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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. 

A regularized Boussinesq equation is studied as a dispersive, long-wave (quasicontinuum) approximation of the Fermi–Pasta–Ulam lattice with a general cubic interaction force. Explicit periodic travelling wave solutions in terms of Jacobi elliptic functions are classified, and their solitary-wave, kink and trigonometric limits are obtained. The Whitham modulation equations describing slow modulations of periodic travelling wave solutions are derived using an averaged variational principle. The convexity (strict hyperbolicity, genuine nonlinearity) of the resulting hydrodynamic-type equations is examined numerically in general and analytically in the solitary-wave and harmonic limits. In particular, the loss of hyperbolicity and the formation of complex conjugate characteristic velocities is shown to lead to modulational instability of periodic travelling waves. The onset of modulational instability is verified by numerical computations of linearized spectra for periodic travelling waves and initial value problems that also reveal additional short-wavelength instabilities.