Three- and four-wave resonances in the nonlinear quadratic Kelvin lattice

In this paper we investigate analytically and numerically the nonlinear Kelvin lattice, namely a chain of masses and nonlinear springs, as in the alpha-Fermi-Pasta-Ulam-Tsingou (FPUT) chain, where, in addition, each mass is connected to a nonlinear resonator, i.e., a second mass free to oscillate. Both nonlinearities are quadratic in the equations of motion. This setup represents the simplest prototype of nonlinear wave propagation on a nonlinear metamaterial. In the linear case, we diagonalize the system, and the two branches of the dispersion relation can be found. Using this result, we derive in the nonlinear case the equations of motion for the normal variables in Fourier space, obtaining a system governed by triad interactions among the two branches of the dispersion relation. We find that the transfer of energy between these two branches is ruled by three- and four-wave resonant interactions. We perform numerical simulations of the primitive equations of motion and highlight the role of resonances as an efficient mechanism for transferring energy. Moreover, as predicted by the theory, we provide direct evidence that four-wave resonances appear on a time scale that is longer than the time scale for three-wave resonances. We also assess the recurrence behaviour (usual in the FPUT system) for the nonlinear Kelvin lattice, and we show that, while recurrence is observed if all the energy is placed, at time t=0, in the lowest mode of the acoustical branch, a non-recurrent behaviour is observed if the initial energy is located in the optical branch.


INTRODUCTION
Waves are ubiquitous in physics and engineering.Advances in technology have created an increasing demand for controlling waves that traditional materials cannot satisfy.In this context, metamaterials offer the possibility to manipulate some of the properties of waves such as their refraction or their dispersion, and are now the subject of many interdisciplinary studies [1], [2].Metamaterials are artificially engineered structures that interact uniquely with waves; their properties depend on the geometrical construction, rather than on the chemical composition of the material.Metamaterials have several practical applications, among which are the absorption of mechanical vibrations [3], the manipulation of acoustic waves [4], and the prevention of damages from earthquakes [5] and coastal erosion [6].For classification and further developments, consult [7,8].
In its simplest setting, a metamaterial can be modelled as a one-dimensional chain (masses and springs) to which one or more masses, acting as resonators, are included.The aim is to understand some basic physical properties of the models, such as the branching out of the dispersion relation, the formation of frequency band gaps (i.e.some frequency ranges are forbidden, or alternatively they have negative effective mass), the nonlinear interactions between modes belonging to the different branches, and new low-order resonances that may arise.Although the governing nonlinear evolution equations can be solved numerically with little effort, the chaos present in the system makes an analytical study necessary for a full comprehension of the properties of the metamaterial and for their exploitation.To this end, in the present paper we consider nonlinearities both in the propagating medium and in the resonator.We give a fully nonlinear treatment to the problem, emphasising the possibility of transferring energies among the normal modes of the dispersive curves.This approach is exploited for the simplest system considered (a dispersion relation with two branches), and it can be extended in a straightforward manner (although algebra becomes cumbersome) to more complicated systems with more than two dispersive curves.
A very first sketch of this model (in its linear version, namely when the springs satisfy Hooke's law) was provided by Lord Kelvin in 1889 to devise a theory of dispersion [9].Kelvin's model consists of a one-dimensional chain of particles of mass m q connected by equal springs of elastic constant χ q , such that each of these masses is attached to another particle of mass m r by a spring of elastic constant χ r (see figure 1 below).This model is explicitly remarked on page 11 of Brillouin's book [10], where the dispersion relation is also shown: it consists of two pass-band branches, the low-frequency one being known as the 'acoustical branch' and the high-frequency one being known as the 'optical branch'.There is a gap of forbidden frequencies between these branches (see figure 2 below).The existence of branches and gaps is ubiquitous in this type of models.Brillouin's book (op.cit.) provides an early account of this and many other models, such as the diatomic chain, and constitutes a pivot point in the history of studies on the structure of one-dimensional lattices, where a particular focus was given to understanding how energy propagates in crystals (see also [11,12]).Interestingly, despite the fact that Brillouin's book is cited by over four thousand publications (most of which deal with metamaterials), and despite the fact that Brillouin calls this model "Kelvin's model", Kelvin's model is rarely called by its name in these publications, and it is usually called with generic names such as "mass-in-mass system".Following the very recent and relevant paper regarding the linear problem [13] (which also contains a very good bibliography), we call this model "Kelvin lattice".See also [14][15][16][17].
We are interested in the nonlinear quadratic version of the Kelvin lattice, namely the case when the springs are nonlinear such that the force contains a small extra quadratic dependence on the displacement: F = χ∆x + α(∆x) 2 .Such a case is reminiscent of the celebrated Fermi-Pasta-Ulam-Tsingou lattice [11] (α-FPUT for short), which corresponds to our nonlinear quadratic Kelvin lattice in the case when the resonators (of masses m r ) are not considered in the picture.The numerical simulations shown in [11] displayed the phenomenon of recurrence: an ordered and reversible nonlinear dynamical exchange of energy amongst the modes of oscillation.The subsequent research on this phenomenon led to the discovery of solitons and integrable nonlinear partial differential equations [18].The recurrence phenomenon led to the apparent paradox that modes of oscillation would not reach equipartition even though they interact nonlinearly due to the α-term.Further research on this paradox led to various approaches to explain how energy equipartition is eventually reached at very late times.The interested reader is directed to these reviews on the subject: [19][20][21][22][23].
Going back to the nonlinear Kelvin lattice and other nonlinear metamaterial models, one could expect the FPUT recurrence phenomenon to be an exception rather than the rule.General nonlinear systems usually display resonances, which by definition break integrability.Although this paper deals with discrete lattices and finite-amplitude effects, where the time scales of the different variables are not necessarily widely separated, it is worth mentioning the strategies related to asymptotic methods that exist in the literature.For example, assuming separation of time scales allows for perturbation-expansion solutions and nonlinear corrections to the dispersion relation [24][25][26].As another example, taking the continuum limit (so the chain is now a one-dimensional continuum) and assuming separation of spatial scales allows for the derivation of nonlinear partial differential equations, a work initiated in 1965 by Zabusky and Kruskal [18] with their celebrated discovery of solitons in the Korteweg-de Vries equation.
This paper is organized as follows: Sec.I introduces the Kelvin lattice and the procedure of diagonalization of the linear problem.Sec.II discusses the natural extension to the nonlinear Kelvin lattice, where, besides the theoretical approach, numerical simulations are used to support our analytical findings on the effect that three-wave and fourwave resonances have on the system.Sec.III discusses an assessment of recurrence behaviour of the nonlinear Kelvin lattice, inspired by the well-known FPUT recurrence [11].Finally, Sec.IV provides discussions and conclusions.Appendix A contains a derivation of the long-wave continuum limit of the Kelvin lattice.

