FRONTS, PULSES AND PERIODIC TRAVELLING WAVES IN TWO-COMPONENT SHALLOW WATER MODELS

This survey is devoted to the travelling-wave solutions to some two-component partial differential equations modelling shallow water waves on irrotational flows as well as on shear flows. Qualitative informations about the travelling-wave solutions are obtained from a general ordinary differential equation for each model considered. The existence and the profile of the travelling waves depend on the values of the constants of integration, and on the existence, the sign and order of multiplicity of the roots of some polynomials of degree 3, 4, 5, 6, depending on the model; fronts, pulses, anti-pulses, multi-pulses, periodic travelling waves will arise. By comparing the effects of the vorticity on the pulse waves in the models with and without vorticity, we find that the right-going waves propagating in the same direction as the underlying shear flow have a higher amplitude and narrower wavelength and the right-going waves for which the underlying shear flow propagates in the opposite direction are wider, their amplitude decreases.