Stability of some stationary solutions for the forced KdV equation

The forced KdV (fKdV) equation has been established by recent studies as a simple mathematical model capable of describing the physics of a shallow layer of fluid subject to external forcing. For a particular one-parameter family of forcings which is characterized by a wave amplitude parameter for supercritical forcing distributions, exact stationary solutions are known. We study the stability of these solutions as the parameter varies. The linear stability analysis is first carried out, and we discuss the structure of the spectrum of the associated eigenvalue problem using a perturbation approach, about isolated parameter values where eigenfunctions can be expressed in closed form and are the fixed-point solutions of the fKdV equation corresponding to zero eigenvalues. The results identify a set of intervals in the parameter space corresponding to different types of manifestation of instability. In the region of the parameter space where the linear stability analysis fails to provide an answer, we have developed a nonlinear analysis to provide a sufficient condition for stability. 13 refs