Modulational stability of weakly nonlinear wave-trains in media with small- and large-scale dispersions

In this paper, we revisit the problem of modulation stability of quasi-monochromatic wave-trains propagating in a media with the double dispersion occurring both at small and large wavenumbers. We start with the shallow-water equations derived by Shrira [Izv., Acad. Sci., USSR, Atmos. Ocean. Phys. (Engl. Transl.) 17, 55-59 (1981)] which describes both surface and internal long waves in a rotating fluid. The small-scale (Boussinesq-type) dispersion is assumed to be weak, whereas the large-scale (Coriolis-type) dispersion is considered as without any restriction. For unidirectional waves propagating in one direction, only the considered set of equations reduces to the Gardner-Ostrovsky equation which is applicable only within a finite range of wavenumbers. We derive the nonlinear Schrödinger equation (NLSE) which describes the evolution of narrow-band wave-trains and show that within a more general bi-directional equation the wave-trains, similar to that derived from the Ostrovsky equation, are also modulationally stable at relatively small wavenumbers k < kc and unstable at k > kc, where kc is some critical wavenumber. The NLSE derived here has a wider range of applicability: it is valid for arbitrarily small wavenumbers. We present the analysis of coefficients of the NLSE for different signs of coefficients of the governing equation and compare them with those derived from the Ostrovsky equation. The analysis shows that for weakly dispersive waves in the range of parameters where the Gardner-Ostrovsky equation is valid, the cubic nonlinearity does not contribute to the nonlinear coefficient of NLSE; therefore, the NLSE can be correctly derived from the Ostrovsky equation.