UCLA

We employ the integrable structure of the Benjamin--Ono equation in order to study its rough solutions. For rough data, our most useful tools are the Lax pair formalism and, as in the inverse scattering transform, the structural information embedded in solutions to the scattering equation. Using these, we prove that Sobolev norms are conserved and locally smoothed for rough initial data. Using the integrable structure, we construct a Hamiltonian that usefully approximates the Benjamin--Ono Hamiltonian. With this we may provide a short new proof that the Cauchy problem is well-posed in L2.