Modulation Instability and Wavenumber Bandgap Breathers in a Time Layered Phononic Lattice

We demonstrate the existence of wavenumber bandgap (q-gap) breathers in a time-periodic phononic lattice. These breathers are localized in time and periodic in space, and are the counterparts to the classical breathers found in spatially-periodic systems. We derive an exact condition for modulation instability that leads to the opening of wavenumber bandgaps. The q-gap breathers become more narrow and larger in amplitude as the wavenumber goes further into the bandgap. In the presence of damping, these structures acquire a non-zero, oscillating tail. The experiment and model exhibit qualitative agreement.


INTRODUCTION
The classical discrete breather is defined as a spatially localized, time-periodic solution of a nonlinear lattice differential equation.They are a fundamental structure found in many platforms, including photonics, phononics, and in electrical systems, see [1][2][3] for comprehensive reviews.One mechanism through which breathers can manifest is the modulation instability (MI) of plane waves in spatially periodic lattices [4].Such breathers have a frequency that falls into a spectral gap [1].
A natural counterpart to the fundamentally important discrete breather is the so-called wavenumber bandgap (q-gap) breather.It is localized in time, periodic in space, and has wavenumber that falls into a q-gap.Q-gap breathers represent a newer class of solutions and are distinct from qbreathers, which are localized in wavenumber and periodic in time [5].Since q-gap breathers have the roles of time and space switched when compared to classical breathers, it is natural to consider lattices that are time varying (instead of space varying).Recent advances in experimental platforms for time varying systems makes the topic particularly relevant, including studies in photonic [6][7][8][9], electric [10][11][12], and phononic systems [13][14][15][16][17][18][19][20][21][22][23][24][25].Time localization can also arise via mechanisms other than a wavenumber bandgap, such as zero-wavenumber gain modulation instability [26].The so-called Akhmediev breathers of the nonlinear Schrödinger equation [27], and its discrete integrable counter-part, the Ablowtiz-Ladik lattice [28], are also localized in time.They do not have wavenumber in a q-gap, and hence are distinct from q-gap breathers.Very recently, it was shown theoretically that time-localized solitons with wavenumbers in the qgap exist in a time-periodic photonic crystal [29].Breathers in time varying phononic systems, however, remain unexplored.
In the present letter, we employ numerical, analytical, and experimental approaches to explore q-gap breathers and the modulation instability leading to the opening of wavenumber bandgaps.Q-gap breathers could be exploited for the creation of phononic frequency combs [30][31][32], energy harvesting applications [33,34], or acoustic signal processing [35].

EXPERIMENTAL AND MODEL SET-UP
The particular phononic lattice under consideration is a system of repelling magnetic masses with grounding stiffness controlled by electrical coils driven applied voltage signals that are given by a time-periodic step function.The experimental setup is adapted form the platform developed in [15,25].The chain is composed of  − 1 ring magnets (K&J Magnetic, Inc., P/N R848) lined with sleeve bearings (McMaster-Carr P/N 6377K2) comprising the uniform masses, arranged with alternating polarity on a smooth rod (McMaster-Carr P/N 8543K28).Electromagnetic coils (APW Company SKU: FC-6489) are fixed concentrically around the equilibrium positions of each of the innermost eight masses, such that they may exert a restoring force on each mass proportional to the current induced by applied voltage stepfunction (Aglient 33220A, Accel Instruments TS250-2).The velocity of each mass is measured using laser Doppler vibrometer (Polytec CLV-2534), repeating experiments to a construct full velocity field for the lattice.Figure 1 shows a schematic of the experimental setup.The system is modeled as Fermi-Pasta-Ulam-Tsingou type lattice [15,25]  ü where   is the displacement of the th ring magnet from its equilibrium position, where the equilibrium distance between adjacent magnets is  = 0.0334 m.The indices run from  = 1 …  − 1 and we consider fixed boundary conditions  0 () =   () = 0.All ring magnets have uniform mass  = 0.0097 kg.Dissipative forces are modeled with a phenomenological viscous damping term     , where the damping coefficient  = 0.15 Ns/m is determined empirically by matching the simulated and experimental spatial decay of the velocity amplitude envelope of waves traveling through the lattice [25].The coupling force term is defined using the repulsive magnetic force between neighboring masses.The experimentally measured force-distance relation between neighboring masses is fit with a dipole-dipole approximation, as in [15], which is given by  () = ( + ) − , where  is the center-to-center distance between masses with  = 9.044 × 10 −7 Nm 4 and  = 4.The current resulting from a periodic step function voltage applied to the electromagnetic coils induces a magnetic field that provides a grounding stiffness modulation of the form The step values   ,   and duty-cycle 0 <  < 1 ([] = s) are parameters.Unless otherwise stated, we use   = 0,   = 150 N/m and  = 0.5 s.We will use the modulation frequency  mod = 1∕ Hz as the main system parameter to be varied.

