Travelling waves for KdV and Boussinesq's system revisited

A simple method is presented for explicitly finding the solitons' wavetrain eta for the Korteweg-de Vries equation in terms of the wave numbers of the solitons, their number N being fully determined by the initial disturbance eta(0) is an element of L-1/2(R) The Boussinesq system is asymptotically examined and explicit formulae are given for the slightly deformed corresponding oscillating waves, the small parameter being epsilon = h(0), where ho is the dimensionless depth of the water layer at rest. The existence of such oscillatory waves is rigorously proved for the Boussinesq system. A different Boussinesq system, yielding solitons, is also revisited.