -F (-λ n ) - ε λ p -λ n -ε ≤ C(δ) √ ε + ε 1-2r (p -n) 2 .
Proof. We use Corollary 2.2 in the small eigenvalue case and deduce that there exists

ξ n,p ∈ [-λ p , -λ n ] such that |(λ p -λ n )F (ξ n,p ) -(p -n)ε| ≤ C(δ)ε √ ε. ( 14
)
As a consequence, we have from Lemma 3.7 that

F (ξ n,p ) p -n - ε λ p -λ n ≤ C(δ)ε √ ε (p -n)(λ p -λ n ) ≤ C(δ) √ ε (p -n) 2 . Using Lemma 3.7 again, we know that | ε λp-λn | ≤ 1 |p-n| , whereas | F (ξn,p) p-n | ≤ 1
|p-n| , and therefore

F (ξ n,p ) 2 (p -n) 2 - ε 2 (λ p -λ n ) 2 ≤ C(δ) √ ε |p -n| 3 .
Then, we know from the Lipschitz properties of F (see Corollary 2.2) that

|F (ξ n,p ) -F (-λ n )| ≤ C(δ)|ξ n,p + λ n | ≤ C(δ)|λ p -λ n |.
However, thanks to [START_REF] Gérard | On the spectrum of the Lax operator of the Benjamin-Ono equation on the torus[END_REF], we have

|λ p -λ n | ≤ C(δ)ε √ ε + |p -n|ε F (ξ n,p ) ≤ C (δ)|p -n|ε, (15)
so that

|F (ξ n,p ) -F (-λ n )| ≤ C(δ)|p -n|ε. (16)
As a consequence, one can replace F (ξ n,p ) by F (-λ n ) up to a small error:

F (-λ n ) 2 (p -n) 2 - ε 2 (λ p -λ n ) 2 ≤ C(δ) √ ε |p -n| 3 + C(δ)ε |p -n| .
Since |p -n| ≤ ε -r with r < 1, we get that ε ≤ 1 |p-n| 1/r and we conclude that

F (-λ n ) 2 (p -n) 2 - ε 2 (λ p -λ n ) 2 ≤ C(δ) √ ε |p -n| 3 + C(δ) √ ε |p -n| 1+1/2r .
Similarly, using inequalities ( 14) and ( 16), we have

|(λ p -λ n -ε)F (-λ n ) -(p -n -F (-λ n ))ε| = |(λ p -λ n )F (-λ n ) -(p -n)ε| ≤ C(δ)ε √ ε + (λ p -λ n )C(δ)|p -n|ε.
Since inequality [START_REF] Gérard | Sharp well-posedness results of the Benjamin-Ono equation in H s (T, R) and qualitative properties of its solution[END_REF]

implies |λ p -λ n | ≤ C(δ)|p -n|ε ≤ C(δ)ε 1-r , we nally get |(λ p -λ n -ε)F (-λ n ) -(p -n -F (-λ n ))ε| ≤ C(δ)(ε √ ε + ε 2-2r ), so that F (-λ n ) (p -n -F (-λ n )) - ε λ p -λ n -ε ≤ C(δ)(ε √ ε + ε 2-2r ) (p -n -F (-λ n ))(λ p -λ n -ε) . Since λ n + ε ∈ Λ -(δ), we know that F (-λ n ) ≤ 1 -1 C(δ)
. Moreover, we make use of Lemma 3.7 and see that actually

F (-λ n ) (p -n -F (-λ n )) - ε λ p -λ n -ε ≤ C(δ) ( √ ε + ε 1-2r ) (p -n) 2 .
Then, we establish an approximation of

1 ε √ a n γ n γ n+1 .
Lemma 3.11 (Approximation of a n in terms of the distribution function). Let 0 < c < r < 1. Then there exist C(δ) > 0, ε 0 (δ) > 0 such that for every δ > 0 and 0 < ε < ε 0 (δ), the following holds. For every n ≥ 1 satisfying

λ n + ε ∈ Λ -(δ), a n γ n γ n+1 ε 2 -sinc(πF (-λ n )) 2 ≤ C(δ)(ε r-c + ε 1-r ).
Proof. We consider the logarithm of a n

log a n γ n γ n+1 ε 2 = log 1 + ε λ n -λ 0 ∞ p=1 p =n log 1 - ε 2 (λ p -λ n ) 2 .
When |p -n| ≥ ε -r , we make use of Lemma 3.7:

ε 2 (λ p -λ n ) 2 ≤ 1 |n -p| 2 ≤ ε r-c |n -p| 1+c/r . When |p -n| ≤ ε -r , since λ n + ε ∈ Λ -(δ), then λ p + ε ∈ Λ -(δ/2
) and using Lemma 3.10 there holds

F (-λ n ) 2 (p -n) 2 - ε 2 (λ p -λ n ) 2 ≤ C(δ)ε 1/2 (p -n) 1+1/2r .
Hence, we get by summation that

log a n γ n γ n+1 ε 2 - ∞ p=1 |p-n|≤ε -r log 1 - F (-λ n ) 2 (p -n) 2 ≤ C(δ)(ε 1/2 + ε r-c ). Note that F (-λ n ) ≤ 1 -1/C(δ), as a consequence, log a n γ n γ n+1 ε 2 ≤ C(δ).
Also note that since λ n + ε ∈ Λ -(δ), we have n ≥ δ Cε , therefore the indexes p ≥ 1 such that |n -p| ≤ ε -r is the same as the indexes p ∈ Z such that |n -p| ≤ ε -r when ε < ε 0 (δ). Moreover, since

F (-λ n ) ≤ 1 -1/C(δ), the change of variable k = p -n for p > n and k = n -p for n > p leads to ∞ p=-∞ |p-n|>ε -r log 1 - F (-λ n ) 2 (p -n) 2 ≤ 2 ∞ k=ε -r +1 log 1 - F (-λ n ) 2 k 2 ≤ C(δ)ε 1-r .
Therefore, we have proven that

log a n γ n γ n+1 ε 2 -2 ∞ k=1 log 1 - F (-λ n ) 2 (p -n) 2 ≤ C(δ)(ε 1/2 + ε r-c + ε 1-r ).
Taking the exponential, since log a n γnγ n+1 ε 2 stays bounded by C(δ) for ε < ε 0 (δ), we deduce

a n γ n γ n+1 ε 2 - ∞ k=1 1 - F (-λ n ) 2 k 2 2 ≤ C(δ)(ε 1/2 + ε r-c + ε 1-r ).
Finally, we use the Weierstrass sine product formula for z ∈ (0, 1)

sinc(πz) = sin(πz) πz = k≥1 1 - z 2 k 2 to deduce that a n γ n γ n+1 ε 2 -sinc(πF (-λ n )) 2 ≤ C(δ)(ε 1/2 + ε r-c + ε 1-r ),
where min(r -c,

1 -2r) < 1/2.
In what follows, the above approximations will lead us to study the series

m 1 ,...,m k ∈Z m 1 +•••+m k =0 k i=1 1 m i -F (-λ n )
.

We rst prove the convergence and nd a formula for this sum.

Lemma 3.12 (Toeplitz identity). Let c ∈ (0, 1). Then

m 1 ,...,m k ∈Z m 1 +•••+m k =0 k i=1 1 |m i -c| < ∞
and there holds

m 1 ,...,m k ∈Z m 1 +•••+m k =0 k i=1 1 m i -c = (-1) k π k-1 sin(kπc) kc sin(πc) k .
Proof. To establish absolute convergence, we use the inequality

|m i -c| ≥ |m i | min(c, 1 -c) to get that m 1 ,...,m k ∈Z m 1 +•••+m k =0 ∃i,|m i |>ε -r k i=1 1 |m i -c| ≤ 1 min(c, 1 -c) k m 1 ,...,m k ∈Z m 1 +•••+m k =0 ∃i,|m i |>ε -r k i=1 1 |m i | ,
where the upper bound is the remainder term of an absolutely convergent series thanks to Lemma 4.7 in Miller and Xu [START_REF] Miller | On the zero-dispersion limit of the Benjamin-Ono Cauchy problem for positive initial data[END_REF], see Remark 3.6.

We now dene

f (x) := m∈Z e imx m -c .
This Fourier series is convergent in L 2 (T), and one can check by direct calculation of the Fourier coecients f

(n) = 1 2π 2π 0 f (x)e -inx dx that it is equal to f (x) = - πe ic(x-π)
sin(πc) .

