Lectures on integrable equations of Benjamin–Ono type

Abstract

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.