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

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Complete Integrability of the Benjamin--Ono Equation on the Multi-Soliton Complete integrability of the Benjamin–Ono equation on the multi-soliton manifolds Ruoci Sun∗† April 22, 2020 Abstract This paper is dedicated to proving the complete integrability of the Benjamin–Ono (BO) equation on the line when restricted to every N-soliton manifold, denoted by N . We construct gener- U alized action–angle coordinates which establish a real analytic symplectomorphism from N onto some open convex subset of R2N and allow to solve the equation by quadrature for any such initialU datum. As a consequence, N is the universal covering of the manifold of N-gap potentials for the BO equation on U the torus as described by G´erard–Kappeler [19]. The global well-posedness of the BO equation in N U is given by a polynomial characterization and a spectral characterization of the manifold N . Besides the spectral analysis of the Lax operator of the BO equation and the shift semigroup actingU on some Hardy spaces, the construction of such coordinates also relies on the use of a generating functional, which encodes the entire BO hierarchy. Keywords Benjamin–Ono equation, generalized action–angle coordinates, Lax pair, inverse spectral transform, multi-solitons, universal covering manifold Throughout this paper, the main results of each section are stated at the beginning. Their proofs are left inside the corresponding subsections. Acknowledgments The author would like to express his sincere gratitude towards his PhD advisor Prof. Patrick G´erard for introducing this problem, for his deep insight, generous advice and continuous encouragement. He also would like to thank warmly Dr. Yang Cao for introducing fiber product method. arXiv:2004.10007v1 [math.AP] 21 Apr 2020 ∗Laboratoire de Math´ematiques d’Orsay, Univ. Paris-Sud XI, CNRS, Universit´eParis-Saclay, F-91405 Orsay, France ([email protected]). †The author is partially supported by the grant ”ANAE”´ ANR-13-BS01-0010-03 of the ’Agence Nationale de la Recherche’. This research is carried out during the author’s PhD studies, financed by the PhD fellowship of Ecole´ Doctorale de Math´ematique Hadamard. 1 Contents 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 classifi- cation 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 Fr´echet 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 G´erard– 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.
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