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

Home , Lax pair

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1 Introduction 2 1.1 Notation...... 5 1.2 Organizationofthispaper...... 6 1.3 Relatedwork ...... 7

2 The Lax operator 8 2.1 Unitaryequivalence ...... 11 2.2 SpectralanalysisI ...... 12 2.3 Conservationlaws ...... 14 2.4 Laxpairformulation ...... 16

3 The action of the shift semigroup 18

4 The manifold of multi-solitons 21 4.1 Diﬀerentialstructure...... 24 4.2 SpectralanalysisII...... 27 4.3 Characterizationtheorem ...... 30 4.4 The stability under the Benjamin–Ono ﬂow ...... 32

5 The generalized action–angle coordinates 34 5.1 Theassociatedmatrix ...... 36 5.2 Inversespectralformulas...... 37 5.3 Poissonbrackets ...... 39 5.4 Thediﬀeomorphismproperty ...... 42 5.5 ALagrangiansubmanifold...... 44 5.6 Thesymplectomorphismproperty...... 46

A Appendices 49 A.1 The simple connectedness of N ...... 50 A.2 Coveringmanifold ...... U .... 51

1 Introduction

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

2 2 ∂tu = H∂ u ∂x(u ), (t, x) R R, (1.1) x − ∈ × 2 2 where u is real-valued and H = isign(D) : L (R) L (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 andc sign(0) = 0, fˆ L2(R) denotes the Fourier–Plancherel transform of f L±2(R). We± adopt the convention Lp(R)= Lp(∈R, C). Its R-subspace consisting of all real-valued Lp- functions∈ is specially emphasized as Lp(R, R) throughout this paper. Equipped with the inner product 2 2 2 (f,g) L (R) L (R) f,g 2 = f(x)g(x)dx C, L (R) is a C-Hilbert space. ∈ × 7→ h iL R ∈ R

2 Derived by Benjamin [4] and Ono [49], this equation describes the evolution of weakly nonlinear internal long waves in a two-layer fluid. The BO equation is globally well-posed in every Sobolev spaces Hs(R, R), 1 s 0. (see Tao [63] for s 1, Burq–Planchon [8] for s> 4 , Ionescu–Kenig [33], Molinet–Pilod [43] and Ifrim–Tataru≥ [29] for s ≥0, etc.) Recall the scaling and translation invariances of equation (1.1): if ≥ 2 u = u(t, x) is a solution, so is uc,y : (t, x) cu(c t,c(x y)). A smooth solution u = u(t, x) is called a 7→ − solitary wave of (1.1) if there exists C∞(R) solving the following non local elliptic equation R ∈ 2 H ′ + =0, (x) > 0 (1.3) R R − R R and u(t, x) = c(x y ct), where c(x) = c (cx), for some c > 0 and y R. The unique (up to translation) solutionR − of equation− (1.3)R is given byR the following formula ∈ 2 (x)= , x R, (1.4) R 1+ x2 ∀ ∈ in Benjamin [4] and Amick–Toland [2] for the uniqueness statement. Inspired from the complete classification of solitary waves of the BO equation, we introduce the main object of this paper. N Definition 1.1. A function of the form u(x) = c (x xj ) is called an N-soliton, for some j=1 R j − positive integer N N+ := Z (0, + ), where cj > 0 and xj R, for every j = 1, 2, ,N. Let 2 ∈ ∞ P ∈ ··· N L (R, R) denote the subset consisting of all the N-solitons. U ⊂ T In the point of view of topology and differential manifolds, the subset N is a simply connected, real analytic, embedded submanifold of the R-Hilbert space L2(R, R). It has realU dimension 2N. The tangent space to N at an arbitrary N-soliton is included in an auxiliary space U := h L2(R, (1 + x2)dx): h(R) R, h =0 , (1.5) T { ∈ ⊂ R } Z ˆ ˆ ω 2 ω i h1(ξ)h2(ξ) in which a 2-covector Λ ( ∗) is well defined by (h1,h2)= 2π R ξ dξ, for every h1,h2 , ∈ T 2 ∈ T by Hardy’s inequality. We define a translation-invariant 2-form ω : u N ω Λ ( ∗), endowed R ∈ U 7→ ∈ T with which N is a symplectic manifold. The tangent space to N at u N is denoted by u( N ). For U U ∈U T U every smooth function f : N R, its Hamiltonian vector field Xf X( N ) is given by U → ∈ U Xf : u N ∂x uf(u) u( N ), ∈U 7→ ∇ ∈ T U where uf(u) denotes the Fréchet derivative of f, i.e. df(u)(h)= h, uf(u) 2 , for every h u( N ). ∇ h ∇ iL ∈ T U The Poisson bracket of f and another smooth function g : N R is defined by U → f,g : u N ωu(Xf (u),Xg(u)) = ∂x uf(u), ug(u) 2 R. { } ∈U 7→ h ∇ ∇ iL ∈ Then the BO equation (1.1) in the N-soliton manifold ( N ,ω) can be written in Hamiltonian form U 1 1 3 ∂tu = XE(u), where E(u)= D u,u − 1 1 u . (1.6) 2h| | iH 2 ,H 2 − 3 R Z The Cauchy problem of (1.6) is globally well-posed in the manifold N (see proposition 4.9). Inspired from the construction of Birkhoff coordinates of the space-periodic BOU equation discovered by Gérard– Kappeler [19], we want to show the complete integrability of (1.6) in the Liouville sense.

1 2 N N j j+1 Let ΩN := (r , r , , r ) R : r < r < 0, j = 1, 2, ,N 1 denote the subset of actions {N j ··· j ∈ ∀ ··· − }RN and ν = j=1 dr dα denotes the canonical symplectic form on ΩN . The main result of this paper is stated as follows.∧ × P

3 N Theorem 1. There exists a real analytic symplectomorphism ΦN : ( N ,ω) (ΩN R ,ν) such that U → × N 1 1 2 N 1 2 N 1 j 2 E Φ− (r , r , , r ; α , α , , α )= r . (1.7) ◦ N ··· ··· −2π | | j=1 X Remark 1.2. A consequence of theorem 1 is that N is simply connected. In fact the manifold N can be interpreted as the universal covering of the manifoldU of N-gap potentials for the Benjamin–OnoU equation on the torus as described by G´erard–Kappeler in [19]. We refer to section A for a direct proof of these topological facts, independently of theorem 1. N Remark 1.3. Then ΦN : u N (I1(u), I2(u), , IN (u); γ1(u),γ2(u), ,γN (u)) ΩN R introduces the generalized action–angle∈ U 7→ coordinates of th···e BO equation in the N···-soliton manifold,∈ × i.e.

Ik(u) Ik, E (u)=0, γk, E (u)= , u N . (1.8) { } { } π ∀ ∈U Theorem 1 gives a complete description of the orbit structure of the ﬂow of equation (1.6) up to real k bi-analytic conjugacy. Let u : t R u(t) N denote the solution of equation (1.6), r (t)= Ik u(t) ∈k 7→ ∈U ◦ denotes action coordinates and α (t)= γk u(t) denotes the generalized angle coordinates, then we have ◦ rk(0)t rk(t)= rk(0), αk(t)= αk(0) , k =1, 2, ,N. (1.9) − π ∀ ···

We refer to deﬁnition 5.1 and theorem 5.2 for a precise description of ΦN . 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 ΦN , denoted by

2πi Ik (u) , if j = k, N N Ik (u) Ij (u) Ij (u) M : u N (Mkj (u))1 j,k N C × ,Mkj (u)= − 6 (1.10) ≤ ≤ πi ∈U 7→ ∈ γj (u)+ q, if j = k,  Ij (u) where Ik,γk : R is given by remark 1.3. Then N has the following polynomial characterization. U → U  Proposition 1.4. A real-valued function u N if and only if there exists a monic polynomial Qu ′ ∈ U Qu ∈ C[X] of degree N, whose roots are contained in the lower half-plane C and u = 2Im . Precisely, Qu − Qu N N − is unique and is the characteristic polynomial of the matrix M(u) C × deﬁned by (1.10). ∈ N An N-soliton is expressed by u(x) = c (x xj ) if and only if its translation–scaling parameters j=1 R j − 1 CN xj cj− i 1 j N are the roots of the characteristic polynomial Qu(X) = det(X M(u)), whose { − } ≤ ≤ ⊂ − P − N coeﬃcients are expressed in terms of the action–angle coordinates (Ij (u),γj (u))1 j N ΩN R . Proposition 1.4 is restated with more details in proposition 4.1, formula (5.11) and≤ theorem≤ ∈ 4.8×which gives a spectral characterization of N . If u : t R u(t) N solves the BO equation (1.1), then we have the following explicit formula U ∈ 7→ ∈ U

t 1 u(t, x) = 2Im M(u ) (x + V(u )) − X(u ), Y (u ) CN , (t, x) R R, (1.11) h 0 − π 0 0 0 i ∈ × N T N N where the inner product of C is X, Y CN = X Y , for every u N , the matrix V(u) C × and the vectors X(u), Y (u) CN are deﬁnedh i by ∈ U ∈ ∈ I (u) T 1 √2πX(u) = ( I1(u) , I2(u) , , IN (u) ), I2(u) | | | | ··· | | V 1 (u)= . . − T 1 1 1  .  √2π Y (u) = (p I1(u) −p, I2(u) − ,p , IN (u) − ), . I u | | | | ··· | |  N ( )  p p p   4 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) = 1 if x A and 2 ⊂ 2 ∈ 1A(x) = 0 if x X A. Recall that H : L (R) L (R) denotes the Hilbert transform given by (1.2). Set ∈ \ 2 →2 2 Id 2 R (f)= f, for every f L (R). Let Π : L (R) L (R) denote the Szeg˝oprojector, deﬁned by L ( ) ∈ →

Id 2 R + iH L ( ) ˆ 2 Π := Πf(ξ)= 1[0,+ )(ξ)f(ξ), ξ R, f L (R). (1.12) 2 ⇐⇒ ∞ ∀ ∈ ∀ ∈ If O is an open subset of C, we denotec 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 and C = z C : Imz < 0 p { ∈ } − { ∈ } respectively. For every p (0, + ], we denote by L+ to be the Hardy space of holomorphic functions on p ∈ ∞ C+ such that L = g Hol(C+): g Lp < + , where + { ∈ k k + ∞}

1 p p g Lp = sup g(x + iy) dx , if p (0, + ), (1.13) k k + y> R | | ∈ ∞ 0 Z and g ∞ = sup g(z) . A function g L is called an inner function if g = 1 on R. When L+ z C+ +∞ k k ∈ | | ∈ 2 2 R| | p = 2, the Paley–Wiener theorem yields the identiﬁcation between L+ and Π[L ( )]:

2 1 2 2 2 L = − [L (0, + )] = f L (R) : suppfˆ [0, + ) = Π(L (R)), + F ∞ { ∈ ⊂ ∞ } where : f L2(R) fˆ L2(R) denotes the Fourier–Plancherel transform. Similarly, we set 2 F ∈ 2 7→R ∈ s 2 s R L = (IdL2(R) Π)(L ( )). Let the filtered Sobolev spaces be denoted as H+ := L+ H ( ) and H−s := L2 Hs−(R), for every s 0. − − ≥ T The domainT of definition of an unbounded operator on some Hilbert space is denoted by D( ) . Given another operator on D( ) such that A(D( )) D( ) and E(D( )) D( ), theirA ⊂E Lie bracket is an operator definedB on BD(⊂) E D( ) ,A whichA is given⊂ B by B B ⊂ A A B ⊂E T [ , ] := . (1.14) A B AB − BA If the operator is self-adjoint, let σ( ) denote its spectrum, σ ( ) denotes the set of its eigenval- A A pp A ues and σ ( ) denotes its continuous spectrum. Then σ ( ) σ ( ) = σ(A) R. Given two cont A cont A pp A ⊂ C-Hilbert spaces 1 and 2, let B( 1, 2) denote the C-Banach space of all bounded C-linear transformations , equippedE E with the uniformE E norm. S E1 →E2

Given a smooth manifold M of real dimension N, let C∞(M) denote all smooth functions f : M R and the set of all smooth vector fields is denoted by X(M). The tangent (resp. cotangent) space to M→ at N R p M is denoted by p(M) (resp. p∗(M)). Given k , the -vector space of smooth k-forms on M is ∈ k T T ∈ k denoted by Ω (M). Given a R-vector space V, we denote by Λ (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. If N is another smooth manifold, F : N M is a smooth map and A is a smooth covariant k-tensor field on M, the pullback of A by F → is denoted by F∗A, which is a smooth k-tensor field on N defined by p N, j =1, 2, , k, ∀ ∈ ∀ ···

(F∗A)p(v , v , , vk)= AF (dF(p)(v ), dF(p)(v ), , dF(p)(vk)) , vj p(N). (1.15) 1 2 ··· (p) 1 2 ··· ∀ ∈ T

5 Given a positive integer N, let C N 1[X] denote the C-vector space of all polynomials with complex ≤ − coefficients whose degree is no greater than N 1 and CN [X] = C N [X] C N 1[X] consists of all R − R ≤ \ C≤ − polynomials of degree exactly N. + = [0, + ) and +∗ = (0, + ). D(z, r) denotes the open disc of radius r> 0, whose center is z C. ∞ ∞ ⊂ ∈ 1.2 Organization of this paper The construction of action–angle coordinates for the BO equation (1.6) mainly relies on the Lax pair formulation ∂tLu = [Bu,Lu], discovered by Nakamura [45] and Bock–Kruskal [6]. Section 2 is dedicated 1 2 to the spectral analysis of the Lax operator Lu : h H+ i∂xh Π(uh) L+ given by definition 2.1 for general symbol u L2(R, R), where Π denotes∈ the Szeg˝oprojector7→ − − given∈ in (1.12) and the Hardy 2 ∈ 2 space L+ is defined in (1.13). Lu is an unbounded self-adjoint operator on L+ that is bounded from 2 below, it has essential spectrum σess(Lu) = [0, + ). If x xu(x) L (R) in addition, every eigenvalue is negative and simple, thanks to an identity firstly∞ found7→ by Wu [∈65]. Then we introduce a generating function which encodes the entire BO hierarchy,

1 λ(u)= (Lu + λ)− Πu, Πu 2 , if λ C σ( Lu), (1.16) H h iL ∈ \ − in deﬁnition 2.9. It provides a sequence of conservation laws controlling every Sobolev norms.

2 In section 3, we study the shift semigroup (S(η)∗)η 0 acting on the Hardy space L+, where S(η)f = eηf iηx ≥ and eη(x) = e . Then a weak version of Beurling–Lax theorem can be obtained by solving a linear 2 differential equation with constant coefficients. Every N-dimensional subspace of L+ that is invariant C d ≤N−1[X] under its infinitesimal generator G = i dη η=0+ S(η)∗ is of the form Q , for some monic polynomial Q whose roots are contained in the lower half-plane C .

− In section 4, the real analytic structure and symplectic structure of the N-soliton subset N are established U at first. Then we continue the spectral analysis of the Lax operator Lu, u N . Lu has N simple eigenvalues λu < λu < < λu < 0 and the Hardy space L2 splits as ∀ ∈ U 1 2 ··· N + C 2 2 ≤N−1[X] L = H (Lu) H (Lu), H (Lu)= H (Lu)=ΘuL , H (Lu)= . (1.17) + cont pp cont ac + pp Qu M Qu where Qu denotes the characteristic polynomial of u given by proposition 1.4 and Θu = is an inner Qu function on the upper half-plane C+. Proposition 1.4 is proved by identifying M(u) in (1.10) as the matrix u u u u u of the restriction G Hpp(Lu) associated to the spectral basis ϕ1 , ϕ2 , , ϕN , where ϕj Ker(λj Lu) u | u { ··· } ∈ − such that ϕ 2 = 1 and uϕ > 0. The generating function λ in (1.16) can be identified as the k j kL R j H Borel–Cauchy transform of the spectral measure of Lu associated to the vector Πu, which yields the in- R variance of N under the BO flow in H∞(R, R). Hence (1.6) is a globally well-posed Hamiltonian system U on N . U Section 5 is dedicated to completing the proof of theorem 1. The generalized angle-variables are the real u u R parts of the diagonal elements of the matrix M(u), i.e. γj : u N Re Gϕj , ϕj L2 and the action- u R ∈U 7→ h i ∈ λ variables are Ij : u N 2πλj . Thanks to the Lax pair formulation dL(u)(X λ (u)) = [Bu ,Lu], ∈ U 7→ 1 ∈2 R λ H 2 where L : u N Lu B(H+,L+) is -affine and Bu is some skew-adjoint operator on L+, we have the following∈U formulas7→ of∈ Poisson brackets,

2π λj ,γk = 1j k, γj ,γk =0 on N , 1 j, k N. (1.18) { } = { } U ≤ ≤

6 N which implies that ΦN : u N (I (u), I (u), , IN (u); γ (u),γ (u), ,γN (u)) ΩN R is a ∈ U 7→ 1 2 ··· 1 2 ··· ∈ × real analytic immersion. The diﬀeomorphism property of ΦN is given by Hadamard’s global inverse ′ Qu theorem. The inverse spectral formula Πu = with Qu(X) = det(X G H (L )), which is re- Qu − | pp u stated as formula (5.11), implies the explicit formula (1.11) of all multi-soliton solutions of the BO equation (1.1) and (5.11) provides an alternative proof of the injectivity of ΦN . Finally, we show that RN ΦN : ( N ,ω) (ΩN ,ν) is a symplectomorphism by restricting the 2-form ω ΦN∗ ν to a special U → × N 1 − Lagrangian submanifold ΛN := γ− (0) N . j=1 j ⊂U T In appendix A, we establish the simple connectedness of N and a covering map from N to the manifold of N-gap potentials from their constructions without usingU the integrability theorems.U

1.3 Related work The BO equation has been extensively studied for nearly sixty years in the domain of partial differential equations. We refer to Saut [60] for an excellent account of these results. Besides the global well-posedness problem, various properties of its multi-soliton solutions has been investigated in details. Matsuno [41] has found the explicit expression of multi-soliton solutions of (1.1) by following the bilinear method of Hirota [26]. The multi-phase solutions (periodic multi-solitons) have been constructed by Satsuma– Ishimori [58] at first. We point out the work of Amick–Toland [2] on the characterization of 1-soliton solutions which can also be revisited by theorem 1 and proposition 1.4. In Dobrokhotov–Krichever [10], the multi-phase solutions are constructed by finite zone integration and they have also established an inversion formula for multi-phase solutions. Compared to their work, we give a geometric description of the inverse spectral transform by proving the real bi-analyticity and the symplectomorphism property of the action–angle map. Furthermore, the inverse spectral formula

Qu′ (x) R Πu(x)= i ,Qu(x) = det(x G Hpp(Lu)) = det(x M(u)), x . (1.19) Qu(x) − | − ∀ ∈ provides a spectral connection between the Lax operator Lu and the inﬁnitesimal generator G. The idea of introducing generating function λ has also been used for the quantum BO equation in Nazarov– Sklyanin [46]. Their method has alsoH been developed by Moll [44] for the classical BO equation. The asymptotic stability of soliton solutions and of solutions starting with sums of widely separated soliton proﬁles is obtained by Kenig–Martel [34].

Concerning the investigation of integrability for the BO equation on R besides the discovery of Lax pair formulation, we mention the pioneering work of Ablowitz–Fokas [1], Coifman–Wickerhauser [9], Kaup– Matsuno [35] and Wu [65, 66] for the inverse scattering transform. In the space-periodic regime, the BO T 2 T 2 T R equation on the torus admits global Birkhoff coordinates on Lr,0( ) := v L ( , ): T v = 0 in Gérard–Kappeler [19]. We refer to Gérard–Kappeler–Topalov [20] to see that{ ∈ the Birkhoff coordinates} s T R s T R of the BO equation on the torus can be extended to a larger Sobolev space Hr,0( ) := v H ( , ): 1 { ∈ T T v = 0 , for every 2

Moreover, the cubic Szeg˝oequation both on T (see G´erard–Grellier [15, 16, 17, 18]) and on R (see Pocov- nicu [51, 52]) admit global (generalized) action–angle coordinates on all ﬁnite-rank generic rational func-

7 T R tion manifolds, denoted respectively by (N)gen and (N)gen. Moreover, the cubic Szeg˝oequation both on T and on R have inverse spectralM formulas whichM permit the Szeg˝oﬂows to be expressed ex- plicitly in terms of time-variables and initial data without using action–angle coordinates. The shift semigroup (S(η)∗)η 0 and its inﬁnitesimal generator G are also used in Pocovnicu [52] to establish the integrability of the≥ cubic Szeg˝oequation on the line.

The BO equation admits an inﬁnite hierarchy of conservation laws controlling every Hs-norm (see N 1 Ablowitz–Fokas [1], Coifman–Wickerhauser [9] in the case 2s and Talbut [62] in the case 2

Throughout this paper, the main results of each section are stated at the beginning. Their proofs are left inside the corresponding subsections.

