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

This paper is a reappraisal of the Hamiltonian model derived by Shrira, Bad ulin and Kharif (BKS) for three-dimensional nonlinear water waves. The mode l was introduced in [J. Fluid Mech. 318 (1996) 375] in an effort to describ e the formation of traveling waves with crescent-shaped features that arise from the instability of the Stokes wave train at moderately large steepnes s. There have been observations of such traveling waves in wave tank experi ments by Su ct 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 l ightly breaking waves, which are asymmetric, with all crescents facing forw ard. Other regimes that they observe apparently give rise to traveling wave s which have asymmetric crescent-shaped features facing both forwards and b ackwards. We show that the BKS model describes the Stokes wave train and it s loss of stability at moderate amplitudes as a Hamiltonian saddle-node bif urcation, 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 s kew wave patterns possessing characteristic hexagonal shaped structures sep arated by quiescent stripes which are produced to the sides of the experime nts in wave tanks. The BKS model has solutions which satisfy two of the thr ee characteristics specified in [J. Fluid Mech. 318 (1996) 375] for nonline ar crescent-shaped waves, avoiding the introduction of a dissipative mechan ism 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 al l of the wave patterns described by the BKS model possess forward and backw ard facing crescent-shaped elements simultaneously, associated with alterna ting crests. These solutions reproduce the features of some but not all of the wave patterns in the observations of Su ct al. [J. Fluid Mech. 124 (198 2) 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 interact ions. While being more complicated and not completely integrable, nonethele ss this model has traveling wave solutions with similar crescent-shaped ele ments, and others with the hexagonal features of the BKS model. (C) 2001 El sevier Science B.V. All rights reserved.