Spectral stability of traveling water waves: Eigenvalue collision, singularities, and direct numerical simulation

In a recent paper (Nicholls (2009) [1]) the author conjectured upon the connection between the onset of dynamic spectral instability of periodic traveling water waves, and singularities present in Taylor series representations of spectral data for the linearized water wave equations. More specifically, he proposed that the onset of instability is always coincident with encountering the smallest singularity in these Taylor series. In this paper we study this connection via a new Direct Numerical Simulation algorithm derived from the surface formulation of the water wave problem due to Zakharov (1968) [5] and Craig & Sulem (1993) [6]. We find compelling evidence that the conjecture is true in the case of deep (as compared to Benjamin & Feir’s (1967) [7] critical depth h ≈ 1.363 ) water, but false for shallow depths as it significantly underpredicts the onset of instability. The utility of the singularity identification strategy advocated in [1], while somewhat lessened in the shallow water case, is nonetheless upheld due to its ability to reliably identify a lower bound of stability and its extremely favorable computational complexity.