The Fermi–Pasta–Ulam–Tsingou problem extended with viscoelastic effects

The Fermi–Pasta–Ulam–Tsingou discrete chain is extended to incorporate viscoelastic effects, where the nonlinear elastic springs connecting the particles are supplemented by dissipative dampers arranged in parallel
with the springs. These dampers are assumed to be nonlinear, and we propose an interparticle force modeled as a polynomial function of both the particle displacements and their time derivatives, capturing the viscous
and damping behavior.
In the continuum limit, we derive partial differential equations governing the dynamics of the chain. By appropriate scaling of the parameters with respect to the lattice constant, we obtain a nonlinear viscoelastic equation and a viscoelastic Boussinesq equation.
Traveling waves in form of shock waves are found for the nonlinear viscoelastic equation, where nonlinearity is balanced by viscous dissipation. These traveling waves are interpreted as propagating expansion waves.