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), discovered by Nakamura [45] and Bock–Kruskal [6]. Then we describe the location and revisit the simplicity of eigenvalues of Lu. At last, we introduce a generating functional λ which encodes the H entire BO hierarchy. The equation ∂tu = ∂x u λ(u) also enjoys a Lax pair structure with the same Lax ∇ H operator Lu. 2 R R 2 Definition 2.1. Given u L ( , ), its associated Lax operator Lu is an unbounded operator on L+, ∈ 1 2 given by Lu := D Tu, where D: h H+ i∂xh L+ and Tu denotes the Toeplitz operator of symbol − 1 ∈ 2 7→ − ∈ 2 R 2 u, defined by Tu : h H+ Π(uh) L+, where the Szeg˝oprojector Π: L ( ) L+ is given by (1.12). ∈ 27→ ∈ → We set Bu := i(T D u Tu ). | | − 2 Both D and Tu are densely defined symmetric operators on L+ and Tu(h) L2 u L2 h L∞ , for every 1 2 R R k k ≤k k k k h H+ and u L ( , ). Moreover, the Fourier–Plancherel transform implies that D is a self-adjoint ∈ 2 ∈ 1 operator on L+, whose domain of definition is H+. 2 R R 2 Proposition 2.2. If u L ( , ), then Lu is an unbounded self-adjoint operator on L+, whose domain ∈ 1 of definition is D(Lu) = H+. Moreover, Lu is bounded from below. The essential spectrum of Lu is C2 2 σess(Lu) = σess(D) = [0, + ) and its pure point spectrum satisfies σpp(Lu) [ 4 u L2 , + ), where 1 ∞ ⊂ − k k ∞ 4 D f L2 C = inf 1 k| | k denotes the Sobolev constant. f H 0 f 4 ∈ +\{ } k kL Thanks to an identity firstly found by Wu [65] in the negative eigenvalue case, we show the simplicity of 2 2 the pure point spectrum σ (Lu), if u L (R, (1 + x )dx) is real-valued. pp ∈ Proposition 2.3. Assume that u L2(R; R) and x xu(x) L2(R). For every λ R and ϕ 1 ∈ 1 7→ ∈ ∈ ∈ Ker(λ Lu), we have uϕ C (R) H (R) and the following identity holds, − ∈ 2 T 2 c uϕ = 2πλ ϕ . (2.1) R − R | | Z Z

Thus σ (Lu) ( , 0) and for every λ σ (Lu), we have pp ⊂ −∞ ∈ pp 1 1 R 1 R 2 R Ker(λ Lu) ϕ H+ :ϕ ˆ R+ C ( +) H ( +) and ξ ξ[ˆϕ(ξ)+ ∂ξϕˆ(ξ)] L ( +) . (2.2) − ⊂{ ∈ | ∈ 7→ ∈ } \ 8 2 2 Corollary 2.4. Assume that u L (R; R) and x xu(x) L (R). Then every eigenvalue of Lu is ∈ 7→ ∈ C2 u 2 L2 simple. If u L∞(R) in addition, then σ (Lu) is a finite subset of [ k k , 0). ∈ pp − 4 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 R Z Then identity (2.1) yields that Ker(A) = 0 . Thus V = V/Ker(A) = Im(A) ֒ C. So we have { } ∼ ∼ → dimC(Vλ) = 1. When u L∞(R) in addition, the finiteness of σpp(Lu) ( , 0) is given by Theorem 1.2 of Wu [65]. ∈ −∞ T We recall some known results of global well-posedness of the BO equation on the line. Proposition 2.5. For every s 0, the Fréchet space C(R,Hs(R)) is endowed with the topology of ≥ uniform convergence on every compact subset of R. There exists a unique continuous mapping u0 Hs(R) u C(R,Hs(R)) such that u solves the BO equation (1.1) with initial datum u(0) = u . ∈ 7→ ∈ 0 Proof. See Tao [63], Burq–Planchon [8], Ionescu–Kenig [33], Molinet–Pilod [43], Ifrim–Tataru [29] etc.

n n Proposition 2.6. For every n N, if u H 2 (R, R), let u : t R u(t) H 2 (R, R) solves equation ∈ 0 ∈ ∈ 7→ ∈ (1.1) with initial datum u(0) = u0, then C( u0 n ) := supt R u(t) n < + . k kH 2 ∈ k kH 2 ∞ Proof. See Ablowitz–Fokas [1], Coifman–Wickerhauser [9]. 2 R R 2 1 When u H ( , ), the Toeplitz operators T D u and Tu are bounded both on L+ and on H+. So Bu is ∈ 2 | | 1 a bounded skew-adjoint operator both on L+ and on H+. Proposition 2.7. Let u : t R u(t) H2(R, R) denote the unique solution of equation (1.1), then ∈ 7→ ∈ 1 2 ∂tL = [B ,L ] B(H ,L ), t R. (2.3) u(t) u(t) u(t) ∈ + + ∀ ∈ Let U : t U(t) B(L2 ) := B(L2 ,L2 ) denote the unique solution of the following equation 7→ ∈ + + + U ′(t)= B U(t),U(0) = Id 2 , (2.4) u(t) L+ if u : t R u(t) H2(R, R) denote the unique solution of equation (1.1). The system (2.4) is globally ∈ 7→ 2∈ well-posed in B(L+), thanks to proposition 2.6, the following estimate 2 2 2 Bu(h) 2 . ( u 2 + u 1 ) h 2 , h L , u H (R, R). k kL k kH k kH k kL ∀ ∈ + ∀ ∈ and a classical Cauchy theorem (see for instance lemma 7.2 of Sun [61]). Since Bu∗ = Bu, the operator U(t) is unitary for every t R. Thus, the Lax pair formulation (2.3) of the BO equation− (1.1) is equivalent ∈ to the unitary equivalence between Lu(t) and Lu(0),

1 2 L = U(t)L U(t)∗ B(H ,L ). (2.5) u(t) u(0) ∈ + + On the one hand, the spectrum of Lu is invariant under the BO ﬂow. In particular, we have σpp(Lu(t))= σpp(Lu(0)). 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 ∈ 2 N 2 f(L )= U(t)f(L )U(t)∗ B(L ), p(L )= U(t)p(L )U(t)∗ B(H ,L ), (2.6) u(t) u(0) ∈ + u(t) u(0) ∈ + + where N is the degree of the polynomial p.

9 n Proposition 2.8. Given n N, let u : t R u(t) H 2 (R, R) denote the solution of equation (1.1), we set ∈ ∈ 7→ ∈ n En(u) := L Πu, Πu − n n . (2.7) h u iH 2 ,H 2 1 Then En(u(t)) = En(u(0)), for every t R. In particular, E1 = E on H 2 (R, R), where the energy functional E is given by (1.6). ∈

2 Deﬁnition 2.9. Given u L (R, R) and λ C σ( Lu), the C-linear transformation λ+Lu is invertible 1 2 ∈ ∈ \ − 1 in B(H+,L+) and the generating function is deﬁned by λ(u) = (Lu + λ)− Πu, Πu L2 . The subset R 2 R R 2 2 H R h R i 2 R R := (λ, u) L ( , ):4λ > C u L2 is open in the -Banach space L ( , ), where X { ∈ × k k } 1 × 2 2 4 C u D f L2 L2 1 k| | k k k the Sobolev constant is given by C = inff H 0 f and we have σ(Lu) [ 4 , + ) by ∈ +\{ } L4 ⊂ − ∞ proposition 2.2. k k

1 The map (λ, u) λ(u)= (Lu + λ)− Πu, Πu 2 R is real analytic. ∈ X 7→ H h iL ∈ Proposition 2.10. Let u : t R u(t) L2(R, R) denote the solution of the BO equation (1.1) and C2 u 2 ∈ 7→ ∈ (0) L2 we choose λ> k k , then λ(u(t)) = λ(u(0)), for every t R. 4 H H ∈ 2 Given (λ, u) , there exists a neighbourhood of u in L (R, R), denoted by u such that the restriction ∈ X V λ : v u λ(v) R is real analytic. The Fr´echet derivative of λ at u is computed as follows, H ∈V 7→ H ∈ H 2 2 d λ(u)(h)= wλ, Πh 2 + wλ, Πh 2 + Thwλ, wλ 2 = h, wλ + wλ + wλ 2 , h L (R, R). H h iL h iL h iL h | | iL ∀ ∈ 1 1 where wλ H is given by wλ wλ(u) wλ(x, u) = [(Lu + λ)− Π]u(x), for every x R. Then ∈ + ≡ ≡ ◦ ∈ 2 u λ(u)= wλ(u) + wλ(u)+ wλ(u). (2.8) ∇ H | |

Given (λ, u ) fixed, the pseudo-Hamiltonian equation associated to Hλ is defined by 0 ∈ X 2 ∂tu = ∂x u λ(u)= ∂x wλ(u) + wλ(u)+ wλ(u) , u(0) = u . (2.9) ∇ H | | 0 2 2 2 There exists an open subset u of L (R, R) such that v u ∂x wλ(v) + wλ(v)+ wλ(v) L is V 0 ∈V 0 7→ | | ∈ + real analytic and u u . Hence (2.9) admits a local solution by Cauchy–Lipschitz theorem. 0 ∈V 0 Remark 2.11. The word ’pseudo-Hamiltonian’ is used here because no symplectic form has been defined 2 on L (R, R) until now. In section 4, we show that ∂x f(u) is exactly the Hamiltonian vector field of the ∇ smooth function f : N R with respect to the symplectic form ω on the N-soliton manifold N defined in (4.2). U → U

Proposition 2.12. Given (λ, u0) ﬁxed, there exists ε > 0 such that (λ, u(t)) , for every t ( ε,ε), where u : t ( ε, +ε) u∈(t X) L2(R, R) denotes the local solution of (2.9) with∈ Xinitial datum ∈ − ∈ − 7→ ∈ u(0) = u0. We have

λ λ ∂tL = [B ,L ], where B := i(T T + T + T ), if (λ, v) . (2.10) u(t) u(t) u(t) v wλ(v) wλ(v) wλ(v) wλ(v) ∈ X λ i.e. (Lu,Bu ) is a Lax pair of equation (2.9).

2 1 Remark 2.13. The Toeplitz operators Twλ(v) and Twλ(v) are bounded both on L+ and on H+, so is the skew-adjoint operator Bλ, if (λ, v) . v ∈ X

10 1 R R 4 ˜ 1 ˜ 1 ǫ For every u H∞( , ) and ǫ (0, 2 2 ), we set ǫ(u) := 1 (u) and Bǫ,u := Bu . Recall that C u ǫ ǫ ǫ ∈ ∈ L2 H H n k k En(u)= L Πu, Πu 2 , we have the following Taylor expansion h u iL M n M 1 1 M ˜ǫ(u)= ( ǫ) En(u) ( ǫ) (Lu + )− Πu,L Πu 2 , M N. (2.11) H − − − h ǫ u iL ∀ ∈ k X=0 Proposition 2.12 then leads to a Lax pair formulation for the equations corresponding to the conservation laws in the BO hierarchy, dn ∂tLu = [ B˜ǫ,u,Lu], dǫn ǫ=0 n n d ˜ where now u evolves according to the pseudo-Hamiltonian ﬂow of En = ( 1) n ǫ. In the case − dǫ ǫ=0H n = 1, we have E = E and B = d B˜ . 1 u dǫ ǫ=0 ǫ,u

This section is organized as follows. In subsection 2.1, we recall some basic facts concerning unitarily equivalent self-adjoint operators on diﬀerent Hilbert spaces. The subsection 2.2 is dedicated to the proofs of proposition 2.2 and 2.3. Proposition 2.8 and 2.10 that concern the conservation laws are proved in subsection 2.3. Proposition 2.7 and proposition 2.12 that indicate the Lax pair structures are proved in subsection 2.4.

2.1 Unitary equivalence

Generally, if 1 and 2 are two Hilbert spaces, let be a self-adjoint operator deﬁned on D( ) 1 and be a self-adjointE operatorE deﬁned on D( ) A. Both and have spectral decompositionsA ⊂E B B ⊂E2 A B = H ( ) H ( ) H ( ), = H ( ) H ( ) H ( ). (2.12) E1 ac A sc A pp A E2 ac B sc B pp B M M M M If and are unitarily equivalent i.e. there exists a unitary operator : such that A B U E1 →E2

= ∗, D( )= D( ), (2.13) B UAU B U A then we have the following identiﬁcation result.

Proposition 2.14. The operators and have the same spectrum and Hxx( )= Hxx( ), for every xx ac, sc, pp . Moreover, for everyA boundedB borel function f : R C, fU( ) isA a boundedB operator on ∈{ } → A , f( ) is a bounded operator on , we have f( )= f( ) ∗. E1 B E2 B U A U Proof. If f is a bounded Borel function, ψ , consider the spectral measure of associated to the ∈ E1 A vector ψ 1, denoted by µψA. Similarly, we denote by µBψ the spectral measure of associated to the vector ψ∈E . Clearly, we have U B U ∈E2 R R supp(µψA) σ( ) , supp(µBψ) σ( ) . ⊂ A ⊂ U ⊂ B ⊂ 1 1 For every λ C σ( ) = C σ( ), formula (2.13) implies that (λ )− ∗ = (λ )− . So the Borel–Cauchy∈ transforms\ A of these\ B two spectral measures are the same.U − A U −B

dµψA(ξ) 1 1 dµBψ(ξ) = (λ )− ψ, ψ 1 = (λ )− ψ, ψ 2 = U . R λ ξ h − A iE h −B U U iE R λ ξ Z − Z −

11 R R 2 Both of these two spectral measures have finite total variations : µψA( ) = µBψ( ) = ψ 1 . Since every finite Borel measure is uniquely determined by its Borel–Cauchy transformU (see Theoremk kE 3.21 of H H Teschl [64] page 108), we have µψA = µBψ. So the restriction Hxx( ) : xx( ) xx( ) is a linear isomorphism, for every xx ac, sc, pp .U Finally, we use the definitionU| A of the spectralA → measuresB to obtain ∈{ }

f( )ψ, ψ 1 = f(ξ)dµψA(ξ)= f(ξ)dµBψ(ξ)= f( ) ψ, ψ 2 h A iE R R U h B U U iE Z Z We may assume that f is real-valued, so that f( ) is self-adjoint. The polarization identity implies that A f( )ψ, φ 1 = f( ) ψ, φ 2 , for every ψ, φ 1. So we obtain f( ) = f( ) ∗ in the case f is hreal-valuedA iE boundedh B BorelU U function.iE In the general∈ E case, it suﬃces to useB f =U RefA+UiImf.

2.2 Spectral analysis I

In this subsection, we study the essential spectrum and discrete spectrum of the Lax operator Lu by proving proposition 2.2 and 2.3. The spectral analysis of Lu such that u is a multi-soliton in deﬁnition 1.1, will be continued in subsection 4.2. Proof of proposition 2.2. For every h L2 , let µD denote the spectral measure of D associated to h, then ∈ + h + ˆ 2 ˆ 2 ∞ h(ξ) D 1[0,+ )(ξ) h(ξ) f(D)h,h 2 = fˆ(ξ)| | dξ = dµ (ξ)= ∞ | | dξ. h iL 2π ⇒ h 2π Z0 2 1 Thus we have σ(D) = σess(D) = σac(D) = [0, + ). If u L (R, R), we claim that u := Tu (D + i)− 2 ∞ ∈ P ◦ is a Hilbert–Schmidt operator on L+.

ˆ Recall that R = (0, + ). In fact, let F : h L2 h L2(R ) denotes the renormalized Fourier– +∗ + √2π +∗ ∞ ∈ 1 7→ ∈ 2 Plancherel transform, then u := F u F − is an operator on L (R∗ ). Then we have A ◦P ◦ + + ∞ uˆ(ξ η) ug(ξ)= Ku(ξ, η)g(η)dη, Ku(ξ, η) := − , ξ, η R∗ . A 2π(η + i) ∀ ∈ + Z0

u L2 Hence its Hilbert–Schmidt norm u (L2(R∗ )) K L2(R∗ R∗ ) k k . Since u is unitarily equiv- kA kHS + ≤k k +× + ≤ 2 P u 2 2 2 2 2 L2 alent to , we have 2 = λ = λ = 2 ∗ k k . u u (L ) λ σ( u) λ σ( u) u (L (R )) 4 A kP kHS + ∈ P ∈ A kA kHS + ≤ P P Then the symmetric operator Tu is relatively compact with respect to D and Weyl’s essential spectrum theorem (Theorem XIII.14 of Reed–Simon [54]) yields that σess(Lu)= σess(D) and Lu is self-adjoint with 1 D(Lu) = D(D) = H+. An alternative proof of the self-adjointness of Lu can be given by Kato–Rellich theorem (Theorem X.12 of Reed–Simon [53]) and the following estimate, for every f H1 , ∈ + 1 2 1 ∂xf L2 2π f L∞ fˆ 1 fˆ 2 √A + ∂xf 2 √A 2 fˆ 2 ∂xf 2 , A = k k . L L L − L L f 2 k k ≤k k ≤k k k k ≤ k k k k k kL r d u 2 d 2 L2 So Tu(f) 2 u 2 f L∞ ∂xf 2 + k k f 2 . k kL ≤k kL k k ≤ π k kL 4 k kL

12 1 2 2 2 Moreover, Tuf,f L2 = R u f u L2 f L4 C u L2 f L2 D f L2 holds by Sobolev embed- |h i 1 | | | | |≤k k 1 k k ≤ k k k k k| | k ding f 4 C D 4 f 2 , for every f H . Then Lu is bounded from below, precisely k kL ≤ k| | kL R ∈ + 1 C2 u 2 f 2 2 L2 L2 Luf,f 2 = D 2 f 2 Tuf,f 2 k k k k . h iL k| | kL − h iL ≥− 4 C2 u 2 2 L2 1 2 C 2 When λ< k k , the map Lu λ : H L is injective. Hence σ (Lu) [ u 2 , + ). − 4 − + → + pp ⊂ − 4 k kL ∞

Before the proof of proposition 2.3, we recall a lemma concerning the regularity of convolutions. Lemma 2.15. For every p (1, + ) and m,n N, we have ∈ ∞ ∈ m,p n, p m+n m+n,+ (W (R) W p−1 (R) ֒ C (R) W ∞(R). (2.14 ∗ → n, p \ m,p R p−1 R α For every f W ( ) W ( ), we have lim x + ∂x f(x)=0, for every α =0, 1, ,m + n. ∈ ∗ | |→ ∞ ··· Proof. In the case m = n = 0, it suffices use Hölder’s inequality and the density argument of the Schwartz class S (R) W m,p(R). In the case m = 0 and n = 1, recall that a continuous function whose weak- ⊂ 1 derivative is continuous is of class C and f, ϕ D(R)′,D(R) = f ϕˇ(0), we use the density argument of the test function class D(R) Lp(R). We concludeh i by induction on∗ n 1 and m N. ⊂ ≥ ∈ Remark 2.16. Identity (2.1) was firstly found by Wu [65] in the case λ < 0. We show that (2.1) still holds in the case λ 0. Hence the operator Lu has no eigenvalues in [0, + ). ≥ ∞ Proof of proposition 2.3. We choose u L2(R;(1+ x2)dx) such that u(R) R, λ R and ϕ L2 such ∈ ⊂ ∈ ∈ + that Lu(ϕ)= λϕ. Applying the Fourier–Plancherel transform, we obtain

uϕ(ξ)1ξ 0 = (ξ λ)ˆϕ(ξ) =: gλ(ξ). (2.15) ≥ − Sinceu ˆ H1(R) andϕ ˆ L2(R), their convolution uϕ = 1 uˆ ϕˆ C1(R) C (R), where C (R) de- ∈ ∈ c 2π ∗ ∈ 0 0 notes the uniform closure of Cc(R) with respect to the L∞(R)-norm, by lemma 2.15. Recall R+ = [0, + ). T ∞ c We claim that if λ< 0, thenϕ ˆ C1(R ); ∈ + if λ 0, thenϕ ˆ C(R ) C1(R λ ). ( ≥ ∈ + +\{ } In fact, if λ 0, we have gλ(λ) = 0. Otherwise, λ would beT a singular point ofϕ ˆ that preventsϕ ˆ from 2 ≥ R 1 2 R 1 R being a L function on +, because ξ ξ λ / L ( +). By using the fact g C ( +) (g is right → − ∈ ∈ diﬀerentiable at ξ = 0 and the derivative g′ is right continuous at ξ = 0), we have

gλ(ξ) gλ(λ) g′ (λ), if λ> 0; ϕˆ(ξ)= − λ ξ λ → g (0+), if λ = 0; − ( λ′ when ξ λ. Soϕ ˆ C(R+) and limξ + ϕˆ(ξ) = 0. Then we derive formula (2.15) with respect to ξ to get the→ following ∈ → ∞

ixu ϕˆ(ξ)= g′ (ξ) = (uϕ)′(ξ)=ϕ ˆ(ξ) + (ξ λ)(ϕ ˆ)′(ξ), ξ [0, + ) λ . (2.16) − ∗ λ − ∀ ∈ ∞ \{ } Thus we have c c d 2 2 2 [(ξ λ) ϕˆ(ξ) ]= ϕˆ(ξ) + 2Re[((ξ λ)(ϕ ˆ)′(ξ))ϕˆ(ξ)] = 2Re[(uϕ)′(ξ)ϕˆ(ξ)] ϕˆ(ξ) . (2.17) dξ − | | | | − − | | c 13 When λ< 0, it suﬃces to integrate equation (2.17) on [0, + ) and use the Plancherel formula ∞ + ∞ 2 (uϕ)′(ξ)ϕˆ(ξ)dξ = 2πi xu(x) ϕ(x) dx. − R | | Z0 Z We also use the fact (ξ λ) ϕˆ(ξ) 2 c= uϕ(ξ)ϕˆ(ξ) 0, as ξ + . Thus, − | | → → ∞ + + 2 ∞ d 2 2 ∞ 2 2 λ ϕˆ(0) = [(ξ λ) ϕˆ(ξ) ]dcξ =4πIm xu(x) ϕ(x) dx ϕˆ(ξ) dξ = 2π ϕ L2(R). | | dξ − | | R | | − | | − k k Z0 Z Z0 + ∞ When λ> 0, there may be some problem of derivability ofϕ ˆ at ξ = λ. We replace the integral 0 by λ ǫ + two integrals − and ∞, for some ǫ (0, λ). Set 0 λ+ǫ ∈ R (ǫ) :=λRϕˆ(0) 2 ǫRϕˆ(λ ǫ) 2 ǫ ϕˆ(λ + ǫ) 2 I | | − | − | − | | + λ+ǫ + λ+ǫ ∞ ∞ 2 2 =2Re (uϕ)′(ξ)ϕˆ(ξ)dξ (uϕ)′(ξ)ϕˆ(ξ)dξ ϕˆ(ξ) dξ + ϕˆ(ξ) dξ 0 − λ ǫ − 0 | | λ ǫ | | Z Z − ! Z Z − 2 2 c R c + Thanks to the continuity ofϕ ˆ on +, we have λ ϕˆ(0) = limǫ 0 (ǫ)= 2π ϕ L2(R). | | → I − k k When λ = 0, we use the same idea and integrate (2.17) over interval [ǫ, + ), for some ǫ> 0. Then ∞

