Complete integrability of the Benjamin--Ono equation on the multi-soliton manifolds

This paper is dedicated to proving the complete integrability of the Benjamin--Ono (BO) equation on the line when restricted to every $N$-soliton manifold, denoted by $\mathcal{U}_N$. We construct generalized action--angle coordinates which establish a real analytic symplectomorphism from $\mathcal{U}_N$ onto some open convex subset of $\mathbb{R}^{2N}$ and allow to solve the equation by quadrature for any such initial datum. As a consequence, $\mathcal{U}_N$ is the universal covering of the manifold of $N$-gap potentials for the BO equation on the torus as described by G\'erard--Kappeler $[19]$. The global well-posedness of the BO equation in $\mathcal{U}_N$ is given by a polynomial characterization and a spectral characterization of the manifold $\mathcal{U}_N$. Besides the spectral analysis of the Lax operator of the BO equation and the shift semigroup acting on some Hardy spaces, the construction of such coordinates also relies on the use of a generating functional, which encodes the entire BO hierarchy.

Derived by Benjamin [4] and Ono [49], this equation describes the evolution of weakly nonlinear internal long waves in a two-layer fluid. The BO equation is globally well-posed in every Sobolev spaces H s (R, R), s ≥ 0. (see Tao [63] for s ≥ 1, Burq-Planchon [8] for s > 1 4 , Ionescu-Kenig [33], Molinet-Pilod [43] and Ifrim-Tataru [29] for s ≥ 0, etc.) Recall the scaling and translation invariances of equation (1.1): if u = u(t, x) is a solution, so is u c,y : (t, x) → cu(c 2 t, c(x − y)). A smooth solution u = u(t, x) is called a solitary wave of (1.1) if there exists R ∈ C ∞ (R) solving the following non local elliptic equation and u(t, x) = R c (x − y − ct), where R c (x) = cR(cx), for some c > 0 and y ∈ R. The unique (up to translation) solution of equation (1.3) is given by the following formula ∀x ∈ R, (1.4) in Benjamin [4] and Amick-Toland [2] for the uniqueness statement. Inspired from the complete classification of solitary waves of the BO equation, we introduce the main object of this paper.
is called an N -soliton, for some positive integer N ∈ N + := Z (0, +∞), where c j > 0 and x j ∈ R, for every j = 1, 2, · · · , N . Let U N ⊂ L 2 (R, R) denote the subset consisting of all the N -solitons.
In the point of view of topology and differential manifolds, the subset U N is a simply connected, real analytic, embedded submanifold of the R-Hilbert space L 2 (R, R). It has real dimension 2N . The tangent space to U N at an arbitrary N -soliton is included in an auxiliary space (1.5) in which a 2-covector ω ∈ Λ 2 (T * ) is well defined by ω(h 1 , h 2 ) = i 2π Rĥ 1(ξ)ĥ2(ξ) ξ dξ, for every h 1 , h 2 ∈ T , by Hardy's inequality. We define a translation-invariant 2-form ω : u ∈ U N → ω ∈ Λ 2 (T * ), endowed with which U N is a symplectic manifold. The tangent space to U N at u ∈ U N is denoted by T u (U N ). For every smooth function f : U N → R, its Hamiltonian vector field X f ∈ X(U N ) is given by where ∇ u f (u) denotes the Fréchet derivative of f , i.e. df (u)(h) = h, ∇ u f (u) L 2 , for every h ∈ T u (U N ). The Poisson bracket of f and another smooth function g : U N → R is defined by {f, g} : u ∈ U N → ω u (X f (u), X g (u)) = ∂ x ∇ u f (u), ∇ u g(u) L 2 ∈ R.
Then the BO equation (1.1) in the N -soliton manifold (U N , ω) can be written in Hamiltonian form where E(u) = 1 2 |D|u, u The Cauchy problem of (1.6) is globally well-posed in the manifold U N (see proposition 4.9). Inspired from the construction of Birkhoff coordinates of the space-periodic BO equation discovered by Gérard-Kappeler [19], we want to show the complete integrability of (1.6) in the Liouville sense.
Let Ω N := {(r 1 , r 2 , · · · , r N ) ∈ R N : r j < r j+1 < 0, ∀j = 1, 2, · · · , N − 1} denote the subset of actions and ν = N j=1 dr j ∧ dα j denotes the canonical symplectic form on Ω N × R N . The main result of this paper is stated as follows. Theorem 1. There exists a real analytic symplectomorphism Φ N : (U N , ω) → (Ω N × R N , ν) such that E • Φ −1 N (r 1 , r 2 , · · · , r N ; α 1 , α 2 , · · · , α N ) = − 1 2π N j=1 |r j | 2 . (1.7) Remark 1.2. A consequence of theorem 1 is that U N is simply connected. In fact the manifold U N can be interpreted as the universal covering of the manifold of N -gap potentials for the Benjamin-Ono equation on the torus as described by . We refer to section A for a direct proof of these topological facts, independently of theorem 1.
In order to establish the link between the action-angle coordinates and the translation-scaling parameters of an N -soliton, we introduce the inverse spectral matrix associated to Φ N , denoted by M : u ∈ U N → (M kj (u)) 1≤j,k≤N ∈ C N ×N , M kj (u) =    2πi I k (u)−Ij (u) where I k , γ k : U → R is given by remark 1.3. Then U N has the following polynomial characterization. An N -soliton is expressed by u(x) = N j=1 R cj (x − x j ) if and only if its translation-scaling parameters {x j − c −1 j i} 1≤j≤N ⊂ C N − are the roots of the characteristic polynomial Q u (X) = det(X − M (u)), whose coefficients are expressed in terms of the action-angle coordinates (I j (u), γ j (u)) 1≤j≤N ∈ Ω N × R N . Proposition 1.4 is restated with more details in proposition 4.1, formula (5.11) and theorem 4.8 which gives a spectral characterization of U N . If u : t ∈ R → u(t) ∈ U N solves the BO equation (1.1), then we have the following explicit formula where the inner product of C N is X, Y C N = X T Y , for every u ∈ U N , the matrix V(u) ∈ C N ×N and the vectors X(u), Y (u) ∈ C N are defined by √ 2πX(u) T = ( |I 1 (u)|, |I 2 (u)|, · · · , |I N (u)|), √ 2π −1 Y (u) T = ( |I 1 (u)| −1 , |I 2 (u)| −1 , · · · , |I N (u)| −1 ), . . .
Given a smooth manifold M of real dimension N , let C ∞ (M) denote all smooth functions f : M → R and the set of all smooth vector fields is denoted by X(M). The tangent (resp. cotangent) space to M at p ∈ M is denoted by T p (M) (resp. T * p (M)). Given k ∈ N, the R-vector space of smooth k-forms on M is denoted by Ω k (M). Given a R-vector space V, we denote by Λ k (V * ) the vector space of all k-covectors on V. Given a smooth covariant tensor field A on M and X ∈ X(M), the Lie derivative of A with respect to X is denoted by L X (A), which is also a smooth tensor field on M. If N is another smooth manifold, F : N → M is a smooth map and A is a smooth covariant k-tensor field on M, the pullback of A by F is denoted by F * A, which is a smooth k-tensor field on N defined by ∀p ∈ N, ∀j = 1, 2, · · · , k, (F * A) p (v 1 , v 2 , · · · , v k ) = A F(p) (dF(p)(v 1 ), dF(p)(v 2 ), · · · , dF(p)(v k )) , ∀v j ∈ T p (N).
(1.15) Given a positive integer N , let C ≤N −1 [X] denote the C-vector space of all polynomials with complex coefficients whose degree is no greater than N − 1 and C N [X] = C ≤N [X]\C ≤N −1 [X] consists of all polynomials of degree exactly N . R + = [0, +∞) and R * + = (0, +∞). D(z, r) ⊂ C denotes the open disc of radius r > 0, whose center is z ∈ C.

