Travelling waves for KdV and Boussinesq’s system revisited

Abstract

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 $\eta_0 \in L_{1/2} ({\NBbbR})$ . The Boussinesq system is asymptotically examined and explicit formulae are given for the slightly deformed corresponding oscillating waves, the small parameter being $\varepsilon = h_0$ , where $h_0$ is the dimonsionless 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.