I. THE LINEAR KELVIN LATTICE
Let us consider a chain of N masses, m q , connected with springs of elastic constant χ q .We connect second masses, m r , to the masses of the chain by means of springs with elastic constant χ r as in Fig. 1.The masses m r are free to oscillate and are not connected to each other.We assume periodic boundary conditions, ı.e., if we denote by j the label of the masses, such that q j (t) is the displacement of the corresponding mass m q and r j (t) is the displacement of the accompanying mass m r , both with respect to their equilibrium positions, then q N +1 (t) = q 1 (t) and r N +1 (t) = r 1 (t).In this section we consider the simplified case of harmonic potentials, whereby all the springs' potential energies are quadratic in the relative displacements.This case leads to linear evolution equations which allow us to construct the normal modes and find two branches for the dispersion relation.The case of anharmonic potentials, namely when the potential energies contain higher-order terms, leads to nonlinear interactions between the normal modes and is considered in the next section.

A. Equation of motion for harmonic potentials
The kinetic energy of the system is given by the sum of the kinetic energies of all particles: while the potential term takes into account the elastic force between the j-th mass of the chain with its nearest neighbours and associated resonator: The Lagrangian of the system is given by L = T − V , then applying the Euler-Lagrange equations the equations of motion follow: qj = χ q m q (q j+1 + q j−1 − 2q j ) + χ r m q (r j − q j ), (4a) rj = χ r m r (q j − r j ).(4b)

B. Equation of motion in Fourier space
To decouple Eq.( 4), the following Discrete Fourier transforms (DFT) are applied: with the completeness of the discrete basis given in terms of the Kronecker delta: Substituting equations ( 5) and ( 6) into equation (4) an equivalent system of equations in terms of the Fourier amplitudes is found: where k = 0, . . ., N − 1, and is the dispersion relation of the classical FPUT monoatomic problem, which can be obtained by disconnecting the masses m r from m q via setting χ r = 0.
The system (7), for each k = 0, . . ., N − 1, consist of independent blocks of two coupled equations.In order to solve these coupled equations, it is useful to consider the Hamiltonian structure.Defining the conjugate momenta the Hamiltonian is prescribed as follows: In terms of the Fourier amplitudes, this Hamiltonian turns out to be with The canonical Poisson bracket is obtained: so the Hamilton equations are Note that the displacements are real quantities, thus Fourier amplitudes and momenta are Hermitian, which translate into the inner property Using equations ( 14) and ( 11), the equations of motion (7) are recovered.

