Transverse instabilities of deep-water solitary waves

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 Schrödinger (NLS) equation. In this paper, the instability of such a wave train with respect to transverse perturbations is examined numerically in the context of the NLS 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. We show analytically that the nature of this cut-off is different from what is claimed in previous works.