On interfacial gravity-capillary solitary waves of the Benjamin type and their stability

Abstract

Symmetric solitary waves on the interface between two fluid layers are discussed for the case when the upper fluid is bounded above and the lower fluid is infinitely deep. In this flow geometry, under the conditions that the fluid densities are nearly equal and interfacial tension is relatively large, Benjamin [J. Fluid Mech. 245, 401 (1992)] derived a weakly nonlinear long-wave evolution equation which admits a new kind of elevation solitary wave that features decaying oscillatory tails. We present computations of solitary waves of this type and examine their stability to small perturbations on the basis of the full Euler equations using numerical methods. Computed solitary-wave profiles are found to be in good agreement with solutions of the Benjamin equation even for Weber and Froude numbers well outside the formal range of validity of weakly nonlinear long-wave analysis. Furthermore, our computations reveal that, unless the upper fluid is very light relative to the lower fluid, moderately steep interfacial solitary waves behave qualitatively as predicted by the Benjamin equation: elevation waves with a tall center crest, akin to those found by Benjamin, are stable and coexist with depression solitary waves (not reported by Benjamin) which are unstable. As expected, for low enough density ratio, these two solution branches exchange stabilities and computed profiles resemble those of free-surface gravity-capillary solitary waves on deep water.