Solitary-wave solutions of the Benjamin equation

Considered here is a model equation put forward by Benjamin that governs ap proximately the evolution of waves on the interface of a two-fluid system i n which surface-tension effects cannot be ignored. Our principal focus is t he traveling-wave solutions called solitary waves, and three aspects will b e investigated. A constructive proof of the existence of these waves togeth er with a proof of their stability is developed. Continuation methods are u sed to generate a scheme capable of numerically approximating these solitar y waves. The computer-generated approximations reveal detailed aspects of t he structure of these waves. They are symmetric about their crests, but unl ike the classical Korteweg0de Vries solitary waves, they feature a finite n umber of oscillations. The derivation of the equation is also revisited to get an idea of whether or not these oscillatory waves might actually occur in a natural setting.