Global Well-Posedness of the Benjamin–ono Equation in H1(r)

We show that the Benjamin–Ono equation is globally well-posed in H s (R) for s≥1. This is despite the presence of the derivative in the nonlinearity, which causes the solution map to not be uniformly continuous in H s for any s 18. The main new ingredient is to perform a global gauge transformation which almost entirely eliminates this derivative.