C. The dispersion relation
For fixed k = 0, . . ., N − 1, a more compact formulation of Eqs.( 7) is achieved using a 2 × 2 matrix form, The eigenfrequencies of this system are obtained from the characteristic equation det(A + Ω 2 I) = 0, leading to the diagonalization of A by similarity (see next subsection).In terms of the associated diagonal matrix these eigenfrequencies appear in two branches (i.e.subspaces): where are the two branches of the dispersion relation, displaying a forbidden frequency band gap (see Fig. 2) of size: Note that so ∆Ω + + ∆Ω − + ∆Ω gap = Ω + max .By varying the parameters χ q , χ r , m q , m r the quantities Eq.( 18) and Eq.( 19) change too, and so do the shapes of the curves in Fig. 2.
FIG. 2. The two branches of the dispersion relation separated by the band gap of forbidden frequencies.We call + the optical branch and − the acoustical branch.The latter contains a degenerate zero-mode, k = 0, whose frequency is zero.

D. Diagonalization
The matrix A is diagonalizable by similarity.Namely, there exists an invertible matrix P such that A is diagonalized: (16).Such a transformation then diagonalizes Eq.( 15).Let us denote by s = ± the two branches.The first step is to solve A⃗ u s = −Ω 2 k(s) ⃗ u s , which returns two eigenvectors: Since , where μ is the reduced mass defined in Eq.(19b), a useful equality follows: From Eq.( 20) the transition matrix is Thus, by means of Eq.( 21), its inverse is written as where we have defined the effective masses such that Now from Eqs.( 15), ( 16), ( 22) and ( 23) we get Ü = AU = P A D P −1 U , from which P −1 Ü = A D P −1 U , or, defining we obtain the diagonal system d 2 , which is a fully decoupled system of two harmonic oscillators: corresponding to the two branches of the dispersion relation.The coordinate transformation (26) can be inverted and simplified using equations (25) to give From Eqs.( 12) and ( 28), using Eqs.( 21) and ( 24), the kinetic term in Eq.( 11) becomes T = N −1 k=0 As for the potential term, using the identities So, in summary we have the canonical transformation, valid for all k = 0, . . ., N − 1 and all s = ±: for which the Hamiltonian Eq.( 11) presenting coupled terms becomes the diagonalized quadratic form with Poisson brackets { Q s k , P s * k ′ } = δ k,k ′ and the Hamilton equations , which are equivalent to Eqs. (27).Note that the acoustical mode corresponding to (k, s) = (0, −) is characterized by only kinetic energy which remains constant in time.Therefore, we introduce the normal variables via the canonical transformation: where (k, s) ̸ = (0, −) .(31) As the above transformation is not applied to the acoustical mode (k, s) = (0, −), the original momentum variable P − 0 is kept, which is conserved in time by virtue of Hamilton equations, as Q − 0 is ignorable.For the normal variables the system (27) becomes decoupled and reads, along with its solution, The above solution implies namely, the wave-action at each normal mode does not evolve in time.In the normal variables, the Hamiltonian Eq.( 30) takes the form: and is the energy associated with the s-branch k-th mode, which does not evolve in time.
E. Analytical solutions to the equations of motion

Periodic boundary conditions
Using equation (32) and recalling the identities k can be written in the form: while the ignorable coordinate evolves as From here on we set C 1 = C 2 = 0, obtained after an appropriate Galilean transformation.In the original variables, using Eq.( 28), the solution in Fourier space reduces to: and in physical space, denoting Rearranging the symmetric contributions from k and N −k in the sums above, we find the solution as a superposition where real amplitudes B s k and phases φ s k are determined by the initial conditions.

