Complete Integrability of the Benjamin–Ono Equation on the Multi-Soliton Manifolds

Commun. Math. Phys. 383, 1051–1092 (2021) Communications in Digital Object Identiﬁer (DOI) https://doi.org/10.1007/s00220-021-03996-1 Mathematical Physics

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

Ruoci Sun Institute for Analysis, Karlsruhe Institute of Technology (KIT), 76131 Karlsruhe, Germany. E-mail: [email protected]

Received: 20 April 2020 / Accepted: 15 January 2021 Published online: 26 March 2021 – © The Author(s) 2021

Abstract: 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 UN . We construct generalized action–angle coordinates which establish a real analytic 2N symplectomorphism from UN onto some open convex subset of R and allow to solve the equation by quadrature for any such initial datum. As a consequence, UN is the universal covering of the manifold of N-gap potentials for the BO equation on the torus as described by Gérard–Kappeler (Commun Pure Appl Math, 2020. https://doi.org/10. 1002/cpa.21896. arXiv:1905.01849). The global well-posedness of the BO equation on UN is given by a polynomial characterization and a spectral characterization of the manifold UN . 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. The inverse spectral formula of an N-soliton provides a spectral connection between the Lax operator and the inﬁnitesimal generator of the very shift semigroup.

Contents

1. Introduction ...... 1052 1.1 Notation ...... 1056 1.2 Organization of the paper ...... 1058 1.3 Related work ...... 1061 2. The Lax Operator ...... 1062 2.1 Spectral analysis I ...... 1062 2.2 Lax pair formulation ...... 1065 2.3 The generating functional ...... 1067 3. The Action of the Shift Semigroup ...... 1070 4. The Manifold of Multi-solitons ...... 1073 4.1 Differential structure and symplectic structure ...... 1073 1052 R. Sun

4.2 Spectral analysis II ...... 1074 4.3 Characterization theorem ...... 1078 4.4 The invariance under the Benjamin–Ono ﬂow ...... 1079 5. The Generalized Action–Angle Coordinates ...... 1082 5.1 The inverse spectral matrix ...... 1083 5.2 Poisson brackets ...... 1084 5.3 The diffeomorphism property ...... 1087 5.4 A Lagrangian submanifold ...... 1087 5.5 The symplectomorphism property ...... 1088 6. Asymptotic Approximation ...... 1090

1. Introduction

The Benjamin–Ono (BO) equation on the line reads as

∂ = ∂2 − ∂ ( 2), ( , ) ∈ R × R, t u H x u x u t x (1.1) where u is real-valued and H =−isign(D) : L2(R) → L2(R) denotes the Hilbert transform, D =−i∂x ,

Hf (ξ) =−isign(ξ) fˆ(ξ), ∀ f ∈ L2(R). (1.2) sign(±ξ) =±1, for all ξ>0 and sign(0) = 0, fˆ ∈ L2(R) denotes the Fourier– Plancherel transform of f ∈ L2(R). We adopt the convention L p(R) = L p(R, C).ItsR- subspace consisting of all real-valued L p-functions is specially emphasized as L p(R, R) 2 2 throughout this paper. Equipped with the inner product ( f, g) ∈ L (R) × L (R) → , = ( ) ( ) ∈ C 2(R) C f g L2 R f x g x dx , L is a -Hilbert space. Derived by Benjamin [3] and Ono [17], the BO equation (1.1) describes the evolution of weakly nonlinear internal long waves in a two-layer ﬂuid. Equation (1.1) is globally well-posed in every Sobolev space H s(R, R),seeTao[23]fors ≥ 1, see Ionescu–Kenig [10]fors ≥ 0, etc. On appropriate Sobolev spaces, equation (1.1) can be written in Hamiltonian form 1 1 3 ∂t u = ∂x ∇u E(u), E(u) = |D|u, u − 1 1 − u , (1.3) 2 H 2 ,H 2 3 R

2 where ∇u E(u) denotes the L (R)-gradient of E, ∂x is the Gardner–Faddeev–Zakharov Poisson structure and X E (u) = ∂x ∇u E(u) is the Hamiltonian vector ﬁeld of E with ∂ ∂ =−∂∗ respect to the Poisson structure x . Since x x is an unbounded operator on L2(R, R) with domain H 1(R, R) and range given by W := ∂ 1(R, R) ={ ∈ 2(R, R) : |ˆu(ξ)|2 ξ< ∞}, x H u L |ξ|2 d + (1.4) R

∂−1 : W → 1(R, R) W ω ∈ its inverse x H is a symplectic structure on . A 2-covector 2(W∗) ω( , ) = ,∂−1 ∀ , ∈ W is deﬁned by h1 h2 h1 x h2 L2 , h1 h2 . Under appropriate conditions on the functionals F and G, ω is the symplectic form corresponding to the Gardner bracket, which is deﬁned by { , }( ) := ∂ ∇ ( ), ∇ ( ) . F G u x u F u u G u L2 (1.5) Complete integrability of the Benjamin–Ono equation on the multi-soliton manifolds 1053

The goal of this paper is to show the complete integrability of equation (1.1) when restricted to every multi-soliton manifold. Recall the scaling and translation invariances of equation (1.1): if u = u(t, x) is a solution, so is the function uc,y : (t, x) → cu(c2t, 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

HR + R − R2 = 0, R(x)>0 (1.6) such that u(t, x) = Rc(x − y −ct), where Rc(x) = cR(cx),forsomec > 0 and y ∈ R. In [2], Amick and Toland have shown that the unique (up to translation) solution of (1.6) is given by R( ) = 2 , ∀ ∈ R. x 2 x (1.7) 1+x Deﬁnition 1.1. For any positive integer N ∈ N+ := Z (0, +∞),thesetUN is deﬁned as follows, N 2 UN := {u ∈ L (R, R) : u(x) = c j R(c j (x − x j )), c j > 0, x j ∈ R, j=1 ∀1 ≤ j ≤ N}. (1.8)

A function u ∈ UN is called an N-soliton of the BO equation (1.1). The set of translation– ( ) := { − −1 , − −1 ,..., − −1 } scaling parameters of u is given by P u x1 c1 i x2 c2 i xN cN i and m(z) denotes the multiplicity of z ∈ P(u) in the expression of u in (1.8). As − ( ) a consequence, u(x) = 2m z Imz and a polynomial characterization of z∈P(u) (x−Rez)2+(Imz)2 each N-soliton is given as follows, ( ) ( ) ( ) u(x) = Im 2m z =−2Im Qu x , Q (X) := (X − z)m z , (1.9) z−x Qu (x) u z∈P(u) z∈P(u) where Qu ∈ C[X] is called the characteristic polynomial of u, ∀u ∈ UN .

The set UN is in one to one correspondance with the set VN that consists of all polynomials of degree N with leading coefﬁcient 1, whose roots are contained in the lower half plane C− ={z ∈ C : Imz < 0}. Moreover, the bijection u ∈ UN → Qu ∈ VN provides the real analytic structure on UN . Proposition 1.2. Equipped with the subspace topology of the R-Hilbert space L2(R, R), 2 the subset UN is a simply connected, real analytic, embedded submanifold of L (R, R) and dimR U N = 2N. For every u ∈ UN , the tangent space to UN at u is given by T (U ) = (Rm(z)( φ ) Rm(z)( φ )) φ ( ) := ( − )−2 ∀ ∈ u N z∈P(u) Re z Im z , where z x x z , z P(u) ⊂ C−. Given u ∈ UN ,wehave R u = 2π N, so the tangent space Tu(UN ) is included in an auxiliary space T := {h ∈ L2(R,(1+x2)dx) : h(R) ⊂ R, hˆ(0) = 0}. (1.10) The Hardy’s inequality yields that T is contained in the auxiliary space W given by (1.4). 2 2 2 ∗ So W L (x dx) = T . We deﬁne a real analytic 2-form ω : u ∈ UN → ω ∈ (W ), i.e. ˆ ˆ +∞ ˆ ˆ i h1(ξ)h2(ξ) h1(ξ)h2(ξ) ωu(h1, h2) = dξ =−Im dξ, 2π R ξ 0 πξ 1054 R. Sun

∀h1, h2 ∈ Tu(UN ). (1.11) Then we show that ω establishes the symplectic structure, which corresponds to the Gardner bracket (1.5), on the N-soliton manifold UN deﬁned by (1.8).

Proposition 1.3. Endowed with ω in (1.11), the real analytic manifold (UN ,ω) is a symplectic manifold. For any smooth function f : UN → R,letXf ∈ X(UN ) denote its Hamiltonian vector ﬁeld, then

X f (u) = ∂x ∇u f (u) ∈ Tu(UN ), ∀u ∈ UN . (1.12) The Gardner bracket in (1.5) coincides with the Poisson bracket associated to the symplectic form ω, i.e. for another smooth function g : UN → R, we have ωu(X f (u), Xg(u)) ={f, g}(u), ∀u ∈ UN . The following result indicates the global well-posedness of the BO equation (1.1)onthe manifold UN .

Proposition 1.4. For every N ∈ N+, the manifold UN is invariant under the BO ﬂow. U ⊂ ∞(R, R) 2(R, 2 ) ∞(R, R) := Remark 1.5. Since N H L x dx with H s≥0 s H (R, R), the energy functional E in (1.3) is well deﬁned on UN . So equations (1.1) and (1.3) are equivalent on UN . Inspired from the construction of Birkhoff coordinates of the space-periodic BO equation in Gérard–Kappeler [8], we want to establish the generalized action–angle coordinates of (1.1)onUN .Let N N := {(r1, r2,...,rN ) ∈ R : r1 < r2 < ···< rN < 0} (1.13) denote the subset of actions. For any j, k = 1, 2,...,N, the Kronecker symbol is denoted by δkj, i.e. δkj = 1if j = k; δkj = 0, if j = k. The main result of this paper is stated as follows. Theorem 1. There exists a real analytic diffeomorphism

N N : u ∈ UN → (I1(u), I2(u),...,IN (u); γ1(u), γ2(u),...,γN (u)) ∈ N × R (1.14) such that the following statements hold: (i) The Poisson brackets (1.5) between the coordinate functions are well deﬁned and

{I j , Ik}=0, {I j ,γk}=δkj, {γ j ,γk}=0onUN , ∀ j, k = 1, 2,...,N. (1.15)

(ii) The energy functional E deﬁned in (1.3), when expressed in the coordinate functions, is given by

N 1 2 E(u) =− I j (u) , ∀u ∈ UN . 2π j=1

The coordinates {I j }1≤ j≤N are referred to as actions and {γ j }1≤ j≤N as (generalized) angles. Complete integrability of the Benjamin–Ono equation on the multi-soliton manifolds 1055

Corollary 1.6. When expressed in the generalized action–angle coordinates I j , γ j , 1 ≤ j ≤ N, the restriction of the BO equation (1.1) to UN reads as

∂t I j ◦ u (t) ={E, I j }(u(t)) = 0,∂t γ j ◦ u (t) ={E,γj }(u(t)) = k j (u(t)), ∀t ∈ R, (1.16)

I j where k j := − π is referred to as the j th frequency and u : t ∈ R → u(t) ∈ UN solves equation (1.1). As a consequence, I j ◦ u(t) = I j ◦ u(0) and γ j ◦ u(t) = γ j ◦ u(0) + (k j ◦ ( )) ∈ −1({ }×RN ) u 0 t. For any r N , N r is a Lagrangian submanifold that is invariant under the ﬂow of (1.1).

Remark 1.7. For any j = 1, 2,...,N, the frequency k j : UN → (0, +∞) is a linear function of the action I j . Hence the motions of the angles are completely decoupled.

Remark 1.8. The image of actions N is a noncompact convex polytope. As a consequence, the manifold UN can be interpreted as the universal covering of the manifold of T T := R/ πZ N-gap potentials UN for the Benjamin–Ono equation on the torus 2 , which is introduced in theorem 7.1 of Gérard–Kappeler [8],

Q ( iy) U T := {v = h ∈ L2(T, R) : h(y) =−eiy e , Q ∈ C+ [X]}, N 2Re Q(eiy) N (1.17)

C+ [ ] Q ∈ C[ ] where N X consists of all polynomials X of degree N with leading coefﬁcient 1, whose roots are contained in the annulus A := {z ∈ C :|z| > 1}. Since the T (Z, ) T fundamental group of UN is + , the manifold UN is mapped real bi-analytically onto UN /Z. We refer to remark 1.13 to see the comparison between the main theorem 1 and theorem 7.1of[8].

A precise description of N is given in Deﬁnition 5.1 and Theorem 5.2. 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 N , denoted by N×N M : u ∈ UN → (Mkj(u))1≤k, j≤N ∈ C , where π π ( ) M (u) := γ (u) + i , ∀1 ≤ j ≤ N; M (u) := 2 i Ik u , jj j I j (u) kj Ik (u)−I j (u) I j (u) ∀1 ≤ j = k ≤ N. (1.18)

Proposition 1.9. Given u ∈ UN , the polynomial Qu in (1.9) is the characteristic poly- ( ) ∈ CN×N nomial of the inverse spectral matrix M u defined by (1.18). As a conse- ( ) = N R( ( − )) quence, an N-soliton is expressed by u x j=1 c j c j x x j if and only if its { − −1 } ⊂ C translation–scaling parameters x j c j i 1≤ j≤N − are eigenvalues with corresponding multiplicities of the matrix M(u), whose coefficients are expressed in terms of N the action–angle coordinates (I j (u), γ j (u))1≤ j≤N ∈ N × R . Proposition 1.9 is restated with more details in Theorem 4.8, Proposition 5.4 and Corol- lary 5.5, which both give a spectral characterization of the N-soliton manifold UN and establish a spectral connection between the inverse spectral matrix M(u) ∈ CN×N and the Lax operator Lu, which is given in Definition 2.2, of the BO equation (1.1), for any u ∈ UN . Then an explicit expression of solutions of equation (1.1) on the multi-soliton manifolds can be deduced by using Corollary 1.6 and Proposition 1.9. 1056 R. Sun

Corollary 1.10. If u : t ∈ R → u(t) ∈ UN solves equation (1.1) such that u(0) = u0, then for any (t, x) ∈ R × R, we have

t −1 u(t, x) = u(t, x; u0) = 2Im M(u0) − (x + π V(u0)) X(u0), Y (u0)CN , (1.19) N T where the inner product of C is X, Y CN = X Y ; and ∀u ∈ UN , the matrix M(u) is given by (1.18), the matrix V(u) ∈ CN×N and the vectors X (u), Y (u) ∈ CN are deﬁned by ⎛ ⎞ √ I (u) π ( )T = ( | ( )|, | ( )|,..., | ( )|), 1 2 X u I1 u I2 u IN u ⎜ I2(u) ⎟ V( ) = ⎜ ⎟ . √ − u ⎝ . ⎠ 1 T −1 −1 −1 .. 2π Y (u) = ( |I1(u)| , |I2(u)| ,..., |IN (u)| ), IN (u) One application of the explicit formula (1.19) is to describe the asymptotic behavior of the multi-soliton solutions of the BO equation (1.1).

∈ U ( , ) = ( , ; ) := N R ( − Corollary 1.11. Given u0 N ,wesetu∞ t x u∞ t x u0 j=1 k j (u0) x γ (u ) − k (u )t), where R (x) = 2c and k =−I j .Ifu: t ∈ R → u(t) ∈ U j 0 j 0 c 1+c2x2 j π N solves (1.1) with u(0) = u0, then > ( ) − ( ) = (i) for any R 0, we have limt→±∞ u t u∞ t L2(−R,R) 0; u(t,x) ∈ R →±∞ = (ii) for any x , we have limt u∞(t,x) 1. When t →±∞,theN-soliton solutions of equation (1.1) can be approximated asymp- totically by the superposition of N solitons such that the j th soliton which starts from the point γ j (u0), moves with constant velocity k j (u0) and constant scaling parameter k j (u0). We refer to Matsuno [14] and the references therein to see another expression of multi-soliton solutions, the soliton interactions, the non linear superposition principle and other asymptotic behaviors of solutions of equation (1.1), which are studied by using Hirota’s bilinear transformation, the pole expansion and the Bäcklund transformation. However, it still remains to solve an algebraic equation (see for instance Proposition 1.9 or formula (3.266) in section 3.3 of Matsuno [14]) by radicals in order to express the velocity & scaling parameter k j (u0) and the starting point γ j (u0) of the asymptotic approximation u∞(u0) in terms of the translation–scaling parameters with corresponding multiplicities of the initial datum u0 ∈ UN . Compared with Matsuno [14], we give I j (u0) a precise and explicit expression of the velocity & scaling parameter k j (u0) =− π of u∞(u0), thanks to the min-max formula (4.8) and deﬁnition 5.1. Remark 1.12. When N = 1, formula (1.19) has been established in Benjamin [3], Ono [17] and Amick–Toland [2]. Moreover, let u : t ∈ R → u(t) ∈ U1 solve the BO equation 2c ( , ) = 1 ∈ R > ∞( , ) = ( , ) = (1.1), if u 0 x 2( − )2 for some x1 and c1 0, then u t x u t x c1 x x1 +1 2c1 ∀( , ) ∈ R2 2( −( ))2 , t x . c1 x x1+c1t +1

1.1. Notation. Before outlining the construction of action–angle coordinates, we introduce some notations used in this paper. The indicator function of a subset A ⊂ X is denoted by 1A, i.e. 1A(x) = 1ifx ∈ A and 1A(x) = 0ifx ∈ X\A. Recall that : 2(R) → 2(R) ( ) = H L L denotes the Hilbert transform given by (1.2). Set IdL2(R) f f , for every f ∈ L2(R).Let : L2(R) → L2(R) denote the Szeg˝o projector, deﬁned by Id +iH := L2(R) ⇔ (ξ) = (ξ) ˆ(ξ), ∀ξ ∈ R, ∀ ∈ 2(R). 2 f 1[0,+∞) f f L (1.20) Complete integrability of the Benjamin–Ono equation on the multi-soliton manifolds 1057

