Nonlinear long waves on water and solitons

The water wave problem has been pivotal in the history of nonlinear wave theory. This problem is one of the most interesting and successful applications of nonlinear hydrodynamics. Waves on the free surface of a body of water (perfect liquid) have always been a fascinating subject, for they represent a familiar yet complex phenomenon, easy to observe but very difficult to describe! The archetypical model equations of Kordeweg and de Vries and of Boussinesq, for example, were originally derived as approximations for water waves, and research into the problem has been sustained vigorously up to the present day. In the present paper, the derivation of the model equations is given in depth and rational use is made of asymptotic methods. Indeed, it is important to understand that in some cases the derivation of these approximate equations is intuitive and heuristic. In fact, it is not clear how to insert the model equation under consideration into a hierarchy of rational approximations, which in turn result from the exact formulation of the selected water wave problem.