Exact Theory of Solitary Waves in a Stratified Fluid with Surface Tension. Part I. Nonoscillatory Case

In this paper we study steady permanent waves in a stratified fluid with surface tension over a flat horizontal bottom, We show that there exists a solitary wave solution of the exact equations, which decays to zero at infinity, if the linearized equations possess no positive eigenvalues. In contrast to the classical case of a fluid with constant density, there are infinitely many critical values τcn of the Bond number τ, and positive eigenvalues of the linearized equations may appear for both τ > τcn and τ < τcn. A positive eigenvalues may cause small-amplitude oscillations and the oscillatory case will be treated in Part II.