  In this paper we investigate analytically and numerically the nonlinear
Kelvin lattice, namely a chain of masses and nonlinear springs, as in the
alpha-Fermi-Pasta-Ulam-Tsingou (FPUT) chain, where, in addition, each mass is
connected to a nonlinear resonator, i.e., a second mass free to oscillate. Both
nonlinearities are quadratic in the equations of motion. This setup represents
the simplest prototype of nonlinear wave propagation on a nonlinear
metamaterial. In the linear case, we diagonalize the system, and the two
branches of the dispersion relation can be found. Using this result, we derive
in the nonlinear case the equations of motion for the normal variables in
Fourier space, obtaining a system governed by triad interactions among the two
branches of the dispersion relation. We find that the transfer of energy
between these two branches is ruled by three- and four-wave resonant
interactions. We perform numerical simulations of the primitive equations of
motion and highlight the role of resonances as an efficient mechanism for
transferring energy. Moreover, as predicted by the theory, we provide direct
evidence that four-wave resonances appear on a time scale that is longer than
the time scale for three-wave resonances. We also assess the recurrence
behaviour (usual in the FPUT system) for the nonlinear Kelvin lattice, and we
show that, while recurrence is observed if all the energy is placed, at time
t=0, in the lowest mode of the acoustical branch, a non-recurrent behaviour is
observed if the initial energy is located in the optical branch.