MODULATION INSTABILITY
To find breathers, we first need to determine the wavenumber bandgap.This is achieved by computing the stability of plane waves (i.e., the modulation stability) of the linearized model where  =  −1 .For time-independent stiffness (() = 0) the undamped ( = 0) linear equation has the dispersion relationship  2 disp () = 4∕ sin 2 (∕2) such that the linear spectrum extends from [0, √ ∕] and all plane waves are stable.In the case of time-dependent stiffness (() ≠ 0) a gap in the wavenumber axis  is possible.For general timeperiodic stiffness () with period  = 2∕ mod , a wavenumber bandgap will open where the dispersion curve  disp () intersects itself when translated by an integer multiple of half the modulation frequency  mod ∕2 [36].The advantage of considering () to be a periodic step function is that the modified dispersion relation can be computed exactly.Making the ansatz   () =   () ⋅ Θ  (), one finds upon substitution into Eq.( 3) and enforcing Dirichlet boundary conditions that the eigenfunctions are   () = sin (    ) where the wavenumber is We can obtain an exact solution of this equation (and dispersion relation and stability condition), by adapting a procedure carried out in the context of an undamped Kronig-Penney photonic lattice [37,38].The general solution of Eq. ( 4) will be a superposition of functions of the form Θ  () =   ()      .The waveform associated to the wavenumber   will be stable if   ≤ 0, or equivalently, if the Floquet multiplier has modulus less than unity, 4) and demanding that   () and Ḣ () are continuous at  = 0 and  =  leads to the following equations (detailed in the Supplemental Material) where  depends on the wavenumber and system parameters, but not the Floquet exponent   : where ).These equations allow for the exact computation of the Floquet exponents   =   +   .An example plot is shown in Fig. 2

(a). If
|| ≤ 1 then   = −∕(2) and   = cos −1 ()∕ .In this case the underlying solution is stable.If ± > 1 then   = (3 ± 1)∕(2 ), implying that the imaginary part of the Floquet exponent is an integer multiple of half the modulation frequency.In this case, the real part of the Floquet exponent is   = ± cosh −1 (∓) ∕ − ∕(2) which implies the following condition for stability, Note that this expression is exact and gives an efficient way to check for stability via direct substitution of the system pa-rameters into   and simply checking the inequality.A plot of   is shown in Fig. 2(b).The linear stability predictions agree well with experimental observations (examples given in the Supplemental Material).The set of wavenumbers where |  | > 1 make up the so-called wavenumber bandgap.The edges can be found by solving   = ±1.See the gray regions of Fig. 2 for example wavenumber bandgaps.