If O is an open subset of C, we denote by Hol(O) all holomorphic functions on O.Let the upper half-plane and the lower half-plane be denoted by C+ ={z ∈ C : Imz > 0} p and C− ={z ∈ C : Imz < 0} respectively. For every p ∈ (0, +∞], we denote by L+ p p to be the Hardy space on C+, which is defined by L+ = L+ (R) := {g ∈ Hol(C+) : g p < +∞}, where L+ 1 p = | ( )|p , ∈ ( , ∞), g L p sup g x + iy dx if p 0 + (1.21) + y>0 R ∞ ∞ = | ( )| ∈ | |= and g L+ supz∈C+ g z . A function g L+ is called an inner function if g 1 R = 2 on . When p 2, the Paley–Wiener theorem yields the identification between L+ and [L2(R)]: 2 = F −1[ 2( , ∞)]={ ∈ 2(R) : ˆ ⊂[ , ∞)}= ( 2(R)), L+ L 0 + f L supp f 0 + L where F : f ∈ L2(R) → fˆ ∈ L2(R) denotes the Fourier–Plancherel transform. 2 = ( − )( 2(R)) Similarly, we set L− IdL2(R) L . Let the filtered Sobolev spaces be s := 2 s(R) s := 2 s(R) ≥ denoted as H+ L+ H and H− L− H , for every s 0. We set ∞(R, R) := s(R, R) H s≥0 H . The domain of definition of an unbounded operator A on some Hilbert space E is denoted by D(A) ⊂ E. Given another operator B on D(B) ⊂ E such that A(D(A)) ⊂ D(B) and B(D(B)) ⊂ D(A), their Lie bracket is an operator defined on D(A) D(B) ⊂ E, which is given by [A, B]:=AB − BA. If the operator A is self-adjoint, let σ(A) σ (A) σ (A) denote its spectrum, pp denotes the set of its eigenvalues and cont denotes its continuous spectrum. Then σcont(A) σpp(A) = σ(A) ⊂ R.GiventwoC-Hilbert spaces E1 and E2,letB(E1, E2) denote the C-Banach space that consists of all bounded C-linear transformations E1 → E2, equipped with the uniform norm. We set B(E1) := B(E1, E1). All manifolds introduced in this paper are smooth manifolds without boundary. Given a smooth manifold M of real dimension N,letC∞(M) denote all smooth functions f : M → R and the set of all smooth vector fields is denoted by X(M). The tangent ∈ T ( ) T ∗( ) (resp. cotangent) space to M at p M is denoted by p M (resp. p M ). Given k k ∈ N+,theR-vector space of smooth k-forms on M is denoted by (M).Givena R-vector space V, we denote by 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 LX (A), which is also a smooth tensor field on M.IfN 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, denoted by F∗A, is a smooth k-tensor field on N that is defined by ∀p ∈ N, ∀ j = 1, 2,...,k, ∗ (F A)p(v1,v2,...,vk) = AF(p) (dF(p)(v1), dF(p)(v2),...,dF(p)(vk)) , ∀v j ∈ Tp(N). (1.22)

Given a positive integer N,letC≤N−1[X] denote the C-vector space of all polynomials with complex coefﬁcients whose degree is no greater than N − 1 and C [ ]=C [ ]\C [ ] N X ≤N X ≤N−1 X consists of all polynomials of degree exactly N. ∈ C [ ] ( ) := N j ( ) = N j Given Q N X ,wesetQ X j=0 a j X ,ifQ X j=0 a j X .We R =[, ∞) R∗ = ( , ∞) C∗ = C\{ } ( , ) ⊂ C set + 0 + , + 0 + and 0 .LetD z r denote the open disc of radius r > 0, whose center is z ∈ C and its boundary is denoted by C (z, r) = ∂ D(z, r) ={η ∈ C :|η − z|=r}. 1058 R. Sun

1.2. Organization of the paper. The construction of action–angle coordinates for the BO equation (1.3)onUN mainly relies on the Lax pair formulation ∂t Lu =[Bu, Lu], discovered by Nakamura [15] and Bock–Kruskal [4]. Section 2 is dedicated to the spectral : ∈ 1 →−∂ − ( ) ∈ 2 analysis of the Lax operator Lu h H+ i x h uh L+ given by definition 2.2 for general symbol u ∈ L2(R, R), where denotes the Szeg˝o projector given in 2 (1.20) and the Hardy space L+ is given in (1.21). Then Lu is an unbounded self-adjoint 2 σ ( ) =[ , ∞) operator on L+ that is bounded from below, it has essential spectrum ess Lu 0 + . 2 2 2 In addition, if u ∈ L (R, x dx) L (R, R), every eigenvalue of Lu is negative and simple, thanks to the following identity, λ|ˆϕ( )|2 =− πϕ2 , λ ∈ R ϕ ∈ (λ − ), 0 2 L2 if and Ker Lu (1.23) which is firstly found by Wu [24] in the case λ<0. Then we introduce a generating functional which encodes the entire BO hierarchy, H ( ) =( λ)−1 , , λ ∈ C\σ(− ), λ u Lu + u u L2 if Lu (1.24) in Definition 2.14. It provides a sequence of conservation laws controlling every Sobolev norm. ( (η)∗) 2 In Sect. 3, we study the shift semigroup S η≥0 acting on the Hardy space L+, η where S(η) f = eη f and eη(x) = ei x . Then a weak version of the Lax Theorem 3.2, which is stated as Lemma 3.3, can be obtained by solving a linear differential equation 2 with constant coefficients. Every N-dimensional subspace of L+ that is invariant under C [ ] = d (η)∗ ≤N−1 X the infinitesimal generator G i dη η=0+ S is of the form Q , for some monic polynomial Q ∈ CN [X] whose roots are contained in the lower half-plane C_. In Sect. 4, the real analytic structure and symplectic structure of the N-soliton subset UN are established at first. Then we continue the spectral analysis of Lu, ∀u ∈ UN .The λu <λu < ··· <λu < Lax operator Lu has N simple eigenvalues 1 2 N 0 and the Hardy 2 space L+ splits as 2 = H ( ) H ( ), H ( ) = H ( ) = 2, L+ cont Lu pp Lu cont Lu ac Lu u L+ C [ ] H (L ) = ≤N−1 X , (1.25) pp u Qu where Q denotes the characteristic polynomial of u given by (1.9) and = Qu is u u Qu an inner function on the upper half-plane C+. Proposition 1.9 is proved by identifying ( ) | M u in (1.18) as the matrix of the restriction G Hpp(Lu ) associated to the spectral basis {ϕu,ϕu,...,ϕu } ϕu ∈ (λu − ) ϕu = ϕu > 1 2 N , where j Ker j Lu such that j L2 1 and R u j 0. The generating function Hλ in (1.24) can be identified as the Borel–Cauchy transform of the spectral measure of Lu associated to u, which yields the invariance of UN under the BO flow in H ∞(R, R). Hence (1.3) is a globally well-posed Hamiltonian system on UN . 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 → ϕu,ϕu ∈ R : ∈ U → πλu ∈ R N Re G j j L2 and the action variables are I j u N 2 j . ( )( ( )) =[ λ, ] : ∈ U → Thanks to the Lax pair formulation dL u XHλ u Bu Lu , where L u N ∈ B( 1, 2) R λ 2 Lu H+ L+ is -affine and Bu is some skew-adjoint operator on L+,wehave N 2π{λ j ,γk}=δkj and {λ j ,λk}=0onUN ,1≤ j, k ≤ N. Then N : UN → N × R Complete integrability of the Benjamin–Ono equation on the multi-soliton manifolds 1059 is a real analytic immersion. The diffeomorphism property of N is given by Hadamard’s N global inverse function theorem. Finally, we show that N : (UN ,ω)→ (N × R ,ν) ω − ∗ ν is a symplectomorphism by restricting N to a special Lagrangian submanifold := N γ −1( ) ⊂ U N j=1 j 0 N . Corollary 1.11 is proved in Sect. 6. Remark 1.13. (Comparison with Gérard–Kappeler [8]) The BO equation on T = R/ 2πZ reads as ∂ v = T∂2v − ∂ (v2), ( , ) ∈ R × T, t H x x t x (1.26) T 2(T, C) T ( ) = where H denotes the Hilbert transform on L that is defined by H f x − |n| ˆ( ) inx ∀ = ˆ( ) inx ∈ 2(T, C) i |n|≥1 n f n e , f n∈Z f n e L . It can be written in Hamil- tonian form on appropriate Sobolev spaces 2π T T 1 1 1 2 1 3 ∂t v = ∂x ∇v E (v), E (v) = (|∂x | 2 v(x)) − v(x) dx, (1.27) 2π 0 2 3 (v) = ∂ ∇ T(v) T where X ET x v E is the Hamiltonian vector field of E with respect to the symplectic form T i ˆ ˆ ˆ inx 2 2 ω ( f , f ) = f (n) f (n), f j = f j (n)e ∈ L , (T) := {v ∈ L (T, R) :ˆv(0) = 0}. 1 2 2πn 1 2 r 0 |n|≥1 |n|≥1 (1.28) 2 (T) . The global Birkhoff coordinates for (1.26)onLr,0 described in theorem 1 1of[8]is denoted by

1 ζ T : v ∈ 2 (T) → (ζ (v)) ∈ h 2 , Lr,0 n n≥1 + (1.29)

1 h 2 := {( ) ⊂ C :( ) 2 := | || |2 < ∞} 2 where + zn n∈N+ zn n∈N+ 1 n≥1 n zn + is a weighted - 2 T sequence space. Thanks to theorem 7.1 of [8], the N-gap potential manifold UN deﬁned ( 2 (T), ωT) by (1.17) is a connected, real analytic, symplectic submanifold of Lr,0 given T by (1.28) and UN is characterized by T ={v ∈ 2 (T) : ζ (v) = ,ζ(v) = , ∀ > }. UN Lr,0 N 0 j 0 j N (1.30) T = . ν˜ := So it is invariant under the ﬂow of equation (1.26) and dimR UN 2N Let N ∧ CN−1 × C∗ i j=1 dz j dz j denote the canonical symplectic form on . The restriction ζ T T of complex Birkhoff coordinates given by (1.29), to the manifold UN establishes a real analytic diffeomorphism ζ T := ζ T : v ∈ T → (ζ (v), ζ (v), . . . , ζ (v)) ∈ CN−1 × C∗, N T UN 1 2 N (1.31) UN

ζ T ζ T ∗ ν˜ = ωT such that N preserves the symplectic structure, i.e. N , and the energy functional ET in (1.27), when expressed in the coordinate functions, is given by N N N 2 T(v) = 2|ζ (v)|2 − |ζ (v)|2 , ∀v ∈ T. E n n k UN n=1 n=1 k=n 1060 R. Sun

The generating functional defined in (1.24) plays a key role in proving the local diffeomorphism property and the symplectomorphism property of action–angle/Birkhoff map in both theorem 1 of this paper and theorem 7.1of[8]. The real analytic structure U T of N (resp. UN ) is constructed by establishing a real analytic embedding from an open CN 2(R, R) 2(T, R) U T subset of to L (resp. L ) with range given by N (resp. UN ). A real analytic covering map from the N-soliton manifold UN to the N-gap potential manifold T UN is established in remark 1.8. However, the construction of the action–angle map N in (1.14) is quite different from the construction of the Birkhoff map ζ T in [8]. . ωT 2 (T) 1 The symplectic form given by (1.28) is well defined on Lr,0 , which is a C T T -Hilbert space that contains every manifold UN .SoUN is a symplectic submanifold ( 2 ,ωT) T of Lr,0 . The BO equation on the torus (1.26), when restricted to UN , is interpreted ( 2 ,ωT) as an integrable subsystem of equation (1.26)on Lr,0 . On the other hand, in the space non-periodic regime, we do not know whether there exists a large submanifold 2 of L (R, R), denoted by L, such that L contains every multi-soliton manifold UN , L is invariant under the flow of (1.1), and there exist action–angle coordinates for the BO equation (1.1)onL, whose restriction to UN is N given in (1.14). Evidently, L can not 1 be chosen as W = ∂x (H (R, R)) given by (1.4), because UN W =∅. However, the ω : ( , ) ∈ W2 → ,∂−1 W 2-covector h1 h2 h1 x h2 L2(R) is defined on . The extension of 2 the symplectic form ω ∈ (UN ), which is defined by (1.11), to the manifold L would be the major difficulty for constructing action–angle coordinates of the BO equation (1.1)onL. Since UN W =∅, we have to use Cartan’s formula (4.2) in order to prove 2 ∗ the closedness of the 2-form ω : u ∈ UN → ωu = ω ∈ (W ), which may not be interpreted as a pullback of ω. Moreover, the simple connectedness of UN is established by a special property of the Viète map (4.1). 2. In any case, the Lax operator for the BO equation is self-adjoint and bounded T from below. The spectrum of the Lax operator Lv in the space-periodic regime consists σ( T) ={λT(v) < λT(v) < ···} ⊂ R of a sequence of simple eigenvalues Lv 0 1 and the gap between each two of them is at least 1. Then the n th action variable is defined |ζ (v)|2 := λT(v) − λT (v) − ∀ ≥ by n n n−1 1in[8], n 1. However, in order to prove the simplicity and negativeness of eigenvalues of the Lax operator Lu in Definition 2.2 for the BO equation on the line (1.1), we have to introduce the auxiliary identity (1.23). The action variables for equation (1.1)onUN are actually the eigenvalues of 2π Lu, ∀u ∈ UN . . T : ∈ 2(T) → ix ( ) ∈ 2(T) 3 The shift operator S f L+ e f x L+ and its adjoint 2(T) := T( 2(T, C)) T : are bounded operators on the Hardy space L+ L , where inx ∈ 2(T, C) → inx ∈ 2(T, C) n∈Z gne L n≥0 gne L denotes the Szeg˝o projector on L2(T, C). So both the inverse formula for v ∈ L2(T, R), which is denoted by formula ( . ) T 4 5 in [8], and the spectral characterization of UN , which is given by formula (1.30)of this paper and (7.2) in [8], can be directly obtained by computing the 0 th Fourier mode T of each eigenfunction of the space-periodic Lax operator Lv without using Beurling’s 2(T) = H ( T) theorem that characterizes all the shift-invariant subspaces of L+ pp Lv .On the other hand, in the case of the BO equation on the line (1.1), the Lax operator Lu in definition 2.2 has not only eigenvalues but also continuous spectrum. In order to deter- mine the characteristic polynomial Qu in (1.9) and prove the spectral characterization Theorem 4.8 for UN , we have to do the spectral decomposition (1.25) and identify each spectral subspace as the corresponding closed shift-invariant (also called translation- ∗ invariant) subspace by both introducing the two shift semigroups (S(η))η≥0, (S(η) )η≥0, Complete integrability of the Benjamin–Ono equation on the multi-soliton manifolds 1061 and using Lax’s scalar representation Theorem 3.2 or its special case stated as Lemma 3.3. In fact, ∀u ∈ UN , the spectral subspace Hac(Lu) is invariant under (S(η))η≥0; H ( ) ( (η)∗) H ( ) = the spectral subspace pp Lu is invariant under S η≥0 and dimC pp Lu N. = d (η)∗ Since the infinitesimal generator G i dη η=0+ S is an unbounded, densely defined 2(R) operator on L+ , given by (3.2), we study its restriction to the N-dimensional spectral H ( ) ( ) = ( − | ) subspace pp Lu . Then Lemma 3.3 yields that Qu X det X G Hpp(Lu ) .

1.3. Related work. Besides the global well-posedness problem of the BO equation (1.1), various properties of its multi-soliton solutions have been investigated in detail. Both the solitary waves for (1.1) and the internal periodic waves for (1.26) are completely classiﬁed in Amick–Toland [2]. The H 1-orbital stability of double solitons of (1.1)is obtained in Neves–Lopes [16]. In Dobrokhotov–Krichever [6], the multi-phase solutions (periodic multi-solitons) for (1.26) are constructed by ﬁnite zone integration and they have also established an inversion formula for multi-phase solutions. Compared with 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

given by (1.14). Furthermore, the inverse spectral formula u =−2Im Qu with N Qu Qu(x) = det(x − G|H (L )) = det(x − M(u)) provides a spectral connection between pp u | ∀ ∈ U the Lax operator Lu and the operator G Hpp(Lu ), u N . Concerning the investigation of the integrability of the BO equations (1.1) and (1.26), besides the discovery of their Lax pair structures, we mention the pioneering work of Ablowitz–Fokas [1], Coifman–Wickerhauser [5], Kaup–Matsuno [12] and Wu [24,25] about the direct and inverse scattering transform of (1.1). Equations (1.1) and (1.26) both admit an infinite hierarchy of conservation laws that control every H s-norm of the ∈ N − 1 < < solutions, see [1] and [5] for the case 2s , see Talbut [22] for the case 2 s 0 and for conservation laws controlling Besov norms, etc. In the space-periodic regime, Gérard and Kappeler have shown in [8] that (1.26) admits global Birkhoff coordinates on 2 (T) Lr,0 , see also remark 1.13 for the comparison between [8] and theorem 1 of this paper. We point out that both Korteweg–de Vries (KdV) equation on T (see Kappeler–Pöschel [11]) and the cubic defocusing Schrödinger (dNLS) equation on T (see Grébert–Kappeler [9]) admit global Birkhoff coordinates. The theory of finite-dimensional Hamiltonian system is transferred to BO, KdV and dNLS equations on T through the submanifolds of finite-gap potentials, which are introduced in order to solve the periodic KdV initial problem. Moreover, the cubic Szeg˝o equations both on T (see Gérard–Grellier [7]) and on R (see Pocovnicu [18]) admit global (generalized) action–angle coordinates on all M( )T finite-rank generic rational function manifolds, denoted respectively by N gen and M( )R M( )R M( )T N gen. A real analytic covering map can be established from N gen to N gen. Moreover, the cubic Szeg˝o equations both on T and on R have inverse spectral formulas which permit the Szeg˝o 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 arealsousedin[18]. Remark 1.14. The BO equation (1.1) can be interpreted as a Schrödinger-type equation, : 2(R) → 2 : ∈ R → ( ) ∈ whichisfilteredbytheSzeg˝o projector L L+.Ifu t u t 2(R, R) w : ∈ R → w( ) := ( ( )) ∈ 2 H solves (1.1) and t t u t H+ , then equation (1.1) reads as an NLS–Szeg˝o equation ∂ w − ∂2w ∂ (w2 (|w|2)) = ,(, ) ∈ R × R. i t x + i x +2 0 t x (1.32) 1062 R. Sun

We refer to Sun [20,21] to see the long time and asymptotic behavior of other NLS–Szeg˝o equations.

2. The Lax Operator

This section is dedicated to studying the Lax operator Lu in the Lax pair formulation of the BO equation (1.1). Then we describe the location of its spectrum and revisit the simplicity of its eigenvalues. At last, we introduce a generating functional Hλ which encodes the entire BO hierarchy. The equation ∂t u = ∂x ∇uHλ(u) also enjoys a Lax pair structure. Now, we recall a basic fact concerning unitarily equivalent self-adjoint operators.

