On the Badulin, Kharif and Shrira model of resonant water waves

Abstract

This paper is a reappraisal of the Hamiltonian model derived by Shrira, Badulin and Kharif (BKS) for three-dimensional nonlinear water waves. The model was introduced in [J. Fluid Mech. 318 (1996) 375] in an effort to describe the formation of traveling waves with crescent-shaped features that arise from the instability of the Stokes wave train at moderately large steepness. There have been observations of such traveling waves in wave tank experiments by Su et al. [J. Fluid Mech. 124 (1982) 45–72] and Su [J. Fluid Mech. 124 (1982) 73–108]. Some of the regimes described in these papers are of lightly breaking waves, which are asymmetric, with all crescents facing forward. Other regimes that they observe apparently give rise to traveling waves which have asymmetric crescent-shaped features facing both forwards and backwards. We show that the BKS model describes the Stokes wave train and its loss of stability at moderate amplitudes as a Hamiltonian saddle-node bifurcation, which corresponds to the formation of a stable three-dimensional wave pattern which exhibits asymmetric crescent-shaped elements. The model also produces a family of solutions homoclinic to the unstable Stokes wave train, which surrounds the orbit of crescent-shaped wave patterns and which provides a mechanism for transition. Other traveling wave solutions of the BKS model having nonzero transverse momentum are good candidates for the skew wave patterns possessing characteristic hexagonal shaped structures separated by quiescent stripes which are produced to the sides of the experiments in wave tanks. The BKS model has solutions which satisfy two of the three characteristics specified in [J. Fluid Mech. 318 (1996) 375] for nonlinear crescent-shaped waves, avoiding the introduction of a dissipative mechanism to describe features of these familiar wave patterns. The one weakness of the BKS model is that the crescent-shaped wave patterns are transformed to themselves under time reversal composed with a phase shift. Therefore all of the wave patterns described by the BKS model possess forward and backward facing crescent-shaped elements simultaneously, associated with alternating crests. These solutions reproduce the features of some but not all of the wave patterns in the observations of Su et al. [J. Fluid Mech. 124 (1982) 45–72] and Su [J. Fluid Mech. 124 (1982) 73–108]. In the deep water case, we introduce and analyze a new and more realistic four degrees of freedom Hamiltonian model of water waves which has two principal five wave interactions. While being more complicated and not completely integrable, nonetheless this model has traveling wave solutions with similar crescent-shaped elements, and others with the hexagonal features of the BKS model.