The solution initially grows exponentially with growth rate given by  5 , due to the modulation instability, but reaches a turning point and then decays with the rate − 5 .This happens uniformly within the lattice, as shown by the intensity plot in Fig. 3(a).Figure 3(b) shows the time series of the velocity of the 6th node, i.e., u6 () =  6 () (solid blue curve).Both panels (a) and (b) demonstrate that the dynamics are localized in time.Spatial periodicity of the solution is imposed by construction due to the finite length of the lattice with zero boundary conditions.The role of space and time have been switched when compared to the classic breathers of space-periodic systems.Thus, the solution shown in Fig. 3 is the so-called called wavenumber bandgap breather.Motivated by the fact that the envelope of a breather of a spaceperiodic FPUT lattice is described by a soliton of the Nonlinear Schrödinger (NLS) equation (in the limit of the temporal frequency approaching the band edge from within the spectral gap), we fit the velocity profile  6 with a function of the form  1 sech where   are fitting parameters and  5 is the real part of the Floquet exponent.See the gray dashed line of Fig. 3(b).The good agreement between the velocity profile and the fit envelope function confirms that the growth/decay rate is indeed given by the real part of the associated Floquet exponent, in this case  5 .To better understand the mechanism behind the formation of the wavenumber bandgap breather, we construct a Poincaré map of the dynamics by sampling the solution with the frequency associated with the breather, namely  mod ∕2.This corresponds to twice the period of the modulation.Thus, the map will be of the form   ( 0 ) = (2 ), where  is an integer,  is the period of (), and is vector valued solution of Eq. ( 1) with initial value  0 .As in the simulation shown in Fig. 3(b), the initial value of the map is given by an unstable plane wave (e.g., with wavenumber  5 ).The red dots of Fig. 3(b) show values of the map  that correspond to  6 .The red dots of Fig. 3(c) show values of  in the ( 6 ,  6 ) phase plane.The eigenvector corresponding to the unstable (stable) Floquet exponent  5 (− 5 ) is shown in red (blue).The gray line of Fig. 3(c) is obtained by repeatedly generating the map  in the ( 6 ,  6 ) plane for various (small) multiples of the initial value  0 .The origin is a saddle type fixed point, and the trajectory forms a near homoclinic orbit, made possible by the nonlinearity of the system.The orbit is not exactly homoclinic, since it does not approach the origin via the stable eigenvector as  → ∞.Indeed, as can also be inferred from Fig. 3(b), the solution does not decay to zero, but rather it experiences small oscillations.This is due to the existence of other modes in the system (e.g., ones with associated multipliers lying on the unit circle), which are excited during the dynamic evolution.The insets of Fig. 3(b) show a normalized spatial Fourier transform of the signal before (the left inset) and after (the right inset) the maximum velocity is attained.In particular, the quantity | v|∕| max v| is shown against the wavenumber, where v() = 2 −1

∑
() sin() where  = 0.66s and  = 1.02s are the times used to compute the transform before and after the turning point, respectively.Before the turning point the only prominent wavenumber is the one associated to the initial value (in this case  5 = 5∕11).After the turning point, there is an additional mode excited that lies outside the wavenumber bandgap (in this case  3 = 3∕11).It is this mode that is primarily responsible for the non-zero oscillations at the tail of the breather.If one imposes the additional criterion that a breather must have tails decaying to zero, then strictly speaking, the structure found here would be a generalized breather, since the orbit is not exactly homoclinic.Classic breathers with tails that do not decay to zero are common in non-integrable systems, and have sometimes been referred to as generalized breathers [39].Over longer time windows, the amplitude of the signal can grow again (leading to a repeated appearance of breathers), but eventually the structure typically breaks down, leading to chaotic type dynamics for long-time simulations.Similar observations have been made for k-gap solitons in photonic systems [29].The existence of perfectly homoclinic solutions in this system is an important question, but it lies beyond the scope of the present article.Inspection of the long time dynamics, as well as breathers for other parameter values, are provided in the Supplemental Material.We now generate a family of wavenumber bandgap breathers parameterized by the distance the underlying wavenumber is from the band edge   .This is a natural parameter to consider, as the distance to the bandedge determines breather width and amplitude in space-periodic systems [1].Keeping all parameters fixed, but gradually varying the modulation frequency  mod has the effect of shifting the bandgap in the wavenumber axis.Thus, we fix the wavenumber ( 5 in this case), whose distance to the right edge will increase as the modulation frequency is increased.For the breather shown in Fig. 3 the distance to the edge is Δ =   −  5 ≈ 0.09.The red dots of Fig. 4(a) show the amplitude of the breather vs. Δ =   ( mod ) −  5 .The amplitude is computed as the