Travelling waves for KdV and Boussinesq’s system revisited

Article type: Research Article

Authors: Frank, Leonid S.

Affiliations: Département de Mathématiques, V.F.R. Sciences Exactes et Naturelles, Moulin de la Hausse, B.P. 1039 Reims Cedex 2, France

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.

Journal: Asymptotic Analysis, vol. 16, no. 1, pp. 1-14, 1998