The theoretical work reported herein studies the free-surface profile, the flow structure,
and the pressure distribution of a finite-amplitude solitary wave on shallow water
with uniform vorticity. The kinematic problem for the stream function is formulated
employing the vertical coordinate and the free surface as the independent variables of
the Poisson equation with variable coefficients that are functions of the Hamiltonian
of the rotational solitary wave. The exact solution of the boundary-value kinematic
problem for the stream function is derived in the form of a power series complemented
by a recurrence relation. The dynamic problems for the Hamiltonian and the free
surface are solved globally in the Boussinesq–Rayleigh approximation. To find angles
enclosed by the branches of the solution at critical points and points of bifurcation the
surface streamline is also treated locally by an exact topological solution. The complete
analysis of the four-dimensional Hamiltonian maps presented in §4 specifies critical
values of the Froude number and the vorticity for five flow regimes: the emergence
of the solitary wave, the flow separation near the bottom, the flow separation near
the crest, the critical regime for an instability, and the formation of a limiting
configuration. The streamlines of the recirculating flow are obtained as a single-eddy
bifurcation that preserves continuity of all derivatives on the boundary streamline.
The eddy separated near the crest forms the limiting configuration by blocking
the upstream current. The results are compared with weakly nonlinear theory, with
numerical simulations and with field observations with satisfactory agreement.