Competition between subharmonic and sideband secondary instabilities on a falling film

Abstract

The secondary instabilities on a falling film which cause a monochromatic wave to evolve into solitary waves are examined with a weakly nonlinear theory. The unique phase speed dispersion relation dictated by inertia and capillarity is found to favor a nonlinear, two-wave subharmonic resonance that causes neighboring crests to coalesce. This occurs when the fundamental wave frequency is below a critical value ωc which can be approximated by a nonlinear resonant frequency. An entire band of secondary waves are excited by this static subharmonic mechanism such that the coalescence occurs nonuniformly. The lower half of this band of secondary waves travels faster than the fundamental and induces the coalesced waves to move faster than the slower fundamental. For monochromatic waves with frequencies above ωc, a three-wave oscillatory sideband instability is triggered which generates two secondary waves that are slower than the fundamental. This sideband instability involves a long-wave modulation that causes several crests to coalesce simultaneously. Both secondary transitions are important intermediate stages of the route to spatiotemporal chaos involving solitary waves.