Proposition 2.1. If E1 and E2 are two Hilbert spaces, let A be a self-adjoint operator deﬁned on D(A) ⊂ E1 and B be a self-adjoint operator deﬁned on D(B) ⊂ E2.BothA and B have spectral decompositions E1 = Hac(A) Hsc(A) Hpp(A), E2 = Hac(B) Hsc(B) Hpp(B). (2.1)

If A and B are unitarily equivalent i.e. there exists a unitary operator U : E1 → E2 such that

BU = UA, D(B) = UD(A), (2.2) then σxx(A) = σxx(B) and UHxx(A) = Hxx(B), for every xx ∈{ac, sc, pp}. Moreover, for every bounded borel function f : R → C,f(A) is a bounded operator on E1,f(B) ∗ is a bounded operator on E2, we have f (B) = U f (A)U .

2.1. Spectral analysis I. In this subsection, we study the essential spectrum and discrete spectrum of the Lax operator Lu. The spectral analysis of Lu such that u is a multi-soliton in definition 1.1, will be continued in Sect. 4.2. 2 Definition 2.2. Given u ∈ L (R, R), its associated Lax operator Lu is an unbounded 2 := − : ∈ 1 →−∂ ∈ 2 operator on L+, given by Lu D Tu, where D h H+ i x h L+ and Tu : ∈ 1 → ( ) ∈ 2 denotes the Toeplitz operator of symbol u, defined by Tu h H+ uh L+, : 2(R) → 2 ∈ 1(R, R) where the Szeg˝o projector L L+ is given by (1.20). If u H in := ( − 2) ∈ B( 1, 2) addition, we define Bu i T|D|u Tu H+ L+ . 2 ( ) ≤ Both D and Tu are densely defined symmetric operators on L+ and Tu h L2 ∞ ∈ 1 ∈ 2(R, R) u L2 h L , for every h H+ and u L . Moreover, the Fourier–Plancherel 2 transform implies that D is a self-adjoint operator on L+, whose domain of definition is 1 H+ . 2 Proposition 2.3. If u ∈ L (R, R), then Lu is an unbounded self-adjoint operator on 2 ( ) = 1 L+, whose domain of definition is D Lu H+ . Moreover, Lu is bounded from below. The essential spectrum of Lu is σess(Lu) = σess(D) =[0, +∞) and its pure point 2 1 u |D| 4 f σ ( ) ⊂[− L2 , ∞) = L2 spectrum satisfies pp Lu 4 + , where C inf f ∈H 1\{ } denotes 4C + 0 f L4 the Sobolev constant. Complete integrability of the Benjamin–Ono equation on the multi-soliton manifolds 1063

∈ 2 μD Proof. For every h L+,let h denote the spectral measure of D associated to h, ˆ 2 (ξ)| ˆ(ξ)|2 ( ) , = +∞ (ξ) |h(ξ)| ξ μD(ξ) = 1[0,+∞) h ξ then we have f D h h L2 0 f 2π d ,sod h 2π d . 2 Thus σ(D) = σess(D) = σac(D) =[0, +∞).Ifu ∈ L (R, R), we claim that Pu := ◦ ( )−1 2 Tu D+i is a Hilbert–Schmidt operator on L+. ˆ In fact, let F : h ∈ L2 → √h ∈ L2(R∗) denotes the renormalized Fourier– + 2π + A := F ◦ P ◦ F −1 2(R∗) Plancherel transform, then u u is an operator on L + . Then we A (ξ) = +∞ (ξ, η) (η) η (ξ, η) := uˆ(ξ−η) ∀ξ,η ∈ R∗ have u g 0 Ku g d , where Ku 2π(η+i) , +.So u L2 A HS( 2(R∗)) ≤K 2(R∗×R∗) ≤ . Since P is unitarily equivalent to A ,we u L + L + + 2 u u u2 P 2 = λ2 = λ2 =A 2 ≤ L2 have u HS( 2 ) λ∈σ(P ) λ∈σ(A ) u HS( 2(R∗)) . L+ u u L + 4 Then the symmetric operator Tu is relatively compact with respect to D and Weyl’s essential spectrum theorem (Theorem XIII.14 of Reed–Simon [19]) yields that σ ( ) = σ ( ) ( ) = ( ) = 1 ess Lu ess D and Lu is self-adjoint with D Lu D D H+ . Moreover, 2 2 −2 1 |T f, f 2 |=| u| f | |≤u 2 f ≤ C u 2 f 2 | | 2 f 2 u L R L L4 L L D L holds by ≤ −1| | 1 ∈ 1 Sobolev embedding f L4 C D 4 f L2 , for every f H+ . Then Lu is bounded 2 2 1 u f 2 L2 L2 from below, precisely L f, f 2 =|D| 2 f −T f, f 2 ≥− . When u L L2 u L 4C4 u2 u2 λ<− L2 −λ : 1 → 2 σ ( ) ⊂[− L2 , ∞) 4 ,themapLu H+ L+ is injective. Hence pp Lu 4 + . 4C 4C

Proposition 2.4. Assume that u ∈ L2(R,(1+x2)dx) and u is real-valued. For every 1 1 λ ∈ R and ϕ ∈ Ker(λ − Lu), we have uϕ ∈ C (R) H (R) and the following identity holds,

| ,ϕ |2 =− πλϕ2 . u L2 2 L2 (2.3)

Thus σpp(Lu) ⊂ (−∞, 0) and for every λ ∈ σpp(Lu), we have (λ − ) ⊂{ϕ ∈ 1 :ˆϕ ∈ 1(R ) 1(R ) Ker Lu H+ |R+ C + H + and 2 ξ → ξ[ˆϕ(ξ) + ∂ξ ϕ(ξ)ˆ ]∈L (R+)}. (2.4)

Before the proof of proposition 2.4, we recall a lemma concerning the regularity of convolutions.

, , p ∈ ( , ∞) m p(R) ∗ n p−1 (R) ⊂ m+n(R) Lemma 2.5. For any p 1 + , we have W W C , ∞ , n, p W m+n + (R), ∀m, n ∈ N. For every f ∈ W m p(R) ∗ W p−1 (R), we have ∂α ( ) = ∀α = , ,..., lim|x|→+∞ x f x 0, 0 1 m + n. Remark 2.6. Identity (2.3) was ﬁrstly found by Wu [24] in the case λ<0. We show that (2.3) still holds in the case λ ≥ 0. Hence the operator Lu has no eigenvalues in [0, +∞). Proof of proposition 2.4. We choose u ∈ L2(R,(1+x2)dx) such that u(R) ⊂ R, λ ∈ R ϕ ∈ 2 (ϕ) = λϕ and L+ such that Lu . Applying the Fourier–Plancherel transform, we obtain

uϕ(ξ)1[0,+∞)(ξ) = (ξ − λ)ϕ(ξ)ˆ =: gλ(ξ). (2.5) 1064 R. Sun 1 2 1 1 Since uˆ ∈ H (R) and ϕˆ ∈ L (R), their convolution uϕ = π uˆ ∗ˆϕ ∈ C (R) C0(R), 2 ∞ where C0(R) denotes the uniform closure of Cc(R) with respect to the L (R)- 1 norm, by Lemma 2.5. We claim that if λ<0, then ϕˆ ∈ C (R+);ifλ ≥ 0, then 1 ϕˆ ∈ C(R+) C (R+\{λ}). In fact, if λ ≥ 0, we have gλ(λ) = 0. Otherwise, λ would be a singular point of ϕˆ 2 1 2 that prevents ϕˆ from being a L function on R+, because ξ → ξ−λ ∈/ L (R+).Byusing 1 the fact g ∈ C (R+) (g is right differentiable at ξ = 0 and the derivative g is right continuous at ξ = 0), we have (ξ) − (λ) (λ), λ> ; ϕ(ξ)ˆ = gλ gλ → gλ if 0 + ξ − λ gλ(0 ), if λ = 0; when ξ → λ.Soϕˆ ∈ C(R+) and limξ→+∞ ϕ(ξ)ˆ = 0. Then we derive (2.5) with respect to ξ to get

− ixu ∗ˆϕ(ξ) = gλ(ξ) = (uϕ) (ξ) =ˆϕ(ξ) + (ξ − λ)(ϕ)ˆ (ξ), ∀ξ ∈[0, +∞)\{λ}. (2.6) Thus we have d [(ξ − λ)|ˆϕ(ξ)|2]=|ˆϕ(ξ)|2 [((ξ − λ)(ϕ)ˆ (ξ))ϕ(ξ)ˆ ] dξ +2Re

= 2Re[(uϕ) (ξ)ϕ(ξ)ˆ ]−|ˆϕ(ξ)|2. (2.7) • λ< +∞(ϕ) (ξ)ϕ(ξ)ˆ ξ = When 0, it sufﬁces to use the Plancherel formula 0 u d − π ( )|ϕ( )|2 [ , ∞) (ξ − 2 i R xu x x dx and to integrate equation (2.7)on 0 + . Since λ)|ˆϕ(ξ)|2 = ϕ(ξ)ϕ(ξ)ˆ → ξ → ∞ λ|ˆϕ( )|2 = +∞ d [(ξ − u 0, as + ,wehave 0 0 dξ ∞ λ)|ˆϕ(ξ)|2] ξ = π xu(x)|ϕ(x)|2 x − + |ˆϕ(ξ)|2 ξ =− πϕ2 d 4 Im R d 0 d 2 L2(R). • λ> ϕˆ ξ = λ When 0, there may be some problem of derivability of at . We replace +∞ λ− +∞ ∈ ( ,λ) I() := the integral 0 by two integrals 0 and λ+ ,forsome 0 .Weset ∞ λ|ˆϕ(0)|2 − |ˆϕ(λ − )|2 − |ˆϕ(λ + )|2, then I() = 2Re + (uϕ) (ξ)ϕ(ξ)ˆ dξ− 0 λ+(ϕ) (ξ)ϕ(ξ)ˆ ξ − +∞ |ˆϕ(ξ)|2 ξ λ+ |ˆϕ(ξ)|2 ξ λ− u d 0 d + λ− d . Thanks to the continuity of 2 2 ϕˆ R λ|ˆϕ( )| = → + I() =− πϕ on +,wehave 0 lim 0 2 L2(R). • When λ = 0, we use the same idea and integrate (2.7) over interval [, +∞),forsome 2 +∞ +∞ 2 >0. Then J () := −|ˆϕ()| = 2Re (uϕ) (ξ)ϕ(ξ)ˆ dξ − |ˆϕ(ξ)| dξ → 0, as → 0. − πϕ2 = λ|ˆϕ( )|2 ϕ ∈ (λ− L ) L So we always have 2 L2(R) 0 ,if Ker u . As a consequence u has only negative eigenvalues, if the real-valued function u ∈ L2(R,(1+x2)dx). Finally we use uϕ(0) =−λϕ(ˆ 0) to get identity (2.3). If λ ∈ σpp(Lu) and ϕ ∈ Ker(λ − Lu)\{0}, we want to prove that

2 ξ → (1+|ξ|)∂ξ ϕ(ξ)ˆ ∈ L (0, +∞). (2.8)

ϕ ∈ 1 → ∞(R) ∈ 2(R,( 2) ) ϕ = uˆ∗ˆϕ ∈ In fact, since H+ L and u L 1+x dx ,wehaveu 2π H 1(R).Formula(2.5) yields that ξ → (|λ| + ξ)ϕ(ξ)ˆ ∈ L2(R) and we have ϕˆ ∈ L1(R). The hypothesis u ∈ L2(R, x2dx) implies that the convolution term xu ∗ˆϕ ∈ L2(R). Since λ<0, we obtain (2.8) by using formula (2.6). Complete integrability of the Benjamin–Ono equation on the multi-soliton manifolds 1065

Corollary 2.7. Assume that u ∈ L2(R,(1+x2)dx) and u is real-valued. Then every ∞ eigenvalue of Lu is simple. If u ∈ L (R) in addition, then σpp(Lu) is a ﬁnite subset of u2 [− L2 , 0). 4C4

Proof. Fix λ ∈ σpp(Lu) and set Vλ = Ker(λ −Lu), then dimC(Vλ) ≥ 1. We deﬁne a linear form A : Vλ → C such that A(ϕ) := uϕ. Then identity (2.3) yields that ∼ ∼R Ker(A) ={0}. Thus we have Vλ = Vλ/Ker(A) = Im(A)→ C.SodimC(Vλ) = 1. ∞ When u ∈ L (R) in addition, the ﬁniteness of σpp(Lu) (−∞, 0) is given by Theorem 1.2 of Wu [24].

2.2. Lax pair formulation. We recall some known results of global well-posedness of the BO equation on the line. Proposition 2.8 (Tao [23], Ionescu–Kenig [10], etc.). Given s ≥ 0, the Fréchet space C(R, H s(R)) is endowed with the topology of uniform convergence on every com- s pact subset of R. There exists a unique continuous mapping u0 ∈ H (R) → u ∈ s C(R, H (R)) such that u solves the BO equation (1.1) with initial datum u(0) = u0. Proposition 2.9 (Ablowitz–Fokas [1], Coifman–Wickerhauser [5], etc.). For every n n n ∈ N := Z [0, +∞),ifu0 ∈ H 2 (R, R),letu : t ∈ R → u(t) ∈ H 2 (R, R) solves equation (1.1) with initial datum u(0) = u0, then we have C(u0 n ) := H 2 sup ∈R u(t) n < +∞. t H 2 ∈ 2(R, R) 2 When u H , the Toeplitz operators T|D|u and Tu are bounded both on L+ and 1 2 1 on H+ .SoBu is a bounded skew-adjoint operator both on L+ and on H+ . Proposition 2.10. Let u : t ∈ R → u(t) ∈ H 2(R, R) denote the unique solution of equation (1.1), then ∂ =[ , ]∈B( 1, 2 ), ∀ ∈ R. t Lu(t) Bu(t) Lu(t) H+ L+ t (2.9) The proof of proposition 2.10 can be found in Gérard–Kappeler [8], Wu [24]etc.In order to make this paper self contained, we recall it here. d Proof. Since (L ◦ u)(t) =−T∂ ( ) =−T ∂2 ( )−∂ ( )2 , it sufﬁces to prove dt t u t H x u t x (u t ) 2 [B , L ] + T ∂2 −∂ 2 = 0 for every u ∈ H (R, R). In fact, we have uˆ(−ξ) = uˆ(ξ), u u H x u x (u ) u = u + u and |D|u = D u − D u. Since both Tu and Bu are bounded both 2 → 2 1 → 1 L+ L+ and H+ H+ ,wehave

[Bu, Lu] f =− ( f ∂x |D|u) + i [u ( f |D|u) −|D|u (uf)] + [∂x u (uf) + u ( f ∂x u)] (2.10) =− ( ∂2 ) I I ∈ 2 , f H x u + 1 + 2 L+ ∈ 1 I I for every f H+ , where the terms 1 and 2 are given by

I1 := i [u ( f |D|u) −|D|u (uf)]

= [ f u∂x u + f u∂x u]− u ( f ∂x u) − ( f u)∂x u + [ ( f u)∂x u − u ( f ∂x u)],

I2 := [∂x u (uf) + u ( f ∂x u)]= ( f u)∂x u + u ( f ∂x u) + ( u ( f ∂x u))

+2f u∂x u + [ f u∂x u + f u∂x u + ( f u)∂x u]. 1066 R. Sun

2 Since ∂x u ∈ L−,wehave [ ( f u)∂x u]= [ f u∂x u]. Thus,

I I = ∂ [ ∂ ∂ ( )∂ ]= [ ∂ ( 2)]∈ 1. 1 + 2 2 f u x u +2 f u x u + f u x u + f u x u f x u H+ (2.11)

[ , ] = [ (∂ ( 2)− ∂2 )] Formulas (2.10) and (2.11) yield that Bu Lu f f x u H x u . Thus equation (2.9) holds along the evolution of equation (1.1). Remark 2.11. As indicated in Gérard–Kappeler [8], there are many choices of the operator Bu. We can replace Bu by any operator of the form Bu + Pu such that Pu is a := 2 skew-adjoint operator commuting with Lu. For instance, we set Cu Bu + iLu and we 2 obtain Cu = iD − 2iDTu +2iTD u.So(Lu, Cu) is also a Lax pair of the BO equation = ( − 2) : 2 → 2 (1.1). The advantage of the operator Bu i T|D|u Tu is that Bu L+ L+ is bounded if u is sufﬁciently regular. For instance, u ∈ H 2(R, R). : → ( ) ∈ B( 2) := B( 2, 2) Let U t U t L+ L+ L+ denote the unique solution of the following equation

U (t) = B ( )U(t), U(0) = Id 2 , (2.12) u t L+ if u : t ∈ R → u(t) ∈ H 2(R, R) denote the unique solution of equation (1.1). The sys- B( 2) tem (2.12) is globally well-posed in L+ , thanks to Proposition 2.9 and the following estimate

( ) ( 2 ) , ∀ ∈ 2, ∀ ∈ 2(R, R). Bu h L2 u H 2 + u H 1 h L2 h L+ u H ∗ Since B =−Bu, the operator U(t) is unitary for every t ∈ R. Thus, the Lax pair u ∗ formulation (2.9) of the BO equation (1.1) is equivalent to Lu(t) = U(t)Lu(0)U(t) ∈ B( 1, 2 ) H+ L+ . On the one hand, the spectrum of Lu is invariant under the BO ﬂow. On the other hand, there exists a sequence of conservation laws controlling every Sobolev n norms H 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 we ( ) = ( ) ( ) ( )∗ ∈ B( 2) ( ) = ( ) ( ) ( )∗ ∈ have f Lu(t) U t f Lu(0) U t L+ and p Lu(t) U t p Lu(0) U t B( N , 2 ) H+ L+ , where N is the degree of the polynomial p.

n Proposition 2.12. Given n ∈ N,letu : t ∈ R → u(t) ∈ H 2 (R, R) solve equation (1.1), we set

n En(u) := L u, u − n n . (2.13) u H 2 ,H 2

1 Then En(u(t)) = En(u(0)), for every t ∈ R. In particular, E1 = EonH2 (R, R), where the energy functional E is given by (1.3). In order to prove Proposition 2.12, we need the following result. Proposition 2.13. If u : t ∈ R → u(t) ∈ H 2(R, R) solve the BO equation (1.1), then we have

∂ ( ) = ( ( )) 2 ( ( )) ∈ 2. t u t Bu(t) u t + iLu(t) u t L+ (2.14) Complete integrability of the Benjamin–Ono equation on the multi-soliton manifolds 1067

∈ 2(R, R) 2 1 ∈ Proof. For every u H , Bu is a bounded operator on both L+ and H+ , u ( ) = 1 ˆ(−ξ) = ˆ(ξ) = | | = − D Lu H+ .Wehaveu u , u u + u and D u D u D u. Since 2 D u ∈ L−,wehave ( uD u) = (uD u). Thus the following two formulas hold,

( ) = ( − 2)( ) = ( )( ) − ( ) − 2( ) Bu u i T|D|u Tu u i u D u i uD u iTu u = ∂ − ( ∂ ) − 2( ), u x u u x u iTu u 2( ) = 2 − ( ) − ◦ ( ) 2( ) iLu u iD u iTu D u iD Tu u + iTu u =−∂2 − (∂ ) − ∂ [ ( )] 2( ). i x u Tu x u x Tu u + iTu u

( ) 2( ) =−∂2 − [ ∂ ∂ ∂ ] Then Bu u + iLu u i x u 2 u x u + u x u + u x u . Finally we replace u by u(t), where u : t ∈ R → u(t) ∈ H 2(R, R) solves equation (1.1)to obtain (2.14). ∞ Proof of proposition 2.12. It sufﬁces to prove it in the case u0 ∈ H (R, R). Then s we use the density argument and the continuity of the ﬂow map u0 ∈ H (R) → u ∈ C([−T, T ]; H s(R)) in proposition 2.8, where ∀T > 0, s ≥ 0. We choose u = ( ) ∈ ∞(R, R) = s(R, R) n ∂ ∂ ( n) =[ , n] u t H s≥0 H ,soLu u, t u and t Lu u Bu Lu u ∞(R, C) ∂ ( ) = n ,∂ ∂ ( n) , are in H . Thus t En u 2Re Lu u t u L2 + t Lu u u L2 . Since 2 n ,∂ =[ n, 2] , = Bu +iLu is skew-adjoint, we have 2Re Lu u t u L2 Lu Bu +iLu u u L2 [ n, ] , ( n, ) Lu Bu u u L2 by (2.14). Since Lu Bu is also a Lax pair of (1.1), we have ∂ ( ) =([ n, ] ∂ ( n)) , = = t En u Lu Bu + t Lu u u L2 0. In the case n 1, we assume that ∈ 1(R, R) = | | = − ( )3 = u H . Since u u + u, D u D u D u and R u 0, we have | | , = , 3 = ( )| |2 = | |2 D u u L2 2 D u u L2 and R u 3 R u + u u 3 R u u .