Fixed boundary conditions
Consider a chain composed by N + 1 masses and as many resonators.Fixed boundary conditions implies that q j (t) = r j (t) = 0, ∀t ∈ R + , for j = 0, N .In this case a real Fourier transform has to be used to write the equation of motion in the Fourier space to get the solutions where Ωs k and then βs k have the same form of Eqs. ( 17), ( 20), but with The fact that Equations ( 38) are a superposition of stationary waves is more clearly visible imposing null initial velocities, so that all B s k are zero, and an initial sinusoidal pattern q j (0) = r j (0) = sin(nπj/N ) with 0 < n < N, n ∈ N. In this case only the n-th harmonic is involved and each of Eq.s( 38) is made by the sum of four waves, two by two of the same amplitude given by the half of and β± n A ± n , and traveling in opposite directions, that is Equating to zero Eqs.(40), the value is found.This means that, if n is a divisor of N , the stationary waves present n + 1 nodes for discrete values of j, i.e. the j * l -th masses and resonators stay still during the time evolution of the system.Otherwise, the chain has only the two trivial nodes at the ends.

II. THE NONLINEAR KELVIN LATTICE A. Equations of motion
An anharmonic potential can be represented via a cubic power term in the Hamiltonian, which leads to three-wave interaction systems.This kind of systems has been widely studied in the past, see [27,28]; the complication here is that, as will be shown below, nonlinear interactions may take place between the two branches of the dispersion relation.Adding the nonlinear contribution to our system gives the following Hamiltonian: where H 0 is given by Eq. (10).From this, the equations of motion are obtained: qj = χ q m q (q j+1 + q j−1 − 2q j ) + χ r m q (r j − q j ) + α m q [(q j+1 − q j ) 2 − (q j − q j−1 ) 2 − (q j − r j ) 2 ] , C. Resonances The system of equations in (51) indicates that there are different types of three-wave interactions that can take place within the same branch of the dispersion relation or even between the two branches of the dispersion relation.By construction, transfer of energy between normal modes takes place only if at least one of the Kronecker deltas is satisfied.Among all these interactions, the most relevant ones are those that satisfy the resonant condition, i.e., those for which, besides the condition on wavenumbers, an analogous condition on frequencies is satisfied.For N → ∞, keeping the distance between the masses fixed, wavenumbers can be treated as continuous variables and three-wave exact resonances exist.For example, if we set m q = m r = χ q = χ r = 1, the resonant interactions between two acoustical and one optical mode can take place if one of the following conditions is satisfied: These interactions correspond to the terms 51).For finite N , resonant triads exist for specific values of the parameters m q , m r , χ q and χ r .In general, non resonant interactions may contribute to higher order resonances.Indeed, there is a well-known asymptotic procedure in analytical mechanics, known as near identity transformation or Lie transform that allows one to remove analytically the non resonant terms; for Hamiltonian systems, the transformation can be canonical [30].The non resonant terms contribute to higher order interactions which may or may not satisfy the resonant condition.A typical example where the technique is employed is the water wave problem where, because of the shape of the dispersion relation, three-wave resonances do not exists and a canonical transformation is employed to remove them and recast them in terms of cubic nonlinearity, which results in four-wave interactions [31].Without going into the details, we mention that, in the model here discussed, four-wave resonant interactions may naturally appear on a time scale larger than the triad time scale.For discrete systems, resonant quartets may be of the form: See [29,32] for details.Numerical simulations of the original equation of motion, displaying three-and four-wave resonant interactions, will be shown in the next section.
We remark that for the standard FPUT lattice (regardless of the boundary conditions), three-wave resonant interactions are forbidden and the lowest order of resonant interactions is the four-wave one, observed in the case of periodic boundary conditions (see [29,33,34] for details).The introduction of the resonators allows for a transfer of energy that takes place on a faster time scale with respect to the standard FPUT one.Moreover, we also recall that from the FPUT lattice, in the limit of large N and small distances between the masses, the KdV equation is recovered.In the Appendix, we report a similar calculation and find that, in the continuum limit, the nonlinear quadratic Kelvin lattice reduces to a Boussinesq equation coupled with a continuum of harmonic oscillators.

