J. Chen, C. Grant, and D.E. Pelinovsky
Nonlinear stability of periodic waves in the Benjamin-Ono equation
Abstract:
We construct the Lyapunov functional for the periodic waves of the Benjamin-Ono equation
based on the conservation of momentum, energy, and a higher-order energy, which exists
due to integrability. Positivity of the second variation of the Lyapunov functional for perturbations
in Sobolev space H1 is established by utilizing the basis properties of eigenfunctions of the linearized
Benjamin-Ono equation in L2. Coercivity of the Lyapunov functional for co-periodic
and multiple-periodic perturbations yields the nonlinear orbital stability of the periodic waves in the
periodic domain.
Keywords:
Benjamin-Ono equation; traveling periodic wave; Lyapunov functional;
completeness of squared eigenfunctions, orbital stability.