Benjamin–-Feir Instability of Interfacial Gravity–Capillary Waves in a Two-Layer Fluid. Part I

This study presents a detailed investigation of the modulational stability of interfacial wave packets in a two-layer inviscid incompressible fluid with finite layer thicknesses and interfacial surface tension. The stability analysis is carried out for a broad range of density ratios and geometric configurations, enabling the construction of stability diagrams in the $(\rho,k)$-plane, where $\rho$ is the density ratio and $k$ is the carrier wavenumber. The Benjamin-–Feir index is used as the stability criterion, and its interplay with the curvature of the dispersion relation is examined to determine the onset of modulational instability.The topology of the stability diagrams reveals several characteristic structures: a localized \emph{loop} of stability within an instability zone, a global \emph{upper} stability domain, an elongated \emph{corridor} bounded by resonance and dispersion curves, and a degenerate \emph{cut} structure arising in strongly asymmetric configurations. Each of these structures is associated with a distinct physical mechanism involving the balance between focusing/defocusing nonlinearity and anomalous/normal dispersion.Systematic variation of layer thicknesses allows us to track the formation, deformation, and disappearance of these regions, as well as their merging or segmentation due to resonance effects. Limiting cases of semi-infinite layers are analyzed to connect the results with known configurations, including the `half-space–layer', `layer–half-space', and `half-space–half-space' systems. The influence of symmetry and asymmetry in layer geometry is examined in detail, showing how it governs the arrangement and connectivity of stable and unstable regions in parameter space. The results provide a unified framework for interpreting modulational stability in layered fluids with interfacial tension, highlighting both global dispersion-controlled regimes and localized stability islands. This work constitutes Part~I of the study; Part~II will address the role of varying surface tension, which is expected to deform existing stability domains and modify the associated nonlinear–dispersive mechanisms.