Solitary-Wave Solutions of the Benjamin Equation

Jerry L. Bona, John P. Albert, Juan Mario Restrepo
1999 SIAM Journal on Applied Mathematics  
Considered here is a model equation put forward by Benjamin that governs approximately the evolution of waves on the interface of a two-uid system in which surface tension e ects cannot be ignored. Our principal focus is the traveling-wave solutions called solitary waves, and three aspects will be investigated. A constructive proof of the existence of these waves together with a proof of their stability is developed. Continuation methods are used to generate a scheme capable of numerically
more » ... ximating these solitary waves. The computer-generated approximations reveal detailed aspects of the structure of these waves. They are symmetric about their crests, but unlike the classical Korteweg-de Vries solitary waves, they feature a nite number 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. This paper was inspired by recent work of Benjamin ( 7], 8]) concerning waves on the interface of a two-uid system. Benjamin was concerned with an incompressible system that, at rest, consists of a layer of depth h 1 of light uid of density 1 bounded above by a rigid plane and resting upon a layer of heavier uid of density 2 > 1 of depth h 2 , also resting on a rigid plane. Because of the density di erence, waves can propagate along the interface between the two uids. In Benjamin's theory, di usivity is ignored, but the parameters of the system are such that capillarity cannot be discarded. Benjamin focused attention upon waves that do not vary with the coordinate perpendicular to the principal direction of propagation. The waves in question are thus assumed to propagate in only one direction, the positive x direction, say, and to have long wavelength and small amplitude a relative to h 1 . The small parameters = a h1 and = h1 are supposed to be of the same order of magnitude, so that nonlinear and dispersive e ects are balanced. Furthermore, the lower layer is assumed to be very deep relative to the upper layer, so that = h2 h1 is large. The coordinate system is chosen so that, at rest, the interface is located at z = 0. Thus, the upper bounding plane is located at z = h 1 and the lower plane at z = ?h 2 . Let (x; t) denote the downward vertical displacement of the interface from its rest
doi:10.1137/s0036139997321682 fatcat:gkpfgeylrjgfpasusaylz4vlye