Fermi–Pasta–Ulam–Tsingou problem

Problem

Let $N \ge 2$ and $\alpha \in \R$. Consider $N$ particles on a line with displacements $q_1, \ldots, q_N$ and momenta $p_1, \ldots, p_N$, with fixed ends $q_0 = q_{N+1} = 0$, and the Hamiltonian system with Hamiltonian $$H = \sum_{k=1}^{N} \frac{p_k^2}{2} + \sum_{k=0}^{N} \left( \frac{1}{2} (q_{k+1} - q_k)^2 + \frac{\alpha}{3} (q_{k+1} - q_k)^3 \right).$$ For $\alpha = 0$ the chain is a set of $N$ independent normal modes, each with a conserved energy. Start with all energy in the lowest normal mode and $\alpha \ne 0$ small. Explain the long-time behaviour of the energies of the normal modes. In particular, explain why the energy does not spread evenly over all modes. Instead it stays in a few low modes and returns almost completely to the initial mode after some time.