We study solutions of the water–wave problem for a fluid layer of finite depth in the presence of gravity and surface tension. We use the canonical Hamiltonian formulation by Zakharov in terms of the surface elevation and the trace of the velocity potential on the surface. With a new continuity result for the Dirichlet–Neumann operator in terms of the surface as a function in H1(ℝ), we show conditional energetic stability of the trivial solution in certain regions of the parameter space. In the same region we obtain stability of solitary waves under the additional assumption that the second variation of the energy has only one negative eigenvalue. The latter assumption is shown to be fulfilled for the small–amplitude solitary waves first constructed by Amick & Kirchgässner.
↵† This paper is dedicated to Klaus Kirchgassner on the occasion of his 70th birthday.