2.3. The generating functional. We introduce a new conservation law of the BO equation (1.1) that encodes the entire BO hierarchy.

2 Deﬁnition 2.14. Given u ∈ L (R, R), λ ∈ C\σ(−Lu), the generating functional of H ( ) =( λ)−1 , X := {(λ, ) ∈ equation (1.1) is deﬁned by λ u Lu + u u L2 . The subset u R × L2(R, R) : C4λ>u2 } R × L2(R, R) 4 L2 is open in , where the Sobolev constant 1 |D| 4 f = L2 is given by C inf f ∈H 1\{ } . + 0 f L4 2 u 2 Since σ(L ) ⊂[− L , +∞),themap(λ, u) ∈ X → Hλ(u) =(L + u 4C4 u λ)−1 , ∈ R u u L2 is real analytic. Proposition 2.15. Let u : t ∈ R → u(t) ∈ H ∞(R, R) denote the solution of the BO equation (1.1) and we choose λ ∈ C\σ(−Lu(0)), then Hλ(u(t)) = Hλ(u(0)), for every t ∈ R. ∞ Proof. Let u : t ∈ R → u(t) ∈ H (R, R) solve equation (1.1). Since σ(−Lu(t)) = σ(− ) λ ∈ B( 1, 2) Lu(0) by Proposition 2.1, the operator + Lu(t) H+ L+ is invertible and we have ∂ H ( ) = ( λ)−1 ,∂ −( λ)−1∂ ( λ)−1 , . t λ u 2Re Lu + u t u L2 Lu + t Lu Lu + u u L2 (2.15) 1068 R. Sun

Formula (2.14) yields that ( λ)−1 ,∂ 2Re Lu + u t u L2 =[( λ)−1, 2] , =[( λ)−1, ] , , Lu + Bu + iLu u u L2 Lu + Bu u u L2 [( λ)−1, ] , Lu + Bu u u L2 =( λ)−1[ , λ]( λ)−1 , . Lu + Bu Lu + Lu + u u L2

Then ∂t Lu =[Bu, Lu] yields that ∂t Hλ(u(t)) = 0. 2 Given (λ, u) ∈ X , there exists a neighbourhood of u in L (R, R), denoted by Vu such that the restriction Hλ : v ∈ Vu → Hλ(v) ∈ R can be expressed by power series. Then H H ( )( ) =w , w , the Fréchet derivative of λ at u is given by d λ u h λ h L2 + λ h L2 + w ,w = ,w w |w |2 ∀ ∈ 2(R, R) w ∈ 1 Th λ λ L2 h λ + λ + λ L2 , h L , where λ H+ is deﬁned −1 by wλ ≡ wλ(u) ≡ wλ(x, u) =[(Lu + λ) ◦ ]u(x), ∀x ∈ R.So 2 ∇uHλ(u) =|wλ(u)| + wλ(u) + wλ(u). (2.16)

Given (λ, u0) ∈ X ﬁxed, we consider the following equation 2 ∂t u = ∂x ∇uHλ(u) = ∂x |wλ(u)| + wλ(u) + wλ(u) , u(0) = u0. (2.17)

V 2(R, R) v ∈ V → ∂ |w (v)|2 w (v) There exists an open subset u0 of L such that u0 x λ + λ + w (v)) ∈ 2(R, R) ∈ V λ L is real analytic and u0 u0 . Hence equation (2.17) admits a unique local L2(R, R)-solution by Cauchy–Lipschitz theorem.

Remark 2.16. In Sect. 4, we show that u ∈ UN → ∂x ∇u f (u) ∈ Tu(UN ) is exactly the Hamiltonian vector ﬁeld of the smooth function f : UN → R with respect to the symplectic form ω on the N-soliton manifold UN deﬁned in (1.11).

Proposition 2.17. Given (λ, u0) ∈ X ﬁxed, there exists ε>0 such that (λ, u(t)) ∈ X , for every t ∈ (−ε, ε), where u : t ∈ (−ε, +ε) → u(t) ∈ L2(R, R) solves (2.17) with initial datum u(0) = u0. Then

∂ =[ λ , ], λ := ( ), (λ, v) ∈ X . t Lu(t) Bu(t) Lu(t) where Bv i Twλ(v)Twλ(v) + Twλ(v) + Twλ(v) if (2.18)

∞ 4C4 ˜ 1 Remark 2.18. For every u ∈ H (R, R) and ∈ (0, ),wesetH(u) := H 1 (u) u2 L2 1 ˜ := 1 ( ) = n , and B,u Bu . Recall that En u Lu u u L2 , we have the following Taylor expansion

K H˜ ( ) = (−)n ( ) − (−)K ( 1 )−1 , K , ∀ ∈ N. u En u Lu + u Lu u L2 K (2.19) k=0 Then Proposition 2.17 leads to a Lax pair formulation for the equations corresponding n ∂ =[d ˜ , ] to the conservation laws in the BO hierarchy, t Lu n B,u Lu , where now u d =0 dn = (− )n H˜ evolves according to the Hamiltonian ﬂow of En 1 dn =0 with respect to the Gardner–Faddeev–Zakharov Poisson structure. In the case n = 1, we have E1 = E d = ˜, and Bu d =0 B u. Complete integrability of the Benjamin–Ono equation on the multi-soliton manifolds 1069

Before proving Proposition 2.17, we introduce the Hankel operators of symbols in L2(R) L∞(R). They are used to calculate the commutators of Toeplitz operators. We notice that the Hankel operators areC-anti-linear and the Toeplitz operators are C-linear. For every symbol v ∈ L2(R) L∞(R), its associated Hankel operator is ( ) = v = (v ) ∀ ∈ 1 v ∈ ∞(R) : 2 → 2 deﬁned by Hv h Th h , h H+ .If L , then Hv L+ L+ is a bounded operator. If v ∈ L2(R), then Hv may be an unbounded operator on 2 1 ∈ 1(R), ∈ 1 L+ whose domain of deﬁnition contains H+ . For any b H h H+ ,wehave ( ) ( ) 2 Tb h H 1 + Hb h H 1 b H 1 h H 1 , so both Tb and Hb are bounded on L+ and on 1 H+ . v,w ∈ 2 ∞(R) ∈ 2(R) Lemma 2.19. For every L+ L and u L , we have [ , ]=− ◦ ∈ B( 2). Tv Tw Hv Hw L+ (2.20) w ∈ 1 (w) ∈ 2 If H+ in addition, then we have Tu L+ and = ◦ ◦ = ◦ ◦ ∈ B( 1, 2). HTu w Tw H u + Hw Tu Tu Hw + H u Tw H+ L+ (2.21) v,w ∈ 2 ∞(R) ∈ 2 w = (w ) (w ) ∈ Proof. For every L+ L and h L+,wehave h h + h 2 [ , ] = (v (w )−w (v )) = (vw −v (w )−vw ) = L+.Thus,wehave Tv Tw h h h h h h − (v (w )) =− ◦ ( ) ∈ 2 ∈ 2(R) w ∈ 1 h Hv Hw h L+.Givenu L and H+ , for every ∈ 1 w = (w ) (w ) ∈ 1(R) ( ), ( ) ∈ 1 h H+ ,wehave h h + h H and Hw h Tw h H+ .So ( (w )) = ( (w ) ) = ◦ ( ) ∈ 2 u h h u H u Tw h L+ and we have ( ) = ( ( w) ) = ( w ) = ( (w ) (w )) HTu w h u h u h u h + u h = ( ◦ ◦ )( ) ∈ 2. Tu Hw + H u Tw h L+

2 Similarly, we have uh = (uh)+ (uh) ∈ L (R) and (uh) = (h u) = H u(h) ∈ 2 ( ) = (w ) = (w ( ) w ( )) = ( ◦ L+. Thus, we have HTu w h uh uh + uh Tw H u + ◦ )( ) ∈ 2 Hw Tu h L+. −1 Lemma 2.20. Given (λ, u) ∈ X given in Deﬁnition 2.14,setwλ(u) = (Lu + λ) ◦ ( ) ∈ 1 u H+ , then [ − , ]= ∈ B( 1, 2). D Tu Twλ(u)Twλ(u) + Twλ(u) + Twλ(u) TD[|wλ(u)|2+wλ(u)+wλ(u)] H+ L+ (2.22) w := w ( ) ∈ 1 w ∈ 1 +, + ∈ 1 Proof. We use abbreviation λ λ u H+ , then λ H−.If f g H+ − − , ∈ 1 [ , ]=[ − , − ]= ∈ 2 and f g H−, then T f + Tg+ T f Tg 0, because for every h L+,we + + − − have T f + [Tg+ (h)]= f g h = Tg+ [T f + (h)] and T f − [Tg− (h)]= ( f (g h)) = − − − − ( ) = ( ( )) = − [ − ( )] ∈ 2 ∈ 2 f g h g f h Tg T f h . Since u L+ and u L−,weuse Leibnitz’s rule and formula (2.20) to obtain that

[D − Tu, Twλ + Twλ ]=TDwλ + TDwλ −[Tu, Twλ ]−[Tu, Twλ ] (2.23) = −[ , ]−[ , ] TDwλ + TDwλ T u Twλ T u Twλ = TDwλ + TDwλ − Hwλ H u + H u Hwλ . 1070 R. Sun

Similarly, formula (2.20)impliesthat

[Tu, Twλ Twλ ]=[Tu, Twλ ]Twλ + Twλ [Tu, Twλ ] =[ , ] [ , ]= − . T u Twλ Twλ + Twλ T u Twλ Hwλ H u Twλ Twλ H u Hwλ (2.24)

∈ 1 w , w ∈ 2 For every h H+ , since λ D λ L−,wehave

[D, Twλ Twλ ]h =[D, Twλ ]Twλ h + Twλ [D, Twλ ]h

= TDwλ (Twλ h) + Twλ (TDwλ h) = [ w (w ) w ( w )]= [(w w w w ) ]∈ 2. D λ λh + λ D λh λD λ + λD λ h L+ [ , ]= ∈ B( 1, 2) So D Twλ Twλ TD|wλ|2 H+ L+ . We use formula (2.20) and Leibnitz’s Rule to obtain that [ , ]=[ , ]−[ , 2 ]= − D Twλ Twλ D Twλ Twλ D Hwλ TD|wλ|2 HDwλ Hwλ + Hwλ HDwλ (2.25)

−1 Recall that wλ = (λ + Lu) u, then we have

Dwλ = Tu(wλ) − λwλ + u. (2.26)

The formulas (2.21) and (2.26) imply the following two identities, − = − λ − = − λ , HDwλ Twλ H u HTu wλ Hwλ + H u Twλ H u Hwλ Tu Hwλ + H u − = − λ − = − λ . HDwλ H u Twλ HTu wλ Hwλ + H u H u Twλ Tu Hwλ Hwλ + H u (2.27)

We use formulas (2.24), (2.25) and (2.27) to get the following formula [ − , ]= − ( − ) ( − ) D Tu Twλ Twλ TD|wλ|2 HDwλ Twλ H u Hwλ + Hwλ HDwλ H u Twλ = − ( − λ 2 ) ( − λ 2 ) TD|wλ|2 Hwλ Tu Hwλ Hwλ + H u Hwλ + Hwλ Tu Hwλ Hwλ + Hwλ H u = − . TD|wλ|2 H u Hwλ + Hwλ H u (2.28)

At last, we combine formulas (2.23) and (2.28) to obtain formula (2.22). End of the proof of proposition 2.17. For every u : t → u(t) ∈ L2(R, R) solving equa- d ( ◦ )( ) =− ∂ ( ) =− w ( ( ))w ( ( )) w ( ( )) w ( ( )) tion (2.17), we have dt L u t T t u t iTD( λ u t λ u t + λ u t + λ u t ). Consequently, the Lax equation (2.18) is obtained by identity (2.22) in Lemma 2.20.

3. The Action of the Shift Semigroup

∗ In this section, we introduce the semigroup of shift operators (S(η) )η≥0 acting on the 2 Hardy space L+ and classify all finite-dimensional translation-invariant subspaces of 2 η ≥ (η) : 2 → 2 (η) = L+. For every 0, we define the operator S L+ L+ such that S f eη f , iηx ∗ ∗ where eη(x) = e . Then, its adjoint is given by S(η) = Te−η .WehaveS(η) ◦ Lu ◦ ∗ ∗ S(η) = Lu + ηId 2 , ∀η ≥ 0. Since S(η) B( 2 ) =S(η)B( 2 ) = 1, (S(η) )η≥ L+ L+ L+ 0 Complete integrability of the Benjamin–Ono equation on the multi-soliton manifolds 1071 is a contraction semi-group. Let −iG denote its infinitesimal generator, i.e. Gf = d (η)∗ ∈ 2 ∀ ∈ ( ) i dη η=0+ S f L+, f D G , where ( ) :={ ∈ 2 : ˆ ∈ 1( , ∞)}, D G f L+ f|R+ H 0 + (3.1) ψ−τ ψ − ∂ ψ = τ ψ( ) = ψ( − ) because lim→0 x L2(0,+∞) 0, where x x and ψ ∈ H 1(0, +∞). Every function f ∈ D(G) has bounded Hölder continuous Fourier transform by Morrey’s inequality and Sobolev extension operator yields the existence ˆ + ˆ of f (0 ) := limξ→0+ f (ξ). The operator G is densely defined and closed. The Fourier transform of Gf is given by ˆ Gf(ξ) = i∂ξ f (ξ), ∀ f ∈ D(G), ∀ξ>0. (3.2) −1 The Hille–Yosida theorem implies that (−∞, 0) ⊂ ρ(iG) and (G − λi) B( 2 ) ≤ L+ λ−1, ∀λ>0. 2 ∞ Lemma 3.1. For every b ∈ L (R) L (R), we have Tb(D(G)) ⊂ D(G) and the following identity [ , ]ϕ = iϕ(ˆ 0+) G Tb 2π b (3.3) holds for every ϕ ∈ D(G). ∗ Proof. For every η>0 and ϕ ∈ D(G), bothS(η) and Tb are bounded operators on 2 1 ∗ ∧ 1 ˆ ˆ L ,sowehave ([S(η) , Tb]ϕ) (ξ) = b ∗ˆϕ(ξ + η) − b ∗ (1R (τ−ηϕ))(ξ)ˆ = + η 2πη + 1 ξ+η ˆ(ζ )ϕ(ξˆ η − ζ) ζ ∀ξ> τ ϕ(ˆ ) =ˆϕ( η) ∀ ∈ R 2πη ξ b + d , 0, where −η x x + , x . Then we change the variable ζ = ξ + tη,for0≤ t ≤ 1, 1 ∗ ∧ 1 1 [ (η) − , ]ϕ (ξ) = ˆ(ξ η)ϕ((ˆ − )η) = η (ξ) φη(ξ), ∀ξ> , η S IdL2 Tb b + t 1 t dt a b + 0 (3.4) + 2π 0 := 1 1 ϕ((ˆ − )η) ∈ C φ ∈ 2 φ (ξ) := 1 1[ ˆ(ξ where aη 2π 0 1 t dt and η L+ such that η 2π 0 b + η) − ˆ(ξ)]ˆϕ(( − )η) ∀ξ> ϕˆ| ∈ 1( , ∞) ϕˆ t b 1 t dt, 0. Since R+ H 0 + , is bounded + and limη→0+ ϕ(η)ˆ =ˆϕ(0 ), Lebesgue’s dominated convergence theorem yields that ϕ(ˆ 0+) 2 ˆ ˆ 2 limη→ + aη = . Since b ∈ L (R), we have lim→ τb − b 2 = 0. So φη 0 2π 0 L L2 2 1 +∞ ˆ ˆ 2 2 1 ˆ ˆ 2 + ˆϕ ∞ |b(ξ tη)−b(ξ)| ξ t =ˆϕ ∞ τ− ηb −b t → η → L 0 0 + d d L 0 t L2 d 0, if 0 . 1 ∗ ϕ(ˆ 0+) 2 Thus (3.4) implies that [S(η) − Id 2 , Tb]ϕ = aη b + φη → b in L , when η L+ 2π + + i ∗ η → 0 . Since ϕ ∈ D(G) and Tb is bounded, we have Tb[(S(η) −Id 2 )ϕ]→(TbG)ϕ η L+ 2 + i ∗ iϕ(ˆ 0+) in L , when η → 0 . Consequently, (S(η) − Id 2 )(Tbϕ) → (TbG)ϕ + b in + η L+ 2π 2 η → + ϕ ∈ ( ) L+ when 0 .SoTb D G and (3.3) holds. The following scalar representation theorem discovered by Lax in [13] allows to classify 2 all translation-invariant subspaces of the Hardy space L+, which plays the same role as the Beurling’s theorem in the case of Hardy space on the circle. 2 Theorem 3.2. (Lax) Every nonempty closed subspace of L+ that is invariant under the ( (η)) 2 semigroup of shift operators S η≥0 is of the form L+, where is a holomorphic function on the upper-half plane C+ ={z ∈ C : Imz > 0}. We have |(z)|≤1, for all z ∈ C+ and |(x)|=1, ∀x ∈ R. Moreover, is uniquely determined up to multiplication by a complex constant of absolute value 1. 1072 R. Sun

