On Solitary Waves and Nonlinear Oscillations in a Stratified Fluid with Surface Tension

In this paper we consider steady permanent waves in a stratified fluid over a flat bottom in the presence of surface tension. Assume that the linearized equations governing the steady flow possess more than two positive eigenvalues, say k21, k22, k23, and k1 > k2 > k3 > 0. The exact equations may be reduced to a partial differential equation subject to boundary conditions coupled with a system of nonlocal ordinary differential equations for an approximate solitary wave solution and nonlinear oscillations. It is shown that there exists a solution to the exact equations consisting of an approximate solitary wave solution plus an oscillatory part with a period near 2π/k1, or near 2π/ki, i = 2 or 3 if |kj − nki| > 0 for any positive integer n and j = 1, 2, 3, j ≠ i, and the equilibrium state of the steady flow satisfies a certain restriction. The results can be extended to the case of n positive eigenvalues for n > 3 without any change.