Indeed, for n ∈ Z, the above expression leads to

f (n) = - πe -icπ sin(πc) 1 2π 2π 0 e i(c-n)x dx = - e -icπ 2 sin(πc) e i(c-n)x i(c -n) 2π 0 = - 1 c -n .
Since f belongs to L 2 and f k also belongs to L 2 for every positive integer k, the convolution theorem implies

m 1 ,...,m k ∈Z m 1 +•••+m k =0 f (m 1 ) . . . f (m k ) = f k (0) = 1 2π 2π 0 f (x) k dx.
As a consequence, we conclude that

m 1 ,...,m k ∈Z m 1 +•••+m k =0 1 (m 1 -c) . . . (m k -c) = (-1) k π k-1 e -ikπc 2 sin(πc) k e ikcx ikc 2π 0 = (-1) k π k-1 sin(kπc) kc sin(πc) k .
Proof of Theorem 3.9. We rst remove the o-diagonal coecients and the eigenvalues which are not in Λ -(δ) thanks to Proposition 3.8 up to adding a remainder term of the form C(δ)ε r + Cδ: we get

u ε 0 (k) -ε n 1 ,...,n k+1 ≥1, n 1 =n k+1 , |n i -n i+1 |≤ε -r , λn i +ε∈Λ -(δ) k i=1 √ a n i γ n i+1 γ n i +1 1 λ n i+1 -λ n i -ε e i(θ n i +1 -θn i+1 ) ≤ C(δ)ε r + Cδ.
A re-indexation implies

u ε 0 (k) -ε n 1 ,...,n k+1 ≥1, n 1 =n k+1 , |n i -n i+1 |≤ε -r , λn i +ε∈Λ -(δ) k i=1 √ a n i γ n i γ n i +1 1 λ n i+1 -λ n i -ε e i(θ n i +1 -θn i ) ≤ C(δ)ε r + Cδ.
We then use the second inequality from Lemma 3.10

F (-λ n ) p -n -F (-λ n ) - ε λ p -λ n -ε ≤ C(δ) √ ε + ε 1-2r (p -n) 2 .
When λ n + ε, λ p + ε ∈ Λ -(δ) and |n -p| ≤ ε -r , we know thanks to the Lipschitz properties of F and inequality [START_REF] Gérard | Sharp well-posedness results of the Benjamin-Ono equation in H s (T, R) and qualitative properties of its solution[END_REF] that

|F (-λ n ) -F (-λ p )| ≤ C(δ)|λ p -λ n | ≤ C (δ)|p -n|ε ≤ C (δ)ε 1-r .
As a consequence, we have that if |n -

n 1 |, |p -n 1 | ≤ ε -r , then F (-λ n 1 ) p -n -F (-λ n 1 ) - ε λ p -λ n -ε ≤ C(δ) √ ε + ε 1-2r (p -n) 2 + C(δ) ε 1-r |p -n| ≤ C (δ) √ ε + ε 1-2r |p -n| .
Using the bound on a n γ n γ n+1 /ε from Lemma 3.5, we get

u ε 0 (k) -ε n 1 ,...,n k+1 ≥1, n 1 =n k+1 |n i -n i+1 |≤ε -r , λn i +ε∈Λ -(δ) k i=1 √ a n i γ n i γ n i +1 ε F (-λ n 1 ) n i+1 -n i -F (-λ n 1 ) e i(θ n i +1 -θn i ) ≤ C(δ)ε r + Cδ + C(δ)ε( √ ε + ε 1-2r ) n 1 ,...,n k+1 ≥1, n 1 =n k+1 |n i -n i+1 |≤ε -r , λn i +ε∈Λ -(δ) k i=1 1 |n i+1 -n i | .
Since there are not more than

C ε indexes n such that λ n + ε ∈ Λ -(δ), this leads to u ε 0 (k) -ε n 1 ,...,n k+1 ≥1, n 1 =n k+1 |n i -n i+1 |≤ε -r , λn i +ε∈Λ -(δ) k i=1 √ a n i γ n i γ n i +1 ε F (-λ n 1 ) n i+1 -n i -F (-λ n 1 ) e i(θ n i +1 -θn i ) ≤ C(δ)(ε r + √ ε + ε 1-2r ) + Cδ.
Next, for every n such that λ n + ε ∈ Λ -(δ), we use the approximation of a n from Lemma 3.11

a n γ n γ n+1 ε 2 -sinc(πF (-λ n )) 2 ≤ C(δ)(ε r + ε 1-2r ).
We also note that x → sinc(x) = sin(x)

x is Lipschitz on R and F is C(δ)-Lipschitz on [-max(u 0 ) + δ, -min(u 0 ) -δ]. Therefore, for every n, p satisfying |n -p| ≤ ε -r and λ n + ε, λ p + ε ∈ Λ -(δ), the use of inequality [START_REF] Gérard | Sharp well-posedness results of the Benjamin-Ono equation in H s (T, R) and qualitative properties of its solution[END_REF]

leads to | sinc(πF (-λ n )) -sinc(πF (-λ p ))| ≤ C(δ)|λ p -λ n | ≤ C (δ)|n -p|ε ≤ C (δ)ε 1-r .
We have proven that for |n -p| ≤ ε -r satisfying λ n + ε, λ p + ε ∈ Λ -(δ), we have

a p γ p γ p+1 ε 2 -sinc(πF (-λ n )) 2 ≤ C(δ)(ε r + ε 1-2r ).
Using the Lipschitz properties of x + and x -in Corollary 2.2 and the formula (2) for the phase constants, we even have

a p γ p γ p+1 ε 2 e i(θ p+1 -θp) -sinc(πF (-λ n )) 2 e i(θ n+1 -θn) ≤ C(δ)(ε r + ε 1-2r ).
Since sinc is nonnegative on [0, π], we get by summation that

u ε 0 (k) -ε n 1 ,...,n k+1 ≥1, n 1 =n k+1 |n i -n i+1 |≤ε -r , λn i +ε∈Λ -(δ) sinc(πF (-λ n 1 )) k e ik(θ n 1 +1 -θn 1 ) k i=1 F (-λ n 1 ) n i+1 -n i -F (-λ n 1 ) ≤ Cδ + C(δ)ε(ε r-c + ε 1-2r ) n 1 ,...,n k+1 ≥1, n 1 =n k+1 |n i -n i+1 |≤ε -r , λn i +ε∈Λ -(δ) k i=1 F (-λ n 1 ) |n i+1 -n i -F (-λ n 1 )| .
We use the absolute convergence from the Toeplitz sum in Lemma 3.12 to treat the sum over indexes n 2 , . . . , n k+1 . Moreover, we know that there are no more than C/ε terms in the sum over n 1 , so that

u ε 0 (k) -ε n 1 ,...,n k+1 ≥1, n 1 =n k+1 |n i -n i+1 |≤ε -r , λn i +ε∈Λ -(δ) sinc(πF (-λ n 1 )) k e ik(θ n 1 +1 -θn 1 ) k i=1 F (-λ n 1 ) n i+1 -n i -F (-λ n 1 ) ≤ C(δ)(ε r-c + ε 1-2r ) + Cδ. (17)
Finally, we make the change of variable n = n 1 and m i = n i+1 -n i for 1 ≤ i ≤ k. It only remains to use the Toeplitz identity from Lemma 3.12 to conclude that

u ε 0 (k) -ε n≥1 λn+ε∈Λ -(δ) sin(πF (-λ n )) π k e ik(θ n+1 -θn) (-1) k π k-1 sin(kπF (-λ n )) kF (-λ n ) sin(πF (-λ n )) k ≤ C(δ)(ε r-c + ε 1-2r ) + R(ε) + Cδ,
where R(ε) is bounded by the remainder term in the Toeplitz identity from Lemma 3.12

R(ε) = Cε n≥1 λn+ε∈Λ -(δ) m 1 ,...,m k+1 ≥1 ∃i,|m i |>ε -r k i=1 1 |m i -F (λ n 1 )| sin(πF (-λ n )) π k π k-1 | sin(kπF (-λ n ))| kF (-λ n ) sin(πF (-λ n )) k ≤ C m 1 ,...,m k+1 ≥1 ∃i,|m i |>ε -r k i=1 1 |m i -F (λ n 1 )| .
We conclude by using that (-1) k e ik(θ n+1 -θn) = e -ik x + (-λn)+x -(-λn) 2 .

Link with Burgers equation and time evolution

In this part, we deduce an approximation of the k-th Fourier coecient of the solution u ε (t) at time t which is coherent with Proposition 3.3.