Organization of this paper
The construction of action-angle coordinates for the BO equation (1.6) mainly relies on the Lax pair formulation ∂ t L u = [B u , L u ], discovered by Nakamura [45] and Bock-Kruskal [6]. Section 2 is dedicated to the spectral analysis of the Lax operator L u : h ∈ H 1 + → −i∂ x h − Π(uh) ∈ L 2 + given by definition 2.1 for general symbol u ∈ L 2 (R, R), where Π denotes the Szegő projector given in (1.12) and the Hardy space L 2 + is defined in (1.13). L u is an unbounded self-adjoint operator on L 2 + that is bounded from below, it has essential spectrum σ ess (L u ) = [0, +∞). If x → xu(x) ∈ L 2 (R) in addition, every eigenvalue is negative and simple, thanks to an identity firstly found by Wu [65]. Then we introduce a generating function which encodes the entire BO hierarchy, (1.16) in definition 2.9. It provides a sequence of conservation laws controlling every Sobolev norms.
In section 3, we study the shift semigroup (S(η) * ) η≥0 acting on the Hardy space L 2 + , where S(η)f = e η f and e η (x) = e iηx . Then a weak version of Beurling-Lax theorem can be obtained by solving a linear differential equation with constant coefficients. Every N -dimensional subspace of L 2 + that is invariant under its infinitesimal generator G = i d dη η=0 + S(η) * is of the form , for some monic polynomial Q whose roots are contained in the lower half-plane C − .
In section 4, the real analytic structure and symplectic structure of the N -soliton subset U N are established at first. Then we continue the spectral analysis of the Lax operator L u , ∀u ∈ U N . L u has N simple eigenvalues λ u 1 < λ u 2 < · · · < λ u N < 0 and the Hardy space L 2 + splits as where Q u denotes the characteristic polynomial of u given by proposition 1.4 and Θ u = Q u Qu is an inner function on the upper half-plane C + . Proposition 1.4 is proved by identifying M (u) in (1.10) as the matrix of the restriction G| Hpp(Lu) associated to the spectral basis {ϕ u 1 , ϕ u 2 , · · · , ϕ u N }, where ϕ u j ∈ Ker(λ u j − L u ) such that ϕ u j L 2 = 1 and R uϕ u j > 0. The generating function H λ in (1.16) can be identified as the Borel-Cauchy transform of the spectral measure of L u associated to the vector Πu, which yields the invariance of U N under the BO flow in H ∞ (R, R). Hence (1.6) is a globally well-posed Hamiltonian system on U N . Section 5 is dedicated to completing the proof of theorem 1. The generalized angle-variables are the real parts of the diagonal elements of the matrix M (u), i.e. γ j : u ∈ U N → Re Gϕ u j , ϕ u j L 2 ∈ R and the actionvariables are I j : u ∈ U N → 2πλ u j ∈ R. Thanks to the Lax pair formulation dL(u)( is R-affine and B λ u is some skew-adjoint operator on L 2 + , we have the following formulas of Poisson brackets, which implies that Φ N : u ∈ U N → (I 1 (u), I 2 (u), · · · , I N (u); γ 1 (u), γ 2 (u), · · · , γ N (u)) ∈ Ω N × R N is a real analytic immersion. The diffeomorphism property of Φ N is given by Hadamard's global inverse theorem. The inverse spectral formula Πu = Q ′ u Qu with Q u (X) = det(X − G| Hpp(Lu) ), which is restated as formula (5.11), implies the explicit formula (1.11) of all multi-soliton solutions of the BO equation (1.1) and (5.11) provides an alternative proof of the injectivity of Φ N . Finally, we show that In appendix A, we establish the simple connectedness of U N and a covering map from U N to the manifold of N -gap potentials from their constructions without using the integrability theorems.

Related work
The BO equation has been extensively studied for nearly sixty years in the domain of partial differential equations. We refer to Saut [60] for an excellent account of these results. Besides the global well-posedness problem, various properties of its multi-soliton solutions has been investigated in details. Matsuno [41] has found the explicit expression of multi-soliton solutions of (1.1) by following the bilinear method of Hirota [26]. The multi-phase solutions (periodic multi-solitons) have been constructed by Satsuma-Ishimori [58] at first. We point out the work of Amick-Toland [2] on the characterization of 1-soliton solutions which can also be revisited by theorem 1 and proposition 1.4. In Dobrokhotov-Krichever [10], the multi-phase solutions are constructed by finite zone integration and they have also established an inversion formula for multi-phase solutions. Compared to their work, we give a geometric description of the inverse spectral transform by proving the real bi-analyticity and the symplectomorphism property of the action-angle map. Furthermore, the inverse spectral formula provides a spectral connection between the Lax operator L u and the infinitesimal generator G. The idea of introducing generating function H λ has also been used for the quantum BO equation in Nazarov-Sklyanin [46]. Their method has also been developed by Moll [44] for the classical BO equation. The asymptotic stability of soliton solutions and of solutions starting with sums of widely separated soliton profiles is obtained by Kenig-Martel [34].
Concerning the investigation of integrability for the BO equation on R besides the discovery of Lax pair formulation, we mention the pioneering work of Ablowitz-Fokas [1], Coifman-Wickerhauser [9], Kaup-Matsuno [35] and Wu [65,66] for the inverse scattering transform. In the space-periodic regime, the BO equation on the torus T admits global Birkhoff coordinates on L 2 r,0 (T) := {v ∈ L 2 (T, R) : T v = 0} in Gérard-Kappeler [19]. We refer to  to see that the Birkhoff coordinates of the BO equation on the torus can be extended to a larger Sobolev space H s r,0 (T) := {v ∈ H s (T, R) : T v = 0}, for every − 1 2 < s < 0. We point out that both Korteweg-de Vries equation on T (see ) and the defocusing cubic Schödinger equation on T (see ) admit global Birkhoff coordinates. The theory of finite-dimensional Hamiltonian system is transferred to the BO, KdV and dNLS equation on T through the submanifolds of corresponding finite-gap potentials, which are introduced to solve the periodic KdV initial problem. We refer to Matveev [42] for details.
Moreover, the cubic Szegő equation both on T (see 16,17,18]) and on R (see Pocovnicu [51,52]) admit global (generalized) action-angle coordinates on all finite-rank generic rational func-tion manifolds, denoted respectively by M(N ) T gen and M(N ) R gen . Moreover, the cubic Szegő equation both on T and on R have inverse spectral formulas which permit the Szegő flows to be expressed explicitly in terms of time-variables and initial data without using action-angle coordinates. The shift semigroup (S(η) * ) η≥0 and its infinitesimal generator G are also used in Pocovnicu [52] to establish the integrability of the cubic Szegő equation on the line.
The BO equation admits an infinite hierarchy of conservation laws controlling every H s -norm (see , Coifman-Wickerhauser [9] in the case 2s ∈ N and Talbut [62] in the case − 1 2 < s < 0 and conservation law controlling Besov norms etc.), so does the KdV equation and the NLS equation (see , Koch-Tataru [36], Faddeev-Takhtajan [11], Gérard [14] and Sun [61] etc.) Throughout this paper, the main results of each section are stated at the beginning. Their proofs are left inside the corresponding subsections.

