Multiple-Soliton Solutions for Extended Shallow Water Wave Equations

Four extended shallow water wave equations are introduced and studied for complete integrability. We show that the additional terms do not kill the integrability of the typical equations. The Hereman’s simplified method and the Cole-Hopf transformation method are used to show this goal. Multiple soliton solutions will be derived for each model. The analysis highlights the effects of the extension terms on the structures of the obtained solutions. KeyWords: Shallow Water Wave Equations; Complete Integrability; Multiple-Soliton Solution

Generalized (2+1)−dimensional breaking soliton equation

In this work, a general (2+1)-dimensional breaking soliton equation is investigated. The Hereman’s simplified method is applied to derive multiple soliton solutions, hence to confirm the model integrability.Publisher's Versio

Theory of small aspect ratio waves in deep water

In the limit of small values of the aspect ratio parameter (or wave steepness) which measures the amplitude of a surface wave in units of its wave-length, a model equation is derived from the Euler system in infinite depth (deep water) without potential flow assumption. The resulting equation is shown to sustain periodic waves which on the one side tend to the proper linear limit at small amplitudes, on the other side possess a threshold amplitude where wave crest peaking is achieved. An explicit expression of the crest angle at wave breaking is found in terms of the wave velocity. By numerical simulations, stable soliton-like solutions (experiencing elastic interactions) propagate in a given velocities range on the edge of which they tend to the peakon solution.Comment: LaTex file, 16 pages, 4 figure