+ + 2 ∞ ∞ 2 (ǫ) := ǫ ϕˆ(ǫ) = 2Re (uϕ)′(ξ)ϕˆ(ξ)dξ ϕˆ(ξ) dξ 0, J − | | − | | → Zǫ Zǫ as ǫ 0. So we always have → c 2 2 2π ϕ 2 R = λ ϕˆ(0) , if ϕ Ker(λ Lu). (2.18) − k kL ( ) | | ∈ − 2 2 As a consequence Lu has only negative eigenvalues, if the real-valued function u L (R, (1 + x )dx). ∈ Finally we use uϕ(0) = λϕˆ(0) to get identity (2.1). If λ σpp(Lu) and ϕ Ker(λ Lu) 0 , we want to prove that − ∈ ∈ − \{ } 2 ξ (1 + ξ )∂ξϕˆ(ξ) L (0, + ). (2.19) c 7→ | | ∈ ∞ 1 R 2 R 2 uˆ ϕˆ 1 R (In fact, since ϕ H+ ֒ L∞( ) and u L ( , (1 + x )dx), we have uϕ = 2∗π H ( ). Formula (2.15 yields that ξ ∈( λ +→ξ)ˆϕ(ξ) L2(R) and∈ we haveϕ ˆ L1(R). The hypothesis∈u L2(R, x2dx) implies that the convolution7→ | | term xu ∈ϕˆ L2(R). Since λ< 0,∈ we obtain (2.19) by using∈ formula (2.16). ∗ ∈ c 2.3 Conservation lawsc Proposition 2.8 and 2.10 are proved in this subsection. We begin with the following proposition. Proposition 2.17. If u : t R u(t) H2(R, R) denotes the unique solution of the BO equation (1.1), then we have ∈ 7→ ∈ 2 2 ∂tΠu(t)= B (Πu(t)) + iL (Πu(t)) L . (2.20) u(t) u(t) ∈ + 2 R R 2 1 Proof. For every u H ( , ) is real-valued, Bu is a bounded operator on both L+ and H+, Πu 1 ∈ 2 ∈ D(Lu)= H+. We haveu ˆ( ξ)= uˆ(ξ), u = Πu + Πu and D u = DΠu DΠu. Since DΠu L , we have − | | − ∈ − Π(ΠuDΠu) = Π(uDΠu). Thus the following two formulas hold,

2 2 2 Bu(Πu)= i(T D u Tu )(Πu)= i(Πu)(DΠu) iΠ(uDΠu) iTu (Πu) = Πu∂xΠu Π(u∂xΠu) iTu (Πu), | | − − − − − 2 2 2 2 2 iL (Πu)= iD Πu iTu(DΠu) iD Tu(Πu)+ iT (Πu)= i∂ Πu Tu(∂xΠu) ∂x[Tu(Πu)] + iT (Πu). u − − ◦ u − x − − u

14 Then we add them together to get the following

2 2 Bu(Πu)+ iL (Πu)= i∂ Πu 2Π[Πu∂xΠu + Πu∂xΠu + Πu∂xΠu] u − x − Finally we replace u by u(t), where u : t R u(t) H2(R, R) solves equation (1.1) to obtain (2.20). ∈ 7→ ∈ Proof of proposition 2.8. It suﬃces to prove (2.7) in the case u0 H∞(R, R). Then we use the density argument and the continuity of the ﬂow map ∈

u Hs(R) u C([ T,T ]; Hs(R)) with T > 0, s 0, 0 ∈ 7→ ∈ − ≥ R R s R R n in proposition 2.5. We choose u = u(t) H∞( , )= s 0 H ( , ), so the functions LuΠu, ∂tΠu and n n ∈ ≥ ∂t(L )Πu = [Bu,L ]Πu are in H∞(R, C). Thus u u T n n ∂tEn(u) = 2Re L Πu, ∂tΠu 2 + ∂t(L )Πu, Πu 2 . h u iL h u iL 2 Since Bu + iLu is skew-adjoint, we use formula (2.20) to get the following

n n 2 n 2Re L Πu, ∂tΠu 2 = [L ,Bu + iL ]Πu, Πu 2 = [L ,Bu]Πu, Πu 2 . h u iL h u u iL h u iL n Since (Lu,Bu) is also a Lax pair of the Benjamin–Ono equation (1.1), we have

n n ∂tEn(u)= ([L ,Bu]+ ∂t(L ))Πu, ∂tΠu 2 =0. h u u iL In the case n = 1, we assume that u H1(R, R). Since u = Πu + Πu, D u = DΠu DΠu and 3 ∈ 3 | | 2 − 2 R(Πu) = 0, we have D u,u L2 =2 DΠu, Πu L2 and R u =3 R(Πu + Πu) Πu =3 R u Πu . In the 1 h| | i h i | | | | general case u H 2 (R, R), we use the density argument. R ∈ R R R

Proof of proposition 2.10. It suﬃces to prove the case u(0) H∞(R, R) and we use the density argument. ∈ Let u : t u(t) H∞(R, R) solve equation (1.1). Since u(t) L2 = u(0) L2 by proposition 2.8 and 2 7→ 2 ∈ k k k k 4λ > C u(0) 2 , we have (λ, u(t)) , ∂tL = [B ,L + λ] and k kL ∈ X u(t) u(t) u(t) 1 1 1 ∂t λ(u) = 2Re (Lu + λ)− Πu, ∂tΠu 2 (Lu + λ)− ∂tLu(Lu + λ)− Πu, Πu 2 . (2.21) H h iL − h iL Formula (2.20) yields that

1 1 2 1 2Re (Lu + λ)− Πu, ∂tΠu 2 = [(Lu + λ)− ,Bu + iL ]Πu, Πu 2 = [(Lu + λ)− ,Bu]Πu, Πu 2 , h iL h u iL h iL 1 1 1 1 [(Lu + λ)− ,Bu]Πu, Πu 2 = BuΠu, (Lu + λ)− Πu 2 + (Lu + λ)Bu(Lu + λ)− Πu, (Lu + λ)− Πu 2 h iL h iL h iL 1 1 = (Lu + λ)− [Bu,Lu + λ](Lu + λ)− Πu, Πu 2 . h iL 2 Then (2.21) yields that ∂tHλ(u(t)) = 0. In the general case u(t) L (R, R), we proceed as in the proof of proposition 2.8 and use the continuity of the generating functional∈

2 2√λ λ : u v L (R, R): v 2 < λ(u) R. H ∈{ ∈ k kL C } 7→ H ∈

15 2.4 Lax pair formulation In this subsection, we prove proposition 2.12 and 2.7. The Hankel operators whose symbols are in 2 L (R) L∞(R) will be used to calculate the commutators of Toeplitz operators. We notice that the Hankel operators are C-anti-linear and the Toeplitz operators are C-linear. For every symbol v 2 R S R 1∈ L ( ) L∞( ), we deﬁne its associated Hankel operator to be Hv(h)= Thv = Π(vh), for every h H+. R 2 2 2 R ∈ If v L∞( ), then Hv : L+ L+ is a bounded operator. If v L ( ), then Hv may be an un- ∈S 2 → 1 ∈ 1 R bounded operator on L+ whose domain of deﬁnition contains H+. For every b H ( ), we have 1 ∈ 2 Tb(h) H1 + Hb(h) H1 . b H1 h H1 , for every h H+, so both Tb and Hb are bounded on L+ and on k 1 k k k k k k k ∈ H+. 2 2 Lemma 2.18. For every v, w L L∞(R) and u L (R), we have ∈ + ∈ 2 [Tv,Tw]= Hv Hw B(L ). (2.22) − ◦ ∈ + 1 2 If w H in addition, then we have Tu(w) L and ∈ + ∈ + 1 2 HT w = Tw H u + Hw Tu = Tu Hw + H u Tw B(H ,L ). (2.23) u ◦ Π ◦ ◦ Π ◦ ∈ + + 2 2 2 Proof. For every v, w L L∞(R) and h L , we have wh = Π(wh)+ Π(wh) L . Thus, ∈ + ∈ + ∈ + T 2 [Tv,Tw]h = Π(vΠ(wh) wΠ(vh)) = Π(vwh vΠ(wh) vwh)= Π(vΠ(wh)) = Hv Hw(h) L . − − − − − ◦ ∈ + 2 R 1 1 1 R Given u L ( ) and w H+, for every h H+, we have wh = Π(wh) + Π(wh) H ( ) and ∈ 1 ∈ ∈ 2 ∈ Hw(h),Tw(h) H . So Π(uΠ(wh)) = Π(Π(wh)Πu)= H u Tw(h) L and we have ∈ + Π ◦ ∈ + 2 HT w(h) = Π(Π(uw)h) = Π(uwh) = Π(uΠ(wh)+ uΠ(wh))=(Tu Hw + H u Tw)(h) L . u ◦ Π ◦ ∈ + 2 2 Similarly, we have uh = Π(uh)+ Π(uh) L (R) and Π(uh) = Π(hΠu)= H u(h) L . Thus, ∈ Π ∈ + 2 HT w(h) = Π(wuh) = Π(wΠ(uh)+ wΠ(uh)) = (Tw H u + Hw Tu)(h) L . u ◦ Π ◦ ∈ +

1 1 Lemma 2.19. Given (λ, u) given in deﬁnition 2.9, set wλ(u) = (Lu + λ)− Π(u) H , then ∈ X ◦ ∈ + 1 2 2 B [D Tu,Twλ(u)Twλ(u) + Twλ(u) + Twλ(u)]= TD[ wλ(u) +wλ(u)+wλ(u)] (H+,L+). (2.24) − | | ∈ 1 1 + + 1 1 Proof. We use abbreviation wλ := wλ(u) H+, then wλ H . If f ,g H+ and f −,g− H , then ∈ ∈ 2− ∈ ∈ − we have [T + ,T + ] = [T − ,T − ] = 0, because for every h L , we have f g f g ∈ + + + 2 T + [T + (h)] = f g h = T + [T + (h)], h L . f g g f ∀ ∈ + 2 and T − [T − (h)] = Π(f −Π(g−h)) = Π(f −g−h) = Π(g−Π(f −h)) = T − [T − (h)]. Since Πu L and f g g f ∈ + Πu L2 , we use Leibnitz’s rule and formula (2.22) to obtain that ∈ −

[D Tu,Tw + Tw ]=T w + T w [Tu,Tw ] [Tu,Tw ] − λ λ D λ D λ − λ − λ =T w + T w [T ,Tw ] [T u,Tw ] (2.25) D λ D λ − Πu λ − Π λ =T w + T w Hw H u + H uHw . D λ D λ − λ Π Π λ

16 Similarly, formula (2.22) implies that

[Tu,Twλ Twλ ] =[Tu,Twλ ]Twλ + Twλ [Tu,Twλ ]

=[TΠu,Twλ ]Twλ + Twλ [TΠu,Twλ ] (2.26) =Hw H uTw Tw H uHw . λ Π λ − λ Π λ 1 2 For every h H+, since wλ, Dwλ L , we have ∈ ∈ −

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

=TDwλ (Twλ h)+ Twλ (TDwλ h) 2 =Π[DwλΠ(wλh)+ wλΠ(Dwλh)] = Π[(wλDwλ + wλDwλ)h] L . ∈ + 1 2 2 B So [D,Twλ Twλ ]= TD wλ (H+,L+). We use formula (2.22) and Leibnitz’s Rule to obtain that | | ∈ 2 2 [D,Twλ Twλ ] = [D,Twλ Twλ ] [D,Hw ]= TD wλ HDwλ Hwλ + Hwλ HDwλ (2.27) − λ | | − 1 Recall that wλ = (λ + Lu)− Πu, then we have

Dwλ = Tu(wλ) λwλ + Πu. (2.28) − The formula (2.23) and (2.28) yield that

H w Tw H u = HT w λHw + H u Tw H u = Hw Tu λHw + H u (2.29) D λ − λ Π u λ − λ Π − λ Π λ − λ Π and H w H uTw = HT w λHw + H u H uTw = TuHw λHw + H u. (2.30) D λ − Π λ u λ − λ Π − Π λ λ − λ Π We use formulas (2.26), (2.27), (2.29) and (2.30) to get the following formula

[D Tu,Tw Tw ] − λ λ 2 =TD w (HDwλ Twλ HΠu)Hwλ + Hwλ (HDwλ HΠuTwλ ) | λ| − − − 2 2 (2.31) 2 =TD w (Hwλ TuHwλ λHw + HΠuHwλ ) + (Hwλ TuHwλ λHw + Hwλ HΠu) | λ| − − λ − λ =TD w 2 HΠuHwλ + Hwλ HΠu | λ| − At last, we combine formulas (2.25) and (2.31) to obtain formula (2.24). 2 R R 1 2 R End of the proof of proposition 2.12. Since L : u L ( , ) Lu = D Tu B(H+,L+) is -aﬃne, for every u L2 , we have ∈ 7→ − ∈ ∈ + d (L u)(t)= T = iT . dt ◦ − ∂tu(t) − D(wλ(u(t))wλ(u(t))+wλ(u(t))+wλ(u(t))) Thus the Lax equation (2.10) is equivalent to identity (2.24) in lemma 2.19.

The proof of proposition 2.7 can be found in G´erard–Kappeler [19], Wu [65] etc. In order to make this paper self contained, we recall it here.

17 2 1 2 Proof of proposition 2.7. Since the Lax map L : u H (R, R) D Tu B(H ,L ) is R-aﬃne, ∈ 7→ − ∈ + + d 2 2 (L u)(t)= T∂tu(t) = TH∂ u(t) ∂x(u(t) ). dt ◦ − − x −

2 2 2 R R It suﬃces to prove [Bu,Lu]+ TH∂ u ∂x(u ) = 0 for every u H ( , ). x − ∈

In fact, u is real-valued, we haveu ˆ( ξ) = uˆ(ξ), u = Πu + Πu and D u = DΠu DΠu. Since both Tu 2 − 2 1| | 1 − and Bu are bounded operators L+ L+ and bounded operators H+ H+ , their Lie Bracket [Bu,Lu] is given by → →

[Bu,Lu]f = Π(f∂x D u)+ iΠ[uΠ(f D u) D uΠ(uf)] + Π[∂xuΠ(uf)+ uΠ(f∂xu)] − | | | | − | | (2.32) = Π(fH∂2u)+ + L2 , − x I1 I2 ∈ + for every f H1 , where the terms and are given by ∈ + I1 I2 :=iΠ[uΠ(f D u) D uΠ(uf)] I1 | | − | | =Π[fΠu∂xΠu + fΠu∂xΠu] ΠuΠ(f∂xΠu) Π(fΠu)∂xΠu + Π[Π(fΠu)∂xΠu ΠuΠ(f∂xΠu)], − − −

:=Π[∂xuΠ(uf)+ uΠ(f∂xu)] = Π(fΠu)∂xΠu + ΠuΠ(f∂xΠu) + Π(ΠuΠ(f∂xΠu)) I2 +2fΠu∂xΠu + Π[fΠu∂xΠu + fΠu∂xΠu + Π(fΠu)∂xΠu].

1 2 2 2 If h1 H and h2 L , then h1h2 L . Since ∂xΠu L , we have Π[Π(fΠu)∂xΠu] = Π[fΠu∂xΠu]. Thus ∈ − ∈ − ∈ − ∈ −

2 1 + =2fΠu∂xΠu + 2Π[fΠu∂xΠu + fΠu∂xΠu + Π(fΠu)∂xΠu] = Π[f∂x(u )] H . (2.33) I1 I2 ∈ + 2 2 Formulas (2.32) and (2.33) yield that [Bu,Lu]f = Π[f(∂x(u ) H∂xu)]. Thus equation (2.3) holds along the evolution of equation (1.1). −

Remark 2.20. As indicated in G´erard–Kappeler [19], 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 skew-adjoint operator commuting 2 2 with Lu. For instance, we set Cu := Bu + iLu and we obtain Cu = iD 2iDTu +2iTDΠu. So (Lu, Cu) − 2 is also a Lax pair of the BO equation (1.1). The advantage of the operator Bu = i(T D u Tu ) is that 2 2 2 | | − Bu : L L is bounded if u is suﬃciently regular. For instance, u H (R, R). + → + ∈

3 The action of the shift semigroup

2 In this section, we introduce the semigroup of shift operators (S(η)∗)η 0 acting on the Hardy space L+ 2≥ and classify all ﬁnite-dimensional translation-invariant subspaces of L+.

2 2 iηx For every η 0, we deﬁne the operator S(η): L L such that S(η)f = eηf, where eη(x)= e . Its ≥ + → + adjoint is given by S(η)∗ = Te−η . We have

S(η)∗ Lu S(η)= Lu + ηIdL2 , η 0. ◦ ◦ + ∀ ≥

18 Since S(η)∗ B(L2 ) = S(η) B(L2 ) =1,(S(η)∗)η 0 is a contraction semi-group. Let iG be its inﬁnites- k k + k k + ≥ − d 2 imal generator, i.e. Gf = i η S(η)∗f L+, f D(G), where d η=0+ ∈ ∀ ∈

2 ˆ 1 D(G) := f L+ : f R+ H (0, + ) , (3.1) { ∈ | ∈ ∞ } ψ τǫψ 1 because limǫ 0 −ǫ ∂xψ L2(0,+ ) = 0, where τǫψ(x)= ψ(x ǫ) and ψ H (0, + ). Every function f D(G) has→ boundedk − H¨olderk continuous∞ Fourier transform by Morrey’s− inequality∈ and∞ Sobolev extension ∈ ˆ + ˆ operator yields the existence of f(0 ) := limξ 0+ f(ξ). The operator G is densely deﬁned and closed. The Fourier transform of Gf is given by →

Gf(ξ)= i∂ξfˆ(ξ), f D(G), ξ > 0. (3.2) ∀ ∈ ∀ In accordance with the Hille–Yosidac theorem, we have 1 1 ( , 0) ρ(iG), (G λi)− B(L2 ) λ− , λ> 0. (3.3) −∞ ⊂ k − k + ≤ ∀

2 Lemma 3.1. For every b L (R) L∞(R), we have Tb(D(G)) D(G) and the following identity ∈ ⊂ T iϕˆ(0+) [G, T ]ϕ = Πb (3.4) b 2π holds for every ϕ D(G). ∈ Proof. For every η> 0 and ϕ D(G), both S(η)∗ and Tb are bounded operators, so we have ∈ ξ+η ∗ ˆ ˆ S(η) b ϕˆ(ξ + η) b [1R+ (τ ηϕˆ)](ξ) 1 ∧ − ˆ [ η ,Tb]ϕ (ξ)= ∗ − ∗ = b(ζ)ˆϕ(ξ + η ζ)dζ, ξ > 0, 2πη 2πη ξ − ∀ Z where τ ηϕˆ(x)=ϕ ˆ(x + η), for every x R. Then we change the variable ζ = ξ + tη, for 0 t 1, − ∈ ≤ ≤ ∗ 1 S(η) IdL2 ∧ − + 1 [ ,Tb]ϕ (ξ)= ˆb(ξ + tη)ˆϕ((1 t)η)dζ = aηb(ξ)+ φη(ξ), ξ > 0, (3.5) η 2π − ∀ Z0 1 1 2 b c where aη := ϕˆ((1 t)η)dζ C and φη L such that 2π 0 − ∈ ∈ + R 1 1 φη(ξ) := [ˆb(ξ + tη) ˆb(ξ)]ϕ ˆ((1 t)η)dt, ξ > 0. 2π − − ∀ Z0 1 + c + Sinceϕ ˆ R+ H (0, + ),ϕ ˆ is bounded and limη 0 ϕˆ(η)=ϕ ˆ(0 ), Lebesgue’s dominated convergence | ∈ ∞ + → ϕˆ(0 ) 2 R ˆ ˆ theorem yields that limη 0+ aη = 2π . Since b L ( ), we have limǫ 0 τǫb b L2 = 0. By using Cauchy–Schwarz inequality→ and Fubini’s theorem, we∈ have → k − k

1 + 1 2 2 ∞ ˆ ˆ 2 2 ˆ ˆ 2 φη L2 . ϕˆ L∞ b(ξ + tη) b(ξ) dξdt = ϕˆ L∞ τ tηb b L2 dt 0, k k k k | − | k k k − − k → Z0 Z0 Z0 when η 0+, by Lebesgue’s dominated convergence theorem. Thus (3.5) implies that → ∗ + S(η) IdL2 − + ϕˆ(0 ) 2 + [ ,Tb]ϕ = aηΠb + φη Πb, in L , when η 0 . η → 2π + →

19 1 2 Since ϕ D(G) and Tb is bounded, we have Tb[(S(η)∗ IdL2 )ϕ] (TbG)ϕ in L , consequently ∈ η − + → + + 1 ϕˆ(0 ) 2 + (S(η)∗ IdL2 )(Tbϕ) (TbG)ϕ + in L+, when η 0 . η − + → 2π →

So Tbϕ D(G) and (3.4) holds. ∈

The following scalar representation theorem of Lax [39] allows to classify all translation-invariant sub- 2 spaces of the Hardy space L+, which plays the same role as Beurling’s theorem in the case of Hardy space on the circle (see Theorem 17.21 of Rudin [56]). 2 Theorem 3.2 (Beurling–Lax). Every nonempty closed subspace of L+ that is invariant under the 2 semigroup of shift operators (S(η))η 0 is of the form ΘL+, where Θ is a holomorphic function in the ≥ upper-half plane C+ = z C : Imz > 0 . We have Θ(z) 1, for all z C+ and Θ(x) =1, x R. Moreover, Θ is uniquely{ determined∈ up to} multiplication| by|≤ a complex con∈stant of absolute| | value∀1. ∈ The following lemma classifies all finite-dimensional subspaces that are invariant under the semi-group (S(η)∗)η 0, which is a weak version of theorem 3.2. ≥ 2 Lemma 3.3. Let M be a subspace of L+ of finite dimension N = dimC M 1 and G(M) M. Then ≥ C ⊂ 1 ≤N−1[X] there exists a unique monic polynomial Q CN [X] such that Q− (0) C and M = , where ∈ ⊂ − Q C N 1[X] denotes all the polynomials whose degrees are at most N 1. Q is the characteristic polynomial ≤ − − of the operator G M . | 2 Proof. We set Mˆ = fˆ L (0, + ): f M , then dimC Mˆ = N. Since Gf = i∂ξfˆ on R 0 , the { ∈ ∞ ∈ } \{ } restriction G M is unitarily equivalent to i∂ξ Mˆ by the renormalized Fourier–Plancherel transformation. | C | C So the characteristic polynomial Q N [X] of i∂ξ Mˆ is well defined, let β1, β2c, , βn denote the ∈ | n { ··· }⊂ distinct roots of Q and mj denote the multiplicity of βj , we have j=1 mj = N and

n N 1 P− mj d k Q(z) = det(z i∂ξ )= (z β ) = z + ckz , ck C. − |Mˆ − j ∈ j=1 k Y X=0 2 The Cayley–Hamilton theorem implies that Q(i∂ξ) = 0 on the subspace Mˆ . If ψ Mˆ L (0, + ), then ψ is a weak-solution of the following diﬀerential equation ∈ ⊂ ∞

N 1 N N − k N k i− Q( D)ψ = ∂ ψ + i − ck∂ ψ =0 on (0, + ), ψ 0 on ( , 0). (3.6) − ξ ξ ∞ ≡ −∞ k X=0 The diﬀerential operator Q( D) is elliptic is on the open interval (0, + ) in the following sense: the − ∞ N symbol of the principal part of Q( D), denoted by aQ : (x, ξ) (0, + ) R ( ξ) , does not vanish except for ξ = 0. Theorem 8.12 of− Rudin [57] yields that ψ is a∈ smooth∞ function.× 7→ − The solution space

ˆ ˆ l iβj ξ Sol(3.6) = SpanC fj,l 0 l mj 1,1 j n, fj,l(ξ)= ξ e− 1R . (3.7) { } ≤ ≤ − ≤ ≤ + n ˆ 2 has complex dimension j=1 mj = N so we have Sol(3.6) = M L+ and Imβj = Re(iβj ) > 0 and ⊂C 1 C ≤N−1[X] Q− (0) . At last, we have M = SpanC fj,l 0 l mj 1,1 j n = , where ⊂ − P { } ≤ ≤ − ≤ ≤ Q l! fj,l(x)= , x R. (3.8) 2π[( i)(x β )]l+1 ∀ ∈ − − j The uniqueness is obtained by identifying all the roots.

