Initial value problem of the linearized Benjamin–Ono equation and its applications

Abstract

We consider the initial value problem of the Benjamin–Ono (BO) equation linearized about the N-soliton solution. By establishing the completeness relation for the eigenfunctions of the linearized BO equation, we construct the explicit solution to this problem. As an application of the above result, we investigate the linear stability of the N-soliton solution. We show that the wave under consideration is stable against infinitesimal perturbations. Thus we have a direct multisoliton perturbation theory for the BO equation without recourse to the inverse scattering transform. In particular, we can handle the first-order solution beyond the adiabatic approximation. The completeness relation established here enables us to give a general scheme for evaluating the first-order correction to the leading-order N-soliton solution. We also demonstrate that the first-order solution satisfies an infinite set of conservation laws modified by the perturbations. Finally, in the one-soliton case, we perform explicit calculations of the first-order corrections for two different dissipative perturbations that arise in real physical systems and analyze their large time asymptotics.