Theorem 3.13 (Fourier coecients and Burgers equation). Let k ∈ Z. Let u ε be the solution to (BO-ε) with initial data u ε 0 . For every T > 0, there exists ε 0 (δ, T ) such that for every ε < ε 0 (δ, T ) and t ∈ [0, T ], there holds

u ε (t)(k) + i 2kπ max(u 0 ) min(u 0 ) e -ik(x + (η)+2ηt) -e -ik(x -(η)+2ηt) dη ≤ Cδ.
Proof. Fix 0 < c < r < 1. We rst consider the function u ε 0 without any time evolution. We justify that we can pass to the limit in the Riemann sum

u ε 0 (k) -ε n≥1 λn+ε∈Λ -(δ) e -ik x + (-λn)+x -(-λn) 2 sinc(kπF (-λ n )) ≤ C(δ)(ε r-c + ε 1-2r ) + R(ε) + Cδ. The function η → sinc(kπF (η)) is C 1 on [-β + δ, β -δ]. Moreover, in the region η + ε ∈ Λ -(δ), there holds F (η) ≥ 1/C(δ). Since λ n + ε ∈ Λ -(δ), Corollary 2.2 implies that |λ n+1 -λ n | ≤ C(δ)ε,
so that the mesh is tending to zero as ε → 0. More precisely, there holds thanks to ( 14), ( 15) and ( 16) that

|(λ n+1 -λ n )F (-λ n ) -ε| ≤ C(δ)ε √ ε,
so that F (-λ) is the distribution function of the η = -λ's. There are at most C ε indexes n such that λ n + ε ∈ Λ -(δ), therefore we get by summation

u ε 0 (k) - n≥1 λn+ε∈Λ -(δ) e -ik x + (-λn)+x -(-λn) 2 sinc(kπF (-λ n ))F (-λ n )(λ n+1 -λ n ) ≤ C(δ)(ε r-c + ε 1-2r ) + R(ε) + Cδ.
Passing to the limit ε → 0, this leads to

u ε 0 (k) + 1 kπ max(u 0 )-δ min(u 0 )+δ e -ik x + (η)+x -(η) 2 sin(kπF (η)) dη ≤ Cδ.
Finally, we use the denition 2πF (η) = x + (η) -x -(η) and simplify

u ε 0 (k) + 1 2ikπ max(u 0 )-δ min(u 0 )+δ e -ikx -(η) -e -ikx + (η) dη ≤ Cδ.
Given that the integrand is bounded by 2, one can remove the δ in the integration bounds up to increasing C, and hence we get the result.

Time evolution Let us now add the time into account. Let u ε be the solution to (BO-ε) with

initial data u ε (t = 0) = u ε 0 . We check that M n,p (u ε (t); ε) becomes M n,p (u ε (t); ε) = M n,p (u ε 0 ; ε) i(ω n+1 (u ε 0 ;ε)-ωp(u ε 0 ;ε))t .
To nd the formula for ω n (u ε 0 ; ε), one can observe that

v(t, x) := 1 ε u ε t ε , x
is the solution to (BO-ε) with parameter ε = 1 and initial data u ε 0 /ε. But f n (v(t); 1) = f n (u ε (t/ε); ε) so that since γ n (u ε 0 ; ε) = εγ n (v ε 0 ; 1) and ζ n (u ε (t); ε) = √ εζ n (v(εt); 1), then Proposition 8.1 from [START_REF] Gérard | On the integrability of the Benjamin-Ono equation on the torus[END_REF] implies

ζ n (u ε (t); ε) = √ εζ n (v ε 0 ; 1)e iωn(v ε 0 ;1)εt .
Therefore,

ω n (u ε 0 ; ε) = εω n (v ε 0 ; 1) = ε n 2 -2 ∞ k=n+1 min(k, n)γ k (v ε 0 ; 1) = εn 2 -2 ∞ k=n+1 min(k, n)γ k (u ε 0 ; ε).
As a consequence, we get the approximate solution at time t by replacing every phase constant θ n by θ n + ω n t, with ω n = ω n (u ε 0 ; ε). Let us establish the Lipschitz properties of these new phase constants. We have

ω n+1 -ω n = ε(2n + 1) -2 ∞ k=n+1 γ k = 2λ n + ε. But when λ n + ε, λ p + ε ∈ Λ -(δ) and |p -n| ≤ ε -r , we have (ω p+1 -ω p ) -(ω n+1 -ω n ) = 2(λ p -λ n ).
Using inequality [START_REF] Gérard | Sharp well-posedness results of the Benjamin-Ono equation in H s (T, R) and qualitative properties of its solution[END_REF], one deduces that

|(ω p+1 -ω p ) -2λ n | ≤ C(δ)ε 1-r .
As a consequence, one has

| exp(i(ω p+1 -ω p )t) -exp(2iλ n t)| ≤ C(δ)ε 1-r t.
We use our above approximation approach by replacing the phase factors e i(θn i+1 -θn i ) by their time evolution e i(θn i+1 -θn i +(ωn i+1 -ωn i )t) in the series. Passing to the limit in the Riemann sum, for every T > 0, there exists ε 0 (δ, T ) such that for every ε < ε 0 (δ, T ) and t ∈ [0, T ], there holds

u ε (t)(k) - i 2kπ max(u 0 )-δ min(u 0 )+δ e -ik(x -(η)+2ηt) -e -ik(x + (η)+2ηt) dη ≤ Cδ.
Given that the integrand is bounded, one can remove the δ in the integration bounds up to increasing C, and hence we get the result.

We now make the link between Theorem 3.13 and the Fourier coecients of the signed sum of branches u B alt for the multivalued solution u B to the Burgers equation obtained with the method of characteristics, see Figure 3. Every point u B is an image of the solution at time t with abscissa x as soon as it solves the implicit equation

u B = u 0 (x -2u B t).
On the real line (see a more detailed description in [START_REF] Miller | On the zero-dispersion limit of the Benjamin-Ono Cauchy problem for positive initial data[END_REF]), new sheets are formed at the breaking points (t ξ , x ξ ) such that ξ is an inection point u 0 (ξ) = 0, u 0 (ξ) = 0, and

(t ξ , x ξ ) = -1 2u 0 (ξ) , ξ - u 0 (ξ) u 0 (ξ) .
In the case of a single well potential, there are two such inection points ξ ± , for which u 0 (ξ ± ) = 0 and u 0 (ξ ± ) = 0. We assume that u 0 (ξ + ) < 0 < u 0 (ξ -). Right after the positive breaking time

t ξ + = -1 2u 0 (ξ + )
, two new branches emerge, so that there are three branches in total. Because of the periodicity, this can lead to more branches as t increases, see Figure 1, that we denote u B 0 (t, x) < • • • < u B 2P (t,x) (t, x). We have dened the signed sum of branches in (1) as

u B alt (t, x) = 2P (t,x) n=0 (-1) n u B n (t, x).
These branches are described by two (actual) functions v B 0 and v B 1 dened on subintervals [X -(t) -2π, X + (t)] and [X -(t), X + (t)] of the real line, see Figure 3.

More precisely, we consider one period of the initial data u 0 , for which we follow the method of characteristics on R. Then the solution to the corresponding multivalued Burgers equation on R has between 0 and 3 branches u(y + 2u 0 (y)t) = u 0 (y)

u(x ± (η) + 2ηt) = η. Let us denote X -(t) = x -(η -(t)) + 2η -(t)t ≤ X + (t) = x + (η + (t)) + 2η + (t)t ∈
[a(v B 1 ), b(v B 1 )] = [X -(t) -2π, X + (t)], the second one v B 1 is well-dened on [a(v B 0 ), b(v B 0 )] = [X -(t), X + (t)
] as in Figure 3.

In the periodic case, this leads to more branches. But as they appear two by two, the odd branches always correspond to the branch v B 1 and the even ones to the branch v B 0 . As a consequence, the graphs of u B 0 , u B 2 , . . . , u B 2P combined are the graph of v B 0 taken modulo 2π in space. The increasing part corresponds to abscissa x = x -(η) for some η whereas the decreasing part corresponds to abscissa x = x + (η) for some η. The graphs of u B 1 , u B 3 , . . . , u B 2P -1 combined are the graph of v B 1 , modulo 2π, they are increasing and correspond to abscissa x = x + (η) for some η. For every η ∈ (min(u 0 ), max(u 0 )), there are exactly two antecedents η

= u B n (t, x -(t, η)) = v B n mod 2 (t, x -(t, η)) and η = u B m (t, x + (t, η)) = v B m mod 2 (t, x + (t, η)).
Proposition 3.14 (Fourier coecients of the multivalued solution to the Burgers equation). Let u 0 be a single well potential. Then there holds

u B alt (t)(k) = - i 2kπ max(u) min(u) e -ik(x + (η)+2ηt) -e -ik(x -(η)+2ηt) dη.
Proof. The union of graphs and the periodicity lead to

u B alt (t)(k) = 2P (t,x) n=0 (-1) n u B n (t, x) = 2P (t,x) n=0 (-1) n b(u B n ) a(u B n ) u B n (x)e -ikx dx = b(v B 0 ) a(v B 0 ) v B 0 (x)e -ikx dx - b(v B 1 ) a(v B 1 )
v B 1 (x)e -ikx dx.

