“Jost Solutions and the Direct Scattering Problem of the Benjamin—Ono equation”
The direct and inverse scattering problems for the Benjamin—Ono equation were formulated by Fokas and Ablowitz in 1983. Rigorous study of these problems involving small general potentials was done by Coifman and Wickerhauser in 1990. Recently, the direct scattering problem for large general potentials was studied by Wu. Certain important spectral properties of the scattering operators, together with existence and uniqueness of the Jost solutions were proven. In this talk, we review these results, and establish certain asymptotic formulas of Jost solutions and scattering coefficients as the spectral parameter varies. These formulas are important for understanding the behavior of the inverse scattering problem. A key difficulty in the proof is to show uniqueness of solutions to certain regularized Fredholm integral equations.