The Benjamin–Ono initial-value problem for rational data with application to long-time asymptotics and scattering

Abstract

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 $ϵ$. 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.