The Lax operator
This section is dedicated to studying the Lax operator L u in the Lax pair formulation of the BO equation (1.1), discovered by Nakamura [45] and Bock-Kruskal [6]. Then we describe the location and revisit the simplicity of eigenvalues of L u . At last, we introduce a generating functional H λ which encodes the entire BO hierarchy. The equation ∂ t u = ∂ x ∇ u H λ (u) also enjoys a Lax pair structure with the same Lax operator L u . Definition 2.1. Given u ∈ L 2 (R, R), its associated Lax operator L u is an unbounded operator on L 2 + , given by L u := D − T u , where D : h ∈ H 1 + → −i∂ x h ∈ L 2 + and T u denotes the Toeplitz operator of symbol u, defined by T u : h ∈ H 1 + → Π(uh) ∈ L 2 + , where the Szegő projector Π : L 2 (R) → L 2 + is given by (1.12). We set B u := i(T |D|u − T 2 u ). Both D and T u are densely defined symmetric operators on L 2 + and T u (h) L 2 ≤ u L 2 h L ∞ , for every h ∈ H 1 + and u ∈ L 2 (R, R). Moreover, the Fourier-Plancherel transform implies that D is a self-adjoint operator on L 2 + , whose domain of definition is H 1 + . Proposition 2.2. If u ∈ L 2 (R, R), then L u is an unbounded self-adjoint operator on L 2 + , whose domain of definition is D(L u ) = H 1 + . Moreover, L u is bounded from below. The essential spectrum of L u is σ ess (L u ) = σ ess (D) = [0, +∞) and its pure point spectrum satisfies denotes the Sobolev constant.
Thanks to an identity firstly found by Wu [65] in the negative eigenvalue case, we show the simplicity of the pure point spectrum Proposition 2.3. Assume that u ∈ L 2 (R; R) and x → xu(x) ∈ L 2 (R). For every λ ∈ R and ϕ ∈ Ker(λ − L u ), we have uϕ ∈ C 1 (R) H 1 (R) and the following identity holds, Thus σ pp (L u ) ⊂ (−∞, 0) and for every λ ∈ σ pp (L u ), we have Corollary 2.4. Assume that u ∈ L 2 (R; R) and x → xu(x) ∈ L 2 (R). Then every eigenvalue of L u is When u ∈ H 2 (R, R), the Toeplitz operators T |D|u and T u are bounded both on L 2 + and on H 1 + . So B u is a bounded skew-adjoint operator both on L 2 + and on H 1 + .
Let U : t → U (t) ∈ B(L 2 + ) := B(L 2 + , L 2 + ) denote the unique solution of the following equation if u : t ∈ R → u(t) ∈ H 2 (R, R) denote the unique solution of equation (1.1). The system (2.4) is globally well-posed in B(L 2 + ), thanks to proposition 2.6, the following estimate . and a classical Cauchy theorem (see for instance lemma 7.2 of Sun [61]). Since B * u = −B u , the operator U (t) is unitary for every t ∈ R. Thus, the Lax pair formulation (2.3) of the BO equation (1.1) is equivalent to the unitary equivalence between L u(t) and L u(0) , On the one hand, the spectrum of L u is invariant under the BO flow. In particular, we have σ pp (L u(t) ) = σ pp (L u(0) ). On the other hand, there exists a sequence of conservation laws controlling every Sobolev norms H n 2 (R), n ≥ 0. Furthermore, the Lax operator in the Lax pair formulation is not unique. If f ∈ L ∞ (R) and p is a polynomial with complex coefficients, then (2.6) where N is the degree of the polynomial p. Then E n (u(t)) = E n (u(0)), for every t ∈ R. In particular, E 1 = E on H 1 2 (R, R), where the energy functional E is given by (1.6). Definition 2.9. Given u ∈ L 2 (R, R) and λ ∈ C\σ(−L u ), the C-linear transformation λ+L u is invertible in B(H 1 + , L 2 + ) and the generating function is defined by , +∞) by proposition 2.2.
Given (λ, u) ∈ X , there exists a neighbourhood of u in L 2 (R, R), denoted by V u such that the restriction Given (λ, u 0 ) ∈ X fixed, the pseudo-Hamiltonian equation associated to H λ is defined by There exists an open subset V u0 of L 2 (R, R) such that v ∈ V u0 → ∂ x |w λ (v)| 2 + w λ (v) + w λ (v) ∈ L 2 + is real analytic and u 0 ∈ V u0 . Hence (2.9) admits a local solution by Cauchy-Lipschitz theorem.
Remark 2.11. The word 'pseudo-Hamiltonian' is used here because no symplectic form has been defined on L 2 (R, R) until now. In section 4, we show that ∂ x ∇f (u) is exactly the Hamiltonian vector field of the smooth function f : U N → R with respect to the symplectic form ω on the N -soliton manifold U N defined in (4.2). Proposition 2.12. Given (λ, u 0 ) ∈ X fixed, there exists ε > 0 such that (λ, u(t)) ∈ X , for every t ∈ (−ε, ε), where u : t ∈ (−ε, +ε) → u(t) ∈ L 2 (R, R) denotes the local solution of (2.9) with initial datum u(0) = u 0 . We have i.e. (L u , B λ u ) is a Lax pair of equation (2.9).
Remark 2.13. The Toeplitz operators T w λ (v) and T w λ (v) are bounded both on L 2 + and on H 1 For every u ∈ H ∞ (R, R) and ǫ ∈ (0, Recall that E n (u) = L n u Πu, Πu L 2 , we have the following Taylor expansioñ (2.11) Proposition 2.12 then leads to a Lax pair formulation for the equations corresponding to the conservation laws in the BO hierarchy, where now u evolves according to the pseudo-Hamiltonian flow of E n = (−1) n d n dǫ n ǫ=0H ǫ . In the case n = 1, we have This section is organized as follows. In subsection 2.1, we recall some basic facts concerning unitarily equivalent self-adjoint operators on different Hilbert spaces. The subsection 2.2 is dedicated to the proofs of proposition 2.2 and 2.3. Proposition 2.8 and 2.10 that concern the conservation laws are proved in subsection 2.3. Proposition 2.7 and proposition 2.12 that indicate the Lax pair structures are proved in subsection 2.4.

Unitary equivalence
Generally, if E 1 and E 2 are two Hilbert spaces, let A be a self-adjoint operator defined on D(A) ⊂ E 1 and B be a self-adjoint operator defined on D(B) ⊂ E 2 . Both A and B have spectral decompositions Proof. If f is a bounded Borel function, ψ ∈ E 1 , consider the spectral measure of A associated to the vector ψ ∈ E 1 , denoted by µ A ψ . Similarly, we denote by µ B U ψ the spectral measure of B associated to the vector Uψ ∈ E 2 . Clearly, we have So the Borel-Cauchy transforms of these two spectral measures are the same.
Both of these two spectral measures have finite total variations : µ A ψ (R) = µ B U ψ (R) = ψ 2 E1 . Since every finite Borel measure is uniquely determined by its Borel-Cauchy transform (see Theorem 3.21 of Teschl [64] page 108), we have µ A ψ = µ B U ψ . So the restriction U| Hxx(A) : H xx (A) → H xx (B) is a linear isomorphism, for every xx ∈ {ac, sc, pp}. Finally, we use the definition of the spectral measures to obtain We may assume that f is real-valued, so that f (A) is self-adjoint. The polarization identity implies that So we obtain f (B) = Uf (A)U * in the case f is real-valued bounded Borel function. In the general case, it suffices to use f = Ref + iImf .

Spectral analysis I
In this subsection, we study the essential spectrum and discrete spectrum of the Lax operator L u by proving proposition 2.2 and 2.3. The spectral analysis of L u such that u is a multi-soliton in definition 1.1, will be continued in subsection 4.2.
Hence its Hilbert-Schmidt norm A u HS(L 2 (R * + )) ≤ K L 2 (R * Since P u is unitarily equivalent to A u , we have P u 2 HS(L 2 Then the symmetric operator T u is relatively compact with respect to D and Weyl's essential spectrum theorem (Theorem XIII.14 of Reed-Simon [54]) yields that σ ess (L u ) = σ ess (D) and L u is self-adjoint with D(L u ) = D(D) = H 1 + . An alternative proof of the self-adjointness of L u can be given by Kato-Rellich theorem (Theorem X.12 of Reed-Simon [53]) and the following estimate, for every f ∈ H 1 + , Before the proof of proposition 2.3, we recall a lemma concerning the regularity of convolutions.
Lemma 2.15. For every p ∈ (1, +∞) and m, n ∈ N, we have (2.14) For every f ∈ W m,p (R) * W n, p p−1 (R), we have lim |x|→+∞ ∂ α x f (x) = 0, for every α = 0, 1, · · · , m + n. Proof. In the case m = n = 0, it suffices use Hölder's inequality and the density argument of the Schwartz class S (R) ⊂ W m,p (R). In the case m = 0 and n = 1, recall that a continuous function whose weakderivative is continuous is of class C 1 and f, ϕ D(R) ′ ,D(R) = f * φ(0), we use the density argument of the test function class D(R) ⊂ L p (R). We conclude by induction on n ≥ 1 and m ∈ N.

