Exact discrete resonances in the Fermi-Pasta-Ulam-Tsingou system - PowerPoint PPT Presentation

Exact discrete resonances in the Fermi-Pasta-Ulam-Tsingou system Miguel D. Bustamante, K. Hutchinson, Y. V. Lvov, M. Onorato Commun. Nonlinear Sci. Numer. Simulat. 73 , 437–471 (2019) Preprint arXiv: http://arxiv.org/abs/1810.06902 School of Mathematics and Statistics University College Dublin

Fermi-Pasta-Ulam-Tsingou system N identical masses connected by anharmonic springs moving in one dimension. Studied in 1953, using numerical simulations (MANIAC). System did not relax to equilibrium; rather, a recurrence behaviour was observed. The result sparked the research field of nonlinear science: integrable systems such as Korteweg de Vries were related to this problem. (Credits: A. L. Burin et al., Entropy 2019 , 21 (1), 51)

Hamiltonian for the p α ` β q FPUT model The Hamiltonian for a chain of N identical particles of mass m , connected by identical anharmonic springs, can be expressed as an unperturbed Hamiltonian, H 0 , plus two perturbative terms, H 3 , H 4 : H “ H 0 ` H 3 ` H 4 (1) with ˆ 1 N j ` κ 1 ˙ ÿ 2 mp 2 2 p q j ´ q j ` 1 q 2 H 0 “ , j “ 1 N H 3 “ α ÿ p q j ´ q j ` 1 q 3 , (2) 3 j “ 1 N H 4 “ β ÿ p q j ´ q j ` 1 q 4 . 4 j “ 1 q j p t q is the displacement of the particle j from its equilibrium position and p j p t q is the associated momentum.

Equations of motion for the original variables q j p t q m : q j “ κ p q j ` 1 ` q j ´ 1 ´ 2 q j q p q j ` 1 ´ q j q 2 ´ p q j ´ q j ´ 1 q 2 ‰ “ ` α p q j ` 1 ´ q j q 3 ´ p q j ´ q j ´ 1 q 3 ‰ “ ` β , j “ 0 , . . . , N ´ 1 This is known as the α ` β -FPUT model. q 0 “ q N q ´ 1 “ q N “ 0 q ´ 1 “ q 0 , q N ´ 1 “ q N We will consider periodic boundary conditions from here on.

Equations in Fourier space: modular momentum condition N ´ 1 N ´ 1 Q k “ 1 q j e ´ i 2 πkj { N , P k “ 1 ÿ ÿ p j e ´ i 2 πkj { N , N N j “ 0 j “ 0 N ´ 1 P 2 H 2 m ` 1 | P k | 2 ` m 2 ω 2 ÿ 0 ` k | Q k | 2 ˘ “ N 2 m k “ 1 N ´ 1 1 ÿ ˜ ` V 1 , 2 , 3 Q 1 Q 2 Q 3 δ 1 ` 2 ` 3 3 k 1 ,k 2 ,k 3 “ 1 N ´ 1 1 ÿ ˜ ` T 1 , 2 , 3 , 4 Q 1 Q 2 Q 3 Q 4 δ 1 ` 2 ` 3 ` 4 , 4 k 1 ,k 2 ,k 3 ,k 4 “ 1 Dispersion relation: c κ ω k “ ω p k q “ 2 m sin p πk { N q , 1 ď k ď N ´ 1 (3) δ 1 ` 2 ` 3 “ δ p k 1 ` k 2 ` k 3 mod N q , (Kronecker δ ), leading to the modular-arithmetic condition k 1 ` k 2 ` k 3 “ 0 mod N .

Equations of motion in Fourier space The equations of motion take the following form: 1 Q 1 “ 1 V 1 , 2 , 3 Q 2 Q 3 δ 1 ` 2 ` 3 ` 1 : ÿ ÿ Q 1 ` ω 2 ˜ ˜ T 1 , 2 , 3 , 4 Q 2 Q 3 Q 4 δ 1 ` 2 ` 3 ` 4 , m m k 2 ,k 3 k 2 ,k 3 ,k 4 where all the sums on k j go from 1 to N ´ 1 . These are exact equations, representing perturbed harmonic oscillators. Normal modes: introduced to diagonalise the Hamiltonian 1 p a k ` a ˚ ? 2 mω k Q k “ N ´ k q . Equations of motion for the normal modes: i B a 1 ÿ p V 123 a 2 a 3 δ 1 ´ 2 ´ 3 ` W 123 a ˚ 2 a 3 δ 1 ` 2 ´ 3 ` Z 123 a ˚ 2 a ˚ B t “ ω k 1 a 1 ` 3 δ 1 ` 2 ` 3 q` k 2 ,k 3 ÿ p R 1234 a 2 a 3 a 4 δ 1 ´ 2 ´ 3 ´ 4 ` S 1234 a ˚ ` 2 a 3 a 4 δ 1 ` 2 ´ 3 ´ 4 k 2 ,k 3 ,k 4 ` T 1234 a ˚ 2 a ˚ 3 a 4 δ 1 ` 2 ` 3 ´ 4 ` U 1234 a ˚ 2 a ˚ 3 a ˚ 4 δ 1 ` 2 ` 3 ` 4 q .

