  The propagation of axisymmetric waves on the surface of a liquid jet under
the action of a radial electric field is considered. The jet is assumed to be
inviscid and perfectly conducting, and a field is set up by placing the jet
concentrically inside a perfectly cylindrical tube whose wall is maintained at
a constant potential. A nontrivial interaction arises between the hydrodynamics
and the electric field in the annulus, resulting in the formation of
electrocapillary waves. The main objective of the present study is to describe
nonlinear aspects of such axisymmetric waves in the weakly nonlinear regime
which is valid for long waves relative to the undisturbed jet radius. This is
found to be possible if two conditions hold: the outer electrode radius is not
too small, and the applied electric field is sufficiently strong. Under these
conditions long waves are shown to be dispersive and a weakly nonlinear theory
can be developed to describe the evolution of the disturbances. The canonical
system that arises is the Kortweg de-Vries equation with coefficients that vary
as the electric field and the electrode radius are varied. Interestingly, the
coefficient of the highest order third derivative term does not change sign and
remains strictly positive, whereas the coefficient $\alpha$ of the nonlinear
term can change sign for certain values of the parameters. This finding implies
that solitary electrocapillary waves are possible; there are waves of elevation
for $\alpha>0$ and of depression for $\alpha<0$. Regions in parameter space are
identified where such waves are found.
