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arXiv · 2109.02489

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

Thomas Kappeler·Riccardo Montalto
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Abstract

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.

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Thomas Kappeler, Riccardo Montalto. 2021-09-06. Normal form coordinates for the Benjamin-Ono equation having expansions in terms of pseudo-differential operators. https://arxiv.org/abs/2109.02489

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