Evolution and modulational instability of interfacial waves in a two-layer fluid with arbitrary layer depths

A nonlinear Schrodinger equation (NLSE) describing the evolution of interfacial waves in a gravitationally stable, inviscid, incompressible, and irrotational two-layer fluid with arbitrary constant layer depths is derived using the multiple scale analysis method. The modulational instability (MI) of the interfacial waves is then analyzed using this NLSE. It is shown that the unstable region shrinks as the density ratio of the two layers increases and as each layer gets thinner. A requirement for unstable waves is that both the upper and lower layers are thicker than the critical depths for those layers. The critical depth of each layer as a function of the density ratio of two layers is obtained by curve fitting and used as a criterion for MI. Moreover, nine cases with various upper- and lower-layer depths are investigated. The relationships of the dark soliton to modulational stability and the bright soliton to MI are discussed in each case. In the unstable regions of the nine cases, it is found that the steepness of the perturbed interface amplitude increases, and the perturbed interface elevation decays more rapidly as the depth of each layer increases. Both the height and the steepness of the perturbed interface elevation increase with increasing density ratio of the two layers.