Capillary-gravity water waves with compactly supported vorticity I: Local constructions and regular drift

This five-part series establishes the first systematic theory of capillary--gravity water waves carrying compactly supported vorticity with fixed positive circulation. Treating the free surface and the vortex as coupled unknowns, we study the well-posedness of traveling-wave equilibria, including the existence and continuation of solution branches with respect to bifurcation parameters, as well as their variational structure, geometric rigidity, canonical classification, and conditional dynamics. In this first paper, we construct small-speed solitary waves carrying a point vortex of fixed circulation by solving the free-surface Bernoulli equation and the regular-drift condition simultaneously. We derive the free-surface expansion through fourth order, including the first capillary correction, and second-order corrections to the conformal and physical vortex depths. We also characterize vortex equilibria in prescribed domains through the Kirchhoff--Routh function, proving existence and uniform localization. A joint Lyapunov--Schmidt reduction then regularizes the point vortex into a smooth, compactly supported vorticity core while preserving its circulation and the prescribed vorticity law. Finally, in the periodic setting, we construct a local branch under a nonresonance condition and show that the vortex depth diverges logarithmically as the wave speed approaches a positive limit, in contrast to its inverse-speed growth in the solitary-wave setting.