Conservation laws
Proposition 2.8 and 2.10 are proved in this subsection. We begin with the following proposition.
We haveû(−ξ) =û(ξ), u = Πu + Πu and |D|u = DΠu − DΠu. Since DΠu ∈ L 2 − , we have Π(ΠuDΠu) = Π(uDΠu). Thus the following two formulas hold, Then we add them together to get the following Proof of proposition 2.8. It suffices to prove (2.7) in the case u 0 ∈ H ∞ (R, R). Then we use the density argument and the continuity of the flow map Since B u + iL 2 u is skew-adjoint, we use formula (2.20) to get the following In the case n = 1, we assume that u ∈ H 1 (R, R). Since u = Πu + Πu, |D|u = DΠu − DΠu and In the general case u ∈ H 1 2 (R, R), we use the density argument.
Proof of proposition 2.10. It suffices to prove the case u(0) ∈ H ∞ (R, R) and we use the density argument.
Then (2.21) yields that ∂ t H λ (u(t)) = 0. In the general case u(t) ∈ L 2 (R, R), we proceed as in the proof of proposition 2.8 and use the continuity of the generating functional

Lax pair formulation
In this subsection, we prove proposition 2.12 and 2.7. The Hankel operators whose symbols are in L 2 (R) L ∞ (R) will be used to calculate the commutators of Toeplitz operators. We notice that the Hankel operators are C-anti-linear and the Toeplitz operators are C-linear. For every symbol v ∈ L 2 (R) L ∞ (R), we define its associated Hankel operator to be , then H v may be an unbounded operator on L 2 + whose domain of definition contains H 1 Similarly, we have uh = Π(uh) + Π(uh) ∈ L 2 (R) and Π(uh) = Π(hΠu) = H Πu (h) ∈ L 2 + . Thus, Since Πu ∈ L 2 + and Πu ∈ L 2 − , we use Leibnitz's rule and formula (2.22) to obtain that . We use formula (2.22) and Leibnitz's Rule to obtain that The formula (2.23) and (2.28) yield that and We use formulas (2.26), (2.27), (2.29) and (2.30) to get the following formula At last, we combine formulas (2.25) and (2.31) to obtain formula (2.24).
End of the proof of proposition 2.12. Since L : Thus the Lax equation (2.10) is equivalent to identity (2.24) in lemma 2.19.
The proof of proposition 2.7 can be found in Gérard-Kappeler [19], Wu [65] etc. In order to make this paper self contained, we recall it here.
Proof of proposition 2.7. Since the Lax map L : .
In fact, u is real-valued, we haveû(−ξ) =û(ξ), u = Πu + Πu and |D|u = DΠu − DΠu. Since both T u and B u are bounded operators L 2 + → L 2 + and bounded operators H 1 for every f ∈ H 1 + , where the terms I 1 and I 2 are given by Thus equation (2.3) holds along the evolution of equation (1.1). Remark 2.20. As indicated in Gérard-Kappeler [19], there are many choices of the operator B u . We can replace B u by any operator of the form B u + P u such that P u is a skew-adjoint operator commuting with L u . For instance, we set C u := B u + iL 2 u and we obtain C u = iD 2 − 2iDT u + 2iT DΠu . So (L u , C u ) is also a Lax pair of the BO equation (1.1). The advantage of the operator

The action of the shift semigroup
In this section, we introduce the semigroup of shift operators (S(η) * ) η≥0 acting on the Hardy space L 2 + and classify all finite-dimensional translation-invariant subspaces of L 2 + .
For every η ≥ 0, we define the operator S(η) : . Every function f ∈ D(G) has bounded Hölder continuous Fourier transform by Morrey's inequality and Sobolev extension operator yields the existence off (0 + ) := lim ξ→0 +f (ξ). The operator G is densely defined and closed. The Fourier transform of Gf is given by In accordance with the Hille-Yosida theorem, we have holds for every ϕ ∈ D(G).
Proof. For every η > 0 and ϕ ∈ D(G), both S(η) * and T b are bounded operators, so we have By using Cauchy-Schwarz inequality and Fubini's theorem, we have The following scalar representation theorem of Lax [39] allows to classify all translation-invariant subspaces of the Hardy space L 2 + , which plays the same role as Beurling's theorem in the case of Hardy space on the circle (see Theorem 17.21 of Rudin [56]).
Moreover, Θ is uniquely determined up to multiplication by a complex constant of absolute value 1.
The following lemma classifies all finite-dimensional subspaces that are invariant under the semi-group (S(η) * ) η≥0 , which is a weak version of theorem 3.2.
denotes all the polynomials whose degrees are at most N −1. Q is the characteristic polynomial of the operator G| M .
The Cayley-Hamilton theorem implies that Q(i∂ ξ ) = 0 on the subspaceM . If ψ ∈M ⊂ L 2 (0, +∞), then ψ is a weak-solution of the following differential equation The differential operator Q(−D) is elliptic is on the open interval (0, +∞) in the following sense: the symbol of the principal part of Q(−D), denoted by a Q : (x, ξ) ∈ (0, +∞) × R → (−ξ) N , does not vanish except for ξ = 0. Theorem 8.12 of Rudin [57] yields that ψ is a smooth function. The solution space The uniqueness is obtained by identifying all the roots.

The manifold of multi-solitons
This section is dedicated to a geometric description of the multi-soliton subsets in definition 1.1. We give at first a polynomial characterization then a spectral characterization for the real analytic symplectic manifold of N -solitons in order to prove the global well-posedness of the BO equation with N -soliton solutions (1.6).
Recall that every N -soliton has the form u( j with x j ∈ R and η j > 0, then we have the following polynomial characterization of the N -solitons.
Moreover, each of the following three properties implies the others: There exists a unique monic polynomial Q u ∈ C N [X] whose roots are contained in the lower half- Proof. We only prove the uniqueness in (a) Since P and Q u are monic polynomials, we have P = Q u . The other assertions are consequences of u = Πu + Πu.
given by proposition 4.1 is called the characteristic polynomial of u. Its roots are denoted by The real analytic structure of U N is given in the next proposition.
Proposition 4.3. Equipped with the subspace topology of L 2 (R, R), the subset U N is a connected, real analytic, embedded submanifold of the R-Hilbert space L 2 (R, R) and dim R U N = 2N . For every u ∈ U N , its translation-scaling parameters are denoted by cl( for some x j ∈ R and η j > 0, then the tangent space to U N at u is given by (4.1) It provides the symplectic structure of the manifold U N .
Proposition 4.4. The nondegenerate real analytic 2-form ω is closed on U N . Endowed with the symplectic form ω, the real analytic manifold (U N , ω) is a symplectic manifold.
For every smooth real-valued function f : U N → R, let X f ∈ X(U N ) denote its Hamiltonian vector field, defined as follows: for every u ∈ U N and h ∈ T u (U N ), Then we have Then, we return back to spectral analysis in order to establish a spectral characterization of the manifold U N . For every monic polynomial Q ∈ C N [X] with roots in C − , we set Θ = Θ Q := Q Q ∈ Hol(C + ), where Then Θ is an inner function on the upper half-plane C + , because |Θ| ≤ 1 on C + and |Θ| = 1 on R. Recall the shift operator S(η) : , for every h ∈ L 2 + , so ΘL 2 + is a closed subspace of L 2 + that is invariant by the semigroup (S(η)) η≥0 (see also the Beurling-Lax theorem 3.2 of the complete classification of the translation-invariant subspaces of the Hardy space L 2 + ). We define K Θ to be the orthogonal complement of ΘL 2 + , thus where the infinitesimal generator G is defined in (3.2). Recall that the C-vector space C ≤N −1 [X] consists of all polynomials with complex coefficients of degree at most N − 1. So Defined on D(L u ) = H 1 + , the unbounded self-adjoint operator L u has the following spectral decomposition The following proposition gives an identification of these subspaces in the spectral decomposition (4.5).
Proposition 4.6. If u ∈ U N , then L u has exactly N simple negative eigenvalues. Let Q u denote the characteristic polynomial of the N -soliton u given in definition 4.2 and Θ u := Θ Qu = Q u Qu denote the associated inner function. Then we have the following identification, For every u ∈ U N , we have the following spectral decomposition of L u : and σ pp (L u ) = {λ u 1 , λ u 2 , · · · , λ u N } consists of all eigenvalues of L u . Proposition 2.2 yields that L u is bounded from below and − C 2 denotes the Sobolev constant. Hence the min-max principle (Theorem XIII.1 of Reed-Simon [54]) yields that where, the above supremum, F describes all subspaces of L 2 + of complex dimension n, 1 ≤ n ≤ N . When n ≥ N + 1, sup dimC F =n I(F, L u ) = inf σ ess (L u ) = 0. Proposition 2.3 and corollary 2.4 yield that there exist eigenfunctions ϕ j : u ∈ U N → ϕ u j ∈ H pp (L u ) such that for every j = 1, 2, · · · , N . Then {ϕ u 1 , ϕ u 2 , · · · , ϕ u N } is an orthonormal basis of the subspace H pp (L u ). We have the following result.
Proposition 4.7. For every j = 1, 2, · · · , N , the j th eigenvalue λ j : u ∈ U N → λ u j ∈ R is real analytic. We refer to proposition 4.14 and formula (4.4) to see that the subspace H pp (L u ) ⊂ D(G) is invariant by G. The matrix representation of G| Hpp(Lu) with respect to the orthonormal basis {ϕ u 1 , ϕ u 2 , · · · , ϕ u N } is given in proposition 5.4. Then the following theorem gives the spectral characterization for N -solitons.

