We investigate steady propagation of subsonic phase transition fronts in a viscoelastic chain with up-down-up elastic interaction forces. Employing a combination of semi-analytical computations, numerical iterations of a nonlinear map and direct numerical simulations, we obtain traveling wave solutions, which are periodic modulo a shift by one lattice spacing, as well as bifurcating solution branches that feature period doubling. For a case of piecewise linear elastic interactions, we systematically investigate linear stability of traveling wave solutions using the Floquet analysis. This analysis is complemented by numerical simulations that explore the fate of unstable solutions perturbed along the corresponding eigenmodes and identify additional bifurcating branches. We show that smooth approximations of the piecewise linear interactions may have a significant effect on stability of low-velocity motion and the form of bifurcating branches.
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