B. Deconinck, D.E. Pelinovsky, and J.D. Carter

Transverse instabilities of deep-water solitary waves

Proceedings of the Royal Society A 462, 2039-2061 (2006)

The dynamics of a one-dimensional slowly modulated, nearly monochromatic localized wave train in deep water is described by a one-dimensional soliton solution of a two-dimensional nonlinear Schrodinger equation. In this paper, the instability of such a wave train with respect to transverse perturbations is examined numerically in the context of the nonlinear Schrodinger equation, using Hill's method. A variety of instabilities are obtained and discussed. Among these, we show that the solitary wave is susceptible to an oscillatory instability (complex growth rate) due to perturbations with arbitrarily short wavelength. Further, there is a cut-off on the instability with real growth rates. This cut-off has been the subject of some discussion in the literature. We show analytically that the nature of this cut-off is different from what is claimed in previous works.

Water waves, solitary wave, transverse instability, Hill method, Evans function, bifurcations of eigenvalues, numerical approximations