Differential structure
The construction of real analytic structure and symplectic structure of U N is divided into three steps. Firstly, we describe the complex structure of Π(U N ). Then the Hermitian metric H for the complex manifold Π(U N ) is introduced in (4.15) and we establish a real analytic diffeomorphism between U N and Π(U N ). The third step is to prove dω = 0 on U N . Since ω = −Π * (ImH), (Π(U N ), H) is a Kähler manifold.
Step I. The Viète map V : (β 1 , β 2 , · · · , β N ) ∈ C N → (a 0 , a 1 , · · · , a N −1 ) ∈ C N is defined as follows is a connected Kähler manifold of complex dimension N . Lemma 4.10. Equipped with the subspace topology of L 2 + , the subset Π(U N ) is a connected topological manifold of complex dimension N and it has a unique complex analytic structure making it into an embedded submanifold of the C-Hilbert space L 2 + . For every u ∈ U N , its translation-scaling parameters are denoted by cl( for some x j ∈ R and η j > 0, then the tangent space to Π(U N ) at Πu is given by (4.12) Proof. We define Γ N : a = (a 0 , a 1 , · · · , a N −1 ) The surjectivity of Γ N is given by the definition of U N . Since the monic polynomial Q is uniquely determined by u ∈ U N , the map Γ N is injective.
is a complex analytic immersion. We claim that Γ N is a topological embedding.
Step II. Given u ∈ U N , the Hermitian metric H Πu is defined as follows We consider the R-linear isomorphism between the Hilbert spaces Then Π • 2Re = Id L 2 + and 2Re • Π = Id L 2 (R,R) and u L 2 = √ 2 Πu L 2 . Then U N = 2Re • Π(U N ) is a real analytic manifold of real dimension 2N . Furthermore we have f u j = 2Reh u j , g u j = 2iReh u j and is an R-linear isomorphism. Since H is Hermitian, the 2-form ω = −Π * (ImH) is nondegenerate on U N .
Step III. We set E : The nondegenerate 2-form ω can be extended to a 2-covector of the subspace T . Recall that If h ∈ T , then we haveĥ(0) = 0 andĥ ∈ H 1 (R). Hence the Hardy's inequality (see Brezis [7], Bahouri-Chemin-Danchin [3] etc.) yields that so the 2-covector ω ∈ Λ 2 (T * ) is well defined and ω u (h 1 , h 2 ) = ω(h 1 , h 2 ). For every smooth vector field X ∈ X(U N ), let X ω ∈ Ω 1 (U N ) denote the interior multiplication by X, i.e. (X ω)(Y ) = ω(X, Y ), for every Y ∈ X(U N ). We shall prove that dω = 0 on U N by using Cartan's formula: Proof of proposition 4.4. For any smooth vector field X ∈ X(U N ), let φ denote the smooth maximal flow of X. If t is sufficiently close to 0, then φ t : u ∈ U N → φ(t, u) ∈ U N is a local diffeomorphism by the fundamental theorem on flows (see Theorem 9.12 of Lee [40]). For every u ∈ U N , h 1 , h 2 ∈ T u (U N ), we compute the Lie derivative of ω with respect to X, Since lim t→0 We choose (V, x i ) a smooth local chart for U N such that u ∈ V and the tangent vector h k has the coordinate expression h k = 2N j=1 h (j) k ∂ ∂x j u , for some h (j) k ∈ R, j = 1, 2 · · · , 2N and k = 1, 2. The tangent vector h k can be identified as some locally constant vector field Y k ∈ X(U N ) defined by Then the vector field [Y 1 , Y 2 ] vanishes in the open subset V . The exterior derivative of the 1-form β = X ω is computed as dβ(Y 1 , Then Cartan's formula (4.19) yields that X (dω) = 0. Since X ∈ X(U N ) is arbitrary, we have dω = 0. As a consequence, the real analytic 2-form ω : u ∈ U N → ω ∈ Λ 2 (T * ) is a symplectic form.

Spectral analysis II
We continue to study the spectrum of the Lax operator L u introduced in definition 2.1. The general cases u ∈ L 2 (R, R) and u ∈ L 2 (R, (1 + x 2 )dx) have been studied in subsection 2.2. We restrict our study to the case u ∈ U N in this subsection. Let Q = Q u denote the characteristic polynomial of u and Θ := Q Q , K Θ = (ΘL 2 + ) ⊥ . Since L u is an unbounded self-adjoint operator of L 2 + , we have the following We shall at first identify those subspaces by proving proposition 4.6 and formula (4.7). Then we turn to study the real analyticity of each eigenvalue λ j : u ∈ U N → λ u j ∈ R.
Proof of proposition 4.6. The first step is to prove . In fact, for every h ∈ L 2 + and Since Recall that L u = L * u , so we have L u (K Θ ) ⊂ K Θ . Since dim C K Θ = N , corollary 2.4 yields that the Hermitian matrix L u|KΘ has exactly N distinct eigenvalues. Hence K Θ ⊂ H pp (L u ).
Before proving the real analyticity of each eigenvalue, we show its continuity at first. Lemma 4.11. For every j = 1, 2, · · · , N , the j th eigenvalue λ j : u ∈ U N → λ u j ∈ R is Lipschitz continuous on every compact subset of U N .
Proof. For every f ∈ H 1 (R), the Sobolev embedding Given j = 1, 2, · · · , N and a subspace F ⊂ L 2 + with complex dimension j − 1, we choose Then L u h, h L 2 = j k=1 |h k | 2 λ u k ≤ λ u j < 0, because λ u k < λ u k+1 . We have the following estimate So estimates (4.21) and (4.22) Since F is arbitrary, the max-min formula (4.8) implies that Every compact subset K ⊂ U N is bounded in L 2 (R, R). Hence u ∈ K → λ u j ∈ R is Lipschitz continuous.
It suffices to show that P j u | Hac(Lu) = 0. In fact the operator P j u = g λ u j (L u ) is self-adjoint by Theorem VIII.6 of Reed-Simon [55], where the real-valued bounded Borel function g λ : R → R is given by a.e. on R, for every λ ∈ R. Since P j u (H pp (L u )) ⊂ Cϕ u j ⊂ H pp (L u ), we have P j u (H ac (L u )) ⊂ H ac (L u ). Let µ ψ = µ Lu ψ denote the spectral measure of L u associated to the function ψ ∈ H ac (L u ), whose support is included in [0, +∞) by formula (4.7), we have For every fixed j = 1, 2, · · · N , we have λ u j = Tr(L u • P j u ). Since every eigenvalue 3 , then λ v j ∈ D(λ u j , ǫ)\D(λ u k , ǫ 0 ), for every v ∈ V and k = j. For example, in the next picture, the dashed circles denote respectively C (λ u j , ǫ 0 ) and C (λ u k , ǫ 0 ); the smaller circles denote respectively C (λ u j , ǫ) and C (λ u k , ǫ) with j < k. The segments inside small circles denote the possible positions of λ v j and λ v k .
Then σ(L v ) D(λ u j , ǫ 0 ) = {λ v j } and C (λ u j , ǫ) is a closed path in D(λ u j , ǫ 0 ) with respect to which λ v j has winding number 1. Thus, denotes the open subset of all bijective bounded C-linear transformations H 1 + → L 2 + , we have the real analyticity of the following map Hence the maps P j : v ∈ V → P j v ∈ B(L 2 + , H 1 + ) and λ j : v ∈ V → Tr(L v • P j v ) ∈ R are both real analytic by composing (4.24) and (4.25).
, where Q u denotes the characteristic polynomial of u ∈ U N whose zeros are contained in C − , so H pp (L u ) ⊂ D(G) is given by (3.7). We have the following consequence.
Proof. For every u, v ∈ U N , we have P j v ϕ u j = ϕ u j , ϕ v j L 2 ϕ v j . Since the Riesz projector P j : v ∈ U N → P j v ∈ B(L 2 + , H 1 + ) is real analytic in the proof of proposition 4.7 and P j u ϕ u j L 2 = 1, there exists a neighbourhood of u, denoted by V, such that P j v ϕ u j L 2 > 1 2 for every v ∈ V and P j : v ∈ V → P j v ∈ B(L 2 + , H 1 + ) can be expressed by power series. Then