The formula for single well functions in Proposition 3.3 becomes

v B 0 (k) = i b(v B 0 ) -a(v B 0 ) k - i 2kπ (x + (t,η),η)∈Gr(v B 0 ) e -ikx + (t,η) dη - (x -(t,η),η)∈Gr(v B 0 ) e -ikx -(t,η) dη . When v = v B 1
, there is only the x + part. Therefore, the formula for increasing functions is written

v B 1 (k) = i b(v B 1 ) -a(v B 1 ) k + i 2kπ (x + (t,η),η)∈Gr(v B 1 )
e -ikx + (t,η) dη.

Consequently, we have

u B alt (t)(k) = i b(v B 0 ) -a(v B 0 ) k - i 2kπ (x + (t,η),η)∈Gr(v B 0 ) e -ikx + (t,η) dη - (x -(t,η),η)∈Gr(v B 0 ) e -ikx -(t,η) dη -i b(v B 1 ) -a(v B 1 ) k - i 2kπ (x + (t,η),η)∈Gr(v B 1 )
e -ikx + (t,η) dη. Now we use that the union of the graphs of v B 0 and v B 1 taken modulo 2π give the graph of u 0 , moreover,

a(v B 0 ) = a(v B 1 )[mod 2π], b(v B 0 ) = b(v B 1 )[mod 2π],
which leads to the result.

We now justify why there exist admissible families for every single well initial data u 0 .

Lemma 3.15 (Existence of admissible approximate initial data). For every single well initial data u 0 , the set of admissible approximate initial data according to Denition 1.5 is non empty. Proof. Let us denote

I := max(u 0 ) min(u 0 ) F (η) dη.
We make the following choice 1. (Small eigenvalues) If 0 ≤ nε ≤ I, then we dene λ n as the solution to max(u 0 ) -λn F (η) dη := nε.

2. (Large eigenvalues) Assume that nε > I, then we dene λ n := nε.

(Phase factors)

We then dene the approximate phase factors by the formula θ 0 = 0 and

θ n+1 -θ n := π - x + (-λ n ) + x -(-λ n ) 2 .
We only need to check that u ε 0 L 2 → u 0 L 2 as ε → 0. For this we use the Parseval formula (4)

1 2 u ε 0 2 L 2 = n≥1 εnγ n (u; ε) = I/ε n=1 εn(λ n (u; ε) -λ n-1 (u; ε) -ε) = I/ε n=1 ελ n (u; ε) -ε 2 I/ε ( I/ε + 1) 2 .
Since we have seen that |(λ n+1 -λ n )F (-λ n ) -ε| ≤ C(δ)ε √ ε, the mesh is tending to zero and the η = -λ n are distributed by F . This leads to

1 2 lim ε→0 u ε 0 2 L 2 = max(u 0 ) min(u 0 ) ηF (η) dη - I 2 2 .
However, there also holds

1 2 u 0 2 L 2 = 1 2π +∞ 0 η Leb(x | |u(x)| > η) dη = max(u 0 ) 0 ηF (η) dη + 0 min(u 0 ) η(1 -F (η)) dη = max(u 0 ) min(u 0 ) ηF (η) dη - (min(u 0 )) 2 2 .
Now, we write

I = max(u 0 ) 0 F (η) dη - 0 min(u 0 ) (1 -F (η)) dη -min(u 0 ).
Since u 0 has mean zero, the rst two terms cancel out, leading to I = -min(u 0 ). This implies the convergence property.

Proof of Theorem 1.2. Let k ∈ Z and T > 0. Let δ > 0. We have established in Theorem 3.13 and Proposition 3.14 that lim sup

ε→0 sup t∈[0,T ] u ε (t)(k) -u B alt (t)(k) ≤ Cδ.
We now pass to the limit δ → 0 and deduce that uniformly on [0, T ], there holds u ε (k) → u B alt (k). Moreover, by conservation of the L 2 norm and the assumption u ε 0 L 2 → u 0 L 2 , all the solutions u ε are bounded in L 2 . This is enough to conclude to the weak convergence of u ε to u B alt in L 2 (T).

Remark 3.16. (Third order Benjamin-Ono equation) Let us now consider the third order equation in the Benjamin-Ono hierarchy

∂ t u = ∂ x -ε 2 ∂ xx u - 3 2 εu|D|u - 3 2 εH(u∂ x u) + u 3 ,
where H is the Hilbert transform, i.e. the Fourier multiplier by -i sgn(n). The spectral parameters are the same as for (BO-ε), and in the time evolution, by considering the rescaled function

v(t) = 1 ε u t ε 2 , x ,
one can see that the frequencies from [START_REF] Gassot | The third order Benjamin-Ono equation on the torus: well-posedness, traveling waves and stability[END_REF]

ω (3) n (v; 1) = n 3 + n p≥1 pγ p (v; 1) -3 p≥1 min(p, n) 2 γ p (v; 1) + 3 p,q≥1
min(p, q, n)γ p (v; 1)γ q (v; 1), should simply be replaced by

ω (3) n (u; ε) = ε 2 ω (3) n (u/ε; 1) = ε 2 n 3 + εn p≥1 pγ p (u; ε) -3ε p≥1 min(p, n) 2 γ p (u; ε) + 3 p,q≥1 min(p, q, n)γ p (u; ε)γ q (u; ε).
As a consequence, the formula λ n (u; ε) = εn -p≥n+1 γ p and the Parseval formula (4) lead to

ω (3) n+1 (u; ε) -ω (3) n (u; ε) = 3ε 2 n 2 + 3ε 2 n + ε 2 + ε p≥1 pγ p (u; ε) -3ε p≥n+1 (2n + 1)γ p (u; ε) + 3 p,q≥n+1 γ p (u; ε)γ q (u; ε),
and nally

ω (3) n+1 (u; ε) -ω (3) n (u; ε) = 3λ 2 n + 3ελ n + ε 2 + u 2 L 2 (T) /2.
As ε → 0, an adaptation of the proof of Theorem 3.13 would lead to the convergence of u ε to the solution u B to the third equation in the inviscid Burgers hierarchy

u B (x, t) = u 0 x -3 u B (x, t) 2 + u 0 2 L 2 /2 t ,
which is comparable to the behavior on the line [START_REF] Miller | The Benjamin-Ono hierarchy with asymptotically reectionless initial data in the zero-dispersion limit[END_REF]. Lemma 3.17 (Weak convergence after the breaking time). Let u 0 (x) = -cos(x). For t right after the breaking time for the Benjamin-Ono equation, there holds

u 0 L 2 (T) > u B alt (t) L 2 (T) .
As a consequence, the solution u ε to equation (BO-ε) with initial data u 0

u ε (t) L 2 (T) = u 0 L 2 (T) > u B alt (t) L 2 (T) ,
and the convergence of u ε (t) to u B alt (t) cannot be strong in L 2 (T).

Eigenvalue equation

In this part, we establish the eigenvalue equation for u(x) = -β cos(x).

Proposition 4.1 (Eigenvalue equation). Let u(x) = -β cos(x). Let λ(u; ε) be an eigenvalue of L u (ε). Then

π 0 cos β ε sin ϕ + 1 + λ ε ϕ dϕ = sin π λ ε ∞ 0 exp - β ε sinh(x) + 1 + λ ε x dx. (18)
A possible further generalization for general trigonometric polynomials, although more technical, would allow us to use the comonotone approximation theorems of continuous functions by trigonometric polynomials (see [START_REF] Lorentz | Degree of approximation by monotone polynomials I[END_REF] in the case of single well potentials and [START_REF] Dzyubenko | Comonotone approximation of periodic functions[END_REF] in more general cases), however, we were not able to push further this approach.

Proof. A scaling argument implies that for the Lax operator L u (ε) associated to the Benjamin-Ono equation with dispersion parameter ε, we have λ n (u; ε) = ελ n ( u ε ; 1). It is therefore enough to only tackle the case ε = 1 and replace later β by β/ε.

