A study of the soliton solutions of the Boussinesq and other nonlinear evolution equations of fluid mechanics

Abstract

After introducing the nonlinear evolution equations of interest: the finite depth fluid (FDF), the Kadomtsev-Petviashvili (KP), the Classical and the ordinary Boussinesq equations, formal asymptotic derivations of the KP and the FDF equations are given for the description of surface and interfacial waves. The N-soliton solution of the FDF equation is reconstructed as a finite sum of Wronskian type determinants. This solution is then shown to reduce to the solutions of the KdV and the Benjamin - Ono equations under specific limiting conditions. Interactions between two solitons of the FDF equation are studied and their interaction properties are shown to reduce to those of the KdV and the Benjamin - Ono equations. Computer plots of the interactions of two-soliton solutions of the FDF and the Benjamin - Ono equations are given. Resonance phenomena in solitons are studied with reference to the KP equation. After discussion of the basic concepts of these phenomena, the N-soliton solution is shown to reduce to the Wronskian of N/2 functions (N-even), each of which represents a triad of solitons when the solitons resonate in pairs. Asymptotic behaviour of the interactions between a triad and a soliton and between two triads are examined and the phase shifts of the triads are obtained directly from the Wronskian representation. The interactions are analysed in detail with reference to numerical computations of the full solutions. After showing that the Classical Boussinesq equations are obtained from Whitham's shallow water wave equations, the basic concept of Hirota's pq=c reduction of the first modified KP hierarchy is outlined. The Classical Boussinesq equations are shown as the pq=O reduction of the same hierarchy. The solution of the hierarchy is manipulated to incorporate the pq=O reduction. As a result of these limiting procedures applied to the problem, Wronskian solutions of the Classical Boussinesq equations in terms of rational functions are produced. Finally the pq=c reduction of the KP hierarchy is applied to the ordinary Boussinesq equation. Using this, the N-soliton solution is expressed as a finite sum of Wronskian type determinants. Analytic verification made for the two-soliton solution shows that a number of Wronskian identities are needed for this purpose. The reason for this behaviour is examined.