Characterization theorem
The characterization theorem 4.8 is proved in this subsection. The direct sense is given by proposition 4.1 and proposition 4.6. Before proving the converse sense of theorem 4.8, we need the following lemmas to prove the invariance of H pp (L u ) under G, if u ∈ L 2 (R, (1 + x 2 )dx) is real-valued, Πu ∈ H pp (L u ) and dim C H pp (L u ) = N ≥ 1. The following lemma gives another version of formula of commutators (see also lemma 3.1).

The stability under the Benjamin-Ono flow
Finally we prove proposition 4.9 in this subsection. Two lemmas will be proved at first in order to obtain the invariance of the property x → xu(x) ∈ L 2 (R) under the BO flow.
Remark 4.16. This result can be strengthened by replacing the assumption u 0 ∈ H 2 (R, R) by a weaker assumption u 0 ∈ H 3 2 + (R, R) = s> 3 2 H s (R, R), because one can construct the conservation law of BO equation controlling the H s -norm for every s > − 1 2 by using the method of perturbation of determinants. We refer to Talbut [62] to see details and Killip-Vişan-Zhang [37] for the KdV and the NLS cases (see also ). It suffices to use lemma 4.15 to prove proposition 4.9.
Then Young's convolution inequality yields that [|D|, χ]g L 2 ∂ x χ| * |ĝ| L 2 ∂ x χ L 1 g L 2 . In order to estimate ∂ x χ L 1 , we divide the integral as two parts. Wet set Then we use the same idea to estimate ∂ 2 Finally, we add them together to get the second estimate in (4.30).
Now we prove the invariance of the property x → xu(x) ∈ L 2 (R) is invariant under the BO flow.
Since the generating function λ ∈ C\σ(−L u ) → H λ (u) ∈ C is the Borel-Cauchy transform of the spectral measure of L u , the invariance of the N −soliton manifold U N under BO flow is obtained by using the inverse spectral transform.
End of the proof of proposition 4.9. If u 0 ∈ U N ⊂ H ∞ (R, R) L 2 (R, x 2 dx), let u = u(t, x) be the unique solution of the BO equation (1.1) with initial datum u(0) = u 0 , then u(t) ∈ H ∞ (R, R) L 2 (R, x 2 dx) by proposition 2.5 and lemma 4.15. Recall the generating function H λ : u ∈ L 2 (R, R) → R defined as where µ Lu ψ denotes the spectral measure of L u associated to the function ψ ∈ L 2 + . So the holomorphic function λ ∈ C\σ(−L u ) → H λ u is the Borel-Cauchy transform of the positive Borel measure m u . We recall that the total variation m u (R) = Πu 2 L 2 is a conservation law of the BO equation (1.1) by proposition 2.8 and formula (2.20). Every finite Borel measure is uniquely determined by its Borel-Cauchy transform (see Theorem 3.21 of Teschl [64] page 108), precisely for every a ≤ b real numbers, we use Stieltjes inversion formula to obtain that For every t ∈ R, proposition 2.10 yields that ) by proposition 4.6 and there exist c 1 , c 2 , · · · , c N ∈ R + such that The spectral measure µ is purely point, so Π[u(t)] ∈ H pp (L u(t) ) for every t ∈ R. The Lax pair structure yields the unitary equivalence between L u(t) and L u(0) . So dim C H pp (L u(t) ) = dim C H pp (L u(0) ) = N is given by proposition 2.14. We conclude by theorem 4.8.

The generalized action-angle coordinates
In this section, we construct the (generalized) action-angle coordinates Φ N in theorem 1 of the BO equation (1.6) with solutions in the real analytic symplectic manifold (U N , ω) of real dimension 2N given in proposition 4.3. The goal of this section is to establish the diffeomorphism property and the symplectomorphism property of Φ N .
Recall that the BO equation with N -soliton solutions is identified as a globally well-posed Hamiltonian system reading as whose energy functional E(u) = L u Πu, Πu L 2 is well defined on U N and the Hamiltonian vector field X E : u ∈ U N → X E (u) = ∂ x (|D|u−u 2 ) ∈ T u (U N ) coincides with the definition (4.3). The Poisson bracket of two smooth functions f, g : U N → R is given by Given u ∈ U N , proposition 4.6 yields that there exist λ u 1 < λ u 2 < · · · < λ u N < 0 and ϕ u j ∈ Ker(λ u j − L u ) ⊂ D(G) such that ϕ u j L 2 = 1 and u, ϕ u j L 2 = 2π|λ u j |, thanks to the spectral analysis in subsection 4.2.
Definition 5.1. For every j = 1, 2, · · · , N , the map I j : u ∈ U N → 2πλ u j ∈ R is called the j th action. The map γ j : u ∈ U N → Re Gϕ u j , ϕ u j L 2 ∈ R is called the j th (generalized) angle.
Theorem 5.2. The map Φ N has following properties: The family (X I1 , X I2 , · · · , X IN ; X γ1 , X γ2 , · · · , X γN ) is linearly independent in X(U N ) and we have The assertion (c) is obtained by a direct calculus: Πu = N j=1 Πu, ϕ u j L 2 ϕ u j , formula (4.9) yields that Thus theorem 5.2 introduces (generalized) action-angle coordinates of the BO equation (5.1) in the sense of (1.8), i.e. {I j , E}(u) = 0 and {γ j , E}(u) = 2λ u j , for every u ∈ U N . This section is organized as follows. The matrix associated to G| Hpp(Lu) is expressed in terms of actions and angles in subsection 5.1. Then the injectivity of Φ N is given by inversion formulas in subsection 5.2. In subsection 5.3, the Poisson brackets of actions and angles are used to show the local diffeomorphism property of Φ N . The surjectivity of Φ N is obtained by Hadamard's global inverse theorem in subsection 5.4. Finally, we use subsection 5.5 and subsection 5.6 to prove that Φ N : (U N , ω) → (Ω N × R N , ν) preserves the symplectic structure.

