A new type of modulation instability of Stokes waves in the framework of an extended NSE system with mean flow

Abstract

Stokes waves on the surface of a layer of an ideal fluid are studied. The nonlinear Schrodinger equation (NSE) for the envelope of the first harmonic and the equation for zero harmonic are extended with allowance for full linear dispersion. To investigate modulational instability (MI) of Stokes waves, we derive a quartic equation for the perturbation frequency without the traditional approximation for the motion of mean current with a group speed on the frequency of fast filling. The interaction of the four roots of this equation is shown to result in the occurrence of MI bands not described by the NSE. The analysis of the obtained expressions demonstrates that the limit kh = 1.363 (where h is the fluid depth and k is the wave number) found by Benjamin and Feir (and also by Whitham and then by Hasimoto and Ono) for the transition between the states of modulationally stable and unstable liquid is valid only in the limiting case of small amplitudes of unperturbed waves and small wave numbers of the perturbation wave.