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

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Complete Integrability of the Benjamin-Ono Equation on the Multi-Soliton Manifolds Ruoci Sun Complete integrability of the Benjamin-Ono equation on the multi-soliton manifolds Ruoci Sun To cite this version: Ruoci Sun. Complete integrability of the Benjamin-Ono equation on the multi-soliton manifolds. 2020. hal-02548712 HAL Id: hal-02548712 https://hal.archives-ouvertes.fr/hal-02548712 Preprint submitted on 20 Apr 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Complete integrability of the Benjamin{Ono equation on the multi-soliton manifolds Ruoci Sun∗y April 20, 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 UN . We construct gener- alized action{angle coordinates which establish a real analytic symplectomorphism from UN onto some open convex subset of R2N and allow to solve the equation by quadrature for any such initial datum. As a consequence, UN is the universal covering of the manifold of N-gap potentials for the BO equation on the torus as described by G´erard{Kappeler [19]. The global well-posedness of the BO equation in UN is given by a polynomial characterization and a spectral characterization of the manifold UN . Besides the spectral analysis of the Lax operator of the BO equation and the shift semigroup acting on some Hardy spaces, the construction of such coordinates also relies on the use of a generating functional, which encodes the entire BO hierarchy. Keywords Benjamin{Ono equation, generalized action{angle coordinates, Lax pair, Hardy space, in- verse 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´erardfor 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. ∗Laboratoire de Math´ematiquesd'Orsay, Univ. Paris-Sud XI, CNRS, Universit´eParis-Saclay, F-91405 Orsay, France ([email protected]). yThe 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´ematiqueHadamard. 1 Contents 1 Introduction 2 1.1 Notation..............................................5 1.2 Organization of this paper....................................6 1.3 Related work...........................................7 2 The Lax operator 8 2.1 Unitary equivalence....................................... 11 2.2 Spectral analysis I........................................ 12 2.3 Conservation laws........................................ 14 2.4 Lax pair formulation....................................... 16 3 The action of the shift semigroup 18 4 The manifold of multi-solitons 21 4.1 Differential structure....................................... 24 4.2 Spectral analysis II........................................ 27 4.3 Characterization theorem.................................... 30 4.4 The stability under the Benjamin{Ono flow.......................... 32 5 The generalized action{angle coordinates 34 5.1 The associated matrix...................................... 36 5.2 Inverse spectral formulas..................................... 37 5.3 Poisson brackets......................................... 39 5.4 The diffeomorphism property.................................. 42 5.5 A Lagrangian submanifold.................................... 44 5.6 The symplectomorphism property................................ 46 A Appendices 49 A.1 The simple connectedness of UN ................................ 50 A.2 Covering manifold........................................ 51 1 Introduction The Benjamin{Ono (BO) equation on the line reads as 2 2 @tu = H@xu − @x(u ); (t; x) 2 R × R; (1.1) 2 2 where u is real-valued and H = −isign(D) : L (R) ! L (R) denotes the Hilbert transform, D = −i@x, 2 Hcf(ξ) = −isign(ξ)f^(ξ); 8f 2 L (R): (1.2) sign(±ξ) = ±1, for all ξ > 0 and sign(0) = 0, f^ 2 L2(R) denotes the Fourier{Plancherel transform of f 2 L2(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 R 2 (f; g) 2 L ( ) × L ( ) 7! hf; giL2 = f(x)g(x)dx 2 , L ( ) is a -Hilbert space. R R R C R C 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) 7! cu(c t; c(x − y)). A smooth solution u = u(t; x) is called a solitary wave of (1:1) if there exists R 2 C1(R) solving the following non local elliptic equation HR0 + R − R2 = 0; R(x) > 0 (1.3) and u(t; x) = Rc(x − y − ct), where Rc(x) = cR(cx), for some c > 0 and y 2 R. The unique (up to translation) solution of equation (1:3) is given by the following formula 2 R(x) = ; 8x 2 ; (1.4) 1 + x2 R 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. PN Definition 1.1. A function of the form u(x) = Rc (x − xj) is called an N-soliton, for some T j=1 j positive integer N 2 N+ := Z (0; +1), where cj > 0 and xj 2 R, for every j = 1; 2; ··· ;N. Let 2 UN ⊂ L (R; R) denote the subset consisting of all the N-solitons. In the point of view of topology and differential manifolds, the subset UN is a simply connected, real analytic, embedded submanifold of the R-Hilbert space L2(R; R). It has real dimension 2N. The tangent space to UN at an arbitrary N-soliton is included in an auxiliary space Z 2 2 T := fh 2 L (R; (1 + x )dx): h(R) ⊂ R; h = 0g; (1.5) R ^ ^ 2 ∗ i R h1(ξ)h2(ξ) in which a 2-covector ! 2 Λ (T ) is well defined by !(h1; h2) = dξ, for every h1; h2 2 T , 2π R ξ 2 ∗ by Hardy's inequality. We define a translation-invariant 2-form ! : u 2 UN 7! ! 2 Λ (T ), endowed with which UN is a symplectic manifold. The tangent space to UN at u 2 UN is denoted by Tu(UN ). For every smooth function f : UN ! R, its Hamiltonian vector field Xf 2 X(UN ) is given by Xf : u 2 UN 7! @xruf(u) 2 Tu(UN ); where ruf(u) denotes the Fr´echet derivative of f, i.e. df(u)(h) = hh; ruf(u)iL2 , for every h 2 Tu(UN ). The Poisson bracket of f and another smooth function g : UN ! R is defined by ff; gg : u 2 UN 7! !u(Xf (u);Xg(u)) = h@xruf(u); rug(u)iL2 2 R: Then the BO equation (1:1) in the N-soliton manifold (UN ;!) can be written in Hamiltonian form Z 1 1 3 @tu = XE(u); where E(u) = hjDju; ui − 1 1 − u : (1.6) 2 H 2 ;H 2 3 R The Cauchy problem of (1:6) is globally well-posed in the manifold UN (see proposition4 :9). Inspired from the construction of Birkhoff coordinates of the space-periodic BO 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 := f(r ; r ; ··· ; r ) 2 R : r < r < 0; 8j = 1; 2; ··· ;N − 1g denote the subset of actions PN j j N and ν = j=1 dr ^ dα denotes the canonical symplectic form on ΩN × R . The main result of this paper is stated as follows. 3 N Theorem 1. There exists a real analytic symplectomorphism ΦN :(UN ;!) ! (ΩN × R ; ν) such that N 1 X E ◦ Φ−1(r1; r2; ··· ; rN ; α1; α2; ··· ; αN ) = − jrjj2: (1.7) N 2π j=1 Remark 1.2. A consequence of theorem 1 is that UN is simply connected. In fact the manifold UN can be interpreted as the universal covering of the manifold of N-gap potentials for the Benjamin{Ono equation on the torus as described by 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 2 UN 7! (I1(u);I2(u); ··· ;IN (u); γ1(u); γ2(u); ··· ; γN (u)) 2 ΩN × R introduces the generalized action{angle coordinates of the BO equation in the N-soliton manifold, i.e. I (u) fI ;Eg(u) = 0; fγ ;Eg(u) = k ; 8u 2 U : (1.8) k k π N Theorem 1 gives a complete description of the orbit structure of the flow of equation (1:6) up to real k bi-analytic conjugacy.
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