The associated matrix
We continue to study the infinitesimal generator G defined in (3.2) when restricted to the invariant subspace H pp (L u ) with complex dimension N . Let M (u) = (M kj (u)) 1≤k,j≤N denote the matrix associated to the operator G| Hpp(Lu) with respect to the basis {ϕ u 1 , ϕ u 2 , · · · , ϕ u N }. Then we state a general linear algebra lemma that describes the location of eigenvalues of the matrix M (u).
Proposition 5.4. For every u ∈ U N , the coefficients of matrix M (u) = (M kj (u)) 1≤k,j≤N are given by Proof. Since L u is a self-adjoint operator on L 2 + and H pp (L u ) ⊂ D(G), we have Since formulas (2.15) and (4.9) imply that −λ u j ϕ u j (0) = uϕ u j (0) = 2π|λ u j |, we use (4.26) to obtain In the case k = j, we use Plancherel formula and integration by parts to calculate Thus we have G * f, g L 2 = Gf, g L 2 + i 2πf (0 + )ĝ(0 + ), for every f, g ∈ H pp (L u ). Then We conclude by γ j (u) = Re℧ j (u) = Gϕ u j , ϕ u j L 2 defined in corollary 4.12.
Then we state a linear algebra lemma that describe the location of spectrum of all matrices of the form defined as (5.5).
So we have Imµ ≤ 0. Assume that µ ∈ R, then formula (5.7) yields that V ⊥ V λ . Moreover, we have We set D λ ∈ C N ×N to be the diagonal matrix whose diagonal Then we have the following formula contradicts the fact that V = 0. Consequently, we have µ ∈ C − .

Inverse spectral formulas
The injectivity of Φ N is proved in this subsection by using inverse spectral formulas. The following lemma describes the relation between the Fourier transform of an eigenfunction ϕ ∈ H pp (L u ) and the inner function associated to u defined by Θ u = Q u Qu with Q u (x) = det(x − M (u)).
At last we show the equivalence between the inversion formulas (4.10) and (5.11).
Revisiting formula (4.10). For every k, j = 1, 2, · · · , N , let K u kj (x) denote the (N − 1) × (N − 1) submatrix obtained by deleting the k th column and j th row of the matrix M (u) − x, for every x ∈ R. So the inversion formula (5.11) and the Cramer's rule imply that by formula (5.5). Using expansion by minors, we have Finally, let Q denote the characteristic polynomial of the operator G| H pp(Lu) , so

Poisson brackets
In this subsection, the Poisson bracket defined in (5.2) is generalized in order to obtain the first two formulas of (5.4). It can be defined between a smooth function from U N to an arbitrary Banach space and another smooth function from U N to R.
The N-soliton subset (U N , ω) is a real analytic symplectic manifold of real dimension 2N , where For every smooth function f : U N → R, its Hamiltonian vector field X f ∈ X(U N ) is given by (4.3).
Since ϕ u j ∈ Ker(λ u j − L u ) and ϕ u j L 2 = 1 by the definition in (4.9), we have So there exists r ∈ R such that B λ u ϕ u j − {H λ , ϕ j }(u) = irϕ u j because Ker(λ u j − L u ) = Cϕ u j by corollary 2.4 and formula (5.21). Recall that B λ u is skew-adjoint and γ j = Re Gϕ u j , ϕ u j L 2 , we have Furthermore, for every (λ, u) ∈ Y, formula (3.4) implies that [G, T w λ (u) ] = 0 and , we replace f by ϕ u j in formula (5.22) to obtain the following Remark 5.12. Recall thatH ǫ = 1 ǫ H 1 ǫ andB ǫ,u := 1 ǫ B 1 ǫ u for every (ǫ −1 , u) ∈ Y. In general, the identity holds for every conservation law E n = (−1) n d n dǫ n ǫ=0H ǫ in the BO hierarchy. Corollary 5.13. For every j, k = 1, 2, · · · , N , we have Proof. Given u ∈ U N , for every λ > then (λ, u) ∈ Y, then (5.16) and (5.19) imply that , for every j = 1, 2, · · · , N . The uniqueness of analytic continuation yields that the following formula holds for every z ∈ C\R, Recall that the actions I j : u ∈ U N → 2πλ u j and the generalized angles γ j : u ∈ U N → Re Gϕ u j , ϕ u j L 2 are both real analytic functions by proposition 4.7 and corollary 4.12.
Proposition 5.14. For every u ∈ U N , the family of differentials Formula of Poisson brackets (5.23) yields that for every j, k = 1, 2, · · · , N , we have We replace h by X I k (u) in (5.24) to obtain that b k = 0, ∀k = 1, 2, · · · , N . Then set h = X γ k (u) As a consequence, Φ N : U N → Ω N × R N is a local diffeomorphism. Moreover, since all the actions (I j ) 1≤j≤N are in evolution by (5.23) and the differentials (dI j (u)) 1≤j≤N are linearly independent for every u ∈ U N , for every r = (r 1 , r 2 , · · · , r N ) ∈ Ω N , the level set where r = (r 1 , r 2 , · · · , r N ) is a smooth Lagrangian submanifold of U N and L r is invariant under the Hamiltonian flow of I j , for every j = 1, 2, · · · , N , by the Liouville-Arnold theorem (see Theorem 5.5.21 of Katok-Hasselblatt [32], see also Fiorani-Giachetta-Sardanashvily [12] and Fiorani-Sardanashvily [13] for the non-compact invariant manifold case).

The diffeomorphism property
This subsection is dedicated to proving the real bi-analyticity of Φ N : U N → Ω N × R N . It remains to show the surjectivity. Its proof is based on Hadamard's global inverse theorem 5.18.
Proposition 5.16. The map Φ N : U N → Ω N × R N is bijective and both Φ N and its inverse Φ −1 N are real analytic.
Proof. The analyticity of Φ N is given by proposition 4.7 and corollary 4.12. The injectivity is given by corollary 5.10. Proposition 5.14 yields that Φ N : U N → Ω N × R N is a local diffeomorphism by inverse function theorem for manifolds. So Φ N is an open map. Since every proper continuous map to locally compact space is closed, Φ N is also a closed map by lemma 5.15. Since the target space Ω N × R N is connected, we have Φ N (U N ) = Ω N × R N and Φ N : U N → Ω N × R N is a real analytic diffeomorphism.
Remark 5.17. We establish the relation between Φ N : U N → Ω N × R N and Γ N : V(C N − ) → Π(U N ) introduced in proposition 4.10. We set M : Ω N × R N → C N ×N to be the matrix-valued real analytic function M(η 1 , η 2 , · · · , η N ; θ 1 , θ 2 , · · · , θ N ) = (M kj ) 1≤k,j≤N with coefficients defined as Then, we set C : M ∈ C N ×N → (a 0 , a 1 , · · · , a N −1 ) ∈ C N such that Since (−1) n−j a j = Tr(Λ n−j M ) is the sum of all principle minors of M of size (N − j) × (N − j), for every j = 1, 2, · · · , N , the map C is real analytic on C N ×N and C • M(Ω N × R N ) ⊂ V(C N − ) by lemma 5.5, where V denotes the Viète map defined as (4.11). In lemma 4.10, we have shown that the map is biholomorphic, where the polynomial Q is defined as (5.25). We conclude by the following identity The smooth manifolds Π(U N ) and V(C N − ) are both diffeomorphic to the convex open subset Ω N × R N , so they are simply connected (see also proposition A.5). At last, we recall Hadamard's global inverse theorem.
Theorem 5.18. Suppose X and Y are connected smooth manifolds, then every proper local diffeomorphism F : X → Y is surjective. If Y is simply connected in addition, then every proper local diffeomorphism F : X → Y is a diffeomorphism.
Proof. For the surjectivity, see Nijenhuis-Richardson [47] and the proof of proposition 5.16. If the target space is simply connected, see Gordon [23] for the injectivity.
Remark 5.19. Since the target space Ω N × R N is convex, there is another way to show the injectivity of Φ N without using the inversion formulas in subsection 5.2. It suffices to use the simple connectedness of Ω N × R N and Hadamard's global inverse theorem 5.18.