Let f be an eigenvector of the Lax operator L u = D -T u with eigenvalue λ. Let α = -β/2. Since u(x + π) = -u(x), the spectrum is unchanged when β becomes -β and we rather study u(x) = 2α cos(x). We expand f and Πu as holomorphic functions on D, and Πu(z) = αz -1 as a holomorphic function on C \ {0}. Then the Szeg® projector has the expression

Π(Πuf ) = 1 2πi ∂D f (ζ) ζ -z αζ -1 dζ.
Applying the residue formula, we get for z ∈ D that

Π(Πuf )(z) = Πuf (z) + α Res ζ=0 f (ζ) ζ -z ζ -1 = Πuf (z) -α f (0) z . The equation Df -Π(uf ) = λf satised by f becomes zf (z) -αz + αz -1 + λ f (z) = -αf (0)z -1 . ( 19
)
Since f is holomorphic, the right hand side does not go to zero as z → 0. As a consequence, we have f (0) = 0, and we can therefore assume that f (0) = 1.

Let us choose the branch of the logarithm corresponding to arg(z) ∈ (0, 2π) and dene

h(z) := z -λ exp -αz + αz -1 .
Then h is solution on C \ R + to zh (z) = -αz + αz -1 + λ h(z).

We deduce that equation ( 19) is equivalent to

(f h) (z) = -αz -λ-2 exp -αz + αz -1 = -αz -2 h(z). (20)
By assumption, we have α > 0. For z ∈ C \ R + , we choose a path γ z joining 0 to z in C \ R + and such that γ z (t) = -t if t ∈ [0, t 0 ] for some t 0 > 0. Since f is holomorphic on D, we get

f (z) = -αz λ exp αz -αz -1 γz ζ -λ-2 exp -αζ + αζ -1 dζ, (21)
and the integral is absolutely convergent.

We rst prove that this expression denes a holomorphic function on C \ R + satisfying the eigenfunction equation, and converging to 1 as z → 0. Indeed, a Taylor expansion of f around 0 of the form S n (z) = 1 + a 1 z + • • • + a n z n with a 0 = 1 and n ≥ 1 transforms equation [START_REF] Lax | The small dispersion limit of the Korteweg-de Vries equation[END_REF] into

((f -S n )h) (z) = -αz -2 h(z) -(a 1 + • • • + na n z n-1 )h(z) + (1 + a 1 z + • • • + a n z n )(α + αz -2 + λz -1 )h(z).
We now dene the coecients a k by induction using this formula in order to cancel all the negative powers of z. We note that the coecient before z -2 h(z) is 0, and the coecient before z -1 h(z) is 0 if αa 1 + λ = 0. Next, the coecient in front of z j h(z), j ≥ 0, is a j+1 -αa j + αa j+2 + λa j+1 , one can therefore choose a j+2 in order to cancel this term. As a consequence, at rank n, one can cancel the terms up to z n-2 h(z). We end up with

((f -S n )h) (z) = R n (z)z n-1 h(z),
where R n = (λ -n)a n + αa n-1 + αa n z is a holomorphic remainder term. We conclude that

f (z) -S n (z) = h(z) -1 γz R n (ζ)ζ n-1 h(ζ) dζ, with h(ζ) = ζ -λ exp -αζ + αζ -1 .
Choosing n large with respect to λ, we see that this denes a holomorphic function on C \ R + , satisfying the dierential ODE [START_REF] Lax | The small dispersion limit of the Korteweg-de Vries equation[END_REF] and converging to 1 as z → 0. Moreover, for every r > 0, the limits f (r + iy) and f (r -iy) exist as y → 0 + , therefore f is an eigenfunction associated to λ if and only if for every r > 0, we have

lim y→0 + f (r + iy) = lim y→0 + f (r -iy). (22)
We now check this property.

We rst assume that λ is not an integer. For r > 0, we integrate (20) over the circle centered at 0 of radius r, starting at r + i0 and ending at r -i0 in the counterclockwise direction. This leads to a second condition (e -2iπλ -1)f (r) exp -αr + αr -1 = -α 2π 0 exp -αre iθ + αr -1 e -iθ r -1 e -i(λ+1)θ i dθ.

By analytic continuation, we obtain that for every z ∈ C \ {0}, we have (e -2iπλ -1)f (z) exp -αz + αz -1 = -iαz -1 2π 0 exp -αze iθ + αz -1 e -iθ e -i(λ+1)θ dθ. [START_REF] Miller | Direct Scattering for the BenjaminOno Equation with Rational Initial Data[END_REF] This expression denes a holomorphic function f outside the origin, solving the ODE (19) of order one. Therefore, f denes an eigenfunction with eigenvalue λ if and only if this expression coincides with [START_REF] Lorentz | Degree of approximation by monotone polynomials I[END_REF] at one non singular point, for instance at the point z = -1:

(e -2iπλ -1) When λ is an integer, the condition ( 22) is satised if and only if the function f given by ( 21) is holomorphic:

1 0 t -λ-2 exp αt -αt -1 dt = iα
0 = C(0,r) ζ -λ-2 exp -αζ + αζ -1 dζ,
where C(0, r) is the circle of radius r centered at 0. Since the integrand is holomorphic outside the origin, this integral does not depend on r so it is enough to calculate it for r = 1: 0 = 2π 0 e iθ(-λ-1) exp -αe iθ + αe -iθ dθ, and this also leads to the result.

Asymptotic expansion for the Lax eigenvalues

In this part, we apply the stationary phase and Laplace methods into identity [START_REF] Hörmander | The analysis of linear partial dierential operators I: Distribution theory and Fourier analysis[END_REF] and get an asymptotic expansion of the Lax eigenvalues λ.

In order to get a uniform bound on the remainder terms, we need do ensure that the stationary point of the phase remains suciently far from the integral boundaries. Therefore, we x a small parameter δ > 0, and we only consider the eigenvalues λ such that ν = λ + ε is inside one of the two intervals Λ -(δ) = [-β + δ, β -δ] and Λ + (δ) = [β + δ, +∞).

We apply the method of stationary phase for the rst term

I 1 (ε, ν) := π 0 exp i β ε sin ϕ + ν ε ϕ dϕ
and the Laplace method for the second term

I 2 (ε, ν) := ∞ 0 exp - β ε sinh(x) + ν ε x dx
that appear in the identity [START_REF] Hörmander | The analysis of linear partial dierential operators I: Distribution theory and Fourier analysis[END_REF] Re(I 1 (ε, ν)) = sin π λ ε I 2 (ε, ν), which we write

I(ε, ν) = 0 (24)
with ν = λ + ε and I(ε, ν) = Re(I 1 (ε, ν)) + sin π ν ε I 2 (ε, ν).

To estimate the large eigenvalues, we choose K(δ) > 0 such that 1 K(δ) u 2 L 2 < 2δ, in order to get thanks to the Parseval formula ( 4)

k≥K(δ)/ε γ k (u; ε) ≤ ε K(δ) k≥K(δ)/ε kγ k (u; ε) ≤ 1 2K(δ) u 2 L 2 < δ.
As a consequence, since λ n (u; ε) = nε -k≥n+1 γ k (u; ε), one can see that for n ≥ K(δ)/ε, there holds

λ n + ε ≥ K(δ) -δ.
Method of stationary phase for the rst term Let us start with I 1 (ε, ν) = π 0 e iS 1 (x,ν)/ε dx,

where the phase is equal to

S 1 (x, ν) = β sin(x) + νx = F 1 (x) + νx, F 1 (x) = β sin(x). Since ∂ x S 1 (x, ν) = β cos(x)
+ ν, we have the following alternative.

1. (Small eigenvalues) If |ν| ≤ β -δ, there is a unique critical point

x 1 (ν) = arccos(-ν/β).

Moreover, there exists δ 1 (δ) > 0 such that

x 1 (ν) ∈ [arccos(1 -δ/β), arccos(-1 + δ/β)] ⊂ [2δ 1 (δ), π -2δ 1 (δ)].
Therefore, there exists δ 2 (δ) > 0 such that F 1 (x) = -sin(x) ≤ -δ 2 (δ) for x ∈ [δ 1 (δ), π -δ 1 (δ)].

