Long-wave transverse instability of interfacial gravity-capillary solitary waves in a two-layer potential flow in deep water

摘要

Interfacial gravity-capillary plane solitary waves, driven by the gravitational force in the presence of interfacial tension in a two-layer deep-water potential flow, bifurcate in the form of wavepackets with a non-zero carrier wavenumber at which the phase speed is minimized. A stability property for the interfacial gravity-capillary plane solitary waves is presented within the framework of the full Euler equations: according to a linear stability analysis based on the perturbation method, such waves are unstable under weak and long-wave disturbances in the transverse direction to the dominant wave propagation. An instability criterion is verified that the total mechanical energy of the solitary waves is a decreasing function of the solitary wavespeed, owing to the fact that the speed of the bifurcating solitary wavepackets is less than the minimum of the phase speed. This result is consistent with an earlier study on the transverse instability of the longitudinally stable interfacial gravity-capillary solitary waves from the Benjamin model equation for weakly nonlinear long interfacial elevations (Kim and Akylas, J Fluid Mech 557:237-256, 2006). The analysis is also applicable to other interfacial gravity-capillary solitary waves that may bifurcate below the minimum of the phase speed, regardless of any restrictions on fluid depths in two-layer potential flows.