  We construct exact solutions to an unusual nonlinear advection--diffusion
equation arising in the study of Taylor--Aris (also known as shear) dispersion
due to electroosmotic flow during electromigration in a capillary. An exact
reduction to a Darboux equation is found under a traveling-wave anzats. The
equilibria of this ordinary differential equation are analyzed, showing that
their stability is determined solely by the (dimensionless) wave speed without
regard to any (dimensionless) physical parameters. Integral curves, connecting
the appropriate equilibria of the Darboux equation that governs traveling
waves, are constructed, which in turn are shown to be asymmetric kink solutions
({\it i.e.}, non-Taylor shocks). Furthermore, it is shown that the governing
Darboux equation exhibits bistability, which leads to two coexisting
non-negative kink solutions for (dimensionless) wave speeds greater than unity.
Finally, we give some remarks on other types of traveling-wave solutions and a
discussion of some approximations of the governing partial differential
equation of electromigration dispersion.
