  We investigate the evolutionary aspects of some integrable soliton models
whose Lagrangians are derived from the pullback of a volume-form to a
two-dimensional target space. These models are known to have infinitely many
conserved quantities and support various types of exact analytic solutions with
nontrivial topology. In particular, we show that, in spite of the fact that
they admit nice smooth solutions, wave propagation about these solutions will
always be ill-posed. This is related to the fact that the corresponding
Euler-Lagrange equations are not of hyperbolic type.
