Validity condition of normal form transformation for the beta$\beta$ β -FPUT system

Source abstract

Abstract In this work, we provide a validity condition for the normal form transformation to remove the non-resonant cubic terms in the beta $\beta$ β -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 much less than 1 slash upper N 1 plus epsilon $\beta \ll 1/N^{1+\epsilon}$ β ≪ 1 / N 1 + ϵ , with upper N $N$ N the number of masses and epsilon $\epsilon$ ϵ 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 upper N $\beta N$ β 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.