  Because the local concentration of vacancies in any material is bounded,
their motion must be accompanied by nonlinear effects. Here we look for such
effects in a simple model for electric field driven vacancy motion in
memristors, solving the corresponding nonlinear Burgers' equation with
impermeable nonlinear boundary conditions analytically. We find non-monotonous
relaxation of the resistance while switching between the stable
(``on''/``off'') states of the memristor; and qualitatively different film
thickness dependencies of switching time (under applied current) and relaxation
time (under no current). Our exact solution can serve as a useful benchmark for
simulations of more complex memristor models.
