# American Institute of Mathematical Sciences

doi: 10.3934/dcds.2022048
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## Normal form coordinates for the Benjamin-Ono equation having expansions in terms of pseudo-differential operators

 1 Institute for Mathematics, University of Zurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland 2 Department of Mathematics, University of Milan, Via Saldini 50, 20133, Milano, Italy

*Corresponding author: Thomas Kappeler

Received  November 2021 Early access April 2022

Fund Project: TK supported in part by the Swiss National Science Foundation, RM supported in part by the Swiss National Science Foundation and INDAM-GNFM and by the ERC starting grant 2021 'Hamiltonian Dynamics Normal Forms and Water Waves' (HamDyWWa), project number 101039762

Near an arbitrary finite gap potential we construct real analytic, canonical coordinates for the Benjamin-Ono equation on the torus having the following two main properties: (1) up to a remainder term, which is smoothing to any given order, the coordinate transformation is a pseudo-differential operator of order 0 with principal part given by a modified Fourier transform (modification by a phase factor) and (2) the pullback of the Hamiltonian of the Benjamin-Ono is in normal form up to order three and the corresponding Hamiltonian vector field admits an expansion in terms of para-differential operators. Such coordinates are a key ingredient for studying the stability of finite gap solutions of the Benjamin-Ono equation under small, quasi-linear perturbations.

Citation: Thomas Kappeler, Riccardo Montalto. Normal form coordinates for the Benjamin-Ono equation having expansions in terms of pseudo-differential operators. Discrete and Continuous Dynamical Systems, doi: 10.3934/dcds.2022048
##### References:
 [1] T. B. Benjamin, Internal waves of permanent form in fluids of great depth, J. Fluid Mech., 29 (1967), 559-592.  doi: 10.1017/S002211206700103X. [2] R. E. Davis and A. Acrivos, Solitary internal waves in deep water, J. Fluid Mech., 29 (1967), 593-607.  doi: 10.1017/S0022112067001041. [3] P. Gérard and T. Kappeler, On the integrability of the Benjamin-Ono equation on the torus, Comm. Pure Appl. Math., 74 (2021), 1685-1747.  doi: 10.1002/cpa.21896. [4] P. Gérard, T. Kappeler and P. Topalov, On the analytic Birkhoff normal form of the Benjamin-Ono equation and applications, Nonlinear Anal., 216 (2022), 32pp. doi: 10.1016/j.na.2021.112687. [5] P. Gérard, T. Kappeler and P. Topalov, On the analyticity of the nonlinear Fourier transform of the Benjamin-Ono equation on $\mathbb T$, preprint, 2021, arXiv: 2109.08988. [6] P. Gérard, T. Kappeler and P. Topalov, On the Benjamin–Ono equation on $\mathbb T$ and its periodic and quasiperiodic solutions, J. Spectr. Theory, 12 (2022), 169-193. [7] P. Gérard, T. Kappeler and P. Topalov, Sharp well-posedness results for the Benjamin-Ono equation in $H^{s}(\mathbb T, \mathbb R)$ and qualitative properties of its solution, to appear in Acta Math., arXiv: 2004.04857. [8] P. Gérard, T. Kappeler and P. Topalov, On smoothing properties and Tao's gauge transform of the Benjamin-Ono equation on the torus, preprint, 2021, arXiv: 2109.00610. [9] P. Gérard, T. Kappeler and P. Topalov, On the spectrum of the Lax operator of the Benjamin-Ono equation on the torus, J. Funct. Anal., 279 (2020), 75pp. doi: 10.1016/j.jfa.2020.108762. [10] B. Grébert and T. Kappeler, The Defocusing NLS Equation and Its Normal Form, EMS Series of Lectures in Mathematics, European Mathematical Society (EMS), Zürich, 2014. doi: 10.4171/131. [11] T. Kappeler and R. Montalto, Canonical coordinates with tame estimates for the defocusing NLS equation on the circle, Int. Math. Res. Not. IMNR, 2018 (2018), 1473-1531.  doi: 10.1093/imrn/rnw233. [12] T. Kappeler and R. Montalto, Normal form coordinates for the KdV equation having expansions in terms of pseudodifferential operators, Comm. Math. Phys., 375 (2020), 833-913.  doi: 10.1007/s00220-019-03498-1. [13] T. Kappeler and R. Montalto, On the stability of periodic multi-solitons of the KdV equation, Comm. Math. Phys., 385 (2021), 1871-1956.  doi: 10.1007/s00220-021-04089-9. [14] T. Kappeler and J. Pöschel, KdV & KAM, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3, Folge, A Series of Modern Surveys in Mathematics, 45, Springer-Verlag, Berlin, 2003. doi: 10.1007/978-3-662-08054-2. [15] T. Kappeler, B. Schaad and P. Topalov, Qualitative features of periodic solutions of KdV, Comm. Partial Differential Equations, 38 (2013), 1626-1673.  doi: 10.1080/03605302.2013.814141. [16] T. Kappeler, B. Schaad and P. Topalov, Semi-linearity of the non-linear Fourier transform of the defocusing NLS equation, Int. Math. Res. Not. IMRN, 2016 (2016), 7212-7229.  doi: 10.1093/imrn/rnv397. [17] I. Krichever, Perturbation theory in periodic problems for two-dimensional integrable systems, Soviet Scientific Reviews C. Math. Phys., 9 (1991), 1-103. [18] S. B. Kuksin, Analysis of Hamiltonian PDEs, Oxford Lecture Series in Mathematics and its Applications, 19, Oxford University Press, Oxford, 2000 [19] S. Kuksin and G. Perelman, Vey theorem in infinite dimensions and its application to KdV, Discrete Contin. Dyn. Syst., 27 (2010), 1-24.  doi: 10.3934/dcds.2010.27.1. [20] G. Métivier, Para-Differential Calculus and Applications to the Cauchy Problem for Nonlinear Systems, Centro di Ricerca Matematica Ennio De Giorgi (CRM) Series, 5, Edizioni della Normale, Pisa, 2008. [21] J.-C. Saut, Benjamin-Ono and intermediate long wave equations: Modeling, IST, and PDE, in Nonlinear Dispersive Partial Differential Equations and Inverse Scattering, Fields Inst. Commun., 83, Springer, New York, 2019, 95–160.