A Lagrangian submanifold
In general, the symplectomorphism property of Φ N is equivalent to its Poisson bracket characterization (5.4), which will be proved in proposition 5.24. The first two formulas of (5.4) given in corollary 5.13, lead us to focusing on the study of a special Lagrangian submanifold of U N , denoted by Λ N := {u ∈ U N : γ j (u) = 0, ∀j = 1, 2, · · · , N }, (5.27) where the generalized angles γ j : u ∈ U N → Re Gϕ u j , ϕ u j L 2 are defined in (5.1). A characterization lemma of Λ N is given at first.
Lemma 5.21. The level set Λ N is a real analytic Lagrangian submanifold of (U N , ω).
Proof. The map γ : u ∈ U N → (γ 1 (u), γ 2 (u), · · · , γ N (u)) ∈ R N is a real analytic submersion by proposition 5.14. So the level set Λ N is a properly embedded real analytic submanifold of U N and dim R Λ N = N . The classification of the tangent space T u (U N ) is given by formula (4.1). If u(x) = N j=1 2ηj x 2 +η 2 j , for some η j > 0, every tangent vector h ∈ Λ N is an even function by lemma 5.20. Soĥ is real valued and we have (5.28) We have (f u j ) ∧ (ξ) = −2π|ξ|e −ηj |ξ| . Then by definition of ω, we have Since the symplectic form ω is real-valued, we have ω u (h 1 , h 2 ) = 0, for every

The symplectomorphism property
Finally, we prove the assertion (b) in theorem 5.2, i.e. the map Φ N : and We set Ψ N = Φ −1 N : Ω N × R N → U N , let Ψ * N ω denote the pullback of the symplectic form ω by Ψ N , i.e. for every p = (r 1 , r 2 , · · · , r N ; The goal is to prove that Recall that the coordinate vectors ∂ ∂r 1 p , ∂ ∂r 2 p , · · · , ∂ ∂r N p ; ∂ ∂α 1 p , ∂ ∂α 2 p , · · · , ∂ ∂α N p form a basis for the tangent space T p (Ω N × R N ). We have the following lemma.
Proof. The proof is divided into three steps. The first step is to prove that for every p ∈ Ω N × R N and every V ∈ T p (Ω N × R N ),ν p ( ∂ ∂α l p , V ) = 0, ∀l = 1, 2, · · · , N.
In fact, let u = Ψ N (p) ∈ U N and p = (r 1 , r 2 , · · · , r N ; α 1 , α 2 , · · · , α N ), so r l = r l (p) = I l • Ψ N (p). Then On the other hand, Sinceν is a smooth 2-form on Ω N × R N , we havẽ for some smooth functions a jk , b jk , c jk ∈ C ∞ (Ω N × R N ), 1 ≤ j < k ≤ N . The second step is to prove Then, let l ∈ {2, · · · , N } be fixed, for every 1 ≤ j < k ≤ N , we have It remains to show that c jk depends on r 1 , r 2 , · · · , r N , for every 1 ≤ j < k ≤ N . The symplectic form ω is closed by proposition 4.4 and ν = dκ is exact, The exterior derivative ofν = 1≤j<k≤N c jk dr j ∧ dr k is computed as following ∂c jk ∂α l dα l ∧ dr j ∧ dr k + ∂c jk ∂r l dr l ∧ dr j ∧ dr k .

A Appendices
We establish several topological properties of the N -soliton manifold U N without using the action-angle map Φ N : U N → Ω N × R N . The Viète map V : (β 1 , β 2 , · · · , β N ) ∈ C N → (a 0 , a 1 , · · · , a N −1 ) ∈ C N is defined by where C + N [X] consists of all monic polynomial Q ∈ C[X] of degree N , whose roots are contained in the annulus A := {z ∈ C : |z| > 1}. The fundamental group of U T N is (Z, +). Remark A.3. The real analytic symplectic manifold U T N is mapped real bi-analytically onto C N −1 × C * by the restriction of the Birkhoff map constructed in Gérard-Kappeler [19]. The union of all finite gap potentials N ≥0 U T N is dense in L 2 r,0 (T) = {v ∈ L 2 (T, R) : T v = 0}. However N ≥1 U N is not dense in L 2 (R, (1 + x 2 )dx). We refer to Coifman-Wickerhauser [9] to see solutions with sufficiently small initial data and the case of non-existence of rapidly decreasing solitons.
The simple connectedness of U N is proved in subsection A.1. Then we establish a real analytic covering map U N → U T N in subsection A.2.

A.1 The simple connectedness of U N
Thanks to the biholomorphical equivalence between the Kähler manifolds Π(U N ) and V(C N − ) established in lemma 4.10, it suffices to prove the simple connectedness of the subset V(C N − ), where V denotes the Viète map defined by (A.1). Since every fiber of the Viète map is invariant under the permutation of components, we introduce the following group action. Equipped with the discrete topology, the symmetric group S N acts continuously on C N by permuting the components of every vector: σ : (β 0 , β 1 , · · · , β N −1 ) ∈ C N → (β σ(0) , β σ(1) , · · · , β σ(N −1) ) ∈ C N , ∀σ ∈ S N . We set ∆ := {(β, β, · · · , β) ∈ C N : ∀β ∈ C}. The goal of this subsection is to prove the following result. The total space P is equipped with the subspace topology of the product space A × E and projections onto the first factor and onto the second factor are denoted respectively by p : (β, e) ∈ P → β ∈ A, W : (β, e) ∈ P → e ∈ E. (A.5) Both p and W are continuous functions on P and the following diagram commutes.
We claim two properties concerning the projections p and W.
i. W : P → E is an open quotient map and p : P → A is a covering map whose model fiber is F. ii. Equipped with the discrete topology, the symmetric group S N acts continuously on P by permuting components of the first factor σ : (β, e) ∈ P → (σ(β), e) ∈ P, ∀σ ∈ S N , where σ ∈ GL N (C) is defined by (A.3). Hence the quotient map W : P → E is closed.
Thanks to the simple connectedness of the base space A, the covering space P is the disjoint union of its connected components (A k ) k∈F and the restriction of the covering map p| A k : A k → A is a homeomorphism. Since P is locally path-connected, every component A k is both open and closed, then W| A k : A k → E an open closed quotient map. So is the lift g k := W| A k • (p| A k ) −1 : A → E. Note that π • g k = V and S N stabilizes every element of ∆. We choose β ∈ A ∆ and b := V(β). Since the fiber V −1 (b) = {β} is a singleton, so is the fiber π −1 (b). Hence |F| = 1 and the universal covering map p : E → B is a homeomorphism. So B is simply connected.
Remark A.6. Let F be a closed submanifold of a smooth connected manifold M without boundary of finite dimension. If dim R M −dim R F ≥ 3, then the inclusion map i : M \F → M induces an isomorphism between the fundamental groups i * : π 1 (M \F, x) → π 1 (M, x), for every x ∈ M \F (see Théorème 2.3 in P.146 of Godbillon [22]). Note that the closed submanifold ∆ ⊂ C N has real dimension 2. When N ≥ 3, the condition A ∆ = ∅ cannot be deduced by the other three conditions in the hypothesis of proposition A.5: A is open, simply connected and stable by S N .
As a consequence, V(C N − ) is open and simply connected because C N − is an open convex subset of C N which is stable under the symmetric group S N and ∆ C N − = {(z, z, · · · , z) ∈ C N : Imz < 0}. Together with lemma 4.10, we finish the proof of proposition A.1.

A.2 Covering manifold
The Szegő projector on L 2 (T, C) is given by Π T v(x) = n≥0 v n e inx , for every v ∈ L 2 (T, C) such that v(x) = n∈Z v n e inx with v n = 1 2π 2π 0 v(x)e −inx dx. Equipped with the subspace topology of Π T (L 2 (T, C)) and the Hermitian form the subset Π T (U T N ) is a Kähler manifold, which is mapped biholomorphically onto V(A N ) with A = C\D(0, 1) = {z ∈ C : |z| > 1} in Gérard-Kappeler [19].
Proposition A.7. There exists a covering map π : V(C N − ) → V(A N ).