We also compute

S 1 (x 1 (ν), ν) = β 2 -ν 2 + νx 1 (ν) = β 2 -ν 2 + ν arccos(-ν/β), ∂ xx S 1 (x 1 (ν), ν) = -F 1 (x 1 (ν)) = -β 2 -ν 2 .
From the stationary phase method (see for instance [START_REF] Hörmander | The analysis of linear partial dierential operators I: Distribution theory and Fourier analysis[END_REF], chapter 7, or adapt directly the proof of [9]), there exists C(δ) > 0 such that for every |ν| ≤ β -δ, there holds

I 1 (ε, ν) - √ 2πε β 2 -ν 2 exp i S 1 (x 1 (ν), ν) ε - π 4 ≤ C(δ)ε, implying Re (I 1 (ε, ν)) - √ 2πε
β 2 -ν 2 cos S 1 (x 1 (ν), ν) ε - π 4 ≤ C(δ)ε. (25)
2. (Large eigenvalues) If K(δ) ≥ ν ≥ β + δ, the phase has no critical point since ∂ x S 1 (0, ν) = β cos(x) + ν ≥ δ for every x. We know that S 1 (0, ν) = 0, therefore, there exists C(δ) > 0 such that for every ν

/β ≥ 1 + δ/β, |I 1 (ε, ν)| ≤ C(δ)ε. (26)
Method of Laplace for the second term Let us now analyze I 2 (ε) = +∞ 0 e S 2 (x,ν)/ε dx, where the phase is equal to

S 2 (x, ν) = -β sinh(x) + νx = F 2 (x) + νx, F 2 (x) = -β sinh(x). Since ∂ x S 2 (
x, ν) = -β cosh(x) + ν, the following holds.

1. (Small eigenvalues) If |ν| ≤ β -δ, we have ∂ x S 2 (x, ν) ≤ -β + ν ≤ -δ for every x, therefore there is no critical point. We get from the Laplace method that

|I 2 (ε, ν)| ≤ C(δ)ε. (27)
2. (Large eigenvalues) If K(δ) ≥ ν ≥ β + δ, then there is a unique critical point

x 2 (ν) = cosh -1 ν β .

There exists δ 1 (δ) > 0 such that x 2 (ν) ≥ 2δ 1 (δ) for every ν ≥ β + δ, and there exists δ 2 (δ) > 0 such that F 2 (x) = -β sinh(x) ≤ -δ 2 (δ) for every x ≥ δ 1 (δ). Moreover, we have

S 2 (x 2 (ν), ν) = -ν 2 -β 2 + νx 2 (ν), F 2 (x 2 (ν)) = -ν 2 -β 2 .
One can therefore apply the adapted Laplace method and get

I 2 (ε, ν) - √ 2πε ν 2 -β 2 e S 2 (x 2 (ν),ν)/ε ≤ C(δ)ε. (28)
Small eigenvalues For every |ν| ≤ β -δ, we conclude that

I(ε, ν) - √ 2πε β 2 -ν 2 cos S 1 (x 1 (ν), ν) ε - π 4 ≤ C(δ)ε. (29)
Therefore, identity [START_REF] Miller | The scattering transform for the BenjaminOno equation in the small-dispersion limit[END_REF] implies that given a small Lax eigenvalue λ, then ν = λ + ε satises √ 2πε

β 2 -ν 2 cos S 1 (x 1 (ν), ν) ε - π 4 ≤ C(δ)ε.
Taking the limit ε → 0, we conclude that cos S 1 (x 1 (ν),ν) ε -π 4 should be close to 0 as soon as ε < ε 0 (δ) is small enough. Therefore there exists an integer N (ε, ν) such that

S 1 (x 1 (ν), ν) ε - 3π 4 -πN (ε, ν) ≤ C(δ) √ ε.
Let us recall the denition of

F (η) = 1 2π Leb{x ∈ [0, 2π] | u(x) ≥ η}.
Then one can see that whenever -β < η < β, we have 2πF (η) = 2 arccos η β . As a consequence,

2π β -ν F (η) dη = 2β x arccos(x) -1 -x 2 1 -ν/β = 2 ν arccos - ν β + β 2 -ν 2 = 2S 1 (x 1 (ν), ν).
We have therefore proven that

β -ν F (η) dη - 3ε 4 -εN (ε, ν) ≤ C(δ)ε √ ε. (30)
The integral term is bounded below since β β-δ F (η) dη ≥ 1 C(δ) . We deduce that when ε < ε 0 (δ) is small enough, we have that necessarily N (ε, ν) ≥ 0 (and even N (ε, ν) ≥ 1 C(δ)ε ).

Large eigenvalues In the case K(δ) ≥ ν ≥ β + δ, we have proven that

I(ε, ν) -sin π ν ε √ 2πε ν 2 -β 2 e S 2 (x 2 (ν),ν)/ε ≤ C(δ)ε. (31)
Then identity [START_REF] Miller | The scattering transform for the BenjaminOno equation in the small-dispersion limit[END_REF] implies that given a large Lax eigenvalue λ,

then ν = λ + ε satises sin π ν ε √ 2πε ν 2 -β 2 e S 2 (x 2 (ν),ν)/ε ≤ C(δ)ε. We introduce the function x → ψ(x) := - √ x 2 -1 + x cosh -1 (x) on [1, ∞). This function satises ψ(1) = 0 and its derivative on (1, ∞) is ψ (x) = - x √ x 2 -1 + cosh -1 (x) + x √ x 2 -1 ≥ 0. This implies that ψ(x) ≥ 0 on [1, ∞). Since ν > β, then S 2 (x 2 (ν), ν) = βψ(ν/β) ≥ 0. We deduce that sin π ν ε ≤ sin π ν ε e S 2 (x 2 (ν),ν)/ε ≤ C(δ) √ ε.
As a consequence, sin(πν/ε) is close to 0 for small ε, and we have the more precise asymptotics

|ν -(1 + N (ε, ν))ε| ≤ C(δ)ε √ ε. (32)
Since ν ≥ β + δ, we know that necessarily, N (ε, ν) ≥ 0 (and even N (ε, ν) ≥ β ε for ε < ε 0 (δ)).

Characterization of the eigenvalues

Conversely, we prove in this part that the asymptotic expansions obtained in part 4.2 actually correspond to eigenvalues for the Lax operator, and therefore we conclude the proof of Theorem 1.6.

More precisely, in each of the two regimes ν ∈ Λ -(δ) = [-β + δ, β -δ] and ν ∈ Λ + (δ) = [β + δ, ∞), we establish that there is exactly one eigenvalue λ such that ν = λ + ε satises N (ε, ν) = N , as soon as N is compatible with the conditions |ν| ≤ β -δ or ν ≥ β + δ.

In this purpose, we x ε small enough and study the variations of the two functions of ν dened as

I 1 (ε, ν) = π 0 exp i β ε sin(x) + ν ε x dx and I 2 (ε, ν) = ∞ 0 exp -β ε sinh(x) + ν ε x dx. We have ∂ ν I 1 (ε, ν) = i ε π 0 x exp i β ε sin(x) + ν ε x dx and ∂ ν I 2 (ε, ν) = 1 ε ∞ 0 x exp - β ε sinh(x) + ν ε x dx.
We also recall that

I(ε, ν) = Re(I 1 (ε, ν)) + sin π ν ε I 2 (ε, ν).
Small eigenvalues First, we assume that |ν| ≤ β -δ. Then the stationary phase method implies that

∂ ν I 1 (ε, ν) -i x 1 (ν) ε √ 2πε β 2 -ν 2 exp i S 1 (x 1 (ν), ν) ε - π 4 ≤ C(δ), so that ∂ ν Re(I 1 (ε, ν)) + x 1 (ν) √ ε √ 2π β 2 -ν 2 sin S 1 (x 1 (ν), ν) ε - π 4 ≤ C(δ), whereas |∂ ν I 2 (ε, ν)| ≤ C(δ).
We conclude that

∂ ν I(ε, ν) + x 1 (ν) √ ε √ 2π β 2 -ν 2 sin S 1 (x 1 (ν), ν) ε - π 4 ≤ C(δ).
Let c 1 > 0 be a small parameter such that if |cos (x)| ≤ c 1 , then d x, πZ + π 2 ≤ π 4 . Using (29), there exist c 2 (δ) > 0 and ε 0 (δ) > 0 such that for ε < ε 0 (δ), then the inequality

|I(ε, ν)| ≤ c 2 (δ) √ ε (33) implies cos S 1 (x 1 (ν), ν) ε - π 4 ≤ c 1 .
Let N ≥ 0 be an integer. Then there exists ν 0