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. ( ) ⊂ 2 = ≥ Lemma 3.3. Let M be a subspace of D G L+ of finite dimension N dimC M 1 and G(M) ⊂ M. Then there exists a unique monic polynomial Q ∈ CN [X] such that C [ ] −1( ) ⊂ C = ≤N−1 X Q 0 − and M Q . Moreover, Q is the characteristic polynomial of the operator G|M . ˆ ˆ ˆ Proof. We set M ={f ∈ L2(0, +∞) : f ∈ M}, then dimC M = N. Since Gf = ∂ ˆ R\{ } | ∂ | i ξ f on 0 , the restriction G M is unitarily equivalent to i ξ Mˆ by the renormalized Fourier–Plancherel transformation. So the characteristic polynomial Q ∈ CN [X] of ∂ | {β ,β ,...,β }⊂C i ξ Mˆ is well defined, let 1 2 n denote the distinct roots of Q and m j β n = , ,..., ∈ denotes the multiplicity of j ,wehave j=1 m j N and there exist c0 c1 cN−1 ! − C ( ) = ( − ∂ | ) = n ( −β )m j = N N 1 k such that Q z det z i ξ Mˆ j=1 z j z + k=0 ck z . The Cayley– ˆ ˆ Hamilton theorem implies that Q(i∂ξ ) = 0 on the subspace M.Ifψ ∈ M ⊂ L2(0, +∞), then ψ is a weak-solution of the following differential equation

N−1 −N N k−N k i Q(−D)ψ = ∂ξ ψ + i ck ∂ξ ψ = 0on(0, +∞), ψ ≡ 0on(−∞, 0), (3.5) k=0 where D =−i∂ξ . The differential operator Q(−D) is elliptic on the open interval (0, +∞) i.e. the symbol of the principal part of Q(−D), denoted by aQ : (x,ξ) ∈ (0, +∞) × R → (−ξ)N , does not vanish except for ξ = 0. So ψ is a smooth function on (0, +∞). The solution space

− β ξ ( . ) = { ˆ } , ˆ (ξ) = ξl i j , Sol 3 5 SpanC f j,l 0≤l≤m j −1,1≤ j≤n f j,l e 1R+ (3.6)

n = ( . ) = ˆ ⊂ 2 β < has complex dimension j=1 m j N,sowehaveSol 3 5 M L+ and Im j 0, C≤ − [X] ∀ = , ,..., = { , } ≤ ≤ − , ≤ ≤ = N 1 j 1 2 N. At last, we have M SpanC f j l 0 l m j 1 1 j n Q , ( ) = l! ∀ ∈ R where f j,l x l+1 , x . The uniqueness is obtained by identifying 2π[(−i)(x−β j )] all the roots. −1 Lemma 3.4. For every monic polynomial Q ∈ CN [X] such that Q (0) ⊂ C−,the = = Q associated inner function is deﬁned by Q Q . The following identity holds for C [ ] ϕ ∈ ≤N−1 X every Q , ϕ(ξ)ˆ = (ξ)∗ϕ, − , ∀ξ> . S 1 L2 0 (3.7) ϕ(ˆ +) =ϕ, − In particular, 0 1 L2 . C [ ] C [ ] C [ ] ≤N−1 X ⊂ ( ) ( ≤N−1 X ) ⊂ ≤N−1 X Proof. Formula (3.6) yields that Q D G , G Q Q and ∗ C≤N−1[X] P ϕˆ ∈ C1(R ), for any ϕ ∈ . Set ϕ = ,forsomeP ∈ C≤ − [X], then we + Q Q! N 1 Q P P 2 N have ϕ = = ∈ L−. Since Q(X) = (X − β ),Imβ < 0, we have Q Q Q j=1 j j β ( ) = N Im j O( 1 ) → ∞ − ∈ 2 x 1+2i = −β + 2 , when x + ,so1 L . As a consequence, j 1 x j x + ϕ(ξ)ˆ = ϕ( )( − ( )) −iyξ = (ξ)∗ϕ, − ∀ξ> we have R y 1 y e dy S 1 L2 , 0. Complete integrability of the Benjamin–Ono equation on the multi-soliton manifolds 1073

4. The Manifold of Multi-solitons This section is dedicated to a geometric description of every multi-soliton subset given in deﬁnition 1.1. Then we give a spectral characterization for the real analytic symplectic manifold UN in order to prove the global well-posedness of the BO equation (1.1)on UN .

4.1. Differential structure and symplectic structure. The real analytic structure of UN is constructed at first. −1 Proof of proposition 1.2. The set VN := {Q ∈ CN [X]:Q (0) ⊂ C−, limx→+∞ ( ) Q x = } V(CN ) V : (β ,β ,...,β ) ∈ CN → x N 1 is identified as − , where 1 2 N N (a0, a1,...,aN−1) ∈ C denotes the Viète map, defined by N N−1 k N (X − β j ) = ak X + X . (4.1) j=1 k=0

Recall that V : CN → CN is a both open and closed quotient map. For any open simply connected subset A ⊂ CN ,ifA is saturated with respect to V and A =∅with := {(β,β,...,β) ∈ CN :∀β ∈ C}, then V(A) is an open simply connected subset N N T of C . With the subspace topology of C and the Hermitian form HCN (X, Y ) = X Y , N the subset (V(C− ), HCN ) is a simply connected Kähler manifold of complex dimension Q N.Themap : (a , a ,...,a − ) ∈ V(CN ) → u = i ∈ L2, where Q is given N 0 1 N 1 − Q + ( ) = N−1 k N by Q X k=0 ak X + X , is both a holomorphic immersion and a topological N embedding. So (UN ) = N ◦ V(C− ) is an embedded complex analytic submanifold 2 ( (U )) = : (CN ) → (U ) of L+ and dimC N N.Themap N V − N is a biholomorphism T ( (U )) = Cm(z)φ φ ( ) = ( − )−2 ∀ ∈ ( ) ∀ ∈ U and u N z∈P(u) z, where z x x z , z P u , u N . √The proof is completed by√ using the isometry property of the R-linear isomorphism 2 2 −1 2 : u ∈ L (R, R) → 2 u ∈ L .Infact,wehave2Re| 2 = ( | 2(R,R)) and √ + L+ L = u L2 2 u L2 . 2 2 2 We set E := L (R, R) L (R, x dx), Ec := {u ∈ E : R u = c}, ∀c ∈ R. Then we have UN ⊂ E2π N , Tu(UN ) ⊂ E0 = T defined in (1.10), ∀u ∈ UN . Moreover, T 1 is included in W = ∂x (H (R, R)), which is defined in (1.4), thanks to the following lemma. | ( )|2 Lemma 4.1 . For every f ∈ H 1(R) such that f ( ) = , we have f x x ≤ (Hardy) 0 0 R |x|2 d ∂ f 2 . 4 x L2 So the 2-form ω in (1.11) is well defined. Then we show that ω is a real analytic symplectic form on UN .

Proof of proposition 1.3. Given any smooth vector ﬁeld X ∈ X(UN ),letXω ∈ 1 (UN ) denote the interior multiplication by X, i.e. (Xω)(Y ) = ω(X, Y ), for every Y ∈ X(UN ). The ﬁrst step is prove that dω = 0onUN by using the following Cartan’s formula:

LX ω = X(dω) +d(Xω). (4.2) 1074 R. Sun

Let φ denote the smooth maximal flow of X.Ift is sufficiently close to 0, then φt : u ∈ UN → φ(t, u) ∈ UN is a local diffeomorphism by the fundamental theorem on flows. For any u ∈ UN , h1, h2 ∈ Tu(UN ), we compute the Lie derivative of ω with respect to X, ω ( φ ( ) , φ ( ) )−ω ( , ) (L ω) ( , ) = φt (u) d t u h1 d t u h2 u h1 h2 X u h1 h2 lim t t→0 dφt (u)h1−h1 dφt (u)h2−h2 = lim ω , dφt (u)h2 + lim ω h1, . t→0 t t→0 t

i So (LX ω)u(h1, h2) = (h1ω(X, h2)) (u) − (h2ω(X, h1)) (u). We choose (V, x ) a U ∈ smooth local chart for N such that u V and the tangent vector hk has the coor- = 2N ( j) ∂ ( j) ∈ R = , ..., dinate expression hk j=1 hk ∂ j ,forsomehk , j 1 2 2N and = , x u k 1 2. The tangent vector hk can be identified as some locally constant vector 2N ( j) ∂ Y ∈ X(U ) Y : v ∈ V → h ∈ Tv(U ) field k N , which is defined by k j=1 k ∂x j v N , Yk : u → (Yk)u = hk, ∀k = 1, 2. Then the vector field [Y1, Y2] vanishes in the open subset V . The exterior derivative of the 1-form β = Xω is computed as dβ(Y1, Y2) = Y1 (β(Y2)) − Y2 (β(Y1)) + β([Y1, Y2]). Thus (d(Xω))u (h1, h2) = (LX ω)u(h1, h2). Then Cartan’s formula (4.2) yields that X(dω) = 0. Since X ∈ X(UN ) is arbitrary, we have dω = 0. ∈ U ϒω : ∈ T (U ) → ω ∈ T ∗(U ) Given u N , we claim that the linear map u h u N h u u N is injective.

∈ ϒω := ( ) ∈ T (U ) In fact, for any h Ker u , we deﬁne h 2Re i h u N . Then the second ˆ = ( ω )( ) = +∞ |h(ξ)|2 ξ = expression of (1.11) yields that 0 h u h 0 πξ d and hence h 2Re ◦ (h) = 0. So ω is nondegenerate and it is a real analytic symplectic form on UN . For any smooth function f : UN → R, its Hamiltonian vector ﬁeld X f ∈ X(U ) ( ) := −(ϒω)−1( ( )) ( )( ) = , ∇ ( ) = N is given by X f u u d f u . Since d f u h h u f u L2 ˆ i h(ξ) ξ(∇ ( ))∧(ξ) ξ ∀ ∈ T (U ) 2π R ξ i u f u d , h u N ,formula(1.12) is obtained.

Corollary 4.2. Endowed with Hermitian form H, which is deﬁned by H u(h1, h2) := ∞ ˆ (ξ) ˆ (ξ) + h1 h2 ξ ∀ , ∈ T ( (U )) ∀ ∈ U ( (U ), H) 0 πξ d , h1 h2 u N , u N , N is a Kähler manifold and ω =− ∗(ImH).

4.2. Spectral analysis II. We continue to study the spectrum of the Lax operator Lu introduced in Deﬁnition 2.2. The general case u ∈ E = L2(R, R) L2(R, x2dx) has been studied in Sect. 2.1. We restrict our study to the case u ∈ UN in this subsection. The operator Lu has the following spectral decomposition 2 = H ( ) H ( ) H ( ). L+ ac Lu sc Lu pp Lu (4.3)

Let Q denote the characteristic polynomial of u given by (1.9) and := = Qu u u Qu Qu denotes the inner function on C+ associated to Qu.WehaveS(η)[uh]=u[S(η)h], ∀ ∈ 2 2 2 h L+,so u L+ is a closed subspace of L+ that is invariant under the semigroup ( (η)) := ( 2)⊥ S η≥0 in section 3. Set Ku u L+ . Thus, 2 = 2 , (η)∗( ) ⊂ ( ( ) ) ⊂ . L+ L+ Ku S Ku Ku and G D G Ku Ku (4.4) Complete integrability of the Benjamin–Ono equation on the multi-soliton manifolds 1075 where G is deﬁned in (3.2). The following proposition identiﬁes the subspaces in (4.3) and (4.4).

Proposition 4.3. If u ∈ UN , then Lu has exactly N simple negative eigenvalues and we have 2 C≤N−1[X] H (L ) = L , H (L ) ={0}, H (L ) = K = . (4.5) ac u u + sc u pp u u Qu Proof. Fix u ∈ UN , we use abbreviation Q := Qu and := u. The ﬁrst step is to prove C≤N−1[X] P C≤N−1[X] K = . In fact, ∀h ∈ L2 and f = ∈ ,forsomeP ∈ C≤ − [X], Q + Q ! Q N 1 P N we have f,h 2 = , h 2 . Since Q(x) = (x − β ) with Im(β )<0, the L Q L j=1 j j P P 2 P meromorphic function has poles in C ,so ∈ L−. Thus f,h 2 = , h 2 = Q + Q L Q L C [ ] ≤N−1 X ⊂ ( 2)⊥ = ∈ −1 , = 0. Thus Q L+ K. Conversely, if f K, then f h L2 2 Q 2 f,h 2 = 0, for every h ∈ L . Thus g := f ∈ L−. It sufﬁces to prove that P := L + Q ˆ ˆ Qf = Qg ∈ C[X]. In fact, Qf = Q(i∂ξ ) f and supp( f ) ⊂[0, +∞) ⇒ supp(Qf) ⊂ [0, +∞). Similarly, we have supp((Qg)∧) ⊂ (−∞, 0]. So supp(Pˆ) ⊂{0} and P is a C [ ] = P ∈ 2(R) ≤ − ⊂ ≤N−1 X ⊂ polynomial. Since f Q L ,wehavedegP N 1. So K Q K. The second step is to show that ( ) = , ∀ ∈ 2. Lu h Dh h L+ (4.6)

C [ ] In fact, we have ≤N−1 X ⊂ L2, = Q and D = DQ − DQ = i Q −i Q = u+ u = Q + Q Q Q Q Q u on R, then L (h) = (D − T )(h) = Dh + h D − i Q + i Q = Dh + u u Q Q

D Q Q ∗ h − i + i = Dh. Recall that L = L ,sowehaveL (K) ⊂ K. Q Q u u u Since dimC K = N, Corollary 2.7 yields that the Hermitian matrix Lu|K has exactly N distinct eigenvalues. Hence K ⊂ Hpp(Lu). : 2 → 2 = −1 = ∗ : ∈ 2 → We set U L+ L+ such that Uh h. Then U U g L+ −1 ∈ 2 : 2 → 2 ( 1) = 1 = g L+, i.e U L+ L+ is unitary. Moreover, we haveU H+ H+ 1 2 [ ( )]= 1 = 1 2 = ( ) H+ L+.Formula(4.6) yields that U D D H+ H+ L+ D Lu|L2 ∗ + and U L | 2 U = D. For every bounded Borel function f : R → C,wehave u L+ Lu f (L | 2 )U = U f (D) by proposition 2.1.Letμψ = μψ denote the spectral u L+ ψ ∈ 2 (ξ) μ (ξ) = ( ) , = measure of Lu associated to L+. Then R f d h f Lu Uh Uh L2 1 +∞ ( ) , = (ξ)| ˆ(ξ)|2 ξ ∀ ∈ 2 π μ (ξ) = R | ˆ(ξ)|2 ξ f D h h L2 2π 0 f h d , h L+.So2 d h 1 + h d . The measure μh is absolutely continuous with respect to the Lebesgue measure on R. 2 ⊂ H ( ) ⊂ H ( ) = (H ( ))⊥ ⊂ 2 Thus L+ ac Lu cont Lu pp Lu L+ and (4.5) is obtained. We (μ ) ⊂[, ∞) ξ> ∈ 2 1(R) have supp h 0 + . For any 0, there exists h L+ L such that ˆ h(ξ) = 0. So we have σess(Lu) = σcont(Lu) = σac(Lu) =[0, +∞). Deﬁnition 4.4. For every u ∈ UN , we have the following spectral decomposition of Lu: " " σ(Lu) = σac(Lu) σsc(Lu) σpp(Lu), where σac(Lu) =[0, +∞), σsc(Lu) =∅ (4.7) σ ( ) ={λu,λu,...,λu } and pp Lu 1 2 N consists of all eigenvalues of Lu. Proposition 2.3 u2 yields that L is bounded from below and − L2 ≤ λu < ··· <λu < 0 with u 4C4 1 N 1 |D| 4 f = L2 C inf f ∈H 1\{ } . + 0 f L4 1076 R. Sun

Hence the min-max principle (Theorem XIII.1 of Reed–Simon [19]) yields that λu = I( , ), I( , ) = { , : ∈ 1 ⊥, = } n sup F Lu F Lu inf Lu h h L2 h H+ F h L2 1 (4.8) dimC F=n−1

2 where, the above supremum, F describes all subspaces of L+ of complex dimension − ≤ ≤ ≥ I( , ) = σ ( ) = n 1, 1 n N. When n N +1,supdimC F=n−1 F Lu inf ess Lu 0. Given j = 1, 2,...,N, Proposition 2.4 and Corollary 2.7 yield that there exists a unique ϕ : ∈ U → ϕu ∈ H ( ) function j u N j pp Lu such that (λu − ) = Cϕu, ϕu = , ϕu, = π|λu|, Ker j Lu j j L2 1 j u L2 2 j (4.9)

= , ,..., {ϕu,ϕu,...,ϕu } for every j 1 2 N. Then 1 2 N is an orthonormal basis of the subspace Hpp(Lu). Before proving the real analyticity of each eigenvalue, we show its continuity at ﬁrst. = , ,..., λ : ∈ U → λu ∈ R Lemma 4.5. For every j 1 2 N, the j th eigenvalue j u N j is Lipschitz continuous on every compact subset of UN . ∈ 1(R) ≤ −1| | 1 Proof. For every f H , the Sobolev embedding f L4 C D 4 f L2 yields that ∀u,v ∈ UN , 2 −2 1 , − v , ≤ − v ≤ − v | | 2 , Luh h L2 L h h L2 u L2 h L4 C u L2 D h L2 h L2 ∀ ∈ 1. h H+ (4.10) = , ,..., ⊂ 2 − Given j 1 2 N and a subspace F L+ whose complex dimension is j 1, ⊥ j u 1 we choose a function h ∈ F Ker(λ − L ) ⊂ H such that h 2 = 1. We k=1 k u + L = j ϕu ∈ C , = j | |2λu ≤ λu < have h k=1 hk k for some hk . Then Luh h L2 k=1 hk k j 0, λu <λu because k k+1. We have the following estimate

1 | 2 2 = , = , , ≤ λu 2 D h L2 Dh h L2 Luh h L2 + uh h L2 j + u L2 h L4 − 1 ≤ 2 | | 2 . C u L2 D h L2 h L2 (4.11) , ≤ λu −4 −v So estimates (4.10) and (4.11) yield that Lvh h L2 j +C u L2 u L2 . Since F |λu −λv|≤ −4( v ) − is arbitrary, the max–min formula (4.8)impliesthat j j C u L2 + L2 u v ⊂ U 2(R, R) ∈ → λu ∈ R L2 . Every compact subset K N is bounded in L . Hence u K j is Lipschitz continuous. = , ,..., λ : ∈ U → λu ∈ R Proposition 4.6. For every j 1 2 N, the j th eigenvalue j u N j is real analytic. Its proof is based on Kato’s perturbation theory for linear operators.

