Energy computations for evolution of class I and II instabilities of Stokes waves

The modified Zakharov equation is used to study the coupled evolution of class I and class II instabilities of surface gravity waves on infinitely deep water. In contrast to single class (I or II) evolution, the coupled behaviour is non-periodic. Except for the very steep waves a dominance of class I modes over those of class II is observed. Energy calculations show that the Hamiltonian of the wave field considered is nearly constant. Thus the Zakharov and the modified Zakharov equations represent consistent approximations of the original water-wave problem.