Dynamical systems approach i B a 1 ÿ p V 123 a 2 a 3 δ 1 ´ 2 ´ 3 ` W 123 a ˚ 2 a 3 δ 1 ` 2 ´ 3 ` Z 123 a ˚ 2 a ˚ B t “ ω k 1 a 1 ` 3 δ 1 ` 2 ` 3 q` k 2 ,k 3 ÿ p R 1234 a 2 a 3 a 4 δ 1 ´ 2 ´ 3 ´ 4 ` S 1234 a ˚ ` 2 a 3 a 4 δ 1 ` 2 ´ 3 ´ 4 k 2 ,k 3 ,k 4 ` T 1234 a ˚ 2 a ˚ 3 a 4 δ 1 ` 2 ` 3 ´ 4 ` U 1234 a ˚ 2 a ˚ 3 a ˚ 4 δ 1 ` 2 ` 3 ` 4 q . There are three interesting regimes: Weakly nonlinear regime: amplitudes are small, so higher-order terms are small. Exact resonances dominates. Normal form theory. (Poincar´ e, Birkhoff, Arnold, etc.) [Bustamante et al., CNSNS 73 , 437 (2019)] Finite amplitudes: the terms of different orders are comparable. Bifurcations and chaos dominate. Precession resonance. [Bustamante et al., PRL 113 , 084502 (2014)] (cf. Critical Balance). Large amplitudes: the higher order term dominates. System recovers re-scaling symmetries. Synchronisation of phases. [Murray & Bustamante, JFM 850 , 624 (2018)]

Weakly nonlinear regime: Dominated by exact resonances i B a 1 ÿ p V 123 a 2 a 3 δ 1 ´ 2 ´ 3 ` W 123 a ˚ 2 a 3 δ 1 ` 2 ´ 3 ` Z 123 a ˚ 2 a ˚ B t “ ω k 1 a 1 ` 3 δ 1 ` 2 ` 3 q` k 2 ,k 3 ÿ p R 1234 a 2 a 3 a 4 δ 1 ´ 2 ´ 3 ´ 4 ` S 1234 a ˚ ` 2 a 3 a 4 δ 1 ` 2 ´ 3 ´ 4 k 2 ,k 3 ,k 4 ` T 1234 a ˚ 2 a ˚ 3 a 4 δ 1 ` 2 ` 3 ´ 4 ` U 1234 a ˚ 2 a ˚ 3 a ˚ 4 δ 1 ` 2 ` 3 ` 4 q . In the limit of small amplitudes, the only relevant interactions are those interactions between wavenumbers that satisfy the momentum equation k 1 ˘ k 2 ˘ . . . ˘ k M “ 0 p mod N q and the frequency resonance equation sin p πk 1 { N q ˘ . . . ˘ sin p πk M { N q “ 0 . These are called M -wave resonances, where M ě 3 is an integer. The unknowns are the integers 1 ď k 1 , . . . , k M ď N ´ 1 . When M “ 3 there are no solutions to these equations. Therefore one can eliminate those interactions via a near-identity transformation.

Normal form variables: Near-identity transformation ´ ¯ A p 1 q 1 , 2 , 3 b 2 b 3 δ 1 ´ 2 ´ 3 ` A p 2 q 2 b 3 δ 1 ` 2 ´ 3 ` A p 3 q ÿ 1 , 2 , 3 b ˚ 1 , 2 , 3 b ˚ 2 b ˚ a 1 “ b 1 ` ` 3 δ 1 ` 2 ` 3 k 2 ,k 3 p B p 1 q 1 , 2 , 3 , 4 b 2 b 3 b 4 δ 1 ´ 2 ´ 3 ´ 4 ` B p 2 q ÿ 1 , 2 , 3 , 4 b ˚ ` 2 b 3 b 4 δ 1 ` 2 ´ 3 ´ 4 ` k 2 ,k 3 ,k 4 B p 3 q 3 b 4 δ 1 ` 2 ` 3 ´ 4 ` B p 4 q 1 , 2 , 3 , 4 b ˚ 2 b ˚ 1 , 2 , 3 , 4 b ˚ 2 b ˚ 3 b ˚ 4 δ 1 ` 2 ` 3 ` 4 q ` . . . and select the matrices A p i q 1 , 2 , 3 , B p i q 1 , 2 , 3 , 4 in order to remove non-resonant interactions. For example, the choice V 1 , 2 , 3 2 V 1 , 2 , 3 V 1 , 2 , 3 A p 1 q , A p 2 q , A p 3 q 1 , 2 , 3 “ 1 , 2 , 3 “ 1 , 2 , 3 “ . ω 3 ` ω 2 ´ ω 1 ω 3 ´ ω 2 ´ ω 1 ´ ω 3 ´ ω 2 ´ ω 1 eliminates the 3 -wave interactions, leading to a system of equations for the normal form variables b 1 , . . . , b N ´ 1 (next slide):

