Stability of some stationary solutions for the forced KdV equation

Abstract

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 in response to external forcing. For a particular one-parameter family of forcings characterized by a wave amplitude parameter for a certain supercritical forcing distribution, the exact stationary solutions are known. We study the stability of these solutions as the parameter varies. The linear stability analysis is first carried out by exploring the structure of the spectrum of the associated eigenvalue problem using a perturbation technique applied to a neighbourhood of isolated parameter values, where eigenfunctions corresponding to eigenvalue zero can be expressed in closed form. 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 discuss a nonlinear analysis that provides a sufficient condition for stability.

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