Travelling waves in shallow water

Abstract

Quasi-stationary surface water waves propagating along a distinguished space direction in an infinite channel of finite depth are considered under the assumption that the liquid is inviscid irrotational and incompressible. Besides the gravitation effect also the capillary phenomenon is taken into consideration. After rescaling the mathematical problem is reduced to a singularly perturbed pseudodifferential equation on the free boundary of the liquid. Depending on the values of two basic dimensionless parameters, the rescaled speed of propagation and the parameter characterizing the surface tension (capillarity), the existence of different kind of quasi-stationary waves (periodic, solitary and their superposition) is established for this exact mathematical model and their asymptotic behavior is investigated when the natural dimensionless small parameter (the ratio of the depth of the undisturbed liquid's layer and the wave's length scale) vanishes.