N = ν 0 N (ε) ≥ -β such that β -ν 0 N F (η) dη - 3ε 4 -εN = 0, ( 34
) implying cos S 1 (x 1 (ν 0 N ), ν 0 N ) ε - π 4 = 0.
As a consequence, inequality [START_REF] Ono | Algebraic solitary waves in stratied uids[END_REF] implies that for ε < ε 0 (δ), there holds

|I(ε, ν 0 N )| ≤ C(δ)ε ≤ c 2 (δ) √ ε/2. Assume that |ν 0 N | ≤ β -2δ. Let [ν * , ν * ] ⊂ [-β + δ, β -
δ] be the largest interval containing ν 0 N and on which inequality [START_REF] Venakides | The Korteweg-de Vries equation with small dispersion: higher order Lax-Levermore theory[END_REF] holds. We prove that the interval [ν * , ν * ] encloses exactly one eigenvalue because of monotonicity of I along the parameter ν, and conversely that this interval is large enough to enclose all the eigenvalues associated to N . On [ν * , ν * ], we have by construction

sin S 1 (x 1 (ν), ν) ε - π 4 ≥ 1 -c 2 1 .
Given that x 1 (ν) = arccos(-ν β ) ≥ 1 C(δ) , we deduce that for ε < ε 0 (δ),

|∂ ν I(ε, ν)| ≥ 1 -c 2 1 C(δ) √ ε -C(δ) ≥ 1 C (δ) √ ε . (35)
By continuity, the ν-derivative of I(ε, ν) stays of the same sign on [ν * , ν * ]. For instance if I(ε, •) is increasing and I(ε, ν 0 N ) < 0, then there holds for ν * ≥ ν ≥ ν 0

N that 1 C (δ) √ ε (ν -ν 0 N ) ≤ I(ε, ν) -I(ε, ν 0 N ). Since |I(ε, ν 0 N )| ≤ c 2 (δ)
√ ε/2, then as long as I(ε, ν) < 0, we know that I(ε, •) is increasing and inequality (33) stays satised. Therefore, there exists ν N ∈ [ν * , ν * ] such that I(ε, ν N ) = 0 and ν N is a Lax eigenvalue. We also know that |ν N -ν 0 N | ≤ δ for ε < ε 0 (δ). An adaptation of this argument applies to the other cases, when I(ε, •) is decreasing or when I(ε, ν 0 N ) ≥ 0. The integer N is thus uniquely dened in the following inequality, which stays true on the (non ordered

) interval [ν 0 N , ν N ] by construction π ε β -ν F (η) dη - 3ε 4 -εN = S 1 (x 1 (ν), ν) ε - π 4 - π 2 -πN ≤ π 4 .
Moreover, this is the same integer N on the whole interval [ν 0 N , ν N ] by continuity. Given N , then ν N is uniquely dened in [ν * , ν * ] because I(ε, •) is strictly monotone in this interval.

Conversely, the function ν →

S 1 (x 1 (ν), ν) = β 2 -ν 2 + ν arccos(-ν/β) is C 1 (δ)-Lipschitz on the interval [-β + δ, β -δ].
As a consequence, let C 0 (δ) be a large constant to be chosen later and let ν

such that |ν -ν 0 N | ≤ c 2 (δ)ε C 0 (δ)C 1 (δ)
.

For ε < ε 0 (δ), the upper bound is less than δ so that ν ∈ [-β + δ, β -δ]. Moreover, the Lipschitz bound implies

cos S 1 (x 1 (ν), ν) ε - π 4 = cos S 1 (x 1 (ν), ν) ε - π 4 -cos S 1 (x 1 (ν 0 N ), ν 0 N ) ε - π 4 ≤ c 2 (δ) C 0 (δ) .
Consequently, inequality [START_REF] Ono | Algebraic solitary waves in stratied uids[END_REF] implies that for ε < ε 0 (δ) chosen small enough and C 0 (δ) chosen large enough, there holds

|I(ε, ν)| ≤ c 2 (δ) √ ε 2 + C(δ)ε ≤ c 2 (δ) √ ε.
Therefore, inequality (33) holds true, and we have proven that when

c 2 (δ)ε C 0 (δ)C 1 (δ) < δ, then ν 0 N - c 2 (δ)ε C 0 (δ)C 1 (δ) , ν 0 N + c 2 (δ)ε C 0 (δ)C 1 (δ) ⊂ [ν * , ν * ].
We deduce that the Lax eigenvalue ν N associated to the integer N , which satises |ν -ν 0 N | ≤ C(δ)ε √ ε thanks to inequality [START_REF] Sun | Complete integrability of the BenjaminOno equation on the multi-soliton manifolds[END_REF], must belong to [ν * , ν * ] and is therefore uniquely dened on [-β + δ, β -δ].

To conclude, if |ν| ≤ β -δ is a small eigenvalue, then there exists |ν 0 N | ≤ β -δ/2 as above. Conversely, if |ν 0 N | ≤ β -δ/2, then one can construct a small eigenvalue |ν| ≤ β -δ/4 such that N = N (ε, ν), and by restriction to the interval [-β + δ, β -δ], we get all the eigenvalues satisfying |ν| ≤ β -δ.

Large eigenvalues We now establish the asymptotics for the eigenvalues satisfying K(δ) ≥ ν ≥ β + δ. Using the Laplace method,

∂ ν I 2 (ε, ν) - x 2 (ν) ε √ 2πε
ν 2 -β 2 e S 2 (x 2 (ν),ν)/ε ≤ C(δ), whereas |∂ ν I 1 (ε, ν)| ≤ C(δ).

We proceed similarly as the small eigenvalue case.

Let N ≥ 0 and ν 0 N = ν 0 N (ε) such that ν 0 N = (N + 1)ε, so that sin π ν 0 N ε = 0. We assume that K(δ) + δ ≥ ν 0 N ≥ β + 2δ. Let c 1 > 0 such that if | sin(x)| ≤ c 1 , then d(x, πZ) ≤ π 4 , and using inequality [START_REF] Venakides | The zero dispersion of the Korteweg-de Vries equation for initial potentials with non-trivial reection coecient[END_REF], let c 2 (δ) > 0 such that if

|I(ε, ν)| ≤ c 2 (δ) √ ε, (36)
then for ε < ε 0 (δ), there holds sin π ν ε ≤ c 1 .

Let [ν * , ν * ] ⊂ [β + δ, K(δ) + 2δ] be the largest interval containing ν 0 N and on which inequality (36) holds.

By denition, on this interval, we have

cos π ν ε ≥ 1 -c 2 1 .
Moreover, recall that on [β + δ, K(δ) + 2δ], we have S 2 (x 2 (ν), ν) ≥ 0 and x 2 (ν) = cosh -1 ( ν β ) ∈ [ 1 C(δ) , C(δ)]. Therefore, when ε < ε 0 (δ), inequality (28) implies As a consequence, we get

∂ ν Re(I 1 (ε, ν)) -sin π ν ε I 2 (ε, ν) = ∂ ν Re(I 1 (ε, ν)) -sin π ν ε ∂ ν I 2 (ε, ν) - π ε cos π ν ε I 2 (ε, ν) ≥ 1 -c 2 1 C(δ) √ ε e S 2 (x 2 (ν),ν)/ε - c 1 C(δ) √ ε -C(δ),
or when we then choose ε < ε 0 (δ) and c 1 such that for instance 1 -c 2 1 ≥ c 1 /2,

|∂ ν I(ε, ν)| ≥ 1 C(δ) √ ε e S 2 (x 2 (ν),ν)/ε ≥ 1 C(δ) √ ε .
Therefore, on [ν * , ν * ], the derivative stays of the same sign. On the other hand, since sin(πν 0 N /ε) = 0, then inequality [START_REF] Venakides | The zero dispersion of the Korteweg-de Vries equation for initial potentials with non-trivial reection coecient[END_REF] implies that |I(ε, ν 0 N )| ≤ C(δ)ε ≤ c 2 (δ) √ ε/2. We deduce by monotonicity that there exists a unique ν N ∈ [ν * , ν * ] such that I(ε, ν) = 0. Using inequality [START_REF] Venakides | The zero dispersion of the Korteweg-de Vries equation for initial potentials with non-trivial reection coecient[END_REF], we deduce that when ε < ε 0 (δ), we have

|I(ε, ν)| ≤ c 2 (δ) √ ε 2 + C(δ)ε ≤ c 2 (δ) √ ε,
and therefore ν ∈ [ν * , ν * ]: we have proven [ν 0 N -c 2 (δ)ε C 0 (δ) , ν 0 N + c 2 (δ)ε C 0 (δ) ] ⊂ [ν * , ν * ] when c 2 (δ)ε C 0 (δ) < δ. The precise asymptotics that a Lax eigenvalue needs to satisfy [START_REF] Venakides | The zero dispersion limit of the Korteweg-de Vries equation with periodic initial data[END_REF] |ν -(N (ε, ν) + 1)ε| ≤ C(δ)ε √ ε ensure that given N such that K(δ) + δ ≥ ν 0 N ≥ β + 2δ, necessarily ν N ∈ [ν * , ν * ], therefore there is exactly one Lax eigenvalue.