Proof. For every u ∈ U ,letP j denotes the Riesz projector of the eigenvalue λu. Then N u j > { (λu, )} { ( , )} there exists 0 0 such that the family of closed discs D j 0 1≤ j≤N D 0 0 , = , ..., u is mutually disjoint and for every j k 1 2 N and any closed path j (piecewise Complete integrability of the Benjamin–Ono equation on the multi-soliton manifolds 1077

1 (λu, ) λu C closed curve) in D j 0 with respect to which the eigenvalue j has winding number 1, we have # P j = 1 (ζ − )−1 ζ, P j ◦ P j = P j , P j ϕu = δ ϕu. u Lu d u u u u k kj k (4.12) 2πi u j u by Theorem XII.5 of Reed–Simon [19]. We choose j to be the counterclockwise- C (λu,) ∈ ( , ) P j = (λu − oriented circle j in (4.12)forsome 0 0 . We claim that Im u Ker j ) = Cϕu Lu j . j j It sufﬁces to show that P |H ( ) = 0. In fact the operator P = gλu (L ) is self-adjoint u ac Lu u j u and bounded, where the bounded Borel function gλ : R → R is given by # 1 −1 gλ(x) := (ζ − x) dζ = 1(λ−,λ+)(x), a.e. on R, 2πi C (λ,)

λ ∈ R P j (H ( )) ⊂ Cϕu ⊂ H ( ) P j (H ( )) ⊂ for every . Since u pp Lu j pp Lu ,wehave u ac Lu L H (L ).Letμψ = μ u denote the spectral measure of L associated to ψ ∈ H (L ), ac u ψ u $ ac u [ , ∞) P j ψ, ψ = 1 (ζ − whose support is included in 0 + by (4.7), so u L2 2πi C (λu ,) $ j )−1ψ, ψ ζ = 1 +∞ (ζ − ξ)−1 ζ μ (ξ) = ψ˜ = P j ψ ∈ Lu L2 d π C (λu ,) d d ψ 0. Set u 2 i 0 j ˜ 2 j ˜ ˜ H (L ) ψ =P ψ,ψ 2 = ac u , then L2 u L 0. So the claim is obtained. = , ,... λu = ( ◦ P j ) λ : For every ﬁxed j 1 2 N,wehave j Tr Lu u . Since every eigenvalue k v ∈ U → λv ∈ R V ⊂ U N k is continuous, there exists an open subset N containing u such |λv − λu| < 0 = 2 0 λv ∈ (λu,)\ (λu, ) that supv∈V sup1≤k≤N k k 3 .Weset 3 , then j D j D k 0 , for every v ∈ V and k = j. For example, in the next picture, the dashed circles denote C (λu, ) C (λu, ) C (λu,) respectively j 0 and k 0 ; the smaller circles denote respectively j C (λu,) < and k with j k. The segments inside small circles denote the possible positions λv λv of j and k .

λv λv j k 0 λu λu j k

σ( ) (λu, ) ={λv} C (λu,) (λu, ) Then Lv D j 0 j and j isaclosedpathinD j 0 with λv respect to which j has winding number 1. Thus, # P j = 1 (ζ − )−1 ζ, λv = ( ◦ P j ), ∀v ∈ V. v Lv d j Tr Lv v (4.13) 2πi C (λu ,) j 1078 R. Sun

v ∈ V → ∈ B( 1, 2) R : A ∈ B ( 1, 2 ) → A−1 ∈ Since Lv H+ L+ is -afﬁne and i I H+ L+ B( 2, 1) B ( 1, 2) ⊂ B( 1, 2 ) L+ H+ is complex analytic, where I H+ L+ H+ L+ denotes the open C 1 → 2 subset of all bijective bounded -linear transformations H+ L+, we have the real analyticity of the following map u 3 u 1 −1 2 1 (ζ, v) ∈ D(λ , )\D(λ , ) × V → (ζ − Lv) ∈ B(L , H ). (4.14) j 4 0 j 2 0 + +

P j : v ∈ V → P j ∈ B( 2, 1) λ : v ∈ V → ( ◦ P j ) ∈ R Hence the maps v L+ H+ and j Tr Lv v are both real analytic by composing (4.13) and (4.14). C [ ] Recall that H (L ) = ≤N−1 X ⊂ D(G) is given by (3.6), ∀u ∈ U .Wehavethe pp u Qu N following consequence. = , ,..., ϕ : ∈ U → ϕu ∈ 1 Corollary 4.7. For every j 1 2 N, both the map j u N j H+ and : ∈ U → ϕu,ϕu ∈ C the map j u N G j j L2 are real analytic. ,v ∈ U P j ϕu =ϕu,ϕv ϕv Proof. Given u N ,wehave v j j j L2 j . Since the Riesz projec- P j : v ∈ U → P j ∈ B( 2, 1) tor N v L+ H+ is real analytic in the proof of proposi- P j ϕu = V tion 4.6 and u j L2 1, there exists a neighbourhood of u, denoted by , such that P j ϕu > 1 v ∈ V P j : v ∈ V → P j ∈ B( 2, 1) v j L2 2 for every and v L+ H+ can be expressed P j ϕu ◦P j (ϕu ),P j (ϕu ) v v j G v j v j L2 ϕ = v (v) = by power series. Then we have j ϕu ,ϕ and j j u 2 . j j L2 Pv (ϕ ) j L2 j u −2 j u j u Hence the restriction : v ∈ V →Pv(ϕ ) G ◦ Pv(ϕ ), Pv(ϕ ) 2 ∈ C is j j L2 j j L P j ϕu,v = − πλvϕu,ϕv real analytic. Since (4.9) yields that v j L2 2 j j j L2 , the restriction − πλv 2 j j ϕ : v ∈ V → Pvϕu ∈ H 1 is real analytic. j P j ϕu ,v j + v j L2

4.3. Characterization theorem. This subsection is dedicated to proving the following spectral characterization theorem for multi-solitons. 2 2 Theorem 4.8. Given N ∈ N+, a function u ∈ UN if and only if u ∈ L (R,(1+x )dx) is real-valued, dimC Hpp(Lu) = N and u ∈ Hpp(Lu). Moreover, we have the following inverse formula ( ) = ( − | )−1 d ( − | ) , ∀ ∈ R. u x i det x G Hpp(Lu ) dx det x G Hpp(Lu ) x (4.15) The direct sense is given by Proposition 4.3. Before proving the converse sense of 2 Theorem 4.8, we need to prove the invariance of Hpp(Lu) under G,ifu ∈ L (R,(1+ 2 x )dx) is real-valued, u ∈ Hpp(Lu) and dimC Hpp(Lu) = N ≥ 1. We give another version of formula of commutators (see also Lemma 3.1). 2 2 Lemma 4.9. For u ∈ L (R,(1+x )dx), u is real-valued, ∀ϕ ∈ Ker(λ − Lu) for some λ ∈ σpp(Lu), then we have ϕ, Tuϕ, Luϕ ∈ D(G) and

+ + [ , ]ϕ = iϕ(ˆ 0 ) , [ , ]ϕ = ϕ − iϕ(ˆ 0 ) . (4.16) G Tu 2π u G Lu i 2π u Complete integrability of the Benjamin–Ono equation on the multi-soliton manifolds 1079

ϕ ∈ 1(R) ( ϕ)∧ = ϕ ∈ Proof. In Proposition 2.4, we have shown that u H ,so Tu u 1R+ 1( , ∞) ϕ ∈ ( ) ϕ ∈ 1 = ( ) = ( ) ϕˆ H 0 + and Tu D G .SoG H+ D Lu D Tu . Moreover, we have is right-continuous at ξ = 0+ and ϕˆ ∈ C1(0, +∞). The weak-derivative of ϕˆ is denoted w w d + ∂ ϕˆ δ { } ∂ ϕˆ = 1R∗ ϕˆ ϕ(ˆ )δ by ξ , 0 denotes the Dirac measure with support 0 , then ξ + dξ + 0 0 w w ∂ξ ( ˆ ∗ˆϕ) = ∂ ( ˆ ∗ˆϕ) =ˆ∗ ∂ ϕˆ ϕˆ = R∗ ϕˆ and u ξ u u ξ by Lemma 2.5. Since 1 + a.e. in 1 R ˆ ∈ (R) ˆ ∗ ϕ(ξ) =ˆ∗[ R∗ ϕ](ξ) ξ> and u H ,wehaveu G u 1 + G , for every 0 and ∧ i i d i + ([G, T ]ϕ) (ξ) = ∂ξ (uˆ ∗ˆϕ)(ξ) − uˆ ∗[1R∗ ϕˆ](ξ) = ϕ(ˆ ) u(ξ) u 2π 2π + dξ 2π 0 . The ﬁrst formula of (4.16) is obtained. Since Lu = D − Tu, we claim that Dϕ ∈ D(G). In fact, ∧ ∂ξ (Dϕ) (ξ) =ˆϕ(ξ) + ξ∂ξ ϕ(ξ)ˆ , ∀ξ>0. Thus (2.4)impliesthatDϕ ∈ H 1(0, +∞). ∧ Then ([G, D]ϕ) (ξ) = i∂ξ (ξϕ)(ξ)ˆ − ξ · i∂ξ ϕ(ξ)ˆ = iϕ(ξ)ˆ , ∀ξ>0. So we have [∂x , G]=Id 2 . The second formula of (4.16) holds. L+ 2 2 Proposition 4.10. If u ∈ L (R,(1+x )dx) is real-valued, dimC Hpp(Lu) = N ≥ 1 and u ∈ Hpp(Lu), then we have Hpp(Lu) ⊂ D(G) and G(Hpp(Lu)) ⊂ Hpp(Lu).

Proof. There exists an orthonormal basis of the vector space Hpp(Lu), denoted by {ψ ,ψ ,...,ψ } ψ = λ ψ σ ( ) ={λ ,λ ,...,λ }⊂ 1 2 N , such that Lu j j j , where pp Lu 1 2 N (−∞, ) λ <λ H ( ) ⊂ −1( 1) ( ) 0 and j j+1. Since (2.4)impliesthat pp Lu G H+ D G , ˆ + iψ j (0 ) formula (4.16) gives that f j := [Lu, G]ψ j =−iψ j + π u ∈ Hpp(Lu).Sowehave ,ψ = ψ , ψ − ψ ,ψ = λ(2 ψ ,ψ − ψ ,ψ ) = f j j L2 G j Lu j L2 GLu j j L2 G j j L2 G j j L2 ,ψ f j k L2 0. For every j = 1, 2,...,N,wesetg j := ≤ ≤ , = ψk. Since f j = 1 k N k j λk −λ j ,ψ ψ ( −λ ) = = ( −λ ) ψ ψ − 1≤k≤N,k= j f j k L2 k,wehave Lu j g j f j Lu j G j . Then G j g j ∈ Ker(Lu − λ j ) = Cψ j and Gψ j ∈ g j + Cψ j ⊂ Hpp(Lu). We conclude by Hpp(Lu) = SpanC{ψ1, ψ2,...,ψN }. Now, we perform the proof of converse sense of Theorem 4.8 and give the explicit formula of Qu.

End of the proof of theorem 4.8. ⇐: Proposition 4.10 yields that G(Hpp(Lu)) ⊂ H ( ) | pp Lu .LetQ denote the characteristic polynomial of the operator G Hpp(Lu ), then C [ ] H ( ) = ≤N−1 X = P0 ∈ C[ ] we have pp Lu Q by Lemma 3.3.So u Q ,forsomeP0 X such that deg P0 ≤ N − 1. It remains to show that P0 = iQ . C [ ] P P DP P0 P (iQ −P0)P ≤N−1 X In fact, we have L ( ) = (D−T P −T )( ) = − ( )+ ∈ , u Q 0 P0 Q Q QQ Q2 Q Q Q for every P ∈ C≤N−1[X], thanks to the invariance of Hpp(Lu) under Lu. Partial-

P P C≤N− [X] (iQ −P )P fraction decomposition implies that ( 0 ) ∈ 1 .So 0 ∈ C≤ − [X] QQ Q Q N 1 for every P ∈ C≤N−1[X]. Choose P = 1, since deg(iQ − P0) ≤ N − 1, we have −1 P0 = iQ ,sou ∈ UN . Since Q ∈ CN [X] is monic and Q (0) ⊂ C−,wehave ( ) = ( ) = ( − | ) Qu x Q x det x G Hpp(Lu ) .

We refer to Proposition 4.10 and formula (4.4) to see the invariance of Hpp(Lu) ⊂ D(G) under G, ∀u ∈ UN . The translation–scaling parameters of u can be identiﬁed as | | the spectrum of G Hpp(Lu ). The matrix representation of G Hpp(Lu ) with respect to the {ϕu,ϕu,...,ϕu } orthonormal basis 1 2 N is given in Proposition 5.4.

4.4. The invariance under the Benjamin–Ono flow. Proposition 1.4 is proved in this subsection. At first, we show the invariance of the property x → xu(x) ∈ L2(R) under 1080 R. Sun the BO flow. Then the spectral characterization Theorem 4.8 is used to establish the global well-posedness of the Hamiltonian system (1.3)onUN . 2 2 2 Lemma 4.11. If u0 ∈ H (R, R) L (R, x dx),letu = u(t, x) solves the BO equation 2 2 (1.1) with initial datum u(0) = u0, then u(t) ∈ L (R, x dx), for every t ∈ R. ∈ Remark 4.12. This result can be strengthened by replacing the assumption u0 2 3 + s H (R, R) by a weaker assumption u0 ∈ H 2 (R, R) = > 3 H (R, R), because one s 2 can construct a conservation law of (1.1), which controls the H s-norm of solution, ∀ > − 1 s 2 , by using the method of perturbation of determinants. We refer to Talbut [22] to see details. It suffices to use Lemma 4.11 to prove Proposition 1.4.

Before proving Lemma 4.11, we need some commutator estimates.

Lemma 4.13. For a general locally Lipschitz function χ : R → R such that ∂ χ,∂3χ,∂5χ ∈ 1(R) x x x L , we have the following commutator estimates

1 [|D|,χ]g + [∂ ,χ]g (∂ χ ∂3χ ) 2 g , ∀g ∈ L2(R), L2 x L2 x L1 x L1 L2 (4.17) 1 1 | |[∂ ,χ] (∂ χ ∂3χ ) 2 ∂ (∂ χ ∂5χ ) 2 , ∀ ∈ 1(R). D x g L2 x L1 x L1 x g L2 + x L1 x L1 g L2 g H ∧ Proof. Since 2π ([|D|,χ]g) (ξ) ≤ η∈R |ξ|−|η| |ˆχ(ξ − η)||g ˆ(η)|dη ≤|∂x χ|∗ |ˆ|(ξ) [| |,χ] ∂ χ g , Young’s convolution inequality yields that D g L2 x L1 g L2 .We − 1 1 ∂3χ ∞ 2 3 2 x L set R =∂ χ ∂ χ , then ∂ χ 1 ≤∂ χ ∞ dξ + 1 x L1 x L1 x L x L |ξ|≤R1 |ξ|>R1 |ξ|2 ∂3χ 1 x L1 3 dξ ∂x χ 1 R + = 2(∂x χ 1 ∂ χ 1 ) 2 . Similarly, we have [∂x ,χ]g 2 L 1 R1 L x L L 3 1 ∂ χ 1 g 2 (∂ χ 1 ∂ χ 1 ) 2 g 2 , so the ﬁrst inequality of (4.17)is x L L x L x L L π (| |[∂ ,χ] )∧ (ξ) ≤|ξ| |ξ −η||χ(ξ ˆ −η)|| ˆ(η)| η ≤|∂2χ|∗ obtained. Since 2 D x g η∈R g d x |ˆ|(ξ) |∂ χ|∗|∂|(ξ) | |[∂ ,χ] ∂2χ ∂ χ ∂ g + x x g , then D x g L2 x L1 g L2 + x L1 x g L2 . − 1 1 4 5 4 2 We set R := ∂ χ ∂ χ , then ∂ χ 1 ≤∂ χ ∞ |ξ|dξ + 2 x L1 x L1 x L x L |ξ|≤R1 5 ∂5χ ∞ ∂ χ 1 1 x L ξ ∂ χ R2 x L = (∂ χ ∂5χ ) 2 |ξ|>R |ξ|3 d x L1 2 + R2 2 x L1 x L1 . Finally, we 1 2 add them together to get the second estimate of (4.17).