20 4 The manifold of multi-solitons

This section is dedicated to a geometric description of the multi-soliton subsets in deﬁnition 1.1. We give at ﬁrst a polynomial characterization then a spectral characterization for the real analytic symplectic manifold of N-solitons in order to prove the global well-posedness of the BO equation with N-soliton solutions (1.6).

N N 2ηj R Recall that every N-soliton has the form u(x)= −1 (x xj )= 2 2 with xj and j=1 η j=1 (x xj ) +η R j − − j ∈ ηj > 0, then we have the following polynomial characterizationP of the NP-solitons. 2 2 Proposition 4.1. The N-soliton subset N H∞(R, R) L (R, x dx) and N M = , for every M = N. Moreover, each of the following threeU ⊂ properties implies the others: U U ∅ 6 T T

(a). u N . ∈U (b). There exists a unique monic polynomial Qu CN [X] whose roots are contained in the lower half- Q′ ∈ plane C such that Πu = i u . Qu − ′ 1 Q (c). There exists Q CN [X] such that Q− (0) C and Πu = i . ∈ ⊂ − Q Q′ ′ Proof. We only prove the uniqueness in (a) (b). If Πu = i u = i P , then we have P ′ 0 on ⇒ Qu P Qu ≡ R. Since P and Qu are monic polynomials, we have P = Qu. The other assertions are consequences of u = Πu + Πu.

Deﬁnition 4.2. For every u N , the unique monic polynomial Qu CN [X] given by proposition 4.1 is ∈U ∈ called the characteristic polynomial of u. Its roots are denoted by zj = xj iηj C , for 1 j N (not N− ∈ − ≤ ≤ necessarily all distinct). The unordered N-uplet cl(z1,z2, ,zN ) C /SN is called the translation– N ··· ∈ − scaling parameters of u, where C /SN denotes the orbit space of the action (A.3) of symmetric group SN on CN . − − The real analytic structure of N is given in the next proposition. U 2 Proposition 4.3. Equipped with the subspace topology of L (R, R), the subset N is a connected, real 2 U analytic, embedded submanifold of the R-Hilbert space L (R, R) and dimR N =2N. For every u N , U ∈U its translation–scaling parameters are denoted by cl(x iη , x iη , , xN iηN ) for some xj R 1 − 1 2 − 2 ··· − ∈ and ηj > 0, then the tangent space to N at u is given by U N 2 2 u u u 2[(x xj ) ηj ] u 4ηj (x xj ) R R − − − u( N )= ( fj gj ), where fj (x)= x x 2 η2 2 , gj (x)= x x 2 η2 2 . (4.1) T U [( j ) + j ] [( j ) + j ] j=1 − − M M

Every tangent space u( N ) is contained in the auxiliary space deﬁned by (1.5) in which the global 2 T U T 2-covector ω Λ ( ∗) is well deﬁned. Recall that the nondegenerate 2-form ω on N is given by ∈ T U

i hˆ1(ξ)hˆ2(ξ) ωu(h1,h2)= ω(h1,h2)= dξ, h1,h2 u( N ). (4.2) 2π R ξ ∀ ∈ T U Z It provides the symplectic structure of the manifold N . U Proposition 4.4. The nondegenerate real analytic 2-form ω is closed on N . Endowed with the sym- U plectic form ω, the real analytic manifold ( N ,ω) is a symplectic manifold. U

21 For every smooth real-valued function f : N R, let Xf X( N ) denote its Hamiltonian vector field, U → ∈ U defined as follows: for every u N and h u( N ), ∈U ∈ T U + i ∞ hˆ(ξ) df(u)(h)= h, uf(u) 2 = iξ( uf(u)) (ξ)dξ = ωu(h,Xf (u)). h ∇ iL 2π ξ ∇ ∧ Z0 Then we have Xf (u)= ∂x uf(u) u( N ), u N . (4.3) ∇ ∈ T U ∀ ∈U Remark 4.5. There are several ways to prove the simple connectedness of N . Firstly, it is irrelevant U to the proof of proposition 5.16. In subsection 5.4, we show that the real analytic manifold N is diffeo- 2N U morphic to some open convex subset of R , hence N is homotopy equivalent to a one-point space. On U the other hand, the simple connectedness of the Kähler manifold Π( N ) can be directly obtained from its construction (see proposition A.5). U Then, we return back to spectral analysis in order to establish a spectral characterization of the manifold Q N . For every monic polynomial Q CN [X] with roots in C , we set Θ=ΘQ := Hol(C+), where U ∈ − Q ∈ N 1 N 1 − j N − j N Q(x) := aj x + x , if Q(x)= aj x + x . j=0 j=0 X X

Then Θ is an inner function on the upper half-plane C+, because Θ 1 on C+ and Θ =1on R. Recall 2 2 | |≤ | | 2 the shift operator S(η): L+ L+ defined in section 3, we have S(η)[Θh] = Θ[S(η)h], for every h L+, 2 → 2 ∈ so ΘL+ is a closed subspace of L+ that is invariant by the semigroup (S(η))η 0 (see also the Beurling–Lax ≥ 2 theorem 3.2 of the complete classification of the translation-invariant subspaces of the Hardy space L+). 2 We define KΘ to be the orthogonal complement of ΘL+, thus

2 2 L =ΘL K , S(η)∗(K ) K and G(D(G) K ) K . (4.4) + + Θ Θ ⊂ Θ Θ ⊂ Θ M \ where the infinitesimal generator G is defined in (3.2). Recall that the C-vector space C N 1[X] consists ≤ − C≤N−1[X] of all polynomials with complex coefficients of degree at most N 1. So Q is an N-dimensional 2 − subspace of L+.

2 1 2 1 The Lax map L : u L (R, R) Lu = D Tu B(H ,L ) is R-aﬃne. Deﬁned on D(Lu)= H , the ∈ 7→ − ∈ + + + unbounded self-adjoint operator Lu has the following spectral decomposition

2 H H H L+ = ac(Lu) sc(Lu) pp(Lu). (4.5) M M The following proposition gives an identiﬁcation of these subspaces in the spectral decomposition (4.5).

Proposition 4.6. If u N , then Lu has exactly N simple negative eigenvalues. Let Qu denote the ∈ U characteristic polynomial of the N-soliton u given in deﬁnition 4.2 and Θ := Θ = Qu denote the u Qu Qu associated inner function. Then we have the following identiﬁcation, C 2 N 1[X] H H H ≤ − ac(Lu)=ΘuL+, sc(Lu)= 0 , pp(Lu)= KΘu = . (4.6) { } Qu

22 For every u N , we have the following spectral decomposition of Lu: ∈U

σ(Lu)= σ (Lu) σ (Lu) σ (Lu), where σ (Lu) = [0, + ), σ (Lu)= (4.7) ac sc pp ac ∞ sc ∅ u u[ u [ and σpp(Lu) = λ1 , λ2 , , λN consists of all eigenvalues of Lu. Proposition 2.2 yields that Lu is { ··· } 1 2 4 C 2 u u D f L2 bounded from below and u 2 λ < < λ < 0, where C = inf 1 k| | k denotes the 4 L 1 N f H 0 f 4 − k k ≤ ··· ∈ +\{ } k kL Sobolev constant. Hence the min-max principle (Theorem XIII.1 of Reed–Simon [54]) yields that

u 1 λn = sup I(F,Lu), I(F,Lu) = inf Luh,h L2 : h H+ F ⊥, h L2 =1 (4.8) dimC F =n 1 {h i ∈ k k } − \ 2 where, the above supremum, F describes all subspaces of L+ of complex dimension n,1 n N. When I ≤ ≤ n N + 1, supdimC F =n (F,Lu) = inf σess(Lu) = 0. Proposition 2.3 and corollary 2.4 yield that there ≥ u exist eigenfunctions ϕj : u N ϕ H (Lu) such that ∈U 7→ j ∈ pp

u C u u u u u Ker(λj Lu)= ϕj , ϕj L2 =1, ϕj ,u L2 = uϕj = 2π λj , (4.9) − k k h i R | | Z q u u u H for every j =1, 2, ,N. Then ϕ1 , ϕ2 , , ϕN is an orthonormal basis of the subspace pp(Lu). We have the following··· result. { ··· }

u Proposition 4.7. For every j =1, 2, ,N, the j th eigenvalue λj : u N λ R is real analytic. ··· ∈U 7→ j ∈ We refer to proposition 4.14 and formula (4.4) to see that the subspace Hpp(Lu) D(G) is invariant by ⊂ u u u G. The matrix representation of G Hpp(Lu) with respect to the orthonormal basis ϕ1 , ϕ2 , , ϕN is given in proposition 5.4. Then the following| theorem gives the spectral characterization{ for N···-solitons.}

2 2 Theorem 4.8. A function u N if and only if u L (R, (1+x )dx) is real-valued, dimC H (Lu)= N ∈U ∈ pp and Πu H (Lu). Moreover, we have the following inversion formula ∈ pp d det(x G H (L )) Πu(x)= i dx − | pp u , x R. (4.10) det(x G H ) ∀ ∈ − | pp(Lu)

Then Qu in definition 4.2 is the characteristic polynomial of G H . The translation–scaling parame- | pp(Lu) ters of u can be identified as the spectrum of G Hpp(Lu). Finally the invariance of N under the BO flow is obtained by its spectral characterization, so| we have the global well-posednessU of the BO equation in the N-soliton manifold (1.6).

Proposition 4.9. If u N , we denote by u : t R u(t) H∞(R, R) the solution of the BO 0 ∈ U ∈ 7→ ∈ equation (1.1) with initial datum u(0) = u . Then u(t) N , for every t R. 0 ∈U ∈ This section is organized as follows. The real analytic structure and the symplectic structure are given in subsection 4.1. Then the spectral decomposition of the Lax operator Lu and the real analyticity of its eigenvalues are given in subsection 4.2, for every u N . The characterization theorem 4.8 is proved in ∈U subsection 4.3. Finally, we show the stability of N under the BO ﬂow in subsection 4.4. U

23 4.1 Diﬀerential structure

The construction of real analytic structure and symplectic structure of N is divided into three steps. U Firstly, we describe the complex structure of Π( N ). Then the Hermitian metric H for the complex U manifold Π( N ) is introduced in (4.15) and we establish a real analytic diffeomorphism between N and U U Π( N ). The third step is to prove dω =0on N . Since ω = Π∗(ImH), (Π( N ), H) is a Kähler manifold. U U − U N N Step I. The Viète map V : (β1,β2, ,βN ) C (a0,a1, ,aN 1) C is defined as follows ··· ∈ 7→ ··· − ∈ N N 1 − k N (X βj )= akX + X . (4.11) − j=1 k Y X=0 Both addition and multiplication of two complex numbers are open continuous maps C2 C, the Viète map V : CN CN is an open quotient map. So V(CN ) is an open connected subset of→CN (see also → − T proposition A.5). With the subspace topology and the Hermitian form HCN (X, Y )= X, Y CN = X Y , N h i the subset (V(C ), HCN ) is a connected Kähler manifold of complex dimension N. − 2 Lemma 4.10. Equipped with the subspace topology of L+, the subset Π( N ) is a connected topological manifold of complex dimension N and it has a unique complex analyticU structure making it into an 2 embedded submanifold of the C-Hilbert space L . For every u N , its translation–scaling parameters + ∈ U are denoted by cl(x iη , x iη , , xN iηN ), for some xj R and ηj > 0, then the tangent space 1 − 1 2 − 2 ··· − ∈ to Π( N ) at Πu is given by U N C u u 1 Πu(Π( N )) = hj , where hj (x)= 2 . (4.12) T U (x xj + ηj i) j=1 M − ′ CN Q 2 Proof. We define ΓN : a = (a0,a1, ,aN 1) V( ) Πu = i Q Π( N ) L+ such that ··· − ∈ − 7→ ∈ U ⊂ N 1 − k N Q(X)= akX + X . k X=0 The surjectivity of ΓN is given by the definition of N . Since the monic polynomial Q is uniquely U N determined by u N , the map ΓN is injective. For every h = (h0,h1, ,hN 1) C , we have ∈U ··· − ∈ N 1 QH′ Q′H − k dΓN (a0,a1, ,aN 1)h = i − , where H(X)= hkX . ··· − Q2 k X=0 H If dΓN (a0,a1, ,aN 1)h = 0, then ( )′ 0. Since deg H deg Q 1, we have H = 0. Thus ··· − Q ≡ ≤ − CN 2 ΓN : V( ) L+ is a complex analytic immersion. We claim that ΓN is a topological embedding. − → (n) (n) (n) (n) CN In fact we set a = (a0 ,a1 , ,aN 1) V( ) such that ··· − ∈ − N 1 ∂xQn ∂xQ 2 − (n) j N R in L+, as n + , where Qn(x)= aj x + x , x . Qn → Q → ∞ ∀ ∈ j=0 X (n) CN (n) R Since a V( ), we have a0 = Qn(0) = 0. For every x , we have ∈ − 6 ∈ Q (x) x ∂ Q (y) x ∂ Q(y) Q(x) n = exp( y n dy) exp( y dy)= , as n + . (4.13) Q (0) Q (y) → Q(y) Q(0) → ∞ n Z0 n Z0

24 Qn Q(x) Every coefficient of converges to the corresponding coefficient of . Since Qn,Q are monic, we Qn(0) Q(0) 1 1 (n) 1 2 CN have limn + = and limn + a = a. Then ΓN− : Π( N ) L+ V( ) is continuous. → ∞ Qn(0) Q(0) → ∞ U ⊂ 2→ − Since ΓN is a complex analytic embedding, with the subspace topology of L+, there exists a unique N complex analytic structure making Π( N )=ΓN V(C ) into an embedded complex analytic submanifold U ◦ − 2 CN N 2ηj of L . The map ΓN : V( ) Π( N ) is biholomorphic. Set u(x) = 2 2 for some + j=1 (x xj ) +η − → U − j xj = xj (u) R and ηj = ηj (u) > 0. Then every h u(Π( N )) is identified as the velocity of the ∈ ∈ TΠ U P smooth curve c : t ( 1, 1) Π( N ) such that c(0) = Πu at t = 0. If we choose ∈ − → U N i c(t, x)= where xj (t) R, ηj (t) > 0. x xj (t)+ ηj (t)i ∈ j=1 X −

Then we have xj (0) = xj , ηj (0) = ηj and

N ηj′ (0) + ixj′ (0) h(x)= ∂t t=0c(t, x)= 2 . (4.14) (x xj + ηj i) j=1 X −

u u u u (ixj (u)+ηj (u))ξ We have hj = Πfj = iΠgj and (hj )∧(ξ)= 2π1ξ 0ξe− . For every h Πu(Π( N )), we ≥ 1 2− − ∈ T U have ξ ξ− hˆ(ξ) L (R) (see also Hardy’s inequality (4.18)). 7→ ∈ Step II. Given u N , the Hermitian metric H u is defined as follows ∈U Π + ∞ hˆ1(ξ)hˆ2(ξ) H u(h ,h )= dξ, h ,h u(Π( N )). (4.15) Π 1 2 πξ ∀ 1 2 ∈ TΠ U Z0 + hˆ(ξ) 2 The sesquilinear form H u is positive definite because H u(h,h) = ∞ | | dξ > 0, if h = 0. Hence Π Π 0 πξ 6 the smooth symmetric covariant 2-tensor field ReH is positive definite on Π( ), so (Π( ), ReH) is a R N N Riemannian manifold of real dimension 2N. U U

We consider the R-linear isomorphism between the Hilbert spaces

Π: u L2(R, R) Πu L2 , f L2 2Ref L2(R, R). ∈ 7→ ∈ + ∈ + 7→ ∈

Then Π 2Re = Id 2 and 2Re Π = Id 2 R R and u 2 = √2 Πu 2 . Then N = 2Re Π( N ) is a L+ L ( , ) L L ◦ ◦ k k k uk u Uu ◦u U real analytic manifold of real dimension 2N. Furthermore we have fj = 2Rehj , gj =2iRehj and

2Re : u(Π( N )) u( N ) (4.16) TΠ U → T U is an R-linear isomorphism. Since H is Hermitian, the 2-form ω = Π∗(ImH) is nondegenerate on N . − U 2 2 2 Step III. We set := L (R, R) L (R, x dx), c := u : u = c , for every c R. Then E E { ∈E R } ∈

N T πN , u( N ) := R , u N . U ⊂E2 T U ⊂ T E0 ∀ ∈U The nondegenerate 2-form ω can be extended to a 2-covector of the subspace . Recall that T

i hˆ1(ξ)hˆ2(ξ) ω(h1,h2)= dξ, h1,h2 . (4.17) 2π R ξ ∀ ∈ T Z

25 If h , then we have hˆ(0) = 0 and hˆ H1(R). Hence the Hardy’s inequality (see Brezis [7], Bahouri– Chemin–Danchin∈ T [3] etc.) yields that ∈

ˆ 2 ˆ h(ξ) ˆ 2 h(ξ) 2 R | | dξ 4 ∂ξh L2 = ξ L ( ), (4.18) R ξ 2 ≤ k k ⇒ 7→ ξ ∈ Z | | 2 so the 2-covector ω Λ ( ∗) is well deﬁned and ωu(h1,h2)= ω(h1,h2). For every smooth vector ﬁeld ∈ T1 X X( N ), let Xyω Ω ( N ) denote the interior multiplication by X, i.e. (Xyω)(Y ) = ω(X, Y ), for ∈ U ∈ U every Y X( N ). We shall prove that dω =0 on N by using Cartan’s formula: ∈ U U

LX ω = Xy(dω) + d(Xyω). (4.19)

Proof of proposition 4.4. For any smooth vector field X X( N ), let φ denote the smooth maximal flow ∈ U of X. If t is sufficiently close to 0, then φt : u N φ(t,u) N is a local diffeomorphism by the ∈ U 7→ ∈ U fundamental theorem on flows (see Theorem 9.12 of Lee [40]). For every u N , h1,h2 u( N ), we compute the Lie derivative of ω with respect to X, ∈ U ∈ T U

ωφt(u)(dφt(u)h1, dφt(u)h2) ωu(h1,h2) (LX ω)u(h1,h2) = lim − t 0 → t dφt(u)h1 h1 dφt(u)h2 h2 = lim ω − , dφt(u)h2 + lim ω h1, − . t 0 t t 0 t → →

dφt(u)hj hj Since limt 0 − = dX(u)hj u( N ), for every j =1, 2, we have → t ∈ T U

(LX ω)u(h ,h )= ω(dX(u)h ,h )+ ω(h , dX(u)h ) = (h ω(X,h )) (u) (h ω(X,h )) (u). 1 2 1 2 1 2 1 2 − 2 1 i We choose (V, x ) a smooth local chart for N such that u V and the tangent vector hk has the 2N (j) ∂ U (j) ∈R coordinate expression hk = h j , for some h , j = 1, 2 , 2N and k = 1, 2. The j=1 k ∂x u k ∈ ··· tangent vector hk can be identified as some locally constant vector field Yk X( N ) defined by ∈ U P 2N (j) ∂ Yk : v V hk v( N ), Yk : u (Yk)u = hk, k =1, 2. ∈ 7→ ∂xj v ∈ T U 7→ j=1 X

Then the vector ﬁeld [Y1, Y2] vanishes in the open subset V . The exterior derivative of the 1-form β = Xyω is computed as dβ(Y , Y )= Y (β(Y )) Y (β(Y )) + β([Y , Y ]). Thus 1 2 1 2 − 2 1 1 2

d(Xyω)u(h ,h )= h ωu(Xu,h ) h ωu(Xu,h )+ ωu(Xu, [Y , Y ]u) = (LX ω)u(h ,h ). 1 2 1 2 − 2 1 1 2 1 2

