Small-Amplitude Solitary Waves for $f$-Plane Capillary-Gravity Flows with Arbitrary Vorticity

We study small-amplitude solitary waves for two-dimensional capillary--gravity flows with arbitrary vorticity on the equatorial $f$-plane. The steady free-boundary problem is formulated as a reversible Hamiltonian spatial-dynamics system in which rotation enters through the speed-dependent effective gravity $g_*=g-2\Omega c$. A center-manifold reduction reduces the local bifurcation problem to finite-dimensional Hamiltonian systems governed by the low-frequency spectrum of a Sturm--Liouville problem with an eigenvalue-dependent boundary condition. We identify the Hamiltonian $0^2$, real $1{:}1$, and Hamiltonian--Hopf resonance curves and obtain corresponding families of symmetric solitary waves under standard non-degeneracy assumptions. We also show that the weak-effective-gravity threshold $g_*=0$ is separated from the $0^2$ and local Hamiltonian--Hopf resonances for uniformly non-stagnant laminar flows, and can be approached only in a near-stagnation regime.