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

TL;DR

This paper constructs real analytic, canonical coordinates for the Benjamin-Ono equation near finite gap potentials, using pseudo-differential operators, to facilitate stability analysis of solutions under perturbations.

Contribution

It introduces a novel coordinate system with pseudo-differential operator structure, normal form properties, and para-differential expansions, advancing the analysis of Benjamin-Ono dynamics.

Findings

Coordinates are real analytic and canonical near finite gap potentials.

The Hamiltonian is in normal form up to order three in these coordinates.

The Hamiltonian vector field admits an expansion in para-differential operators.

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.