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This dissertation studies two mathematical problems concerning waves in a stratified body of water governed by the incompressible Euler equations. In the first part, we present a large-amplitude existence theory for two-dimensional solitary waves by means of a global bifurcation theoretic approach. That is, for any piece-wise smooth upstream density distribution and laminar background current, we construct a global curve of solutions. This curve bifurcates from the background current and, following along the curve, we find waves that are arbitrarily close to having horizontal stagnation points.
The second part of the work tackles the problem of orbital stability of solitary waves in a two-layered body of water under the effect of capillary and gravity.  The flow in each region is taken to be incompressible with constant vorticity. Using a spatial dynamics technique, we establish the existence of small-amplitude spatially localized waves of this type for sufficiently strong surface tension. Then, we prove an orbital stability result through a variant of the Grillakis--Shatah--Strauss method.

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