Patrick Gérard

We give a proof of the soliton resolution conjecture for the Benjamin--Ono equation, namely every solution with sufficiently regular and decaying initial data can be written as a finite sum of soliton solutions with different velocities up to a radiative remainder term in the long--time asymptotics. We provide a detailed correspondence between the spectral theory of the Lax operator associated to the initial data and the different terms of the soliton resolution expansion. The proof is based on a new use of a representation formula of the solution due to the second author, and on a detailed analysis of the distorted Fourier transform associated to the Lax operator.

These lectures are devoted to two integrable PDE on the line enjoying similar structures: the Benjamin–Ono equation and the Calogero–Moser derivative nonlinear Schrödinger equation. For both equations, a Lax pair of operators is introduced on the Hardy space of the upper-half plane, and is used to prove conservation laws and explicit formulae, and to study soliton and multisoliton solutions. In the special case of the Benjamin–Ono equation, the small dispersion limit with general initial data is proved to exist and is identified. These lectures were presented at the 2024 PDE Days, Centre Paul Langevin, Aussois, France.

We show that the initial-value problem for the Benjamin–Ono equation on \mathbb{R} with L^{2}(\mathbb{R}) rational initial data with only simple poles can be solved in closed form via a determinant formula involving contour integrals. The dimension of the determinant depends on the number of simple poles of the rational initial data only and the matrix elements depend explicitly on the independent variables (t,x) and the dispersion coefficient \epsilon . This allows for various interesting asymptotic limits to be resolved quite efficiently. As an example, and as a first step towards establishing the soliton resolution conjecture, we prove that the solution with initial datum equal to minus a soliton exhibits scattering.

This work makes analytic progress in the deterministic study of turbulence in Hamiltonian systems by identifying two types of energy cascade solutions and the corresponding large- and small-scale structures they generate. The first cascade represents condensate formation via a highly coherent process recently uncovered, while the second cascade, which has not been previously observed, leads to the formation of other large-scale structures. The concentration of energy at small scales is characterized in both cases by the development of a power-law spectrum in finite time, causing the blow-up of Sobolev norms and the formation of coherent structures at small scales. These structures approach two different types of singularities: a point discontinuity in one case and a cusp in the other. The results are fully analytic and explicit, based on two solvable families of Hamiltonian systems identified in this study.

In this paper, we prove that the Benjamin–Ono equation is globally in time C^{0} -well-posed in the Hilbert space H^{-1/2,\sqrt{\log}}(\mathbb{T},\mathbb{R}) of periodic distributions in H^{-1/2}(\mathbb{T},\mathbb{R}) with \sqrt{\log} -weights. The space H^{-1/2,\sqrt{\log}}(\mathbb{T},\mathbb{R}) can thus be considered as a maximal low regularity phase space for the Benjamin–Ono equation corresponding to the scale H^{s}(\mathbb{T},\mathbb{R}) , s>-1/2 .