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Normal form coordinates for the Benjamin-Ono equation having expansions in terms of pseudo-differential operators

  • *Corresponding author: Thomas Kappeler

    *Corresponding author: Thomas Kappeler 

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

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  • 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.

    Mathematics Subject Classification: Primary: 37K10; Secondary: 35Q55.


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