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We consider two-dimensional inviscid irrotational flow in a two layer fluidunder the effects of gravity and surface tension. The upper fluid is boundedabove by a rigid lid, and the lower fluid is bounded below by a rigid bottom.We use a spatial dynamics approach and formulate the steady Euler equations asa Hamiltonian system, where we consider the unbounded horizontal coordinate $x$as a time-like coordinate. The linearization of the Hamiltonian system isstudied, and bifurcation curves in the $(\beta,\alpha)$-plane are obtained,where $\alpha$ and $\beta$ are two parameters. The curves depend on twoadditional parameters $\rho$ and $h$, where $\rho$ is the ratio of thedensities and $h$ is the ratio of the fluid depths. However, the bifurcationdiagram is found to be qualitatively the same as for surface waves. Inparticular we find that a Hamiltonian-Hopf bifurcation and a Hamiltonian real1:1 resonance occur for certain values of $(\beta,\alpha)$. Of particularinterest are solitary wave solutions of the Euler equations. Such solutionscorrespond to homoclinic solutions of the Hamiltonian system. We investigatethe parameter regimes where the Hamiltonian-Hopf bifurcation and theHamiltonian real 1:1 resonance occur. In both these cases we perform a centermanifold reduction of the Hamiltonian system and show that homoclinic solutionsof the reduced system exist. In contrast to the case of surface waves we findparameter values $\rho$ and $h$ for which the leading order nonlinear term inthe reduced system vanishes. We make a detailed analysis of this phenomenon inthe case of the real 1:1 resonance.

Internal gravity-capillary solitary waves in finite depth