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##### References:
 [1] T. B. Benjamin, Internal waves of permanent form in fluids of great depth, J. Fluid Mech., 29 (1967), 559-592.  doi: 10.1017/S002211206700103X. [2] R. E. Davis and A. Acrivos, Solitary internal waves in deep water, J. Fluid Mech., 29 (1967), 593-607.  doi: 10.1017/S0022112067001041. [3] P. Gérard and T. Kappeler, On the integrability of the Benjamin-Ono equation on the torus, Comm. Pure Appl. Math., 74 (2021), 1685-1747.  doi: 10.1002/cpa.21896. [4] P. Gérard, T. Kappeler and P. Topalov, On the analytic Birkhoff normal form of the Benjamin-Ono equation and applications, Nonlinear Anal., 216 (2022), 32pp. doi: 10.1016/j.na.2021.112687. [5] P. Gérard, T. Kappeler and P. Topalov, On the analyticity of the nonlinear Fourier transform of the Benjamin-Ono equation on $\mathbb T$, preprint, 2021, arXiv: 2109.08988. [6] P. Gérard, T. Kappeler and P. Topalov, On the Benjamin–Ono equation on $\mathbb T$ and its periodic and quasiperiodic solutions, J. Spectr. Theory, 12 (2022), 169-193. [7] P. Gérard, T. Kappeler and P. Topalov, Sharp well-posedness results for the Benjamin-Ono equation in $H^{s}(\mathbb T, \mathbb R)$ and qualitative properties of its solution, to appear in Acta Math., arXiv: 2004.04857. [8] P. Gérard, T. Kappeler and P. Topalov, On smoothing properties and Tao's gauge transform of the Benjamin-Ono equation on the torus, preprint, 2021, arXiv: 2109.00610. [9] P. Gérard, T. Kappeler and P. Topalov, On the spectrum of the Lax operator of the Benjamin-Ono equation on the torus, J. Funct. Anal., 279 (2020), 75pp. doi: 10.1016/j.jfa.2020.108762. [10] B. Grébert and T. Kappeler, The Defocusing NLS Equation and Its Normal Form, EMS Series of Lectures in Mathematics, European Mathematical Society (EMS), Zürich, 2014. doi: 10.4171/131. [11] T. Kappeler and R. Montalto, Canonical coordinates with tame estimates for the defocusing NLS equation on the circle, Int. Math. Res. Not. IMNR, 2018 (2018), 1473-1531.  doi: 10.1093/imrn/rnw233. [12] T. Kappeler and R. Montalto, Normal form coordinates for the KdV equation having expansions in terms of pseudodifferential operators, Comm. Math. Phys., 375 (2020), 833-913.  doi: 10.1007/s00220-019-03498-1. [13] T. Kappeler and R. Montalto, On the stability of periodic multi-solitons of the KdV equation, Comm. Math. Phys., 385 (2021), 1871-1956.  doi: 10.1007/s00220-021-04089-9. [14] T. Kappeler and J. Pöschel, KdV & KAM, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3, Folge, A Series of Modern Surveys in Mathematics, 45, Springer-Verlag, Berlin, 2003. doi: 10.1007/978-3-662-08054-2. [15] T. Kappeler, B. Schaad and P. Topalov, Qualitative features of periodic solutions of KdV, Comm. Partial Differential Equations, 38 (2013), 1626-1673.  doi: 10.1080/03605302.2013.814141. [16] T. Kappeler, B. Schaad and P. Topalov, Semi-linearity of the non-linear Fourier transform of the defocusing NLS equation, Int. Math. Res. Not. IMRN, 2016 (2016), 7212-7229.  doi: 10.1093/imrn/rnv397. [17] I. Krichever, Perturbation theory in periodic problems for two-dimensional integrable systems, Soviet Scientific Reviews C. Math. Phys., 9 (1991), 1-103. [18] S. B. Kuksin, Analysis of Hamiltonian PDEs, Oxford Lecture Series in Mathematics and its Applications, 19, Oxford University Press, Oxford, 2000 [19] S. Kuksin and G. Perelman, Vey theorem in infinite dimensions and its application to KdV, Discrete Contin. Dyn. Syst., 27 (2010), 1-24.  doi: 10.3934/dcds.2010.27.1. [20] G. Métivier, Para-Differential Calculus and Applications to the Cauchy Problem for Nonlinear Systems, Centro di Ricerca Matematica Ennio De Giorgi (CRM) Series, 5, Edizioni della Normale, Pisa, 2008. [21] J.-C. Saut, Benjamin-Ono and intermediate long wave equations: Modeling, IST, and PDE, in Nonlinear Dispersive Partial Differential Equations and Inverse Scattering, Fields Inst. Commun., 83, Springer, New York, 2019, 95–160.
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