Then Cartan’s formula (4.19) yields that Xy(dω) = 0. Since X X( N ) is arbitrary, we have dω = 0. 2 ∈ U As a consequence, the real analytic 2-form ω : u N ω Λ ( ∗) is a symplectic form. ∈U 7→ ∈ T Since ImH = ( 2Re)∗ω, where 2Re : Π( N ) N is a real analytic diﬀeomorphism, the associated − − U → U 2-form ImH is closed. So (Π( N ), H) is a K¨ahler manifold. The simple connectedness of Π( N ) is proved in subsection A.1. U U

26 4.2 Spectral analysis II

We continue to study the spectrum of the Lax operator Lu introduced in deﬁnition 2.1. The general cases u L2(R, R) and u L2(R, (1 + x2)dx) have been studied in subsection 2.2. We restrict our study to ∈ ∈ Q the case u N in this subsection. Let Q = Qu denote the characteristic polynomial of u and Θ := Q , 2∈ U 2 KΘ = (ΘL+)⊥. Since Lu is an unbounded self-adjoint operator of L+, we have the following

2 2 H H H L+ =ΘL+ KΘ = ac(Lu) sc(Lu) pp(Lu). M M M We shall at first identify those subspaces by proving proposition 4.6 and formula (4.7). Then we turn to u study the real analyticity of each eigenvalue λj : u N λ R. ∈U 7→ j ∈ C ≤N−1[X] 2 Proof of proposition 4.6. The first step is to prove KΘ = Q . In fact, for every h L+ and C ∈ P ≤N−1[X] f = , for some P C N 1[X], we have Q ∈ Q ∈ ≤ − P (x)Θ(x)h(x) P (x)h(x) P f, Θh L2 = dx = dx = ,h L2 . h i R Q(x) R Q(x) hQ i Z Z Since Q(x)= N (x α ) with Im(α ) > 0, the meromorphic function P has poles in C , so P L2 . j=1 j j Q + Q − C ∈ − P ≤N−1[X] 2 Thus f, Θh LQ2 = ,h L2 = 0. Thus (ΘL+)⊥ = KΘ. h i h Q i Q ⊂ 1 2 Q 2 Conversely, if f KΘ, then Θ− f,h L2 = f, Θh L2 = 0, for every h L+. Thus g := f L . It ∈ h i h i ∈ Q ∈ − suffices to prove that P := Qf = Qg C[X]. In fact, ∈

Qf = Q(i∂ξ)fˆ and supp(fˆ) [0, + )= supp(Qf) [0, + ). ⊂ ∞ ⇒ ⊂ ∞ ˆ P 2 R Similarly, supp((cQg)∧) ( , 0]. Thus supp(P ) 0 and P is a polynomial.c Since f = Q L ( ), ⊂ −∞ C ⊂ { } ∈ we have deg P N 1. So K ≤N−1[X] . ≤ − Θ ⊂ Q 2 2 The second step is to prove Lu(ΘL ) ΘL . Precisely, we have + ⊂ + 2 Lu(Θh)=ΘDh, h L . (4.20) ∀ ∈ + C ′ ′ Since ≤N−1[X] L2 ,Θ= Q and DΘ = DQ DQ = i Q i Q = Πu + Πu = u on R, we have Q ⊂ + Q Θ Q − Q Q − Q

′ ′ Q′ Q DΘ Q′ Q Lu(Θh) = (D Tu)(Θh)=ΘDh + h DΘ i Θ+ i = ΘDh + hΘ i + = ΘDh. − − Q Q Θ − Q Q Recall that Lu = L∗ , so we have Lu(K ) K . Since dimC K = N, corollary 2.4 yields that the u Θ ⊂ Θ Θ Hermitian matrix Lu K has exactly N distinct eigenvalues. Hence KΘ Hpp(Lu). | Θ ⊂ 2 2 On the other hand, we set UΘ : L ΘL such that UΘh =Θh. Thus UΘ B(L2 ,ΘL2 ) = 1 and + → + k k + + 1 2 1 2 U − = U ∗ : g ΘL Θ− g L . Θ Θ ∈ + 7→ ∈ + So U : L2 ΘL2 is a unitary operator. U (H1 )=ΘH1 = H1 ΘL2 . Formula (4.20) yields that Θ + → + Θ + + + + 1 1 T 2 UΘ∗ Lu ΘL2 UΘ = D,UΘ[D(D)] = ΘH+ = H+ ΘL+ = D(Lu ΘL2 ). | + | + \ 27 For every bounded Borel function f : R C, we have f(Lu)UΘ = UΘf(D) by proposition 2.14. We Lu → 2 2 denote by µψ = µ the spectral measure of Lu associated to ψ L , then h L , we have ψ ∈ + ∀ ∈ + + 1 ∞ 2 f(ξ)dµΘh(ξ)= f(Lu)UΘh,UΘh L2 = Θf(D)h, Θh L2 = f(D)h,h L2 = f(ξ) hˆ(ξ) dξ. R h i h i h i 2π | | Z Z0 2 1R hˆ(ξ) + | | So dµΘh(ξ) = 2π dξ. The spectral measure µΘh is absolutely continuous with respect to the R 2 H H H 2 Lebesgue measure on . Thus ΘL+ ac(Lu) cont(Lu) = ( pp(Lu))⊥ ΘL+ and (4.6) is ⊂ ⊂ 2 ⊂ 2 obtained. We have supp(µ h) [0, + ), for every h L . ξ 0, there exists h L such that Θ ⊂ ∞ ∈ + ∀ ≥ ∈ + hˆ(ξ) = 0. So we have σ (Lu)= σ (Lu)= σ (Lu) = [0, + ). 6 ess cont ac ∞ Before proving the real analyticity of each eigenvalue, we show its continuity at ﬁrst.

u Lemma 4.11. For every j = 1, 2, ,N, the j th eigenvalue λj : u N λ R is Lipschitz ··· ∈ U 7→ j ∈ continuous on every compact subset of N . U 1 1 Proof. For every f H (R), the Sobolev embedding f 4 C D 4 f 2 yields that u, v N , ∈ k kL ≤ k| | kL ∀ ∈U 2 1 1 Luh,h 2 Lvh,h 2 u v 2 h 4 C u v 2 D 2 h 2 h 2 , h H . (4.21) h iL − h iL ≤k − kL k kL ≤ k − kL k| | kL k kL ∀ ∈ + Given j =1, 2, ,N and a subspace F L2 with complex dimension j 1, we choose ··· ⊂ + − j j u 1 u h F ⊥ Ker(λ Lu) H , h 2 =1, h = hkϕ . ∈ k − ⊂ + k kL k k k \ M=1 X=1 j 2 u u u u Then Luh,h 2 = hk λ λ < 0, because λ < λ . We have the following estimate h iL k=1 | | k ≤ j k k+1 1 2 P u 2 1 D 2 h 2 = Dh,h 2 = Luh,h 2 + uh,h 2 λ + u 2 h 4 C u 2 D 2 h 2 h 2 . (4.22) k | kL h iL h iL h iL ≤ j k kL k kL ≤ k kL k| | kL k kL u 2 So estimates (4.21) and (4.22) yield that Lvh,h L2 λj + C u L2 u v L2 . Since F is arbitrary, the max–min formula (4.8) implies that h i ≤ k k k − k

u v 2 λ λ C ( u 2 + v 2 ) u v 2 . | j − j |≤ k kL k kL k − kL 2 u Every compact subset K N is bounded in L (R, R). Hence u K λ R is Lipschitz continuous. ⊂U ∈ 7→ j ∈

Proof of proposition 4.7. For every u N , the Lax operator Lu has N negative simple eigenvalues, denoted by λu < λu < < λu < 0. Let∈ UPj denotes the Riesz projector of the eigenvalue λu and 1 2 ··· N u j D(z,ǫ)= η C : η z <ǫ , C (z,ǫ)= ∂D(z,ǫ)= η C : η z = ǫ , z C, ǫ> 0. { ∈ | − | } { ∈ | − | } ∀ ∈ u Then there exists ǫ0 > 0 such that the family of closed discs D(λj ,ǫ0) 1 j N D(0,ǫ0) is mutually {u } ≤ ≤1 { } u disjoint and for every j, k =1, 2 ,N and any closed path Γj (piecewise C closed curve) in D(λj ,ǫ0) ··· u S with respect to which the eigenvalue λj has winding number 1, we have

Pj 1 1 Pj Pj Pj Pj u u u = (ζ Lu)− dζ, u u = u, uϕk = 1j=kϕk . (4.23) 2πi Γu − ◦ I j

28 u C u by Theorem XII.5 of Reed–Simon [54]. We choose Γj to be the counterclockwise-oriented circle (λj ,ǫ) j u u in (4.23) for some ǫ (0,ǫ ). We claim that ImP = Ker(λ Lu)= Cϕ . ∈ 0 u j − j j j It suﬃces to show that P H L = 0. In fact the operator P = gλu (Lu) is self-adjoint by Theorem u| ac( u) u j VIII.6 of Reed–Simon [55], where the real-valued bounded Borel function gλ : R R is given by →

1 1 gλ(x) := (ζ x)− dζ = 1(λ ǫ,λ+ǫ)(x), a.e. on R, 2πi C − − I (λ,ǫ)

j u j Lu for every λ R. Since P (H (Lu)) Cϕ H (Lu), we have P (H (Lu)) H (Lu). Let µψ = µ ∈ u pp ⊂ j ⊂ pp u ac ⊂ ac ψ denote the spectral measure of Lu associated to the function ψ Hac(Lu), whose support is included in [0, + ) by formula (4.7), we have ∈ ∞ + Pj 1 1 1 ∞ 1 uψ, ψ L2 = (ζ Lu)− ψ, ψ L2 dζ = (ζ ξ)− dζ dµψ(ξ)=0. h i 2πi C (λu,ǫ)h − i 2πi 0 C (λu,ǫ) − I j Z I j ! ˜ j ˜ 2 j ˜ ˜ Set ψ = P ψ H (Lu), then ψ 2 = P ψ, ψ 2 = 0. So the claim is obtained. u ∈ ac k kL h u iL u Pj v R For every ﬁxed j =1, 2, N, we have λj = Tr(Lu u). Since every eigenvalue λk : v N λk is ··· ◦ ∈U v7→ u ∈ ǫ0 continuous, there exists an open subset N containing u such that supv sup1 k N λk λk < 3 . ∈V ≤ ≤ 2ǫ0 v u V⊂Uu | − | We set ǫ = 3 , then λj D(λj ,ǫ) D(λk ,ǫ0), for every v and k = j. For example, in the ∈ \ C u ∈ VC u 6 next picture, the dashed circles denote respectively (λj ,ǫ0) and (λk ,ǫ0); the smaller circles denote C u C u respectively (λj ,ǫ) and (λk ,ǫ) with j < k. The segments inside small circles denote the possible v v positions of λj and λk.

v v λj λk 0 u u λj λk

u v C u u v Then σ(Lv) D(λj ,ǫ0)= λj and (λj ,ǫ) is a closed path in D(λj ,ǫ0) with respect to which λj has winding number 1. Thus, { } T Pj 1 1 v Pj v = (ζ Lv)− dζ, λj = Tr(Lv v), v . (4.24) 2πi C (λu,ǫ) − ◦ ∀ ∈V I j 1 2 R 1 2 1 2 1 Since v Lv B(H+,L+) is -aﬃne and i : BI(H+,L+) − B(L+,H+) is complex ∈ V 7→ ∈ 1 2 1 2 A ∈ 7→ A ∈ C analytic, where BI(H+,L+) B(H+,L+) denotes the open subset of all bijective bounded -linear transformations H1 L2 , we⊂ have the real analyticity of the following map + → +

u 3 u 1 1 2 1 (ζ, v) D(λ , ǫ ) D(λ , ǫ ) (ζ Lv)− B(L ,H ). (4.25) ∈ j 4 0 \ j 2 0 × V 7→ − ∈ + +

29 Pj Pj 2 1 Pj R Hence the maps : v v B(L+,H+) and λj : v Tr(Lv v) are both real analytic by composing (4.24) and∈ V (4 7→.25). ∈ ∈ V 7→ ◦ ∈

C≤N−1[X] Recall that Hpp(Lu)= , where Qu denotes the characteristic polynomial of u N whose zeros Qu ∈U are contained in C , so Hpp(Lu) D(G) is given by (3.7). We have the following consequence. − ⊂ u u Corollary 4.12. For every j =1, 2, ,N, the map ℧j : u N Gϕ , ϕ 2 C is real analytic. ··· ∈U 7→ h j j iL ∈ Pj u u v v Pj Pj Proof. For every u, v N , we have vϕj = ϕj , ϕj L2 ϕj . Since the Riesz projector : v N v 2 1 ∈U h i Pj u ∈U 7→ ∈ B(L+,H+) is real analytic in the proof of proposition 4.7 and uϕj L2 = 1, there exists a neighbourhood j u 1 k kj j 2 1 P 2 P P B of u, denoted by , such that vϕj L > 2 for every v and : v v (L+,H+) can be expressed by powerV series. Thenk k ∈ V ∈ V 7→ ∈

j u j u j u P ϕ G P (ϕ ), P (ϕ ) 2 v v j ℧ v j v j L ϕj = u v , j (v)= h ◦ j i . 2 u 2 ϕj , ϕj L Pv(ϕ ) 2 h i k j kL ℧ Pj u 2 Pj u Pj u C Hence the restriction j : v (ϕ ) −2 G (ϕ ), (ϕ ) 2 is real analytic. ∈ V 7→ k v j kL h ◦ v j v j iL ∈

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

2 2 Lemma 4.13. For u L (R, (1 + x )dx), ϕ Ker(λ Lu) for some λ σ (Lu), then we have ∈ ∈ − ∈ pp ϕ, Tuϕ, Luϕ D(G) and ∈ iϕˆ(0+) iϕˆ(0+) [G, Tu]ϕ = Πu, [G, Lu]ϕ = iϕ Πu. (4.26) 2π − 2π where Θ=Θ = Qu with Q the characteristic polynomial of u. u Qu u 1 1 Proof. In proposition 2.3, we have shown that uϕ H (R), so (Tuϕ)∧ = uϕ1R H (0, + ) and ∈ + ∈ ∞ Tuϕ D(G). We recall the regularity of eigenfunctions (2.2) ∈ c c 1 1 R 1 R 2 R Ker(λ Lu) ϕ H+ :ϕ ˆ R+ C ( +) H ( +) and ξ ξ[ˆϕ(ξ)+ ∂ξϕˆ(ξ)] L ( +) . (4.27) − ⊂{ ∈ | ∈ 7→ ∈ } 1 \ + 1 So Gϕ H+ = D(Lu)= D(Tu). Moreover, we haveϕ ˆ is right-continuous at ξ =0 andϕ ˆ C (0, + ). The weak-derivative∈ ofϕ ˆ is denoted by ∂wϕˆ, δ denotes the Dirac measure with support ∈0 , then ∞ ξ 0 { }

w d + w w ∂ ϕˆ = 1R∗ ϕˆ +ϕ ˆ(0 )δ0, ∂ξ(û ϕˆ)= ∂ (û ϕˆ)=û ∂ ϕˆ (4.28) ξ + dξ ∗ ξ ∗ ∗ ξ

1 by lemma 2.15. Sinceϕ ˆ = 1R∗ ϕˆ a.e. in R andu ˆ H (R), we haveu ˆ Gϕ(ξ)=ˆu [1R∗ Gϕ](ξ), for + ∈ ∗ ∗ + i i d i + every ξ > 0 and ([G, T ]ϕ) (ξ) = ∂ (ˆu ϕˆ)(ξ) uˆ [1R∗ fˆ](ξ) = ϕˆ(0 )u(ξ). Together with u ∧ 2π ξ 2π + dξ 2π ∗ − ∗ c c

30 b (5.9), the ﬁrst formula of (4.26) is obtained. Since Lu = D Tu, we claim that Dϕ D(G). In fact, − 1 ∈ ∂ξ(Dϕ)∧(ξ)=ϕ ˆ(ξ)+ ξ∂ξϕˆ(ξ), ξ > 0. Thus (4.27) implies that Dϕ H (0, + ). Then ∀ ∈ ∞

([G, D]ϕ)∧ (ξ)= i∂ξ(ξϕˆ)(ξ) ξ i∂ξϕˆ(ξ)= iϕˆ(ξ), ξ > 0. (4.29) − · c ∀

So we have [∂x, G]=Id 2 . The second formula of (4.26) holds. L+

2 2 Proposition 4.14. If u L (R, (1 + x )dx) is real-valued, dimC H (Lu)= N 1 and Πu H (Lu), ∈ pp ≥ ∈ pp then we have H (Lu) D(G) and G(H (Lu)) H (Lu). pp ⊂ pp ⊂ pp Proof. There exists an orthonormal basis of H (Lu), denoted by ψ , ψ , , ψN , such that pp { 1 2 ··· }

Luψj = λj ψj , where σ (Lu)= λ , λ , , λN ( , 0), λj < λj . pp { 1 2 ··· }⊂ −∞ +1 1 1 Since (4.27) implies that H (Lu) G− (H ) D(G), formula (4.26) gives that pp ⊂ + T+ iψˆj(0 ) fj := [Lu, G]ψj = iψj + Πu H (Lu), j =1, 2, ,N. − 2π ∈ pp ∀ ···

So we have fj , ψj 2 = Gψj ,Luψj 2 GLuψj , ψj 2 = λ( Gψj , ψj 2 Gψj , ψj 2 ) = 0. h iL h iL − h iL h iL − h iL

fj ,ψk L2 For every j = 1, 2, ,N, we set g := h i ψ . Since f = f , ψ 2 ψ , j 1 k N,k=j λ λj k j 1 k N,k=j j k L k ··· ≤ ≤ 6 k − ≤ ≤ 6 h i we have (Lu λj )gj = fj = (Lu λj )Gψj . Then Gψj gj Ker(Lu λj )= Cψj and − − P − ∈ − P

Gψj gj + Cψj H (Lu). ∈ ⊂ pp

We conclude by H (Lu) = SpanC ψ , ψ , , ψN . (see also formulas (4.4) and (4.6)) pp { 1 2 ··· } Now, we perform the proof of converse sense of theorem 4.8 give the explicit formula of Qu.

End of the proof of theorem 4.8. : Proposition 4.14 yields that G(Hpp(Lu)) Hpp(Lu). Let Q denote ⇐ ⊂ C≤N−1[X] the characteristic polynomial of the operator G H , then we have H (Lu) = by lemma | pp(Lu) pp Q P0 3.3. So Πu = , for some P C[X] such that deg P N 1. It remains to show that P = iQ′. Since Q 0 ∈ 0 ≤ − 0 Hpp(Lu) is invariant under Lu, for every P C N 1[X], we have ∈ ≤ −

P P DP P0P (iQ′ P0)P C N 1[X] P ≤ − Lu( ) = (D T 0 T P0 )( )= Π( )+ −2 . Q − Q − Q Q Q − QQ Q ∈ Q

C ′ P0P ≤N−1[X] (iQ P0)P Partial-fraction decomposition implies that Π( ) . So − C N 1[X] for every QQ ∈ Q Q ∈ ≤ − P C N 1[X]. Choose P = 1, since deg(iQ′ P0) N 1, we have P0 = iQ′, so u N . Since ∈ ≤ − 1 − ≤ − ∈ U Q CN [X] is monic and Q− (0) C , we have Qu(x)= Q(x) = det(x G H (L )). ∈ ⊂ − − | pp u

31 4.4 The stability under the Benjamin–Ono flow Finally we prove proposition 4.9 in this subsection. Two lemmas will be proved at first in order to obtain the invariance of the property x xu(x) L2(R) under the BO flow. 7→ ∈ 2 2 2 Lemma 4.15. If u0 H (R, R) L (R, x dx), let u = u(t, x) solves the BO equation (1.1) with initial ∈ 2 2 datum u(0) = u0, then u(t) L (R, x dx), for every t R. ∈ T ∈ 2 Remark 4.16. This result can be strengthened by replacing the assumption u0 H (R, R) by a weaker 3 ∈ 2 + R R s R R assumption u0 H ( , ) = s> 3 H ( , ), because one can construct the conservation law of BO ∈ 2 equation controlling the Hs-norm for every s> 1 by using the method of perturbation of determinants. S 2 We refer to Talbut [62] to see details and Killip–Vi¸san–Zhang− [37] for the KdV and the NLS cases (see also Koch–Tataru [36]). It suffices to use lemma 4.15 to prove proposition 4.9. Before proving lemma 4.15, we need some commutator estimates used in Gérard–Lenzmann–Pocovnicu– Raphaël [21], we recall it here. R R 3 5 1 R Lemma 4.17. For a general locally Lipschitz function χ : such that ∂xχ, ∂xχ, ∂xχ L ( ), then we have the following commutator estimates → ∈

1 2 3 2 R [ D ,χ]g L2 + [∂x,χ]g L2 . ( ∂xχ L1 ∂xχ L1 ) g L2 , g L ( ), k | | k k k k k k k k k ∀ ∈ (4.30) 3 1 5 1 1 D [∂x,χ]g 2 . ( ∂xχ 1 ∂ χ 1 ) 2 ∂xg 2 + ( ∂xχ 1 ∂ χ 1 ) 2 g 2 , g H (R). k| | kL k kL k x kL k kL k kL k x kL k kL ∀ ∈ Proof. We use ξ η ξ η to estimate the Fourier modes of [ D ,χ]g. | | − | | ≤ | − | | |

2π ([ D ,χ]g)∧ (ξ) ξ η χˆ(ξ η) gˆ(η) dη ξ η χˆ(ξ η) gˆ(η) dη = ∂xχ gˆ (ξ). | | ≤ η R | | − | | | − || | ≤ η R | − || − || | | | ∗ | | Z ∈ Z ∈

d Then Young’s convolution inequality yields that [ D ,χ]g L2 . ∂xχ gˆ L2 . ∂xχ L1 g L2 . In order k | | k k | ∗ | |k 1k k 1 k k − 2 3 2 to estimate ∂xχ L1 , we divide the integral as two parts. Wet set 1 = ∂xχ L1 ∂xχ L1 , so k k dR k k kd k 3 3 d ∂xχ L∞ ∂xχ L1 3 1 1 ∞ 1 1 1 2 ∂xχ L ∂xχ L dξ + k k2 dξ . ∂xχ L 1 + k k = ( ∂xχ L ∂xχ L ) . k k ≤k k ξ 1 ξ > 1 ξ k k R 1 k k k k Z| |≤R Z| | R d| | R d d 3 1 Similarly, we have [∂x,χ]g 2 . ∂xχ 1 g 2 . ( ∂xχ 1 ∂ χ 1 ) 2 . Thus (4.30) is obtained. k kL k kL k kL k kL k x kL

2π ( D [∂x,χ]g)∧ (ξ) ξ d ξ η χˆ(ξ η) gˆ(η) dη | | ≤| | η R | − || − || | Z ∈

ξ η 2 χˆ(ξ η) gˆ(η) dη + ξ η χˆ(ξ η) η gˆ(η) dη ≤ η R | − | | − || | η R | − || − || || | Z ∈ Z ∈ 2 = ∂ χ gˆ (ξ)+ ∂xχ ∂xg (ξ) | x | ∗ | | | | ∗ | | 2 2 So we have D [∂x,χ]g L2 . ∂dxχ gˆ L2 + ∂dxχ ∂dxg L2 . ∂xχ L1 g L2 + ∂xχ L1 ∂xg L2 . Then k| | k k| |∗| |k k| |∗| |k k1 k 1k k k k k k 2 − 4 5 4 we use the same idea to estimate ∂xχ L1 , we set 2 := ∂xχ L1 ∂xχ L1 . Thus, dk k dR dk k dk k d 5 5 d ∂xχ L∞ 2 ∂xχ L1 5 1 2 1 ∞ 1 1 1 2 ∂xχ L ∂xχ L ξ dξ + k k3 dξ . ∂xχ L 2 + k 2k = ( ∂xχ L ∂xχ L ) . k k ≤k k ξ 1 | | ξ > 1 ξ k k R 2 k k k k Z| |≤R Z| | R d| | R Finally,d we addd them together to get the second estimate in (4.30).