Normal form equations of motion i B b 1 ÿ p R 1234 b 2 b 3 b 4 δ 1 ´ 2 ´ 3 ´ 4 ` S 1234 b ˚ B t “ ω k 1 b 1 ` 2 b 3 b 4 δ 1 ` 2 ´ 3 ´ 4 k 2 ,k 3 ,k 4 ` T 1234 b ˚ 2 b ˚ 3 b 4 δ 1 ` 2 ` 3 ´ 4 ` U 1234 b ˚ 2 b ˚ 3 b ˚ ` O p| b | 5 q . 4 δ 1 ` 2 ` 3 ` 4 q Does the transformation converge? Open question in general. See “On the convergence of the normal form transformation in discrete Rossby and drift wave turbulence” by Walsh & Bustamante, arXiv:1904.13272 Can we eliminate some of the 4 -wave interactions? Yes, provided they are not resonant. The transformation created extra interactions: 5 -wave, 6 -wave, etc. Therefore the question about resonances is relevant for all possible M waves.

FPUT exact resonances: Diophantine equations Definition ( M -wave resonance) Let N be the number of particles of the FPUT system. An M -wave resonance is a list (i.e., a multi-set) t k 1 , . . . , k S ; k S ` 1 , . . . , k S ` T u with S, T ą 0 , S ` T “ M and 1 ď k j ď N ´ 1 for all j “ 1 , . . . , M , that is a solution of the momentum conservation and frequency resonance conditions k 1 ` . . . ` k S “ k S ` 1 ` . . . ` k S ` T p mod N q , ω p k 1 q ` . . . ` ω p k S q “ ω p k S ` 1 q ` . . . ` ω p k S ` T q , where ω p k q “ 2 sin p πk { N q . Physically, this corresponds to the conversion process of S waves into T waves. The Hamiltonian term is proportional to b k 1 ¨ ¨ ¨ b k S b ˚ k S ` 1 ¨ ¨ ¨ b ˚ k S ` T .

Preliminary: Forbidden M -wave resonances Theorem (Forbidden Processes) Resonant processes converting 1 wave to M ´ 1 waves or M ´ 1 waves to 1 wave do not exist, for any M ‰ 2 . Also, resonant processes converting 0 wave to M waves or M waves to 0 wave do not exist, for any M ą 0 . Proof. The function ω p k q “ 2 | sin p πk { N q| is strictly subadditive for k P R , k R N Z : ω p k 1 ` k 2 q ă ω p k 1 q ` ω p k 2 q , k 1 , k 2 P R z N Z . Therefore, for example, resonant processes converting 2 waves into 1 wave (or vice versa) are not allowed because this would require ω p k 1 ` k 2 q “ ω p k 1 q ` ω p k 2 q , which is not possible. Similarly, resonant processes converting M ´ 1 waves into 1 wave (or vice versa) are not allowed because subadditivity implies ω p k 1 ` . . . ` k p q ă ω p k 1 q ` . . . ` ω p k p q , k 1 , . . . , k p P R z N Z , for any p ě 2 .

What is new about 4 -wave, 5 -wave and 6 -wave resonances? New methods to construct M -wave resonances: Pairing-off method & Cyclotomic method. The appropriate method to be used depends on properties of the number of particles N and the number of waves M . The case of 4 -wave resonances has been studied extensively and all solutions are known. 4 -wave resonances are integrable and thus they do not produce energy mixing across the Fourier spectrum: one needs to go to higher orders. The case of 5 -wave resonances is completely new and relies on the existence of cyclotomic polynomials (to be defined below). The case of 6 -wave resonances is also new and both methods (pairing-off and cyclotomic) are used to construct them. We do not need to search for M -wave resonances with M ą 6 because these will provide less relevant corrections to the system’s behaviour.