Summary
We consider the normal form for the celebrated Fermi-Pasta-Ulam-Tsingou (FPUT) lattice for an arbitrary number N of particles, with any number of nonlinear terms in a polynomial expansion ('alpha', 'beta', etc.), and with either periodic or fixed boundary conditions. The normal-form equations of motion truncated up to 5-wave interactions are considered. When 3 divides N and N > 8, this normal form contains non-trivial resonances (Bustamante et al. 2019), and is thus generally non-integrable. This is in contrast to what happens with the normal form truncated up to 4-wave interactions, which is known to be integrable for all values of N. Based on (Harper et al. 2013) we construct new independent quadratic invariants for the normal form truncated up to 5-wave interactions. We show that there is always a significant number of these invariants, no matter how large N is. We provide explicit examples for relatively small N cases. We then move on to compute numerically the number of positive Lyapunov exponents of our FPUT normal form truncated up to 5-wave interactions, for relatively small values of N, and show that it is consistent with the above result on new independent invariants. We then construct a new thermalization theory (with constraints) for the normal-form energies, and validate it numerically, finding very good agreement. Finally, we show numerically that at low enough nonlinearity the normal form truncated up to 5-wave resonances provides a good approximation to the original system.
