  We consider a propagation of transition fronts in one-dimensional chains with
bi-stable nondegenerate on-site potential. If one adopts linear coupling in the
chain and piecewise linear on-site force, then it is possible to develop
well-known exact solutions for the front and accompanying oscillatory tail. We
demonstrate that these solutions are essentially non-robust. Various
approximations for the on-site potential with the same basic parameters (height
and coordinate of the potential barrier, energy effect and distance between the
potential wells) lead to substantially different front velocities. Besides,
inclusion of even weak nearest neighbor nonlinearity drastically modifies the
front structure and parameters. The energy concentration in the front zone
leads to a dominance of the nonlinear term. It turns out that the dynamics can
be efficiently studied in terms of an equivalent model with a single degree of
freedom. This estimation leads to accurate prediction of the front velocity and
parameters of the oscillatory tail. Moreover, it turns out that the solution is
robust - exact shape of the on-site potential weakly effects the front
parameters. This finding also conforms to the simplified model, since the
latter invokes only the general shape characteristics of the on-site potential.