D. Numerical simulations
To support our analytical findings, we perform numerical simulations of the equations of motion (43).We use a symplectic integrator scheme [35] for the time-marching, ensuring an error around the eleventh digit in the Hamiltonian over our integration time.
FIG. 3. A visual scheme of the modes interactions.The modes are labelled k s , where k = 0, 1, 2, 3 denotes the wavenumber and s = ± denotes the branch (−: acoustical, +: optical).The orange continuous line connects two resonant triads.The vertical dashed line separates the frequencies belonging to the optical and acoustical modes.
For simplicity, we consider the case N = 4, α = 0.2.All other parameters are fixed to one, except for χ r which is selected in such a way to allow exact three-wave resonances.Eight normal modes are accessible: four belonging to the optical branch and four to the acoustical branch; however, the k = 0 acoustical mode remains constant in time.Therefore, only seven modes are active.These are shown schematically in Fig. 3.For each of the two triads k3 is equal to zero if χ r satisfies the following equation: The approximated value χ r ≈ 0.3522011287389576 is the solution to Eq.(56) that will be used in simulations.Initial conditions are supplied by prescribing the value of a s k for some specific modes related to the chosen resonant triad.More specifically, the initial conditions satisfy the following relations: therefore, at t = 0, only normal modes {1 − , 1 + } have energy.In Fig. 4a), we show the evolution in time of the energy, (with initial condition given in equation ( 57)), as a function of time for each normal mode, for the case N = 4, α = 0.2 and all other parameters set to one except for χr.In (a), χr is selected to solve (56), so that three-wave exact resonances are allowed.In (b), χr has been increased by 15% with respect to the value in a).
To further support our analytical results, tests are conducted changing χ r , so that exact three-wave resonant interactions are detuned, i.e., the resonant condition on frequencies is only approximately satisfied.The results are shown in Fig. 4b), where clearly the exchange of energy between modes {1 − , 1 + } and mode {2+} has been drastically reduced.Interestingly, four-wave resonant interactions are still active, involving the same quartet (1 − , 1 + , 3 − , 3 + ) as in the previous case.

III. THE RECURRENCE IN THE NONLINEAR KELVIN LATTICE
As mentioned earlier, when χ r is set to zero in equations (4), the system reduces to the standard α-FPUT chain [11].Such a model has been widely studied, and there are excellent reviews on the subject [19,[21][22][23]36].As is well known, the numerical simulations described in [11] showed some unexpected results: instead of observing the equipartition of energy among the degrees of freedom of the system, the authors observed the quasi-recurrence to the initial state.Therefore, an interesting question to be answered is what happens to the recurrence in the presence of resonators.In this context, although not being the main focus of the present paper, we have performed some numerical computations with the aim of assessing the recurrence behaviour in the presence of resonators attached to the chain.As a first step, we have reproduced numerically the original result in [11] by using our model and setting to zero the parameter χ r and the initial values of the resonator variables r j (figure not shown).Next, for the numerical simulations of the nonlinear Kelvin lattice, we keep the parameter choices of the original FPUT problem, namely we take N = 32, α = 0.25, χ q = χ r = 1 and m q = m r = 1, consider fixed boundary conditions, and characterize the initial condition by exciting at time t = 0 only the longest mode of the chain, namely q j ∝ sin(πj/N ); in the presence of resonators an initial condition has to be provided also for the variables r j .Among the infinite number of choices for these initial conditions that can lead to different results, we have picked representative initial conditions using what we believe to be the simplest strategy: to keep exciting only the longest mode (namely r j ∝ sin(πj/N )), and to ensure all initial conditions have the same linear part H 0 of the initial energy.Now, to assess the recurrence behaviour, we propose two types of initial conditions based on physical considerations regarding our acoustical/optical splitting of the normal modes.The first type of initial condition is defined so that only the longest acoustical mode is initially excited, namely r j = β − 1 q j , referring to equation (28).For this choice, cf.figure 5(a), the evolution of the system displays a clear recurrent behaviour, strikingly similar to the classical α-FPUT system [11].For our choice of parameters, this gives initially r j ≈ 1.005 q j , which makes the cubic term in the Hamiltonian relatively small initially, which could be an explanation for the recurrent behaviour.Analogously, the second type of initial condition is defined so that only the longest optical mode is initially excited, namely r j = β + 1 q j , again referring to equation (28).For this choice, cf.figure 5(b), the system displays a recurrent behaviour only for a few periods, to then succumb to a gradual loss of linear energy to the cubic term of the conserved Hamiltonian (not shown in the figure), accompanied by bursting behaviour of the other modes' energies.For our choice of parameters, this gives initially r j ≈ −0.9952 q j , which makes the cubic term in the Hamiltonian relatively larger initially than in the recurrent case, while still smaller than the linear energy H 0 , so that during the first stages (before t = 1000) the linear energy H 0 is the main contributor to the Hamiltonian.
Because of the number of parameters involved, the dynamics of our system is very rich, and a detailed study on the recurrence (or non-recurrence) should be performed; however, we find that this interesting research activity is outside the scope of the present paper, and it will be part of future research.

