The bifurcation of interfacial capillary-gravity waves under \(\text{O}(2)\) symmetry (Q652030)

The uniform steady stratified laminar motion of two ideal fluids with different densities in a layer parallel to their interface is considered. The problem of determination of small disturbances that arise on the interface of two fluids is reduced to the system of two nonlinear integro-differential equations (IDEs) with respect to two unknown functions \(\Theta\) and \(S\) parameterizing the interface and depending on three parameters (speed of upper and lower fluid and capillarity at the interface). The disturbance waves of small amplitude arise at the nonlinear interaction between two higher harmonics in the linearization. In two appropriate Banach spaces the system of IDEs is solved by bifurcation theory methods and \(\text{O}(2)\)-symmetry representation conditions.