To conclude, if K(δ) ≥ ν ≥ β + δ is a large eigenvalue, then there exists K(δ) + δ ≥ ν 0 N ≥ β + δ/2 as above. Conversely, if K(δ) + δ/2 ≥ ν 0 N ≥ β + δ/4, then there exists exactly one large eigenvalue associated to N such that K(δ) + δ/2 ≥ ν ≥ β + δ/2, and by restriction we get all the Lax eigenvalues satisfying K(δ) ≥ ν ≥ β + δ.

Other eigenvalues We now establish an upper bound on the number of eigenvalues which do not t into any of the two categories listed above. But we also have the asymptotic expansion as n → ∞ λ n -nε → 0, more precisely, by denition of K(δ), for n ≥ K(δ)/ε, one has |λ n -nε| < δ.

When δ < 1/4, since N is uniquely dened, this implies that the n-th Lax eigenvalue satises λ n = ν n -ε.

We now establish a lower bound on the number of small Lax eigenvalues λ satisfying λ + ε ∈ F (η) dη.

We deduce that there are at least Proof of Theorem 1.6. When λ n + ε, λ p + ε ∈ Λ + (δ), we have already seen from the study of the large eigenvalues that if n ≤ K(δ)/ε, then |λ n -nε| ≤ C(δ)ε √ ε, whereas if n > K(δ)/ε, then |λ n -nε| ≤ Cδ.

In the former part, we have seen that there are at most for every λ n + ε ∈ Λ -(δ). This implies the small eigenvalues point of the Theorem.

Figure 1 :

 1 Figure 1: Multivalued solution of the Burgers equation obtained by the method of characteristics, with initial data u 0 (x) = -β cos(x)

Figure 2 :

 2 Figure 2: Denition of Λ ± (δ) (on the left) and distribution of the eigenvalues in the zero-dispersion limit (on the right), with initial data u 0 (x) = -β cos(x)

section 4 ,

 4 we establish an asymptotic expansion of the Lax eigenvalues associated to equation (BO-ε) for the cosine initial data. Acknowledgments The author is grateful to Patrick Gérard for relevant advice on this problem, in particular, for providing notes on Proposition 4.1 about the cosine function. This material is based upon work supported by the National Science Foundation under Grant No. DMS-1439786 while the author was in residence at the Institute for Computational and Experimental Research in Mathematics in Providence, RI, during the Hamiltonian Methods in Dispersive and Wave Evolution Equations program.

Figure 3 :

 3 Figure 3: Sketch of the multivalued solution of the Burgers equation obtained by the method of characteristics, with initial data u 0 (x) = -β cos(x)

2π 0 exp 1 0t

 01 αe iθ -αe -iθ e -i(λ+1)θ dθ, or e -iπλ 2 sin(-πλ) -λ-2 exp αt -αt -1 dt = α 2π 0 exp (i(2α sin(θ) -(λ + 1)θ)) dθ.We set the change of variable t = e -x and θ = ϕ + π, and get that this is equivalent to sin(πλ) ∞ 0 exp ((λ + 1)x -2α sinh(x)) dx = π 0 cos ((λ + 1)ϕ + 2α sin(ϕ)) dϕ.

|I 2

 2 (ε, ν)| ≥ √ ε C(δ) e S 2 (x 2 (ν),ν)/ε -C(δ)ε ≥ √ ε C (δ)e S 2 (x 2 (ν),ν)/ε .

4 .

 4 Moreover, since inequality[START_REF] Zabusky | Interaction of solitons in a collisionless plasma and the recurrence of initial states[END_REF] holds on [ν * , ν * ] by denition, we have d(ν, εZ) ≤ ε 4 on this interval, and by continuity, the same integer N as for ν 0 N appears in the inequality |ν N -(N + 1)ε| ≤ 1Conversely, let C 0 (δ) > 0 be a large constant, let ε < ε 0 (δ), and let ν such that|ν -ν 0 N | ≤ c 2 (δ)ε C 0 (δ) .Since sin is 1-Lipschitz, then

First

  ) + δ ≥ ν 0 N := (N + 1)ε ≥ β + 2δso that we get a Lax eigenvalue ν N ∈ [β + δ, K(δ) + 2δ] which satises |ν N -(N + 1)ε| ≤ 1 4 .

0 NF (η) dη - 3ε 4 -

 04 [-β + δ, β -δ]. Condition (34) β -ν εN = 0 is true for some ν 0 N ∈ [-β + 2δ, β -2δ] as soon as

FF

  (η) dη -1 suitable integers N . But since β -β F (η) dη = β and F (η) ≤ 1 for every η, we have 1 ε β-2δ -β+2δ (η) dη -β ≤ βδ ε .As a conclusion, among the integers N which do not lead to large eigenvalues ν N ≥ β + δ, that is, lead to small eigenvalues |ν N | ≤ β -δ. The remaining eigenvalues consist of no more than Cδ ε indexes.

Cδε-β+δ/ 4 Fβ-β+δ/ 2 F-β+δ/ 2 -β+δ/ 4 F

 4224 eigenvalues such that λ n + ε ∈ Λ -(δ) ∪ Λ + (δ). Reasoning with δ/2 instead of δ, the indexes counting argument from the former part implies the index N leads to an eigenvalue ν N = λ n + ε ∈ [β -δ, β -δ/2] as soon asβ (η) dη ≤ εN + 3ε 4 ≤ (η) dη.Since F ≥ 1/C on (-∞, 0], we deduce that that there are at least1 ε (η) dη ≥ δ Cεsuch indexes, or at least δ Cε Lax eigenvalues λ n + ε ∈ [β -δ, β -δ/2]. Similarly, one knows that there are at least δ Cε Lax eigenvalues such that λ n + ε ∈ [-β + δ/2, -β + δ]. Finally, in the region λ n +ε ∈ Λ -(δ) = [-β +δ, β -δ], the counting of the indexes leads to the same conclusion, except that only the lower bound F (η) ≥ 1/C(δ) holds on (-∞, β -δ/4], so that there are between 1 C(δ)ε and Cδ ε eigenvalues in the region [-β, -β + δ]. Therefore there exists 1C(δ)ε ≤ N 0 ≤ Cδ ε such that for every n, if λ n + ε = ν N ∈ Λ -(δ), then n -N = N 0 . Inequality (30) becomes β -λ-ε F (η) dη -3ε 4 -ε(n -N 0 ) ≤ C(δ)ε √ ε

  R the branching points at time t such that η -(t) ≤ 0 and η + (t) ≥ 0. Then one can express these branches in terms of two branches. The rst branch v B 0 is well-dened on

Proof. We have x -(η, t) = π -arccos(η) + 2ηt and x + (η, t) = π + arccos(η) + 2ηt. We rst study x + as a function of η. We get ∂ η x + (η, t) = -1 1 -η 2 + 2t.

When 2t > 1, there is exactly one solution η + = 1 -1/2t in (0, 1] and one solution η -= -η

increasing from (0, u B (t, 0)) to (π/2 + 2t, 1), and decreasing from (π/2 + 2t, 1) to (X + , η + ). Similarly, v B 2 is decreasing from (X -, η -) to (2π -2t, -1) and increasing from (2π -2t, -1) to (2π, u B (t, 2π)).

We note that X + = π + arccos(η

2 and similarly,

, and as a consequence,

We conclude that when η ∈ [0, η + ],

Similarly,

Finally, we write

By splitting the integrals between the zones η > η + and 0 ≤ η ≤ η + , we conclude that

4 Lax eigenvalues for initial data u 0 (x) = -β cos(x)

The aim of this part is to establish the asymptotic expansion on the Lax eigenvalues from Theorem 1.6.

Using the fact that any eigenfunction f n (u 0 ; ε) ∈ L 2 + (T) for the Lax operator admits an analytic expansion on the complex unit disc, we derive an integral identity in part 4.1. Then we apply the method of stationary phase and the Laplace method to deduce an asymptotic expansion of the Lax eigenvalues in part 4.2. Conversely, we justify that this method enables us to list all the Lax eigenvalues in the two regions Λ + (δ) and Λ -(δ) in part 4.3.