The long-wave limit for the water wave problem I. The case of zero surface tension

The Korteweg-de Vries equation, Boussinesq equation, and many other equations can be formally derived as approximate equations for the two-dimensional water wave problem in the limit of long waves. Here we consider the classical problem concerning the validity of these equations for the water wave problem in an infinitely long canal without surface tension. We prove that the solutions of the water wave problem in the long-wave limit split up into two wave packets, one moving to the right and one to the left, where each of these wave packets evolves independently as a solution of a Korteweg-de Vries equation. Our result allows us to describe the nonlinear interaction of solitary waves.