IV. CONCLUSIONS
Starting from the pioneering work of Enrico Fermi and collaborators, [11], masses and springs have been considered non trivial toy models suitable for understanding fundamental aspects of propagation of waves in nonlinear dispersive systems.The one dimensional chain with quadratic and cubic nonlinearity has been widely studied in the literature [22,36,37] and many results are nowadays available.On the other hand, the field of metamaterials is rapidly evolving and it finds application in many different physical problems as for example in electromagnetic, acoustic, seismic, phononic, elastic, water waves [5,6,8,38,39].
In this paper we have tried to understand the propagation properties of a nonlinear chain coupled with extra resonators that act as a metamaterial.Once the nonlinearity is introduced, the problem becomes much more complicated from a theoretical point of view; the idea here adopted, and originally developed in the field of Wave Turbulence [40], is to diagonalize the linear system and use the normal variables to describe the nonlinear problem.The resulting model corresponds to two equations, written in Fourier space, which are decoupled at the linear order and the coupling appears in the nonlinear terms.Nonlinearities are expressed as interactions characterized by rules enforced by the presence of Kronecker deltas, which select the interacting wave numbers via the so-called momentum conditions.In this framework, exact resonances, i.e. those for which a resonant condition on frequencies is also fulfilled, play a dominant role in transporting energy between normal modes.Because of the quadratic nonlinearity in the equations 1 qj ≈ 1.005 qj, which implies that only spectral mode a − 1 (the longest acoustical mode) is nonzero initially.The system displays recurrence, very much like the classical α-FPUT system [11].In (b), the initial condition is characterized by qj = N sin(πj/N ) ≈ 0.2782 sin(πj/N ), rj = β + 1 qj ≈ −0.9952 qj, which implies that only spectral mode a + 1 (the longest optical mode) is nonzero initially.The system displays non-recurrent behaviour, as can be seen by the gradual loss of the linear energy to the cubic term of the conserved Hamiltonian (not shown), and the bursting behaviour of the modes' energies.
of motion in this specific case, the interactions are characterized by resonant triads at the leading order in nonlinearity; however, a deeper and accurate analysis reveals that non-resonant triads can generate resonant quartets; this result is well known in the field of surface gravity waves where triads are not resonant [41].In addition, we performed numerical simulations of the primitive equations of motion with some specific initial conditions characterized by resonant triads; however, as expected, on a longer time scale, resonant quartets appear naturally in the simulations.Also, in the spirit of the original report by Fermi and collaborators, we investigated the recurrence behaviour of the nonlinear Kelvin lattice and showed that if only the lowest mode of the acoustical branch is perturbed, recurrence is observed; however, if only the optical branch is perturbed, the recurrence is destroyed.Finally, we calculated (see appendix) the long-wave limit of the nonlinear Kelvin lattice, obtaining a Boussinesq equation that is coupled nonlinearly to a continuum of harmonic oscillators.
As far as we are aware of, the field of nonlinear metamaterials is only recently being developed.The present approach, innovative in this field, could open up new strategies and new applications in the field.

FIG. 1 .
FIG.1.A graphic representation of the Kelvin lattice: a second mass free to oscillate is connected to each mass of the monoatomic chain by means of a spring.

FIG. 5 .
FIG. 5. Recurrence assessment of the Kelvin lattice for N = 32, α = 0.25 fixed boundary condition case, and initial energy (linear part) H0 = 2.5511.All other parameters are set to one.Evolution of the linear part of the spectral energy,E + k +E − k , with E s k = Ω s k |a s k | 2 ,as a function of time for the first 5 modes.In (a), the initial condition is characterized by qj = √ N sin(πj/N ) ≈ 5.657 sin(πj/N ), rj = β − 1 qj ≈ 1.005 qj, which implies that only spectral mode a − 1 (the longest acoustical mode) is nonzero initially.The system displays recurrence, very much like the classical α-FPUT system[11].In (b), the initial condition is