A new kind of solitary wave

Abstract

The investigation focuses on solitary-wave solutions of an approximate pseudo-differential equation governing the unidirectional propagation of long waves in a two-fluid system where the lower fluid with greater density is infinitely deep and the interface is subject to capillarity. The validity of this model equation is shown to depend on the assumption that T / g (ρ 2 -ρ 1 ) h 2 [Gt ] 1, where T is the interfacial surface tension, ρ 2 − ρ 1 the difference between the densities of the fluids and h the undisturbed thickness of the upper layer. Various properties of solitary waves are demonstrated. For example, they have oscillatory outskirts and their velocities of translation are less than the minimum velocity of infinitesimal waves. Also, they realise respective minima of an invariant functional for fixed values of another such functional, being in consequence orbitally stable. Explicit non-trivial solutions of the equation in question are unavailable, but an existence theory is presented covering both periodic and solitary waves of permanent form.