Generalized Solitary Waves in a Stratified Fluid

Abstract

The problems considered in this paper deal with generalized solitary waves in a stratified fluid over a flat bottom in the presence of surface tension. By a generalized solitary wave we mean a solitary wave with a small amplitude oscillation at infinity. First let us briefly discuss some recent progress in the exact theory of solitary waves on water of constant density with surface tension. An approximate equation governing solitary waves with surface tension was originally derived by Korteweg and de Vries1 and has been named after them. Let τ be the Bond number, a nondimensional surface tension coefficient, and F be the Froude number, the square of a nondimensional wave speed. The critical values of τ and F are respectively 1/3 and 1. For τ > 1/3, F 1 but near 1, the solution denotes a wave of elevation. The existence of a solitary wave of depression, which decays to zero at infinity, was proved by Amick and Kirchgassner2. For 0 < τ < 1/3, the linearized governing equations possess one positive eigenvalue and the corresponding eigenfunction is oscillatory. Numerical studies and other physical models due to Hunter and Vanden-Broeck3, Zuferia4 and Boyd5 indicate that a generalized solitary wave may take place. The main difficulties are to isolate the oscillatory part and to estimate its amplitude. The existence proof of a generalized solitary wave has been given by Beale6 and Sun7 independently using different methods.