Benjamin--Feir Instability of Interfacial Gravity--Capillary Waves in a Two-Layer Fluid. Part II. Surface-Tension Effects

This second part of the study develops a complete geometric and asymptotic description of how surface tension governs the modulational stability of interfacial waves in a two-layer fluid. Extending the analytical framework of Part~I, surface tension is treated as a freely adjustable parameter, making it possible to trace the nonlinear and dispersive properties of the system across the full range of depth ratios and density contrasts. Using the nonlinear Schrödinger reduction together with long-wave asymptotics, the mechanisms that shape the boundaries between stable and unstable regimes are identified and their dependence on surface tension is quantified. The long-wave structure is controlled by two special density values that mark the bases of the loop and the corridor on the stability diagrams. Their ordering switches at a threshold that exists only when the lower layer is deeper, and loop-type structures occur only in this regime. A second organising parameter is the classical Bond threshold, at which the dispersive and nonlinear singularities coincide. When surface tension exceeds this value and the upper layer is sufficiently deep, the interaction between resonant and dispersive effects produces a capillary cut that replaces the corridor and characterises strongly capillary, upper-layer-dominated configurations. To unify these observations, the full three-dimensional critical surfaces that separate different types of nonlinear and dispersive behaviour are computed. The familiar loop, corridor, and cut appear as planar sections of these surfaces, and their transitions follow directly from the deformation of the intersection between the resonant and dispersive sheets. Two depth ratios correspond to genuine geometric degeneracies: equal layer depths, where the intersection reduces to a straight line, and the golden-ratio configuration, where the critical surface becomes horizontally tangent at the Bond threshold. Overall, Part~II completes the geometric and physical classification of modulational stability in two-layer interfacial waves and provides a framework for future extensions incorporating shear, external forcing, flexible boundaries, or variable bathymetry.