  While several articles have been written on water waves on flows with
constant vorticity, little is known about the extent to which a nonconstant
vorticity affects the flow structure, such as the appearance of stagnation
points. In order to shed light on this topic, we investigate in detail the flow
beneath solitary waves propagating on an exponentially decaying sheared
current. Our focus is to analyse numerically the emergence of stagnation
points. For this purpose, we approximate the velocity field within the fluid
bulk through the classical Korteweg-de Vries asymptotic expansion and use the
Matlab language to evaluate the resulting streamfunction. Our findings suggest
that the flow beneath the waves can have zero, one or two stagnation points in
the fluid body. We also study the bifurcation between these flows. Our
simulations indicate that the stagnation points emerge from a streamline with a
sharp corner.