32 Now we prove the invariance of the property x xu(x) L2(R) is invariant under the BO flow. 7→ ∈ R Proof of lemma 4.15. We choose a cut-off function χ Cc∞( ) such that χ decreases in [0, + ), χ is even and ∈ ∞ 0 χ 1, χ 1 on [ 1, 1], supp(χ) [ 2, 2]. (4.31) ≤ ≤ ≡ − ⊂ − 2 2 2 If u H (R) L (R, x dx), we claim that there exists a constant = ( u(0) 1 ) such that 0 ∈ C C k kH T 2 x 2 2 t 2 2 R I(R,t) := χ ( R ) x u(t, x) dx e| |( x u(0, x) dx + 1), t , R> 1, (4.32) R | | | | ≤C R | | | | ∀ ∈ ∀ Z Z 2 2 if u solves the BO equation ∂tu = H∂ u ∂x(u )= D ∂xu 2u∂xu. x − | | − x x In fact, we define ρ(x) := xχ(x). For every R> 0, we set ρR(x) := Rρ( R )= xχ( R ). Thus

2 2 2 ∂tI(R,t) = 2Re ρ ∂tu(t),u(t) 2 = 2Re ρ D ∂xu(t) 2ρ u(t)∂xu(t),u(t) 2 = (u(t)) + (u(t)), h R iL h R| | − R iL J1 J2 where for every u H2(R), we deﬁne ∈ 2 2 2 (u) := 4Re ρ u∂xu,u 2 = (u) 4 ∂xu L∞ ρRu 2 . u 2 ρRu 2 (4.33) J1 − h R iL ⇒ |J1 |≤ k k k kL k kH k kL and 2 2 (u) := 2Re ρ D ∂xu,u 2 = [ρ , D ∂x]u,u 2 , J2 h R| | iL h R | | iL 2 because D ∂x = ( D ∂x)∗ is an unbounded skew-adjoint operator on L (R), whose domain of deﬁnition 2 | | − | | s is H (R), u ρRu is a bounded self-adjoint operator on H (R), for every s 0. Since 7→ ≥ 2 [ρ , D ∂x]= ρR[ρR, D ∂x] + [ρR, D ∂x]ρR, [ρR, D ∂x] = [ρR, D ∂x]∗ = [ρR, D ]∂x + D [ρR, ∂x], R | | | | | | | | | | | | | | we have

(u)= ρR[ρR, D ∂x]u + [ρR, D ∂x]ρRu,u 2 J2 h | | | | iL =2Re [ρR, D ∂x]u,ρRu 2 (4.34) h | | iL =2Re [ρR, D ]∂xu,ρRu 2 + 2Re D [ρR, ∂x]u,ρRu 2 . h | | iL h| | iL 3 1 5 3 Since ∂xρR L1 = R ∂xρ L1 , ∂xρR L1 = R− ∂xρ L1 and ∂xρR L1 = R− ∂xρ L1 , the commutator estimatesk (4.k30) yieldk thatk if uk H2(kR), then k k k k k k ∈ 2 2 2 (u) 2 ρRu 2 + [ρR, D ]∂xu 2 + D [ρR, ∂x]u 2 |J2 |≤ k kL k | | kL k| | kL 2 3 2 5 2 . ρRu 2 + ∂xρR 1 ∂ ρR 1 ∂xu 2 + ∂xρR 1 ∂ ρR 1 u 2 k kL k kL k x kL k kL k kL k x kL k kL 2 3 2 2 5 2 (4.35) . ρRu 2 + ∂xρ 1 ∂ ρ 1 ∂xu 2 + R− ∂xρ 1 ∂ ρ 1 u 2 k kL k kL k x kL k kL k kL k x kL k kL 2 2 . ρRu 2 + u 1 k kL k kH for every R 1. Proposition 2.6 and 2.8 yield that there exists a conservation law of (1.1) controlling H2-norm of≥ the solution. Let u : t R u(t) H2(R) denote the solution of the BO equation (1.1). ∈ 7→ ∈ 2 2 Then supt R u(t) H . u0 2 1. Since I(R,t)= ρRu(t) L2 , estimates (4.33) and (4.35) imply that ∈ k k k kH k k

∂tI(R,t) (I(R,t)+1), t R, | |≤C ∈ for some constant = ( u0 H2 ). Thus (4.32) is obtained by Gronwall’s inequality. Let R + , we conclude by using Lebesgue’sC C k k monotone convergence theorem. → ∞

33 Since the generating function λ C σ( Lu) λ(u) C is the Borel–Cauchy transform of the spectral ∈ \ − 7→ H ∈ measure of Lu, the invariance of the N soliton manifold N under BO ﬂow is obtained by using the inverse spectral transform. − U

2 2 End of the proof of proposition 4.9. If u0 N H∞(R, R) L (R, x dx), let u = u(t, x) be the unique ∈U ⊂ 2 2 solution of the BO equation (1.1) with initial datum u(0) = u0, then u(t) H∞(R, R) L (R, x dx) by T ∈2 proposition 2.5 and lemma 4.15. Recall the generating function λ : u L (R, R) R deﬁned as H ∈ → T dm (ξ) 1 u Lu C λ(u)= (λ + Lu)− Πu, Πu L2 = , mu := µΠu, λ σ( Lu), (4.36) H h i R ξ + λ ∀ ∈ \ − Z Lu 2 where µ denotes the spectral measure of Lu associated to the function ψ L . So the holomorphic ψ ∈ + function λ C σ( Lu) λu is the Borel–Cauchy transform of the positive Borel measure mu. We ∈ \ − 7→ H R 2 recall that the total variation mu( ) = Πu L2 is a conservation law of the BO equation (1.1) by proposition 2.8 and formula (2.20). Everyk ﬁnitek Borel measure is uniquely determined by its Borel– Cauchy transform (see Theorem 3.21 of Teschl [64] page 108), precisely for every a b real numbers, we use Stieltjes inversion formula to obtain that ≤ 1 1 1 b mu((a,b)) + mu([a,b]) = lim Im x+iǫ(u)dx. 2 2 −π ǫ 0+ H → Za For every t R, proposition 2.10 yields that λ[u(t)] = λ[u(0)], λ C σ (L ) = C σ (L ). ∈ H H ∀ ∈ \ pp u(0) \ pp u(t) Since u(0) N , we have Π[u(0)] Hpp(Lu(0)) by proposition 4.6 and there exist c1,c2, ,cN R+ such that ∈ U ∈ ··· ∈ N Lu(t) Lu(0) µ = mu(t) = mu(0) = µ = cj δ u(0) . Π[u(t)] Π[u(0)] λj j=1 X L The spectral measure µ u(t) is purely point, so Π[u(t)] H (L ) for every t R. The Lax pair struc- Π[u(t)] ∈ pp u(t) ∈ ture yields the unitary equivalence between Lu(t) and Lu(0). So dimC Hpp(Lu(t)) = dimC Hpp(Lu(0))= N is given by proposition 2.14. We conclude by theorem 4.8.

5 The generalized action–angle coordinates

In this section, we construct the (generalized) action–angle coordinates ΦN in theorem 1 of the BO equation (1.6) with solutions in the real analytic symplectic manifold ( N ,ω) of real dimension 2N given in proposition 4.3. The goal of this section is to establish the diﬀeomorphismU property and the symplectomorphism property of ΦN .

Recall that the BO equation with N-soliton solutions is identified as a globally well-posed Hamiltonian system reading as ∂tu(t)= XE(u(t)), u(t) N , (5.1) ∈U whose energy functional E(u) = LuΠu, Πu L2 is well defined on N and the Hamiltonian vector field h 2 i U XE : u N XE(u)= ∂x( D u u ) u( N ) coincides with the definition (4.3). The Poisson bracket ∈U 7→ | | − ∈ T U of two smooth functions f,g : N R is given by U → f,g : u N ωu(Xf (u),Xg(u)) = ∂x uf(u), ug(u) 2 R. (5.2) { } ∈U 7→ h ∇ ∇ iL ∈

34 u u u u u Given u N , proposition 4.6 yields that there exist λ < λ < < λ < 0 and ϕ Ker(λ Lu) ∈U 1 2 ··· N j ∈ j − ⊂ u u u D(G) such that ϕ 2 = 1 and u, ϕ 2 = 2π λ , thanks to the spectral analysis in subsection 4.2. k j kL h j iL | j | q u R Deﬁnition 5.1. For every j = 1, 2, ,N, the map Ij : u N 2πλj is called the j th action. u u ··· ∈ U 7→ ∈ The map γj : u N Re Gϕ , ϕ 2 R is called the j th (generalized) angle. ∈U 7→ h j j iL ∈ 1 2 N N 1 2 N N Set ΩN := (r , r , , r ) R : r < r < < r < 0 R , the canonical symplectic form on { ··· ∈ ··· } ⊂ R2N = (r1, r2, , rN ; α1, α2, , αN ): rj , αj R is given by ν = N drj dαj . Endowed with { ··· ··· ∀ ∈ } j=1 ∧ the subspace topology and the embedded real analytic structure of R2N , the submanifold (Ω RN ,ν) P N is a symplectic manifold of real dimension 2N. The action–angle map is deﬁned by ×

N ΦN : u N (I (u), I (u), , IN (u); γ (u),γ (u), ,γN (u)) ΩN R . (5.3) ∈U 7→ 1 2 ··· 1 2 ··· ∈ × Theorem 1 is restated here.

Theorem 5.2. The map ΦN has following properties:

N (a). The map ΦN : N ΩN R is a real analytic diﬀeomorphism. U → × (b). The pullback of ν by ΦN is ω, i.e. ΦN∗ ν = ω. 1 1 2 N 1 2 N N 1 N j 2 (c). We have E Φ− : (r , r , , r ; α , α , , α ) ΩN R r ( , 0). ◦ N ··· ··· ∈ × 7→ − 2π j=1 | | ∈ −∞ P N Remark 5.3. The real analyticity of ΦN : N ΩN R is given by proposition 4.7 and corollary 4.12. The symplectomorphism property (b) is equivalentU → to× the following Poisson bracket characterization (see proposition 5.24)

Ij , Ik =0, Ij ,γk = 1j k, γj,γk =0 on N , j, k =1, 2, ,N. (5.4) { } { } = { } U ∀ ···

The family (XI ,XI , ,XI ; Xγ ,Xγ , ,Xγ ) is linearly independent in X( N ) and we have 1 2 ··· N 1 2 ··· N U ∂ ∂ dΦN (u): XIk (u) k , dΦN (u): Xγk (u) k . 7→ ∂α ΦN (u) 7→ −∂r ΦN (u)

N u u The assertion (c) is obtained by a direct calculus: Πu = Πu, ϕ 2 ϕ , formula (4.9) yields that j=1h j iL j

N P N 2 u 2 u Ij (u) E(u)= Lu(Πu), Πu 2 = Πu, ϕ 2 λ = . h iL |h j iL | j − 2π j=1 j=1 X X Thus theorem 5.2 introduces (generalized) action–angle coordinates of the BO equation (5.1) in the sense u of (1.8), i.e. Ij , E (u)=0 and γj, E (u)=2λ , for every u N . { } { } j ∈U This section is organized as follows. The matrix associated to G H is expressed in terms of actions | pp(Lu) and angles in subsection 5.1. Then the injectivity of ΦN is given by inversion formulas in subsection 5.2. In subsection 5.3, the Poisson brackets of actions and angles are used to show the local diﬀeomorphism property of ΦN . The surjectivity of ΦN is obtained by Hadamard’s global inverse theorem in subsection N 5.4. Finally, we use subsection 5.5 and subsection 5.6 to prove that ΦN : ( N ,ω) (ΩN R ,ν) preserves the symplectic structure. U → ×

35 5.1 The associated 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. Let M(u) = (Mkj (u))1 k,j N denote the matrix associated u u ≤u ≤ to the operator G Hpp(Lu) with respect to the basis ϕ1 , ϕ2 , , ϕN . Then we state a general linear algebra lemma that| describes the location of eigenvalues{ of the··· matrix}M(u).

Proposition 5.4. For every u N , the coeﬃcients of matrix M(u) = (Mkj (u))1 k,j N are given by ∈U ≤ ≤

u i λk u u | u| , if j = k, u u λk λj λj Mkj (u)= Gϕj , ϕk L2 = − | | 6 (5.5) h i  r i γj (u) 2 λu , if j = k.  − | j | 2  Proof. Since Lu is a self-adjoint operator on L and H (Lu) D(G), we have + pp ⊂ u u u u u u u u (λ λ )Mkj (u)= GLuϕ , ϕ 2 Gϕ ,Luϕ 2 = [G, Lu]ϕ , ϕ 2 . j − k h j k iL − h j k iL h j k iL Since formulas (2.15) and (4.9) imply that λuϕu(0) = uϕu(0) = 2π λu , we use (4.26) to obtain − j j j | j | q c d u u u u i u + u i u + u λk (λj λk )Mkj (u)= iϕj ϕj (0 )Πu, ϕk L2 = ϕj (0 )uϕk (0) = i |λu| . − h − 2π i −2π − j r | | In the case k = j, we use Plancherel formulac and integration by partsc to calculated

+ + i ∞ i + + ∞ G∗f,g 2 = f,Gg 2 = fˆ(ξ)∂ξ gˆ(ξ)dξ = fˆ(0 )gˆ(0 )+ ∂ξfˆ(ξ)gˆ(ξ)dξ h iL h iL − 2π 2π Z0 Z0 i + + Thus we have G∗f,g 2 = Gf,g 2 + fˆ(0 )gˆ(0 ), for every f,g H (Lu). Then h iL h iL 2π ∈ pp u 2 1 u u u u ϕj (0) 1 ImMjj (u)= ( Gϕj , ϕj L2 G∗ϕj , ϕj L2 )= | | = u . 2i h i − h i − 4π −2 λj c | | u u We conclude by γj (u)=Re℧j (u)= Gϕ , ϕ 2 deﬁned in corollary 4.12. h j j iL

Then we state a linear algebra lemma that describe the location of spectrum of all matrices of the form deﬁned as (5.5).

Lemma 5.5. For every N N+, we choose N negative numbers λ1 < λ2 < < λN < 0 and N real ∈ N N ··· numbers γ1,γ2, ,γN R. The matrix = ( kj )1 k,j N C × is deﬁned as ··· ∈ M M ≤ ≤ ∈

∗ Then Im = M−M is negative semi-deﬁnite and σpp( ) C . Furthermore, the map M 2i M ⊂ −

(λ1, λ2, , λN ; γ1,γ2, ,γN ) = ( kj )1 k,j N ··· ··· 7→ M M ≤ ≤ N deﬁned as (5.6) is real analytic on ΩN R . ×

36 N T 1 1 1 Proof. The vector Vλ R is deﬁned as V := ((2 λ )− 2 , (2 λ )− 2 , , (2 λN )− 2 ). So we have ∈ λ | 1| | 2| ··· | |

1 T Im = = Vλ Vλ . 2√ λj λk M − | || | 1 k,j N − · ≤ ≤ T 2 Recall that X, Y CN := X Y , thus (Im )X,X CN = X, Vλ CN 0. So Im is a negative semi-deﬁniteh matrix.i If µ σ · ( ) andh V MKer(µ i ) −|h0 , it suﬃcesi | to≤ show thatM Imµ< 0. ∈ pp M ∈ −M \{ } 2 2 2 V, Vλ CN = (Im )V, V CN = Imµ V CN , where V CN = V, V CN > 0. (5.7) − |h i | h M i k k k k h i

So we have Imµ 0. Assume that µ R, then formula (5.7) yields that V Vλ. Moreover, we have ≤ ∈ λ N N ⊥ ( ∗)V = 2i V, Vλ CN Vλ = 0. We set D C × to be the diagonal matrix whose diagonal M−M − h i λ1 ∈ λ2 λ elements are λ1, λ2, , λN , i.e. D = . . Then we have the following formula ···  ..  λN   λ λ T [ ,D ]= i(IN +2D VλV ). (5.8) M λ λ So [ ,D ]V = iV by (5.8). Recall that ∗V = V = µV . Finally, M M M 2 λ λ λ i V CN = [ ,D ]V, V CN = ( µ)D V, V CN = D V, ( ∗ µ)V CN =0 k k h M i h M− i h M − i contradicts the fact that V = 0. Consequently, we have µ C . 6 ∈ − N N Corollary 5.6. For every u N , let M(u) = (Mkj (u))1 k,j N C × denote the matrix deﬁned by ∈UM(u) M(u)∗ ≤ ≤ ∈ formula (5.5), then ImM(u)= − is negative semi-deﬁnite and σpp(M(u)) C . 2i ⊂ − Remark 5.7. The fact σpp(M(u)) C can also be given by using the inversion formula (4.10) and ⊂ − proposition 4.1. The characteristic polynomial Qu(x) = det(x M(u)) has zeros in C . − −

5.2 Inverse spectral formulas

The injectivity of ΦN is proved in this subsection by using inverse spectral formulas. The following lemma describes the relation between the Fourier transform of an eigenfunction ϕ H (Lu) and the ∈ pp Qu inner function associated to u deﬁned by Θu = with Qu(x) = det(x M(u)). Qu − 1 Lemma 5.8. For every monic polynomial Q CN [X] such that Q− (0) C , the associated inner ∈ C ⊂ − function is deﬁned by Θ= Q . The following identity holds for every ϕ ≤N−1[X] , Q ∈ Q

ϕˆ(ξ)= S(ξ)∗ϕ, 1 Θ 2 . (5.9) h − iL + In particular, ϕˆ(0 )= ϕ, 1 Θ 2 . h − iL n P C 1 C mj Proof. Since ϕ = Q , for some P N 1[X] and Q− (0) , recall that Q(x)= j=1(x zj) with ∈ ≤ − n ⊂ − − Imzj < 0, z ,z , ,zN are all distinct and mj = N. Formulas (3.7) and (3.8) imply that 1 2 ··· j=1 Q P l! l iz ξ ˆ j 1R fj,l(x)= l+1 = fj,l(ξ)= ξ e− + (ξ). 2π[( i)(x zj)] ⇒ − −

37 1 R Since ϕ SpanC fj,l 1 j mj ,1 j n, partial-fractional decomposition implies thatϕ ˆ C ( +∗ ), and the ∈ { } ≤ ≤ ≤ ≤ ∈ + Q Q P P 2 right limitϕ ˆ(0 ) = limξ 0+ ϕˆ(ξ) exists. Recall that Θ = Q , so we have Θϕ = Q = L . Since → Q Q ∈ − N Imzj 1 2 Θ(x)=1+2i + ( 2 ), when x + , we have 1 Θ L . Then j=1 x zj x + − O → ∞ − ∈ P iyξ ϕˆ(ξ)= ϕ(y)(1 Θ(y))e− dy = ϕ, S(ξ)(1 Θ) L2 = S(ξ)∗ϕ, 1 Θ L2 , ξ 0. R − h − i h − i ∀ ≥ Z

Proposition 5.9. For every u N , we set Qu CN [X] to be the characteristic polynomial of u and ∈U ∈ Qu we deﬁne the associated inner function as Θu = . Then the following inversion formula holds, Qu

1 1 f(z)= (G z)− f, 1 Θu 2 , f H (Lu), z C . (5.10) 2πih − − iL ∈ pp ∀ ∈ +

C≤N−1[X] Proof. If f Hpp(Lu)= , then formula (5.9) yields that ∈ Qu iξG fˆ(ξ)= S(ξ)∗f, 1 Θu 2 = e− f, 1 Θu 2 . h − iL h − iL G G∗ H Since ImG := −2i is a negative semi-deﬁnite operator on pp(Lu) by proposition 5.4 and lemma 5.5, C the operator Re(i(z G)) H (Lu) = (ImG Imz) H (Lu) is negative deﬁnite, for every z +. So − | pp − | pp ∈ + 1 ∞ iξ(z G) 1 1 f(z)= e − f, 1 Θu 2 dξ = (G z)− f, 1 Θu 2 . 2π h − iL 2πih − − iL Z0

u u u 2π Recall that Πu, ϕj L2 = 2π λj and 1 Θ, ϕj L2 = λu , for every j = 1, 2, ,N, by (2.15) and h i | | h − i | j | ··· (4.9). Since Πu Hol( z qC : Imz > ǫ ), for some ǫ>q0, we have the following inversion formula ∈ { ∈ − } 1 1 1 Πu(x)= (G x)− Πu, 1 Θ 2 = i (M(u) x)− X(u), Y (u) CN , x R, (5.11) 2πi h − − iL − h − i ∀ ∈ where the two vectors X(u), Y (u) RN are defined as ∈ X(u)T = ( λu , λu , , λu ), Y (u)T = ( λu 1, λu 1, , λu 1), (5.12) | 1 | | 2 | ··· | N | | 1 |− | 2 |− ··· | N |− q q q q q q and M(u) is the N N matrix of the infinitesimal generator G associated to the orthonormal basis u u u × ϕ1 , ϕ1 , , ϕN , defined in (5.4). A consequence of the inverse spectral formula (5.11) is the explicit formula{ of··· the BO} flow with N-soliton solutions as described by formula (1.11).

