Stability of solitary waves in shallow water

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Abstract
The stability properties of the solitary wave solutions of a Boussinesq equation and the Korteweg–de Vries equation are studied. In both cases the eigenfunctions of the linear equations for the perturbation are obtained directly through a series of transformations. For the Korteweg–de Vries equation, the results of Jeffrey and Kakutani are reproduced. Furthermore, it is demonstrated that localized perturbations can be expanded in terms of certain linear combinations of these eigenfunctions, contrary to the statement of Jeffrey and Kakutani. It is shown that the Korteweg–de Vries soliton is stable whereas the Boussinesq solitary wave is unstable to infinitesimal perturbations. The instability of the Boussinesq solitary wave is attributed to deficiencies of the long wavelength approximations used by Boussinesq in deriving his equation. No physical instability of water waves should be anticipated as a consequence of this result. A short discussion of a nonlinear stability analysis for the Boussinesq solitary wave is given. The evidence for nonlinear stability is inconclusive.

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