Now we prove the invariance of the property x → xu(x) ∈ L2(R) is invariant under the BO ﬂow. χ ∈ ∞(R) χ Proof of lemma 4.11. We choose a cut-off function Cc such that decreases in [0, +∞), χ is even, 0 ≤ χ ≤ 1, χ ≡ 1on[−1, 1] and supp(χ) ⊂[−2, 2].If 2 2 2 2 u0 ∈ H (R, R) L (R, x dx),letu : t ∈ R → u(t) ∈ H (R, R) solves the BO equation (1.1) with initial datum u(0) = u0, we claim that there exists a constant C = C( ( ) ) u 0 H 1 such that ( , ) := χ 2( x )| |2| ( , )|2 I R t R x u t x dx R | | ≤ Ce t ( |x|2|u(0, x)|2dx +1), ∀t ∈ R, ∀R > 1, (4.18) R Complete integrability of the Benjamin–Ono equation on the multi-soliton manifolds 1081

ρ( ) := χ( ) > ρ ( ) := ρ( x ) = χ( x ) In fact, we deﬁne x x x . For every R 0, we set R x R R x R . Thus ∂ ( , ) = ρ2 | |∂ ( ) − ρ2 ( )∂ ( ), t I R t 2Re R D x u t 2 Ru t x u t ( ) = J ( ( )) J ( ( )), u t L2 1 u t + 2 u t where for every u ∈ H 2(R), we deﬁne J ( ) := − ρ2 ∂ , ⇒ 1 u 4Re Ru x u u L2 |J ( )|≤ ∂ ∞ ρ 2 ρ 2 1 u 4 x u L Ru L2 u H 2 Ru L2 (4.19) 2 2 2 and J2(u) := 2Reρ |D|∂x u, u 2 =[ρ , |D|∂x ]u, u 2 . Since [ρ , |D|∂x ]= R L R L ∗ R ρR[ρR, |D|∂x ] + [ρR, |D|∂x ]ρR and [ρR, |D|∂x ]=[ρR, |D|∂x ] =[ρR, |D|]∂x + |D|[ρR,∂x ],wehave J ( ) = [ρ , | |]∂ ,ρ | |[ρ ,∂ ] ,ρ . 2 u 2Re R D x u Ru L2 +2Re D R x u Ru L2 ∂ ρ = ∂ ρ ∂3ρ = −1∂ ρ ∂5ρ = Since x R L1 R x L1 , x R L1 R x L1 and x R L1 −3∂ ρ ∈ 2(R) R x L1 , the commutator estimates (4.17) yield that if u H , then |J ( )|≤ ρ 2 [ρ , | |]∂ 2 | |[ρ ,∂ ] 2 2 u 2 Ru L2 + R D x u L2 + D R x u L2 ρ 2 ∂ ρ ∂3ρ ∂ 2 ∂ ρ ∂5ρ 2 Ru L2 + x R L1 x R L1 x u L2 + x R L1 x R L1 u L2 ρ 2 ∂ ρ ∂3ρ ∂ 2 −2∂ ρ ∂5ρ 2 Ru L2 + x L1 x L1 x u L2 + R x L1 x L1 u L2 ρ 2 2 Ru L2 + u H 1 (4.20) for every R ≥ 1. Proposition 2.9 and 2.12 yield that there exists a conservation law of (1.1) controlling H 2-norm of the solution. Let u : t ∈ R → u(t) ∈ H 2(R) denote the solution of the BO equation (1.1). Then sup ∈R u(t) 2 1. Since I (R, t) = t H u0 H2 ρ ( )2 |∂ ( , )|≤C( ( , ) ) ∀ ∈ R Ru t 2 , estimates (4.19) and (4.20) imply that t I R t I R t +1 , t , L C = C( ) for some constant u0 H 2 . Thus (4.18) is obtained by Gronwall’s inequality. Let R → +∞, we conclude by Lebesgue’s monotone convergence theorem.

Since the generating function λ ∈ C\σ(−Lu) → Hλ(u) ∈ C is the Borel–Cauchy transform of the spectral measure of Lu, the invariance of UN under the BO ﬂow is obtained by the inverse spectral transform. ∞ 2 2 End of the proof of proposition 1.4. If u0 ∈ UN ⊂ H (R, R) L (R, x dx),letu = u(t, x) denote the solution of the BO equation (1.1) with initial datum u(0) = u0, then u(t) ∈ H ∞(R, R) L2(R, x2dx) by Proposition 2.8 and Lemma 4.11.Given λ ∈ C\R, the generating function Hλ : u ∈ L2(R, R) → R reads as Hλ(u) = (ξ) (λ )−1 , = dmu := μLu μLu + Lu u u L2 R ξ+λ with mu u, where ψ denotes the spectral ψ ∈ 2 λ ∈ measure of Lu associated to the function L+. So the holomorphic function C\R → Hλu is the Borel–Cauchy transform of the positive Borel measure mu.The m (R) = u2 total variation u L2 is a conservation law of the BO equation (1.1)by Proposition 2.12 and formula (2.14). Thanks to the Stieltjes inversion formula, every ﬁnite Borel measure is uniquely determined by its Borel–Cauchy transform. For every t ∈ R,wehaveHλ[u(t)]=Hλ[u(0)] by proposition 2.15. Since u(0) ∈ UN ,wehave [u(0)]∈Hpp(Lu(0)) by Proposition 4.3. Consequently, there exist c1, c2,...,cN ∈ 1082 R. Sun

R\{ } μLu(t) = = = μLu(0) = N δ [ ( )]∈ 0 such that [u(t)] mu(t) mu(0) [u(0)] j=1 c j λu(0) . Then u t j Hpp(Lu(t)), ∀t ∈ R. The Lax pair structure yields the unitary equivalence between Lu(t) and Lu(0).SodimC Hpp(Lu(t)) = dimC Hpp(Lu(0)) = N by Proposition 2.1.We conclude by Theorem 4.8.

5. The Generalized Action–Angle Coordinates

In this section, we construct the global (generalized) action–angle coordinates N in Theorem 1 of the Hamiltonian system (1.3) with solutions in the real analytic symplectic manifold (UN ,ω)of real dimension 2N given in Proposition 1.2. The goal of this section is to establish the diffeomorphism property and the symplectomorphism property of N . Proposition 1.3 yields that the Poisson bracket of two smooth functions f, g : UN → R is given by

{ , }: ∈ U → ω ( ( ), ( )) =∂ ∇ ( ), ∇ ( ) ∈ R. f g u N u X f u Xg u x u f u u g u L2 (5.1) ∈ U λu <λu < ··· <λu < Given u N , Proposition 4.3 yields that there exist 1 2 N 0 and ϕu ∈ (λu − ) ⊂ ( ) ϕu = ,ϕu = π|λu| j Ker j Lu D G such that j L2 1 and u j L2 2 j , thanks to the spectral analysis in Sect. 4.2. = , ,..., : ∈ U → πλu ∈ R Deﬁnition 5.1. For every j 1 2 N,themapI j u N 2 j is γ : ∈ U → ϕu,ϕu ∈ R called the j th action. The map j u N Re G j j L2 is called the j th (generalized) angle.

The set N is deﬁned by (1.13) and we adopt the superscript instead of the subscript 1 2 N N 1 2 N in this section: N ={(r , r ,...,r ) ∈ R : r < r < ··· < r < 0}. Then ( × RN ,ν) the real analytic manifold N is a symplectic manifold of real dimension ν = N j ∧ α j : ∈ U → 2N, where j=1 dr d . The action–angle map is given by N u N N (I1(u), I2(u),...,IN (u); γ1(u), γ2(u), . . . , γN (u)) ∈ N × R . Theorem 1 is restated here.

Theorem 5.2. The map N has following properties: N (a). The map N : UN → N × R is a real analytic diffeomorphism. ( ) ν ω ∗ ν = ω b . The pullback of by N is , i.e. N . ( ) ◦ −1 : ( 1, 2,..., N ; α1,α2,...,αN ) ∈ × RN → c . We have E N r r r N − 1 N | j |2 ∈ (−∞, ) 2π j=1 r 0 .

N Remark 5.3. The real analyticity of N : UN → N × R is given by Proposition 4.6 and Corollary 4.7. The symplectomorphism property (b) is equivalent to the Poisson ( , ,..., ; , ,..., ) bracket characterization (1.15). The family X I1 X I2 X IN Xγ1 Xγ2 XγN is linearly independent in X(UN ) and we have ∂ ∂ dN (u) : X I (u) → , dN (u) : Xγ (u) →− . k ∂αk N (u) k ∂rk N (u)

( ) = N ,ϕu ϕu The assertion c is obtained by a direct calculus: u j=1 u j L2 j ,formula ( )2 ( ) = ( ), = N | ,ϕu |2λu =− N I j u (4.9) yields that E u Lu u u L2 j=1 u j L2 j j=1 2π . Complete integrability of the Benjamin–Ono equation on the multi-soliton manifolds 1083

| This section is organized as follows. The matrix associated to G Hpp(Lu ) is expressed in terms of actions and angles in Sect. 5.1. In Sect. 5.2, the Poisson brackets of actions and angles are used to prove the local diffeomorphism property of N . The bijectivity of N is obtained by Hadamard’s global inverse function theorem in Sect. 5.3. Finally, N we use Sects. 5.4 and 5.5 to prove that N : (UN ,ω) → (N × R ,ν)preserves the symplectic structure.

5.1. The inverse spectral matrix. We continue to study the infinitesimal generator G defined in (3.2) when restricted to the invariant subspace Hpp(Lu) with complex dimension N. Then we state a general linear algebra lemma that describes the location of | eigenvalues of the operator G Hpp(Lu ). N×N Proposition 5.4. For every u ∈ UN ,letM(u) = (Mkj(u))1≤k, j≤N ∈ C denote the inverse spectral matrix defined by (1.18) and Definition 5.1. Then M(u) is the matrix | {ϕu,ϕu,...,ϕu } associated to the operator G Hpp(Lu ) with respect to the basis 1 2 N , i.e. ( ) = ϕu,ϕu ≤ , ≤ Mkj u G j k L2 , 1 k j N. = ∗ H ( ) ⊂ ( ) (λu − λu) ϕu,ϕu = Proof. Since Lu Lu and pp Lu D G ,wehave j k G j k L2 [ , ]ϕu,ϕu −λuϕu( ) = ϕu( ) = G Lu j k L2 . Since formulas (2.5) and (4.9) imply that j j 0 u j 0 π|λu| 2 j , we use formula (4.16) to obtain that if k and j are different, then u u u u u i u + u i u + u (λ − λ )Gϕ ,ϕ 2 =iϕ − ϕ ( ) u,ϕ 2 =− ϕ ( )uϕ ( ) = j% k j k L j 2π j 0 k L 2π j 0 k 0 λu ∞ − k = ∗ , =−i + ˆ(ξ)∂ ˆ(ξ) ξ = i λu . In the case k j,wehave G f g L2 2π 0 f ξ g d & j ' i ˆ( +) ˆ( +) +∞ ∂ ˆ(ξ) ˆ(ξ) ξ ∗ , = , i ˆ( +) ˆ( +) 2π f 0 g 0 + 0 ξ f g d and G f g L2 Gf g L2 + 2π f 0 g 0 , , ∈ H ( ) ϕu,ϕu = for any f g pp Lu by using formula (3.2). Consequently, we have Im G j j L2 − 1 |ϕu( )|2 =− 1 = ( ) π 0 |λu | ImM jj u . 4 j 2 j N Corollary 5.5. For every u ∈ UN , we define two vectors X (u), Y (u) ∈ R as ( )T = ( |λu|, |λu|,..., |λu |), ( )T = ( |λu|−1, |λu|−1,..., |λu |−1), X u 1 2 N Y u 1 2 N (5.2) Then we have the following inverse spectral formula −1 u(x) =−i(M(u) − x) X(u), Y (u)CN , ∀x ∈ R. (5.3) N Hence, the map N : UN → N × R is injective. , = , ,..., u ( ) ( − ) × ( − ) Proof. For any k j 1 2 N,letKkj x denote the N 1 N 1 sub- matrix obtained by deleting the k th row and j th column of the matrix M(u) − x, ∈ R ( ( ) − )−1 ( ), ( ) = for all x . The Cramer’s% rule yields that M u x X u Y u CN (−1)k+ j det(K u (x)) λu N det(K u (x))+R kj k = j=1 jj := (− )k+ j 1≤k, j≤N det(M(u)−x) λu det(M(u)−x) , where R 1≤k= j≤N 1 % j λu ( u ( )) k = ( N λu − N λu) ( ( ) − ) = det Kkj x λu i j=1 j k=1 k det M u x 0by(1.18) and Defini- j ( ) = ( − ( )) ( ) = (− )N−1 N ( u ( )) tion 5.1.IfQu x det x M u , then Qu x 1 j=1 det K jj x . Then formula (5.3) is obtained by formula (4.15). 1084 R. Sun

The next lemma describes the location of spectrum of all matrices of the form deﬁned as (1.18).

Lemma 5.6. For every N ∈ N+, we choose N negative numbers λ1 <λ2 < ··· < λ < γ ,γ ,...,γ ∈ R M = (M ) ∈ CN×N N 0 and 1 2 N . The matrix kj 1≤k, j≤N is deﬁned λ M−M∗ as M = γ + i and M = i k ,ifk = j. Then (M) := is jj j 2λ j kj λk −λ j λ j 2i negative semi-deﬁnite and σpp(M) ⊂ C−.

∈ RN T := (( |λ |)− 1 ,( |λ |)− 1 ,...,( |λ |)− 1 ) Proof. The vector Vλ is defined as Vλ 2 1 2 2 2 2 2 N 2 . √ 1 T So we have (M) = − =−Vλ · Vλ . Thus ( (M))X, XCN = 2 |λ ||λ | j k 1≤k, j≤N 2 −|X, VλCN | ≤ 0. So (M) is negative semi-definite. If μ ∈ σpp(M) and V ∈ Ker(μ − M)\{0}, it suffices to show Imμ<0. Since

−| , |2 = (M) , = μ 2 , V Vλ CN V V CN Im V CN 2 = , > , where V CN V V CN 0 (5.4) we have Imμ ≤ 0. Assume that μ ∈ R, then formula (5.4) yields that V ⊥ Vλ. Moreover, ∗ λ N×N we have (M − M )V =−2iV, VλCN Vλ = 0. We set D ∈ C to be the diagonal λ matrix whose diagonal elements are λ1,λ2,...,λN , i.e. D = (λkδ jk)1≤k, j≤N . Then we have the following formula

λ λ T [M, D ]=i(IN +2D VλVλ ). (5.5)

[M, λ] = 2 =(M − μ) λ , = So D V iV by (5.5). Finally, i V CN D V V CN λ ∗ D V,(M − μ)V CN = 0 contradicts the fact that V = 0. Consequently, we have μ ∈ C−.

5.2. Poisson brackets. In this subsection, the Poisson bracket defined in (5.1) is generalized in order to obtain the first two formulas of (1.15). It can be defined between a smooth function from UN to an arbitrary Banach space and another smooth function from UN to R. For every smooth function f : UN → R, its Hamiltonian vector field X f ∈ X(UN ) is given by (1.12). For any Banach space E and any smooth map F : u ∈ UN → F(u) ∈ E, we define the Poisson bracket of f and F as follows

{ f, F}:u ∈ UN →{f, F}(u) := dF(u)(X f (u)) ∈ TF(u)(E) = E. (5.6) E = R ∈ U If , then the deﬁnition in formula (5.6) coincides with (5.1). For every u N λ ∈ C\σ(− ) = N ,ϕu ϕu and Lu , since u j=1 u j L2 j , the generating functional

N 2πλu −1 j Hλ(u) =(Lu + λ) u, u 2 =− (5.7) L λ + λu j=1 j is well deﬁned. The analytical continuation allows to extend the map λ → Hλ(u) to the C\σ (− ) λ =−λu domain pp Lu , and it has simple poles at every j . Proposition 2.3 yields Complete integrability of the Benjamin–Ono equation on the multi-soliton manifolds 1085

2 1 u |D| 4 f − L2 ≤ λu < ··· <λu < = L2 that 4 0, where C inf f ∈H 1\{ } denotes the 4C 1 N + 0 f L4 Sobolev constant. So we introduce Y ={(λ, ) ∈ R × U : 4λ> 2 }=X (R × U ) , u N 4C u L2 N (5.8) where X is given by Deﬁnition 2.14. Then Y is open in R × UN and H : (λ, u) ∈ Y → 2πλu − N j ∈ R H j=1 λ λu is real analytic by Proposition 4.6. The Fréchet derivative of λ is + j given by (2.16), so

( ) = ∂ ∇ H ( ) = ∂ (|w ( )|2 w ( ) w ( )), ∀(λ, ) ∈ Y, XHλ u x u λ u x λ u + λ u + λ u u (5.9)

−1 by formula (1.12), where wλ(u) = (Lu + λ) ( u). The following proposition restates the Lax pair structure of the Hamiltonian equation associated to Hλ. Even though the stability of UN under the Hamiltonian flow of Hλ remains as an open problem, the Poisson bracket defined in (5.6) provides an algebraic method to obtain the first two formulas of (1.15). (λ, ) ∈ Y {H , }( ) =[ λ, ] Proposition 5.7. Given u defined by (5.8), we have λ L u Bu Lu and

{H ,λ }( ) = , {H ,γ }( ) = [ , λ]ϕu,ϕu =− λ , λ j u 0 λ j u Re G Bu j j L2 (λ λu )2 (5.10) + j

= , ,..., λ = ( ) for every j 1 2 N, where Bu i Twλ(u)Twλ(u) + Twλ(u) + Twλ(u) . : ∈ 2(R, R) → = − ∈ B( 1, 2) ∀ ∈ 2 Proof. Since L u L Lu D Tu H+ L+ , u L+,we ( )( ) =− ∀ ∈ 2 (λ, ) ∈ Y C have dL u h Th, h L+.If u , then the -linear transforma- λ ∈ B( 1, 2 ) {H , }( ) = tion Lu + H+ L+ is bijective. So formula (5.9) yields that λ L u ( )( H ( )) =− 2 dL u X λ u T∂x (|wλ(u)| +wλ(u)+wλ(u)). Then identity (2.22) yields the Lax equation for the Hamiltonian ﬂow of the generating function Hλ, i.e.

{H , }( ) =[ λ, ]∈B( 1, 2). λ L u Bu Lu H+ L+ (5.11)

ϕ : ∈ U → ϕu = λuϕu ∈ 1 (λ, ) ∈ Y Consider the map L j u N Lu j j j H+ , for every u ,we have

{H , }( )ϕu {H ,ϕ }( ) = λu{H ,ϕ }( ) {H ,λ }( )ϕu ∈ 1. λ L u j + Lu λ j u j λ j u + λ j u j H+ λ Then (5.11) yields (λu − L ) B ϕu −{Hλ,ϕ }(u) ={Hλ,λ }(u)ϕu. Since ϕu ∈ j u u j j j j j u u u λ u Ker(λ − L ) and ϕ 2 = 1by(4.9), we have {Hλ,λ }(u) =(λ − L ) B ϕ − j u j L j j u u j u 2 2 {Hλ,ϕ }(u) ,ϕ 2 = N : ϕ ∈ L →ϕ N ◦ ϕ ≡ U j j L 0. We deﬁne 2 L2 , then 2 j 1on N . Then we have

= (N ◦ ϕ )( ) ( ) = ϕu, {H ,ϕ }( ) . 0 d 2 j u XHλ u 2Re j λ j u L2 (5.12) 1086 R. Sun

∈ R λϕu −{H ,ϕ }( ) = ϕu (λu − ) = So there exists r such that Bu j λ j u ir j because Ker j Lu Cϕu λ j by corollary 2.7 and formula (5.12). Since Bu is a skew-adjoint operator on 2 u u u L and γ = ReGϕ ,ϕ 2 ,wehave{Hλ,γ }(u) = Re G{Hλ,ϕ }(u), ϕ 2 + + j j j L j j j L ϕu, {H ,ϕ }( ) = [ , λ]ϕu,ϕu (λ, ) ∈ Y G j λ j u L2 Re G Bu j j L2 . Furthermore, for every u , formula (3.3) implies that [G, Twλ(u)]=0 and

[ , λ] = [ , ]( ( ) ) =− 1 [(w ( ) )∧( +) ˆ( +)]w ( ), ∀ ∈ ( ). G Bu f i G Twλ(u) Twλ(u) f + f 2π λ u f 0 + f 0 λ u f D G (5.13)

u ∧ + u u −1 u u Since (wλ(u)ϕ ) (0 ) =ϕ ,wλ(u) 2 = (λ + λ ) u,ϕ and u,ϕ = j j L j j L2 j L2 −λuϕu( +) ϕu [ , λ]ϕu,ϕu = j j 0 , we replace f by j in formula (5.13) to obtain G Bu j j L2 − λ ∀(λ, ) ∈ Y (λ λu )2 , u . + j

1 ˜ 1 ˜ 1 −1 Remark 5.8. Recall that H = H 1 and B,u := Bu in Remark 2.18, ∀( , u) ∈ Y. n dn ˜ u u In general, the identity (−1) {E ,γ }(u) = Re[G, B, ]ϕ ,ϕ 2 holds for n j dn =0 u j j L dn = (− )n H˜ ∀ ≤ ≤ every conservation law En 1 dn =0 in the BO hierarchy, 1 j N.

