The Lyapunov stability of the N-soliton solutions in the Lax hierarchy of the Benjamin-Ono equation

The Lyapunov stability is established for the N-soliton solutions in the Lax hierarchy of the Benjamin-Ono (BO) equation. We characterize the N-soliton profiles as critical points of certain Lyapunov functional. By using several results derived by the inverse scattering transform of the BO equation, we demonstarate the convexity of the Lyapunov functional when evaluated at the N-soliton profiles. From this fact, we deduce that the N-soliton solutions are energetically stable.