Steady dark solitary waves emerging from wave-generated meanflow: the role of modulation equations

Various classes of steady and unsteady dark solitary waves (DSWs) are known to exist in modulation equations for water waves in finite depth. However, there is a class of steady DSWS of the full water-wave problem which are missed by the classical modulation equations such as the Hasimoto-Ono, Benney-Roskes, and Davey-Stewartson. These steady DSWs, recently discovered by Bridges and Donaldson, are pervasive in finite depth, arise through secondary criticality of Stokes gravity waves, and are synchronized with the Stokes wave. In this paper, the role of DSWs in modulation equations for water waves is reappraised. The intrinsic unsteady nature of existing modulation equations filters out some interesting solutions. On the other hand, the geometry of DSWs in modulation equations is very similar to the full water wave problem and these geometrical properties are developed. A model equation is proposed which illustrates the general nature of the emergence of steady DSWs due to wave-generated mean flow coupled to a periodic wave. Although the existing modulation equations are intrinsically unsteady, it is shown that there are also important shortcomings when one wants to use them for stability analysis of DSWs.