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• W2897152869 abstract "In systems of N coupled anharmonic oscillators, exact resonant interactions play an important role in the exchange of energy between normal modes. In the weakly nonlinear regime, those interactions may be responsible for the equipartition of energy in Fourier space. Here we consider analytically resonant wave-wave interactions for the celebrated Fermi-Pasta-Ulam-Tsingou (FPUT) system. Using a number-theoretical approach based on cyclotomic polynomials, we show that the problem of finding exact resonances for a system of N particles is equivalent to a Diophantine equation whose solutions depend sensitively on the number of particles N, in particular on the set of divisors of N. We provide an algorithm to construct all possible resonances for N particles, based on two basic methods: pairing-off and cyclotomic, which we introduce and use to build up explicit solutions to the 4-wave, 5-wave and 6-wave resonant conditions. Our results shed some light in the understanding of the long-standing FPUT paradox, regarding the sensitivity of the resonant manifolds with respect to the number of particles N and the corresponding time scale of the interactions leading to an eventual thermalisation. In this light we demonstrate that 6-wave resonances always exist for any N, while 5-wave resonances exist if N is divisible by 3 and N ⩾ 9. It is known that in the discrete case 4-wave resonances do not produce energy mixing across the spectrum, so we investigate whether 5-wave resonances can produce energy mixing across a significant region of the Fourier spectrum by looking at the structure of the interconnected network of Fourier modes that can interact nonlinearly via resonances. We obtain that the answer depends on the set of odd divisors of N which are not divisible by 3: the size of this set determines the number of dynamically independent components, corresponding to independent constants of motion (energies). We show that 6-wave resonances connect all these independent components, providing in principle a restoring mechanism for full-scale thermalisation." @default.
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• W2897152869 date "2019-07-01" @default.
• W2897152869 modified "2023-10-05" @default.
• W2897152869 title "Exact discrete resonances in the Fermi-Pasta-Ulam–Tsingou system" @default.
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• W2897152869 doi "https://doi.org/10.1016/j.cnsns.2019.03.004" @default.
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