In this work, we provide a validity condition for the normal form transformation to remove the non-resonant cubic terms in the $\beta$-FPUT (Fermi–Pasta–Ulam–Tsingou) system. We show that for a wave field with random phases, the normal form transformation is valid by dominant probability if $\beta \ll 1/N^{1+\epsilon}$, with $N$ the number of masses and $\epsilon$ an arbitrarily small constant. To obtain this condition, a bound is needed for a summation in the transformation equation, which we prove rigorously in the paper. The condition also suggests that the importance of the non-resonant terms in the evolution equation is governed by the parameter $\beta N$. We design numerical experiments to demonstrate that this is indeed the case for spectra at both thermal-equilibrium and out-of-equilibrium conditions. The methodology developed in this paper is applicable to other Hamiltonian systems with cubic non-linearities where a normal form transformation needs to be applied.

We prove the existence of small-amplitude periodic travelling waves in dimer Fermi–Pasta–Ulam–Tsingou (FPUT) lattices without assumptions of physical symmetry. Such lattices are infinite, one-dimensional chains of coupled particles in which the particle masses and/or the potentials of the coupling springs can alternate. Previously, periodic travelling waves were constructed in a variety of limiting regimes for the symmetric mass and spring dimers, in which only one kind of material data alternates. The new results discussed here remove the symmetry assumptions by exploiting the gradient structure and translation invariance of the travelling wave problem. Together, these features eliminate certain solvability conditions that symmetry would otherwise manage and facilitate a bifurcation argument involving a two-dimensional kernel and cokernel.