Corollary 5.9. For any j, k = 1, 2,...,N, we have 2π{λ j ,γk}(u) = δkj, {λk,λj }(u) = 0, ∀u ∈ UN .

u2 Proof. Given u ∈ U , ∀λ> L2 ,wehave(λ, u) ∈ Y, then formula (5.7) and N 4C4 − λ ={H ,γ }( ) = π N { λ ,γ }( ) = formula (5.10) imply that (λ λu )2 λ j u 2 k=1 λ+λ j u + j k {λ ,γ }( ) {λ ,λ }( ) − πλ N k j u ={H ,λ }( ) = πλ N k j u ∀ = , ,..., 2 k=1 (λ λu )2 and 0 λ j u 2 k=1 (λ λu )2 , j 1 2 N. + k + k {λ ,γ }( ) − z =− π N k j u The uniqueness of analytic continuation yields that ( λu )2 2 z k=1 ( λu )2 z+ j z+ k {λ ,λ }( ) N k j u = ∀ ∈ C\R and k=1 ( λu )2 0, z . z+ k : ∈ U → πλu γ : ∈ U → Recall that the actions I j u N 2 j and the generalized angles j u N ϕu,ϕu Re G j j L2 are both real analytic functions by Proposition 4.6 and Corollary 4.7.

Proposition 5.10. Given u ∈ UN , the family {dI1(u), dI2(u),...dIN (u); dγ1(u), dγ2(u), ... γ ( )} T ∗(U ) d N u is linearly independent in the cotangent space u N . As a consequence, N N : UN → N × R is a local diffeomorphism.

, ,..., , , ,..., ∈ R ( N ( ) γ ( )) Proof. Given a1 a2 aN b1 b2 bN such that j=1 a j dI j u +b j d j u (h) = 0, ∀h ∈ Tu(UN ). Corollary 5.9 yields that ∀ j, k = 1, 2,...,N,wehave ( )( ( )) ={ , }( ) = γ ( )( ( )) ={ ,γ }( ) = δ dI j u X Ik u Ik I j u 0 and d j u X Ik u Ik j u jk.We ( ) = = ( ) = replace h by X Ik u to obtain that bk 0. Then set h Xγk u ,wehaveak 0.

Since all the actions (I j )1≤ j≤N are in evolution by Corollary 5.9 and the differen- ( ( )) ∈ U L := tials dI j u 1≤ j≤N are linearly independent for any u N , the level set r N −1( j ) U ∀ = ( 1, 2,..., N ) ∈ j=1 I j r is a real analytic Lagrangian submanifold of N , r r r r N . Moreover, Lr is invariant under the Hamiltonian ﬂow of I j , ∀ j = 1, 2,...,N,by the Arnold–Liouville theorem. Complete integrability of the Benjamin–Ono equation on the multi-soliton manifolds 1087

5.3. The diffeomorphism property. This subsection is dedicated to proving the real bi- N analyticity of N : UN → N × R . It remains to show its surjectivity. The proof is based on Hadamard’s global inverse function theorem. Theorem 5.11. (Hadamard) 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. N Lemma 5.12. The map N : UN → N × R is proper. × RN ∈ −1( ) Proof. If K is compact in N , we choose un N K ,so ( ) = ( πλun , πλun ,..., πλun ; γ ( ), γ ( ),...,γ ( )) ∈ , ∀ ∈ N. N un 2 1 2 2 2 N 1 un 2 un N un K n

We assume that there exists (2πλ1, 2πλ2,...,2πλN ; γ1,γ2,...,γN ) ∈ K such that λun → λ γ ( ) → γ ( ( )) j j and j un j up to a subsequence. So M un n∈N converges = ( ) ∈ CN×N to some matrixM Mkj 1≤k, j≤N whose coefﬁcients are deﬁned by |λ | M = i k ,ifk = j; M = γ − i , ∀1 ≤ j, k ≤ N. Lemma 5.6 kj λk −λ j |λ j | jj j 2|λ j |

Q Q yields that σ (M) ⊂ C−.WesetQ(x) := det(x − M) and u = i − i ∈ pp Q Q N N UN . The Viète map V is deﬁned in (4.1) and V(C− ) is open in C . Then there ( ) (n) (n) (n) exists a n = (a , a ,...,a ), a = (a , a ,...,a − ) ∈ V(CN ) such that 0 1 N−1 0 1 N 1 − ( ) = ( − ( )) = N−1 (n) j N ( ) = N−1 j N Qn x det x M un j=0 a j x + x and Q x j=0 a j x + x . ( ) We have lim → ∞ Q (x) = Q(x), ∀x ∈ R. So lim → ∞ a n = a. The continu- n + n n + : = ( , ,..., ) ∈ (CN ) → = Q ∈ 2 ity of N a a0 a1 aN−1 V − u i Q L+ yields that → 2 → ∞ U 2(R, R) un u in L+,asn + . Since N inherits the subspace topology of L , we have (un)n∈N converges to u in UN . The continuity of the map N shows that N (u) = (2πλ1, 2πλ2,...,2πλN ; γ1,γ2,...,γN ) ∈ K . N Proposition 5.13. The map N : UN → N × R is a real analytic diffeomorphism.

5.4. A Lagrangian submanifold. In general, the symplectomorphism property of N is equivalent to its Poisson bracket characterization (1.15). The ﬁrst two formulas of (1.15), which are given in Corollary 5.9, lead us to focusing on the study of a special Lagrangian submanifold of UN , denoted by

N := {u ∈ UN : γ j (u) = 0, ∀ j = 1, 2,...,N}. (5.14)

Lemma 5.14. For every u ∈ UN , then each of the following four properties implies the others:

(a) The N-soliton u ∈ N . (b) For every x ∈ R, we have u(x) = u(−x). (c) The N-soliton u is an even function R → R. (d) The Fourier transform uˆ is real-valued. Proof. (a) ⇒ (b) is obtained by (5.3) and (5.2). (b) ⇒ (c) is given by the formula u = u + u. (c) ⇒ (d) is given by u(x) = u(x) = u(−x). Finally, ( ) ⇒ ( ) λ ∈ σ ( ) ={λu,λu,...,λu } ϕ ∈ (λ − ) d a :ﬁx pp Lu 1 2 N and Ker Lu . Since both u and its Fourier transform uˆ are real-valued, we have [(ϕ)∨]∧(ξ) = ϕ(ξ)ˆ , where 1088 R. Sun

∨ ∨ ∨ ∨ (ϕ) (x) := ϕ(−x), ∀x,ξ ∈ R. Since Tu((ϕ) ) = (Tuϕ) ,wehave(ϕ) ∈ Ker(λ−Lu). {ϕu,ϕu,...,ϕu } H ( ) We choose the orthonormal basis 1 2 N in pp Lu as in formula (4.9). ˜ Corollary 2.7 yields that dimC Ker(λ − Lu) = 1. There exists θ j ∈ R such that ∨ θ˜ θ˜ ∧ (ϕu) = i j ϕu ⇔ (ϕu)∧(ξ) = i j (ϕu) (ξ) ∀ξ ∈ R ∀ = , ,..., j e j j e j , , j 1 2 N.Sowe θ˜ φu := ( i j )ϕu (φu)∧ set j exp 2 j , then its Fourier transform j is a real-valued function. Hence γ ( ) = φu,φu =− 1 ∂ [(φu)∧],(φu)∧ = j u Re G j j L2(R) 2π Im ξ j j L2(0,+∞) 0.

Lemma 5.15. The level set N is a real analytic Lagrangian submanifold of (UN ,ω). N Proof. The map γ : u ∈ UN → (γ1(u), γ2(u),...,γN (u)) ∈ R is a real analytic submersion by Proposition 5.10. So the level set N is a properly embedded real analytic submanifold of UN and dimR N = N. The classiﬁcation of the tan- η T (U ) ( ) = N 2 j gent space u N is given by Proposition 1.2.Ifu x j=1 2 η2 ,forsome x + j 2[x2−η2] η > T ( ) = N R u u( ) = j j 0, then we have u N j=1 f j , where f j x [ 2 η2]2 .We x + j ∧ −η |ξ| ( u) (ξ) =−π|ξ| j ω ω ( , ) = have f j 2 e . Then by deﬁnition of ,wehave u h1 h2 ˆ (ξ) ˆ (ξ) ˆ (ξ) ˆ (ξ) i h1 h2 ξ = i h1 h2 ξ ∈ R ∀ , ∈ T ( ) 2π R ξ d 2π R ξ d i , h1 h2 u N . Since the symplectic form ω is real-valued, we have ωu(h1, h2) = 0, for every h1, h2 ∈ Tu(N ). ( ) = = 1 U U Since dimR N N 2 dimR N , N is a Lagrangian submanifold of N .

5.5. The symplectomorphism property. Finally, we prove the assertion (b) in Theorem : (U ,ω) → ( × RN ,ν) # = −1 : 5.2, i.e. the map N N N is symplectic. We set N N × RN → U #∗ ω ω # N N ,let N denote the pullback of the symplectic form by N which is deﬁned by (1.22). The goal of this subsection is to prove that ν˜ := #∗ ω − ν = . N 0 (5.15)

N Lemma 5.16. For every u ∈ UN ,setp= N (u) ∈ N × R . Then we have ( )( ( )) = ∂ , ∀ = , ,..., . d N u X Ik u ∂αk p k 1 2 N (5.16)

Proof. Fix u ∈ UN and p = N (u), ∀h ∈ Tu(UN ),wehavedN (u)(h) ∈ N 1 2 N 1 2 N Tp(N × R ). For every smooth function f : p = (r , r ,...,r ; α ,α ,...,α ) ∈ N N × R → f (p) ∈ R,wehave(dN (u)(h)) f = d( f ◦ N )(u)(h) = ∂ ∂ N (dI (u)(h) f +dγ (u)(h) f ). For every k = 1, 2,...,N, we replace h j=1 j ∂r j p j ∂αj p ( ) ∈ T (U ) ∂ f = ( )( ( )) by X Ik u u N , thus Corollary 5.9 yields that ∂αk p d N u X Ik u f . ∞ Lemma 5.17. For every 1 ≤ j < k ≤ N, there exists a smooth function c jk ∈ C (N × RN ) such that ∂ j k c jk ν˜ = c jkdr ∧ dr , = 0, ∀ j, k, l = 1, 2,...,N, (5.17) ∂αl p 1≤ j

1 2 N 1 2 N N for every p = (r , r ,...,r ; α ,α ,...,α ) ∈ N × R . Complete integrability of the Benjamin–Ono equation on the multi-soliton manifolds 1089

Proof. The proof is divided into three steps. The ﬁrst step is to prove that for every N N p ∈ N × R and every V ∈ Tp(N × R ), ν˜ ( ∂ , ) = , ∀ = , ,..., . p ∂αl p V 0 l 1 2 N (5.18)

= # ( ) ∈ U = ( 1, 2,..., N ; α1,α2,...,αN ) l = In fact, let u N p N and p r r r ,so r ∂ ∂ rl (p) = I ◦ # (p). Then we have (#∗ ω) ( , V ) = ω (d# (p) , l N N p ∂αl p u N ∂αl p ∂ d# (p)(V )) = ω (X (u), d# (p)(V )) by (5.16). Thus (#∗ ω) ( , V ) = N u Il N N p ∂αl p ∂ −dI (u)(d# (p)(V )) =−d(I ◦ # )(p)(V ). On the other hand, ν ( , V ) = l N l N p ∂αl p N ( j ∧ α j )( ∂ , ) =− l ( )( ) ν˜ = #∗ ω − ν j=1 dr d ∂αl p V dr p V . Thus (5.18) is obtained by N .

ν˜ = ( α j ∧ αk j ∧ αk j ∧ k) Since we have 1≤ j

1 2 N It remains to show that c jk depends only on r , r ,...,r , for every 1 ≤ j < k ≤ N. The symplectic form ω is closed by Proposition 1.3 and ν = dκ is exact, where κ = ∂c N r j dα j .Sodν˜ = #∗ (dω) = 0. Precisely, we have N jk dαl ∧ j=1 N 1≤ j

Since the 2-form ν˜ is independent of α1,α2,...,αN , it sufﬁces to consider points N p = (r, α) ∈ N × R with α = 0. We shall prove ν˜ = 0 by introducing the Lagrangian submanifold N ×{0RN }.

End of the proof of formula (5.15). We have N ×{0RN }=N (N ), where N is the Lagrangian submanifold of (UN ,ω) deﬁned by (5.14). If q ∈ N ×{0RN },set v = #N (q) ∈ N ,wehave

N ∂ Tq (N ×{0RN }) = R = dN (v)(Tv(N )). (5.20) ∂r j q j=1 1090 R. Sun

1 2 N 1 2 N N For any point p = (r , r ,...,r ;α ,α ,...,α ) ∈ N × R and ∀V1, V2 ∈ ( ) ( ) ( ) ( ) T ( × RN ) V = N a m ∂ b m ∂ a m , b m ∈ R m = p N , where m j=1 j ∂r j p + j ∂αj p , j j , , = ( 1, 2,..., N ; , ,..., ) ∈ ×{ } , ∈ 1 2, we choose q r r r 0 0 0 N 0RN and W1 W2 N (m) ∂ T ( ×{0RN }), where W = = a , m = 1, 2. We set v = # (q) ∈ . q N m j 1 j ∂r j q N N ( ) = ( ) ν˜ ( , ) = ( (1) (2) − Since c jk p c jk q , then (5.17) yields that p V1 V2 1≤ j

6. Asymptotic Approximation

This section is dedicated to describing the asymptotic behavior of the multi-soliton solutions of (1.1).

Proof of corollary 1.11. Given u ∈ UN , we deﬁne M(u) = (M jj(u)δkj)1≤k, j≤N , where 2 t M jj is given in (1.18). Given (t, x) ∈ R ,wesetA = A(u, t, x) := M(u)−x − π V(u), −1 where V is given in Corollary 1.10. Then A(u, t, x) = (a j (u, t, x)δkj)1≤k, j≤N , where π a (x, t, u)−1 := γ (u) − x − t I (u) + i .WesetK(u) := M(u) − M(u), then j j π j I j (u) −1 ∀u0 ∈ UN ,wehaveu∞(t, x, u0) = 2ImA(u0, t, x) X(u0), Y (u0)CN .Ifu : t ∈ R → u(t) ∈ UN solves the BO equation (1.1) such that u(0) = u0 and |t| is large, then

−1 u(t, x) =u(t, x; u0) = 2Im(A(u0, t, x) + K(u0)) X(u0), Y (u0)CN −1 n −1 (6.1) =u∞(t, x; u0) +2Im −A(u0, t, x) K(u0) A(u0, t, x) X(u0), Y (u0)CN . n≥2

− N |I j (u0)| > A( , ) 1 ∞(− , ) ≤ ( | |− −|γ ( )|)2 Given R 0, we have u0 t Lx R R j=1 π t R j u0 + − 1 2 π2 → | |→ ∞ T( , , )> 2 0, when t + . So there exists u0 R N 0 such I j (u0) 2 −1 that 2N A(u , t) ∞(− , )K(u )CN×N < 1, if |t|≥T(u , R, N). Moreover, 0 Lx R R 0 0 A( , )−12 ≤ π N ( ) u0 t 2 (R) j=1 k j u0 . Then (6.1) yields that Lx −1 n −1 u(t) − u∞(t) 2 u ,N −A(u0, t) K(u0) A(u0, t) 2 Lx (−R,R) 0 Lx (−R,R) n≥2 −1 2 2 −1 u ,N A(u0, t) ∞ K(u0) × A(u0, t) 2 → 0 0 Lx (−R,R) CN N Lx (R) | |→ ∞ ∈ R T ( , , )> as t + .Givenx , similarly, there exists u0 x N 0 such that the ∈[T ( , , ), ∞) → 2 −A( , , )−1K( ) n series of functions t u0 x N + 2t Im n≥2 u0 t x u0 A( , , )−1 ( ), ( ) ∈ C 2 ( , ) = u0 t x X u0 Y u0 CN converges uniformly. Since limt→±∞ t u∞ t x N 2 A( , , )−1 =−πV( )−1 u(t,x) = j=1 ( )3 and limt→±∞ t u0 t x u0 ,wehave ( , ) 1+ k j u0 u∞ t x 2 −A( , , )−1K( ) n A( , , )−1 ( ), ( ) ( 2 ( , ))−1 2t Im n≥2 u0 t x u0 u0 t x X u0 Y u0 CN t u∞ t x → 1, as |t|→+∞ by formula (6.1). Complete integrability of the Benjamin–Ono equation on the multi-soliton manifolds 1091

Acknowledgments. The author would like to express his sincere gratitude towards his PhD advisor Prof. Patrick Gérard for his deep insight, generous advice and continuous encouragement. He is grateful to the editor and the anonymous referee for their careful reading of this manuscript and for their very useful and valuable suggestions. He also would like to thank warmly Prof. Yang Cao and Prof. Dr. Thomas Kappeler for useful thematic and extended discussions. The author thanks the hospitality of Université Paris-Saclay, where much of this research work was carried out during the last year of his PhD studies. He acknowledges the financial support by the grant ”ANAÉ” ANR-13-BS01-0010-03 of the ’Agence Nationale de la Recherche’ and by the PhD fellowship of École Doctorale de Mathématique Hadamard. The final version of this manuscript was completed while the author was visiting Institute for Analysis and Collaborative Research Center (CRC) 1173 of Karlsruhe Institute of Technology in November 2020. The author is grateful to Prof. Dr. Xian Liao for her welcoming invitation, warm hospitality and valuable comments which make the exposition of this manuscript gain considerably in clarity. It is also a pleasure to acknowledge the financial support by the Deutsche Forschungsgemeinschaft (DFG) through CRC 1173.

Funding Open Access funding enabled and organized by Projekt DEAL. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional afﬁliations.

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Communicated by H.-T. Yau