  Electrophoretic separation of a mixture of chemical species is a fundamental
technique of great usefulness in biology, health care and forensics. In
capillary electrophoresis the sample migrates in a microcapillary in the
presence of a background electrolyte. When the ionic concentration of the
sample is sufficiently high, the signal is known to exhibit features
reminiscent of nonlinear waves including sharp concentration `shocks'.In this
paper we consider a simplified model consisting of a single sample ion and a
background electrolyte consisting of a single co-ion and a counterion in the
absence of any processes that might change the ionization states of the
constituents. If the ionic diffusivities are assumed to be the same for all
constituents the concentration of sample ion is shown to obey a one dimensional
advection diffusion equation with a concentration dependent advection
velocity.If the analyte concentration is sufficiently low in a suitable
non-dimensional sense, Burgers' equation is recovered, and thus, the time
dependent problem is exactly solvable with arbitrary initial conditions. In the
case of small diffusivity either a leading edge or trailing edge shock is
formed depending on the electrophoretic mobility of the sample ion relative to
the background ions. Analytical formulas are presented for the shape, width and
migration velocity of the sample peak and it is shown that axial dispersion at
long times may be characterized by an effective diffusivity that is exactly
calculated. These results are consistent with known observations from physical
and numerical simulation experiments.
