Exact Theory of Solitary Waves in a Stratified Fluid with Surface Tension. Part II. Oscillatory Case

In this paper we study two-dimensional waves of permanent type in a stratified fluid with free surface in the presence of surface tension over a flat horizontal bottom. For a rather arbitrary equilibrium state of density stratification and horizontal velocity shear we prove that the exact governing equations possess a solution which consists of an approximate solitary wave solution and a small oscillatory part, if the linearized governing equations have one positive eigenvalue only. The amplitude of the oscillation at infinity can be either of the order smaller than any positive power of a small parameter ε > 0 or of the order of a fixed positive power of ε. Furthermore, we are also able to deal with the case of two positive eigenvalues and prove that there exist two solutions, each of which consists of an approximate solitary wave solution plus an oscillatory part with a different period and an amplitude of the order of a fixed positive power of ε.