N Corollary 5.10. The map ΦN : N ΩN R is injective. U → × u v Proof. If ΦN (u)=ΦN (v) for some u, v N , then λ = λ and γj (u)= γj (v), for every j. So ∈U j j M(u)= M(v), X(u)= X(v), Y (u)= Y (v).

Then the inversion formula (5.11) gives that Πu = Πv. Thus, u = 2ReΠu = 2ReΠv = v. At last we show the equivalence between the inversion formulas (4.10) and (5.11).

38 u Revisiting formula (4.10). For every k, j =1, 2, ,N, let Kkj (x) denote the (N 1) (N 1) subma- trix obtained by deleting the k th column and j··· th row of the matrix M(u) x, for− every× x− R. So the inversion formula (5.11) and the Cramer’s rule imply that − ∈

k+j u u N u ( 1) det(K (x)) λ det(Kjj (x)) + R iΠu(x)= − kj k = j=1 , (5.13) det(M(u) x) λu det(M(u) x) 1 k,j N s j P ≤X≤ − −

u k+j u λk where R := 1 k=j N ( 1) det(Kkj (x)) λu . The coeﬃcients of the matrix M(u) x satisﬁes that ≤ 6 ≤ − j − P q u i λk (M(u) x)kj = Mkj (u)= λu λu λu , if 1 j = k N, − k j j ≤ 6 ≤ − r by formula (5.5). Using expansion by minors, we have

N N k+j u u u u u iR = ( 1) (λ λ )(M(u) x)kj det(K (x)) = ( λ λ ) det(M(u) x)=0. − k − j − kj k − j − 1 k,j N k=1 j=1 ≤X≤ X X Finally, let Q denote the characteristic polynomial of the operator G H , so | pp(Lu) N N u Q(x) = det(x G H ) = det(x M(u)),Q′(x) = ( 1) det(K (x)). − | pp(Lu) − − jj j=1 X

5.3 Poisson brackets In this subsection, the Poisson bracket defined in (5.2) is generalized in order to obtain the first two formulas of (5.4). It can be defined between a smooth function from N to an arbitrary Banach space U and another smooth function from N to R. U

The N-soliton subset ( N ,ω) is a real analytic symplectic manifold of real dimension 2N, where U

i hˆ1(ξ)hˆ2(ξ) ωu(h1,h2)= dξ, h1,h2 u(UN ), u N . 2π R ξ ∀ ∈ T ∀ ∈U Z

For every smooth function f : N R, its Hamiltonian vector ﬁeld Xf X( N ) is given by (4.3). Recall U → ∈ U that Xf (u) = ∂x uf(u) and df(u)(h) = ω(h,Xf (u)), h u( N ). For any Banach space and any ∇ ∀ ∈ T U E smooth map F : u N F (u) , we deﬁne the Poisson bracket of f and F as follows ∈U 7→ ∈E

f, F : u N f, F (u) := dF (u)(Xf (u)) ( )= . (5.14) { } ∈U 7→ { } ∈ TF (u) E E If = R, then the deﬁnition in formula (5.14) coincide with (5.2) and we recall it here, E

f, F (u) = dF (u)(Xf (u)) = ωu(Xf (u),XF (u)). (5.15) { }

39 1 For every λ C σ( Lu), the generating function λ(u)= (Lu + λ)− Πu, Πu L2 is well deﬁned. Since N ∈ \ u − u H h i Πu = Πu, ϕ 2 ϕ , we have j=1h j iL j P N u 2 N u Πu, ϕj L2 2πλj λ(u)= |h i | = . (5.16) H λ + λu − λ + λu j=1 j j=1 j X X

The analytical continuation allow to extend the generating function λ λ(u) to the domain C σpp( Lu), u 7→ HC2 2 u \ u − and it has simple poles at every λ = λj . Proposition 2.2 yields that 4 u L2 λ1 < < λN < 0, 1 − − k k ≤ ··· 4 D f L2 where C = inf 1 k| | k denotes the Sobolev constant. So we introduce f H 0 f 4 ∈ +\{ } k kL 2 2 = (λ, u) R N :4λ > C u 2 = (R N ) , (5.17) Y { ∈ ×U k kL } X ×U \ where is given by definition 2.9. Then the subset is open in R N and the map : (λ, u) X u Y ×U H ∈ Y 7→ N 2πλj R j=1 λ+λu is real analytic by proposition 4.7. Recall that the Fréchet derivative (2.8) is given by − j ∈ P 2 d λ(u)(h)= wλ, Πh 2 + wλ, Πh 2 + Thwλ, wλ 2 = h, wλ + wλ + wλ 2 , h u( N ). H h iL h iL h iL h | | iL ∀ ∈ T U 1 1 where wλ H is given by wλ wλ(u) wλ(x, u) = [(Lu + λ)− Π]u(x), for every x R. Thus ∈ + ≡ ≡ ◦ ∈ 2 X (u)= ∂x u λ(u)= ∂x( wλ(u) + wλ(u)+ wλ(u)), (λ, u) . (5.18) Hλ ∇ H | | ∀ ∈ Y 1 2 by (4.3). The Lax map L : u N Lu = D Tu B(H ,L ) is R-affine, hence real analytic. The ∈ U 7→ − ∈ + + following proposition restates the Lax pair structure of the Hamiltonian equation associated to λ. Even H though the stability of N under the Hamiltonian flow of λ remains as an open problem, the Poisson bracket defined in (5.14U) provides an algebraic method to obtainH the first two formulas of (5.4).

u Proposition 5.11. Given (λ, u) defined by (5.17), we have λ,L (u) = [B ,Lu] and ∈ Y {H } λ λ u u λ 2 λ, λj (u)=0, λ,γj (u)=Re [G, Bu ]ϕj , ϕj L = u 2 , (5.19) {H } {H } h i −(λ + λj ) for every j =1, 2, ,N, where Bu = i(T T + T + T ). ··· λ wλ(u) wλ(u) wλ(u) wλ(u) 2 1 2 2 Proof. Since L : u L (R, R) Lu = D Tu B(H ,L ), for every u L , we have ∈ 7→ − ∈ + + ∈ + 2 dL(u)(h)= Th, h L . − ∀ ∈ + 1 2 If (λ, u) , then the C-linear transformation Lu + λ B(H ,L ) is bijective. So formula (5.18) ∈ Y ∈ + + 2 yields that λ,L (u) = dL(u)(X λ (u)) = TD( w (u) +w (u)+w (u)). Then identity (2.24) yields the {H } H − | λ | λ λ Lax equation for the Hamiltonian flow of the generating function λ, i.e. H u 1 2 λ,L (u) = [B ,Lu] B(H ,L ). (5.20) {H } λ ∈ + + u u u 1 Consider the map Lϕj : u N Luϕ = λ ϕ H , for every (λ, u) , we have ∈U 7→ j j j ∈ + ∈ Y u u u λ,L (u)ϕ + Lu ( λ, ϕj (u)) = λ λ, ϕj (u)+ λ, λj (u)ϕ {H } j {H } j {H } {H } j 1 with λ, ϕj (u) H and λ, λj (u) R. Then (5.20) yields that {H } ∈ + {H } ∈ u λ u u (λ Lu) B ϕ λ, ϕj (u) = λ, λj (u)ϕ . j − u j − {H } {H } j 40 u u u Since ϕ Ker(λ Lu) and ϕ 2 = 1 by the definition in (4.9), we have j ∈ j − k j kL u λ u u λ, λj (u)= (λ Lu) B ϕ λ, ϕj (u) , ϕ 2 =0. {H } h j − u j − {H } j iL 2 2 Let : ϕ L ϕ 2 , then we have ϕj 1 on N . Then we have N2 ∈ 7→ k kL N2 ◦ ≡ U u 0 = d( ϕj )(u) = 2Re ϕ , λ, λj (u) 2 . (5.21) N2 ◦ h j {H } iL R λ u u u C u So there exists r such that Bu ϕj λ, ϕj (u)= irϕj because Ker(λj Lu) = ϕj by corollary ∈ λ− {H } u u − 2.4 and formula (5.21). Recall that B is skew-adjoint and γj = Re Gϕ , ϕ 2 , we have u h j j iL u u λ u u λ,γj (u)=Re G λ, ϕj (u), ϕ 2 + Gϕ , λ, ϕj (u) 2 = Re [G, B ]ϕ , ϕ 2 . {H } h {H } j iL h j {H } iL h u j j iL Furthermore, for every (λ, u) , formula (3.4) implies that [G, T ] = 0 and ∈ Y wλ(u) λ 1 + + [G, B ]f = i[G, T ](T (f)+ f)= [(wλ(u)f)∧(0 )+ fˆ(0 )]wλ(u), f D(G). (5.22) u wλ(u) wλ(u) − 2π ∀ ∈ u + u u 1 u u u u + Since (w (u)ϕ ) (0 )= ϕ , w (u) 2 = (λ + λ ) u, ϕ and u, ϕ = λ ϕ (0 ), we replace f λ j ∧ j λ L j − j L2 j L2 j j u h i h i h i − by ϕj in formula (5.22) to obtain the following c u, ϕu λ u u j L2 1 1 u λ 2 h i 2 [G, Bu ]ϕj , ϕj L = ( u u ) wλ(u), ϕj L = u 2 , (λ, u) . h i 2π λj − λ + λj h i −(λ + λj ) ∀ ∈ Y

1 1 1 ǫ 1 Remark 5.12. Recall that ˜ǫ = 1 and B˜ǫ,u := Bu for every (ǫ− ,u) . In general, the identity H ǫ H ǫ ǫ ∈ Y dn ˜ u u En,γj (u)=Re [G, n Bǫ,u]ϕ , ϕ 2 , 1 j N { } h dǫ ǫ=0 j j iL ≤ ≤ n dn ˜ holds for every conservation law En = ( 1) n ǫ in the BO hierarchy. − dǫ ǫ=0H Corollary 5.13. For every j, k =1, 2, ,N, we have ···

2π λj ,γk (u)= 1j k, λk, λj (u)=0, u N . (5.23) { } = { } ∀ ∈U C2 u 2 L2 Proof. Given u N , for every λ> k k then (λ, u) , then (5.16) and (5.19) imply that ∈U 4 ∈ Y N N λ λ λk,γj (u) = λ,γj (u)=2π ,γj (u)= 2πλ { } , −(λ + λu)2 {H } {λ + λu } − (λ + λu)2 j k k k k X=1 X=1

N λ ,λj (u) { k } and 0 = λ, λj (u)=2πλ k=1 (λ+λu)2 , for every j = 1, 2, ,N. The uniqueness of analytic {H } k ··· continuation yields that the following formula holds for every z C R, P ∈ \ N N z λk,γj (u) λk, λj (u) = 2πz { } , { } =0. −(z + λu)2 − (z + λu)2 (z + λu)2 j k k k k X=1 X=1

41 u u u Recall that the actions Ij : u N 2πλj and the generalized angles γj : u N Re Gϕj , ϕj L2 are both real analytic functions by∈U proposition7→ 4.7 and corollary 4.12. ∈U 7→ h i

Proposition 5.14. For every u N , the family of diﬀerentials ∈U

dI (u), dI (u), dIN (u); dγ (u), dγ (u), dγN (u) { 1 2 ··· 1 2 ··· } is linearly independent in the cotangent space ∗( N ). Tu U Proof. For every a ,a , ,aN ,b ,b , ,bN R such that 1 2 ··· 1 2 ··· ∈ N aj dIj (u)+ bj dγj (u) (h)=0, h u( N ). (5.24)   ∀ ∈ T U j=1 X   Formula of Poisson brackets (5.23) yields that for every j, k =1, 2, ,N, we have ···

dIj (u)(XI (u)) = Ik, Ij (u)=0, dγj (u)(XI (u)) = Ik,γj (u)= 1j k k { } k { } =

We replace h by XI (u) in (5.24) to obtain that bk = 0, k =1, 2, ,N. Then set h = Xγ (u) k ∀ ··· k N N ak = aj γk, Ij (u)= aj dIj (u) (Xγ (u))=0, k =1, 2, ,N. − { }   k ∀ ··· j=1 j=1 X X  

N As a consequence, ΦN : N ΩN R is a local diffeomorphism. Moreover, since all the actions U → × (Ij )1 j N are in evolution by (5.23) and the differentials (dIj (u))1 j N are linearly independent for ≤ ≤ 1 2 N ≤ ≤ every u N , for every r = (r , r , , r ) ΩN , the level set ∈U ··· ∈ N 1 j 1 2 N r = I− (r ), where r = (r , r , , r ) L j ··· j=1 \ is a smooth Lagrangian submanifold of UN and r is invariant under the Hamiltonian flow of Ij , for every j = 1, 2, ,N, by the Liouville–Arnold theoremL (see Theorem 5.5.21 of Katok–Hasselblatt [32], see also Fiorani–Giachetta–Sardanashvily··· [12] and Fiorani–Sardanashvily [13] for the non-compact invariant manifold case).

5.4 The diﬀeomorphism property N This subsection is dedicated to proving the real bi-analyticity of ΦN : N ΩN R . It remains to show the surjectivity. Its proof is based on Hadamard’s global inverse theoremU → 5.18×.

N Lemma 5.15. The map Φ: N ΩN R is proper. U → × N 1 Proof. If K is compact in ΩN R , we choose un Φ− (K), so × ∈ N

un un un ΦN (un)=(2πλ , 2πλ , , 2πλ ; γ (un),γ (un), ,γN (un)) K, n N. 1 2 ··· N 1 2 ··· ∈ ∀ ∈

42 un We assume that there exists (2πλ1, 2πλ2, , 2πλN ; γ1,γ2, ,γN ) K such that λj λj and ··· ··· ∈ N N → γj (un) γj up to a subsequence. So (M(un))n N converges to some matrix M C × whose co- eﬃcients→ are deﬁned as follows ∈ ∈

i λk | | , if k = j, λk λj λj Mkj = − | | 6 i γj q , if k = j. 2 λj  − | |  Q′ Q′ Lemma 5.5 yields that σpp(M) C . We set Q(x) := det(x M) and u = i i N . The Vi`ete ⊂ − − Q − Q ∈U map V is deﬁned in (4.11) and V(CN ) is open in CN . Then there exists − (n) (n) (n) (n) CN a = (a0 ,a1 , ,aN 1), a = (a0,a1, ,aN 1) V( ) ··· − ··· − ∈ − N 1 (n) j N N 1 j N such that Qn(x) = det(x M(un)) = − a x + x and Q(x)= − aj x + x . We have − j=0 j j=0 P P(n) lim Qn(x)= Q(x), x R = lim a = a n + n + → ∞ ∀ ∈ ⇒ → ∞ ′ CN Q 2 The continuity of the map ΓN : a = (a0,a1, ,aN 1) V( ) Πu = i Q L+ yields that ··· − ∈ − 7→ ∈

Qn′ (n) Q′ 2 Πun = i =ΓN (a ) ΓN (a)= i = Πu in L+, as n + . Qn → Q → ∞

2 Since N inherits the subspace topology of L (R, R), we have (un)n N converges to u in N . The U ∈ U continuity of the map ΦN shows that ΦN (u)=(2πλ , 2πλ , , 2πλN ; γ ,γ , ,γN ) K. 1 2 ··· 1 2 ··· ∈

RN 1 Proposition 5.16. The map ΦN : N ΩN is bijective and both ΦN and its inverse ΦN− are real analytic. U → ×

Proof. The analyticity of ΦN is given by proposition 4.7 and corollary 4.12. The injectivity is given by N corollary 5.10. Proposition 5.14 yields that ΦN : N ΩN R is a local diﬀeomorphism by inverse U → × function theorem for manifolds. So ΦN is an open map. Since every proper continuous map to locally N compact space is closed, ΦN is also a closed map by lemma 5.15. Since the target space ΩN R is N N × connected, we have ΦN ( N )=ΩN R and ΦN : N ΩN R is a real analytic diﬀeomorphism. U × U → ×

N N Remark 5.17. We establish the relation between ΦN : N ΩN R and ΓN : V(C ) Π( N ) N U N→ N × − → U introduced in proposition 4.10. We set : ΩN R C × to be the matrix-valued real analytic M × → function (η1, η2, , ηN ; θ1,θ2, ,θN ) = ( kj )1 k,j N with coeﬃcients deﬁned as M ··· ··· M ≤ ≤ 2πi ηk , if k = j, ηk ηj ηj kj = − 6 πi M θj + q, if k = j.  ηj

N N N Then, we set : M C × (a0,a1, ,aN 1) C such that C ∈ 7→ ··· − ∈ N 1 − j N Q(x) := aj x + x = det(x M). (5.25) − j=0 X

43 n j n j Since ( 1) − aj = Tr(Λ − M) is the sum of all principle minors of M of size (N j) (N j), for − N N N − ×N − every j = 1, 2, ,N, the map is real analytic on C × and (ΩN R ) V(C ) by lemma 5.5, where V denotes··· the VièteC map defined as (4.11). In lemmaC◦M4.10, we× have shown⊂ that− the map CN Q′ ΓN : a = (a0,a1, ,aN 1) V( ) Πu = i Q Π( N ) is biholomorphic, where the polynomial Q ··· − ∈ − 7→ ∈ U is defined as (5.25). We conclude by the following identity

1 Φ− = 2Re ΓN (5.26) N ◦ ◦C◦M N N The smooth manifolds Π( N ) and V(C ) are both diffeomorphic to the convex open subset ΩN R , so they are simply connectedU (see also− proposition A.5). At last, we recall Hadamard’s global inverse× theorem. Theorem 5.18. Suppose X and Y are connected smooth manifolds, then every proper local diffeomorphism F : X Y is surjective. If Y is simply connected in addition, then every proper local diffeomorphism F :→X Y is a diffeomorphism. → Proof. For the surjectivity, see Nijenhuis–Richardson [47] and the proof of proposition 5.16. If the target space is simply connected, see Gordon [23] for the injectivity.

N Remark 5.19. Since the target space ΩN R is convex, there is another way to show the injectivity × of ΦN without using the inversion formulas in subsection 5.2. It suﬃces to use the simple connectedness N of ΩN R and Hadamard’s global inverse theorem 5.18. ×

5.5 A Lagrangian submanifold

In general, the symplectomorphism property of ΦN is equivalent to its Poisson bracket characterization (5.4), which will be proved in proposition 5.24. The ﬁrst two formulas of (5.4) given in corollary 5.13, lead us to focusing on the study of a special Lagrangian submanifold of N , denoted by U

ΛN := u N : γj (u)=0, j =1, 2, ,N , (5.27) { ∈U ∀ ··· } u u where the generalized angles γj : u N Re Gϕ , ϕ 2 are deﬁned in (5.1). A characterization ∈ U 7→ h j j iL lemma of ΛN is given at ﬁrst.

Lemma 5.20. For every u N , then each of the following four properties implies the others: ∈U

(a). u ΛN . (b). For∈ every x R, we have Πu(x) = Πu( x). (c). u is an even∈ function R R. − (d). The Fourier transform uˆ→is real-valued.

R N N 2ηj Then every element u ΛN has translation–scaling parameter in (i ) /SN i.e. u(x)= j=1 x2+η2 , for ∈ j some η > 0. j P

Proof. (a) (b): If u ΛN , then the matrix M(u) defined in (5.5) is an N N matrix with purely imaginary coefficients.⇒ Recall∈ the definition of X(u), Y (u) RN in (5.12): × ∈ X(u)T = ( λu , λu , , λu ), Y (u)T = ( λu 1, λu 1, , λu 1). | 1 | | 2 | ··· | N | | 1 |− | 2 |− ··· | N |− q q q q q q 44 The inversion formula (5.11) yields that

1 1 Πu(x)= i (M(u) x)− X(u), Y (u) CN = i (M(u)+ x)− X(u), Y (u) CN = Πu( x). h − i − h i − (b) (c) is given by the formula u = Πu + Πu. (c) (d) is given by u(x)= u(x)= u( x). ⇒ ⇒ − u u u (d) (a): Choose λ σ (Lu)= λ , λ , , λ and ϕ Ker(λ Lu). Since both u and its Fourier ⇒ ∈ pp { 1 2 ··· N } ∈ − transformu ˆ are real-valued, we have [(ϕ)∨]∧(ξ)= ϕˆ(ξ), where (ϕ)∨(x) := ϕ( x), x, ξ R. Thus, − ∀ ∈

Tu((ϕ)∨) = (Tuϕ)∨ = (ϕ)∨ Ker(λ Lu). ⇒ ∈ − u u u We choose the orthonormal basis ϕ , ϕ ,, , ϕ in H (Lu) as in formula (4.9). Proposition 2.4 { 1 2 ··· N } pp yields that dimC Ker(λ Lu) = 1. For every j =1, 2, ,N, there exists θ˜j R such that − ··· ∈

u iθ˜j u u iθ˜j u (ϕ )∨ = e ϕ (ϕ ) (ξ)= e (ϕ )∧(ξ), ξ R. j j ⇐⇒ j ∧ j ∀ ∈ ˜ u iθj u u So we set φj := exp( 2 )ϕj , then its Fourier transform (φj )∧ is a real-valued function. Recall the deﬁnition of G in (3.2) and γj in (5.5), then we have

u u u u 1 u u γj (u)=Re Gϕj , ϕj L2(R) = Re Gφj , φj L2(R) = Im ∂ξ[(φj )∧], (φj )∧ L2(0,+ ) =0. h i h i −2π h i ∞ by using Plancherel formula.

