On the stability of solitary waves with decaying oscillatory tails

In dispersive wave systems with dispersion relations such that the phase speed attains an extremum at a finite wavenumber, a rich variety of solitary waves that feature decaying oscillatory tails is known to arise. Here we use the fifth-order Korteweg-de Vries (KdV) equation, a model for small-amplitude gravity-capillary waves on water of finite depth when the Bond number is close to 5, to examine the stability of the two symmetric solitary-wave solution branches that bifurcate at the minimum phase speed. In the vicinity of the bifurcation point, these solitary waves take the form of modulated wave packets with envelopes that can be approximated by the same soliton solution of the nonlinear Schrödinger (NLS) equation, suggesting that both branches would be stable in the small-amplitude limit. It is shown, however, that the branch of the so-called elevation waves is unstable while the branch of depression waves is stable, consistent with numerical results. The coupling between the carrier oscillations and their envelope, an effect beyond all orders of the expansion underlying the NLS equation, is essential to this behaviour: the dimensionless growth rates of the instability modes found for elevation waves are exponentially small with respect to the solitary-wave steepness. The asymptotic procedure followed here would be useful in discussing the stability of solitary waves with decaying oscillatory tails in other settings as well, and details are worked out for a nonlinear beam equation and a modified fifth-order KdV equation.