Lemma 5.21. The level set ΛN is a real analytic Lagrangian submanifold of ( N ,ω). U N Proof. The map γ : u N (γ (u),γ (u), ,γN (u)) R is a real analytic submersion by proposi- ∈U 7→ 1 2 ··· ∈ tion 5.14. So the level set ΛN is a properly embedded real analytic submanifold of N and dimR ΛN = N. U N 2ηj The classiﬁcation of the tangent space u( N ) is given by formula (4.1). If u(x)= j=1 x2+η2 , for some T U j ˆ ηj > 0, every tangent vector h ΛN is an even function by lemma 5.20. So h is realP valued and we have ∈ N 2[x2 η2] R u u − j u(ΛN )= fj , where fj (x)= [x2+η2]2 . (5.28) T j j=1 M

u ηj ξ We have (f )∧(ξ)= 2π ξ e− | |. Then by deﬁnition of ω, we have j − | |

i hˆ1(ξ)hˆ2(ξ) i hˆ1(ξ)hˆ2(ξ) ωu(h1,h2)= dξ = dξ iR, h1,h2 u(ΛN ). (5.29) 2π R ξ 2π R ξ ∈ ∀ ∈ T Z Z

Since the symplectic form ω is real-valued, we have ωu(h1,h2) = 0, for every h1,h2 u(ΛN ). Since 1 ∈ T dimR(ΛN )= N = dimR N ,ΛN is a Lagrangian submanifold of N . 2 U U

45 5.6 The symplectomorphism property N Finally, we prove the assertion (b) in theorem 5.2, i.e. the map ΦN : ( N ,ω) (ΩN R ,ν) is U → × i hˆ1(ξ)hˆ2(ξ) symplectic, where ω(h ,h ) := dξ, for every h ,h u( N ) and 1 2 2π R ξ 1 2 ∈ T U R N N 1 2 N 1 2 N 2N 1 2 N j j ΩN R = (r , r , , r ; α , α , , α ) R : r < r < < r < 0 , ν = dr dα . × { ··· ··· ∈ ··· } ∧ j=1 X 1 RN We set ΨN =ΦN− :ΩN N , let ΨN∗ ω denote the pullback of the symplectic form ω by ΨN , i.e. 1 2 × N →U1 2 N N for every p = (r , r , , r ; α , α , , α ) ΩN R , set u =ΨN (p) N , ··· ··· ∈ × ∈U

(ΨN∗ ω)p (V1, V2)= ωu(dΨN (p)(V1), dΨN (p)(V2)), (5.30)

N for every V , V p(ΩN R ). The goal is to prove that 1 2 ∈ T ×

ν˜ := Ψ∗ ω ν =0. (5.31) N − Recall that the coordinate vectors ∂ , ∂ , , ∂ ; ∂ , ∂ , , ∂ form a basis for the ∂r1 p ∂r2 p ··· ∂rN p ∂α1 p ∂α2 p ··· ∂αN p RN tangent space p(ΩN ). We have the following lemma. T × N Lemma 5.22. For every u N , set p =ΦN (u) ΩN R . Then we have ∈U ∈ × ∂ dΦN (u)(XIk (u)) = , k =1, 2, ,N. (5.32) ∂αk p ∀ ···

N Proof. Fix u N and p =ΦN (u), for every h u( N ), we have dΦN (u)(h) p(ΩN R ). For every ∈U 1 2 N 1 2 ∈ T UN N ∈ T × smooth function f : p = (r , r , , r ; α , α , , α ) ΩN R f(p) R, then ··· ··· ∈ × 7→ ∈ N ∂f ∂f (dΦN (u)(h)) f = d(f ΦN )(u)(h)= dIj (u)(h) + dγj (u)(h) . (5.33) ◦ ∂rj p ∂αj p j=1 X

For every k = 1, 2, ,N, we replace h by XI (u), where XI denotes the Hamiltonian vector ﬁeld of ··· k k the k th action Ik deﬁned in (5.1), thus the Poisson bracket formulas (5.23) yield that

N ∂f ∂f ∂f = Ik, Ij (u) + Ik,γj (u) = (dΦN (u)(XIk (u))) f. ∂αk p { } ∂rj p { } ∂αj p j=1 X

N Lemma 5.23. For every 1 j < k N, there exists a smooth function cjk C∞(ΩN R ) such that ≤ ≤ ∈ ×

j k ∂cjk ν˜ = cjkdr dr , =0, j,k,l =1, 2, ,N, (5.34) ∧ ∂αl p ∀ ··· 1 j

46 N Proof. The proof is divided into three steps. The ﬁrst step is to prove that for every p ΩN R and N ∈ × every V p(ΩN R ), ∈ T × ∂ ν˜p( , V )=0, l =1, 2, ,N. (5.35) ∂αl p ∀ ··· 1 2 N 1 2 N l l In fact, let u =ΨN (p) N and p = (r , r , , r ; α , α , , α ), so r = r (p)= Il ΨN (p). Then ∈U ··· ··· ◦ ∂ ∂ (ΨN∗ ω)p( , V )= ωu(dΨN (p) , dΨN (p)(V )) = ωu(XIl (u), dΨN (p)(V )) ∂αl p ∂αl p

∂ by (5.32). Thus (ΨN∗ ω)p( l , V )= dIl(u)(dΨN (p)(V )) = d(Il ΨN )(p)(V ). On the other hand, ∂α p − − ◦

N ∂ j j ∂ l νp( , V )= (dr dα ) , V = dr (p)(V ) ∂αl p ∧ ∂αl p − j=1 X

Thus (5.35) is obtained byν ˜ =Ψ∗ ω ν. N − N Sinceν ˜ is a smooth 2-form on ΩN R , we have × j k j k j k ν˜ = (ajkdα dα + bjkdr dα + cjkdr dr ), ∧ ∧ ∧ 1 j

j k ∂ j j k ∂ k j dr dα ( , V )= 1k=ldr (p)(V ) and dα dα ( , V )= 1j=ldα (p)(V ) 1k=ldα (p)(V ). ∧ ∂αl p − ∧ ∂αl p −

Then, let l 2 , ,N be ﬁxed, for every 1 j < k N, we have ∈{ ··· } ≤ ≤ k j j ∂ alkdα (p)(V ) (ajldα (p)(V )+ bjldr (p)(V ))=ν ˜p( , V )=0. (5.36) − ∂αl p 1 l

··· − 1 2 N It remains to show that cjk depends on r , r , , r , for every 1 j < k N. The symplectic form ω ··· N ≤j j ≤ is closed by proposition 4.4 and ν = dκ is exact, where κ = j=1 r dα . So

d˜ν = d(Ψ∗ ω) dν =Ψ∗ P(dω)=0. N − N j k The exterior derivative ofν ˜ = 1 j

47 1 2 N N Since the 2-formν ˜ is independent of α , α , , α , it suﬃces to consider points p = (r, α) ΩN R ··· ∈ × N with α = 0. We shall prove thatν ˜ = 0 by introducing the following Lagrangian submanifold of ΩN R , × 1 2 N 2N 1 2 N ΩN 0RN = (r , r , , r ;0, 0, , 0) R : r < r < < r < 0 . ×{ } { ··· ··· ∈ ··· }

End of the proof of formula (5.31). The submersion level set theorem implies that ΩN 0RN is a prop- N ×{ } erly embedded N-dimensional submanifold of ΩN R . We have ΩN 0RN =ΦN (ΛN ), where ΛN is × ×{ } the Lagrangian submanifold of ( N ,ω) deﬁned by (5.27). For every q ΩN 0RN , set v =ΨN (q) ΛN , we claim at ﬁrst that U ∈ ×{ } ∈ N ∂ q(ΩN 0RN )= R = dΦN (v)( v (ΛN )). (5.37) T ×{ } ∂rj q T j=1 M In fact, every tangent vector V q(ΩN 0RN ) is the velocity at t = 0 of some smooth curve ∈ T × { } ξ : t ( 1, 1) ξ(t) = (ξ (t), ξ (t), , ξN (t); 0, 0, , 0) ΩN 0RN such that ξ(0) = q, i.e. ∈ − 7→ 1 2 ··· ··· ∈ ×{ } N d ∂f RN Vf = (f ξ)= ξj′ (0) , f C∞(ΩN ). (5.38) dt t=0 ◦ ∂rj q ∀ ∈ × j=1 X

So the ﬁrst equality of (5.37) is obtained. Then we set η(t)=ΨN ξ(t), t ( 1, 1). For every N ◦ ∀ ∈ − g C∞( N ), we replace f by g ΨN C∞(ΩN R ) in (5.38) to obtain that ∈ U ◦ ∈ × d dΨN (q)(V )g = V (g ΨN )= (g η)= η′(0)g. ◦ dt t=0 ◦

Since η is a smooth curve in the Lagrangian section ΛN such that η(0) = v, we have dΨN (q)(V )= η′(0) N j j ∈ v(ΛN ). So formula (5.37) holds. Since ν = dr dα , the submanifold ΩN 0RN is Lagrangian. T j=1 ∧ ×{ } 1 2 N 1 2 NP N N For every p = (r , r , , r ; α , α , , α ) ΩN R and every V , V p(ΩN R ), where ··· ··· ∈ × 1 2 ∈ T × N (m) ∂ (m) ∂ (m) (m) R Vm = aj + bj , aj ,bj , m =1, 2, ∂rj p ∂αj p ∈ j=1 X 1 2 N we choose q = (r , r , , r ;0, 0, , 0) ΩN 0RN and W , W q(ΩN 0RN ), where ··· ··· ∈ ×{ } 1 2 ∈ T ×{ } N (m) ∂ Wm = aj , m =1, 2. ∂rj p j=1 X

We set v =ΨN (q) ΛN . We have proved that cjk(p)= cjk(q), then (5.34) yields that ∈ (1) (2) ν˜p(V1, V2)= aj ak cjk(p)=˜νq(W1, W2)= ωv(dΨN (v)(W1), dΨN (v)(W2)), 1 j

48 N Proposition 5.24. If Φ˜ N : ( N ,ω) (ΩN R ,ν) is a diﬀeomorphism, U → ×

Φ˜ N (u) = (I˜ (u), I˜ (u), , I˜N (u);γ ˜ (u), γ˜ (u), , γ˜N (u)), u N , 1 2 ··· 1 2 ··· ∀ ∈U for some smooth functions I˜j , γ˜j on N , then each of the following three properties implies the others: U N (a). Φ˜ N : ( N ,ω) (ΩN R ,ν) is a symplectomorphism, i.e. Φ˜ ∗ ν = ω. U → × N (b). For every j, k =1, 2, ,N, we have I˜j , I˜k = γ˜j , γ˜k =0 and I˜j , γ˜k = 1j=k on N . (c). For every k =1, 2, ···,N, we have { } { } { } U ··· ∂ ∂ dΦ˜ (u)(X (u)) = , dΦ˜ (u)(X (u)) = , u . N I˜k k N γ˜k k N ∂α Φ˜ N (u) −∂r Φ˜ N (u) ∀ ∈U

N Proof. (a) (b). For any smooth function f :ΩN R R, its Hamiltonian vector ﬁeld is given by ⇒ × → N ∂f ∂ ∂f ∂ N Xf (p)= (p) (p) , p ΩN R . (5.39) ∂rj ∂αj p − ∂αj ∂rj p ∀ ∈ × j=1 X 1 If Φ˜ ν = ω, then X (u) = dΨ˜ (p) X (p), if p = Φ˜ (u), where Ψ˜ = Φ˜ − . The Poisson bracket of N∗ f Φ˜ N N f N N N ◦ N ◦ two smooth functions f,g on ΩN R is given by × N ν ∂f ∂g ∂f ∂g f,g (p) = (Xf g) = νp(Xf (p),Xg(p)) = (p) (p) (p) (p). (5.40) { } p ∂rj ∂αj − ∂αj ∂rj j=1 X ν Then f Φ˜ N ,g Φ˜ N = f,g Φ˜ N on N . It suﬃces to choose f,g I˜j Ψ˜ N , γ˜j Ψ˜ N 1 j N . { ◦ ◦ } { } ◦ U ∈{ ◦ ◦ } ≤ ≤ (b) (c). We do the same calculus as in lemma 5.22 to obtain that ⇒ ∂ ∂ dΦ˜ (u)(X (u)) = , dΦ˜ (u)(X (u)) = , u . (5.41) N I˜k k N γ˜k k N ∂α Φ˜ N (u) −∂r Φ˜ N (u) ∀ ∈U

(c) (a). Formula (5.41) implies that X ,X , ,X ; X ,X , ,X forms a basis in X( ). I˜1 I˜2 I˜N γ˜1 γ˜2 γ˜N N ⇒ ˜ { ··· ··· } U Since the 2-covectors (ΦN∗ ν)u and ωu coincide at every couple of elements of this basis, they are the same, ˜ so ΦN∗ ν = ω.

A Appendices

We establish several topological properties of the N-soliton manifold N without using the action–angle N U N N map ΦN : N ΩN R . The Viète map V : (β1,β2, ,βN ) C (a0,a1, ,aN 1) C is defined by U → × ··· ∈ 7→ ··· − ∈ N N 1 − k N (X βj )= akX + X . (A.1) − j=1 k Y X=0 Proposition A.1. Endowed with the Hermitian form H introduced in (4.15), (Π( N ), H) is a simply connected Kähler manifold which is biholomorphically equivalent to V(CN ). U −

49 Proposition A.2. The N-soliton manifold N is a universal covering manifold of the following N-gap potential manifold for the BO equation on theU torus T := R/2πZ as described by Gérard–Kappeler [19], iy T Q (e ) U = v = h + h L2(T, R): h : y T eiy ′ C, Q C+ [X] , (A.2) N { ∈ ∈ 7→ − Q(eiy) ∈ ∈ N } C+ C where N [X] consists of all monic polynomial Q [X] of degree N, whose roots are contained in the annulus A := z C : z > 1 . The fundamental∈ group of U T is (Z, +). { ∈ | | } N T CN 1 C Remark A.3. The real analytic symplectic manifold UN is mapped real bi-analytically onto − ∗ by the restriction of the Birkhoff map constructed in Gérard–Kappeler [19]. The union of all finite× gap T 2 2 potentials U is dense in L (T)= v L (T, R): v =0 . However N is not dense in N 0 N r,0 { ∈ T } N 1 U L2(R, (1 + x2)d≥ x). We refer to Coifman–Wickerhauser [9] to see solutions with sufficiently≥ small initial S R S data and the case of non-existence of rapidly decreasing solitons.

The simple connectedness of N is proved in subsection A.1. Then we establish a real analytic covering T U map N U in subsection A.2. U → N

A.1 The simple connectedness of N U N Thanks to the biholomorphical equivalence between the Kähler manifolds Π( N ) and V(C ) established in lemma 4.10, it suffices to prove the simple connectedness of the subset VU(CN ), where V− denotes the Viète map defined by (A.1). Since every fiber of the Viète map is invariant under− the permutation of components, we introduce the following group action. Equipped with the discrete topology, the symmetric N group SN acts continuously on C by permuting the components of every vector: N N σ : (β0,β1, ,βN 1) C (βσ(0),βσ(1), ,βσ(N 1)) C , σ SN . (A.3) ··· − ∈ 7→ ··· − ∈ ∀ ∈ A subset A CN is said to be stable under S if σ(A)= A. We recall the basic property of the N σ SN ⊂ ∈ Viète map V and the action of symmetric group SN . S Lemma A.4. The Viète map V : CN CN is a both open and closed quotient map. For every N → A C , A is stable under SN if and only if A is saturated with respect to V, the quotient space A/SN is⊂ homeomorphic to V(A). We set ∆ := (β, β, ,β) CN : β C . The goal of this subsection is to prove the following result. { ··· ∈ ∀ ∈ } Proposition A.5. For every open simply connected subset A CN , if A is stable under the symmetric ⊂ N group SN and A ∆ = , then V(A) is an open simply connected subset of C . 6 ∅ CN Proof. Let A T be a nonempty open simply connected subset that is stable by SN . The subset B := V(A) is open,⊂ connected and locally simply connected, then it admits a universal covering space E and a covering map π : E B. The triple (E,π,B) is identified as a fiber bundle over B whose model fiber F is discrete. The target→ is to show that F has cardinality 1.

Let (P,q,B) denote the fiber product (Husemöller [28]) of bundles (A, V,B) and (E,π,B), defined by

P = A B E := (β,e) A E : π(e)= V(β) , q : (β,e) P V(β)= π(e) B. (A.4) × { ∈ × } ∈ 7→ ∈ The total space P is equipped with the subspace topology of the product space A E and projections onto the ﬁrst factor and onto the second factor are denoted respectively by × p : (β,e) P β A, W : (β,e) P e E. (A.5) ∈ 7→ ∈ ∈ 7→ ∈ Both p and W are continuous functions on P and the following diagram commutes.

50 W P E

p q π

A B V

We claim two properties concerning the projections p and W. i. W : P E is an open quotient map and p : P A is a covering map whose model ﬁber is F. → → ii. Equipped with the discrete topology, the symmetric group SN acts continuously on P by permuting components of the ﬁrst factor

σ : (β,e) P (σ(β),e) P, σ SN , ∈ 7→ ∈ ∀ ∈ where σ GLN (C) is defined by (A.3). Hence the quotient map W : P E is closed. ∈ → Thanks to the simple connectedness of the base space A, the covering space P is the disjoint union of its connected components (Ak)k F and the restriction of the covering map p A : Ak A is a ∈ | k → homeomorphism. Since P is locally path-connected, every component Ak is both open and closed, then 1 W A : Ak E an open closed quotient map. So is the lift gk := W A (p A )− : A E. Note | k → | k ◦ | k → that π gk = V and SN stabilizes every element of ∆. We choose β A ∆ and b := V(β). Since the ◦ 1 1 ∈ fiber V− (b)= β is a singleton, so is the fiber π− (b). Hence F = 1 and the universal covering map p : E B is a homeomorphism.{ } So B is simply connected. | | T → Remark A.6. Let F be a closed submanifold of a smooth connected manifold M without boundary of finite dimension. If dimR M dimR F 3, then the inclusion map i : M F M induces an isomorphism − ≥ \ → between the fundamental groups i : π1(M F, x) π1(M, x), for every x M F (see Théorème 2.3 in P.146 of Godbillon [22]). Note that∗ the closed\ submanifold→ ∆ CN has real∈ dimension\ 2. When N 3, the condition A ∆ = cannot be deduced by the other three⊂ conditions in the hypothesis of proposition≥ 6 ∅ A.5: A is open, simply connected and stable by SN . T As a consequence, V(CN ) is open and simply connected because CN is an open convex subset of CN − N − N which is stable under the symmetric group SN and ∆ C = (z, z, ,z) C : Imz < 0 . Together with lemma 4.10, we finish the proof of proposition A.1. − { ··· ∈ } T

A.2 Covering manifold 2 T C T inx 2 T C The Szeg˝oprojector on L ( , ) is given by Π v(x) = n 0 vne , for every v L ( , ) such π ≥ ∈ that v(x) = v einx with v = 1 2 v(x)e inxdx. Equipped with the subspace topology of n Z n n 2π 0 − P ΠT(L2(T, C)) and∈ the Hermitian form P R 2π T 1 T T 1 1 T T H (v , v )= D− Π v , Π v 2 T = D− Π v (x)Π v (x)dx, 1 2 h 1 2iL ( ) 2π 1 2 Z0 T T A N A the subset Π (UN ) is a K¨ahler manifold, which is mapped biholomorphically onto V( ) with = C D(0, 1) = z C : z > 1 in G´erard–Kappeler [19]. \ { ∈ | | } Proposition A.7. There exists a covering map π : V(CN ) V(A N ). − →

51 Remark A.8. Consider the cubic Szeg˝oequation on the torus (see G´erard–Grellier [15, 16, 17, 18])

T T T 2 T i∂tw = Π ( w w ), (t, x) R T, (A.6) | | ∈ × and the cubic Szeg˝oequation on the line (see Pocovnicu [51, 52]), we set ΠR := Π in (1.12),

R R R 2 R i∂tw = Π ( w w ), (t, x) R R. (A.7) | | ∈ × The manifold of N-solitons for the cubic Szeg˝oequation on the line is not simply connected. Let (N)R R P (x) M denote all rational functions of the form w : x R C where P C N 1[X] and Q CN [X] Q(x) ≤ − 1 ∈ 7→ ∈ ∈ ∈ R is a monic polynomial such that Q− (0) C and P,Q have no common factors. Then (N) is a Kähler manifold of complex dimension 2N⊂. So− is the subset (N)T consisting of all rationalM functions ix M T T P (e ) C C C of the form w : x ix where P N 1[X] and Q N [X] is a monic polynomial such Q(e ) ≤ − 1 ∈ 7→ ∈ ∈ ∈ that Q− (0) A and P,Q have no common factors. Both of them have rank characterization of Hankel operators by⊂ Kronecker-type theorem (see Lemma 8.12 in Chapter 1 of Peller [50], p. 54). So the manifold (N)R (resp. (N)T) is invariant under the flow of equation (A.7) (resp. of equation (A.6)) and the (generalized)M action–angleM coordinates of equation (A.7) (resp. of equation (A.6)) are defined in some open dense subset of (N)R (resp. of (N)T). Moreover, if N 2 then (N)R is simply connected by proposition A.5 andM remark A.6. ThereM exists a holomorphic≥ covering mapM (N)R (N)T by following the construction in proposition A.7. The manifold of N-solitons for theM cubic Szeg˝oequation→ M on the line is an open dense subset of (N)R M

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