maximum velocity of the 6th node, i.e., max  ‖ 6 ()‖.For Δ > 0.4 the breathers do not form a coherent localization, like in Fig. 3. Similar simulations with wavenumber near the left edge of the bandgap ( 5 ≈   ) did not lead to the robust formation of breathers.The amplitude data is fit with a function of the form  1 Δ  2 , with the best fit values being  1 = 3.34 and  2 = 0.57 (the solid line in Fig. 4(a)).This is consistent with the trend found for classic breathers in space-periodic systems where it is well known that the breather amplitude grows like  ( √ Δ ) , where Δ is the difference between the breather frequency and the edge of the frequency spectrum [1].Now that we have established the existence of wavenumber bandgap breathers in Eq. ( 1), we now consider the role of damping, which will bring us closer to the experimentally relevant situation.Breathers in experiments will always be a dissipative analog of breathers in lossless models.Thus, we modify our definition of a breather to account for dissipative effects.To motivate our modified definition, we repeat the simulations shown in Fig. 3 but with a nonzero damping parameter.The yellow square (blue circle) markers of Fig. 3(c) show the orbit with a damping parameter of  = 0.075 ( = 0.15) Ns/m.The orbit starts close to being homoclinic, but the dynamics are attracted to a stable fixed point (i.e., a timeperiodic orbit of the original system with period 2 ).Note that the time-dependent stiffness term in the model (with the () coefficient) acts as an effective gain term to balance loss.The orbit in the damped system experiences an initial exponential growth and a turning point, like the lossless breather, but rather than approaching near zero amplitude, the dynamics tend to a stable fixed point.Thus, the left tail of the "damped breather" (in the time domain) is much like a lossless breather, whose amplitude is slightly lower due to the presence of damping.The right tail of the "damped breather" approaches a periodic oscillation, whose amplitude is not necessarily small relative to the amplitude of the breather.See Fig. 4(b) for an example time series with  = 0.15 Ns/m.An alternative classification for the structure found in the damped system would be a wavenumber bandgap "front", since the solution is heteroclinic, as it connects the unstable zero fixed point to a non-zero stable fixed point.Important for the present study, however, is that the transition between the zero and nonzero state is approximately described by the lossless breather (i.e., the near homoclinic orbit).We measure the amplitude of the structure in the same way we measured the breather amplitude in the lossless system, i.e., max  ‖ 6 ()‖.We repeat this for various modulation frequencies (and hence Δ values) for the damping value  = 0.15 Ns/, which is shown as the blue circle markers in Fig. 4(a).The qualitative amplitude trend is similar to the lossless case, but the amplitude is decreased.The amplitude data in the damped case is also fit with a function of the form  1 Δ  2 , with the best fit values being  1 = 2.62 and  2 = 0.59.
We now turn to the experimental construction of wavenumber bandgap breathers (using the dissipative definition of a breather defined above).To excite a plane wave with a particular wavenumber (in this case  5 ) the unmodulated system is driven with the frequency  disp ( 5 ).Once the desired plane wave is excited, the initial driving is turned off and the modulation is turned on simultaneously.Like in the damped simulation, the amplitude will initially grow, reach a turning point, decay, but eventually approach a periodic orbit (or exhibit chaotic behavior).An example experimental time series is shown in Fig. 4(c).Notice the qualitative agreement to the theoretical prediction shown in Fig. 4(b).Since the band edge can only be computed in the infinite lattice, we use the relation Δ =   ( mod ) −  5 , where   ( mod ) is found analytically.In general, there will be a mistuning between the experimental and model dynamics for a fixed modulation frequency.We account for this mistuning by measuring the modulation frequency the mode  5 becomes unstable in the model and experiment.The difference between the experimental and theoretical critical modulation frequencies,  mod , is then added to the experimental modulation frequencies when calculating the distance to the band edge, namely Δ =   ( mod − mod )− 5 .Thus, Δ = 0 corresponds to the frequency at which  5 becomes unstable for both experiment and model.The mode associated to  5 is considered unstable when its corresponding amplitude in Fourier space exceeds a noise threshold (details given in the Supplemental Material).With these definitions in place, we now measure the amplitude of the breather as a function of Δ, see the markers with error bars in Fig. 4(a).The qualitative amplitude trend agrees with the theoretical prediction.Although the time series between the theoretical prediction for  = 0.15 Ns/m and the experiment agree qualitatively (compare panels (b) and (c) of Fig. 4) the model underestimates the amplitude.This can be partially explained by the fact that other modes (including unstable ones) in the experiment besides  5 are excited.Inspection of the spatial Fourier transform of the experiment shows there are always traces of additional modes, even before the turning point of the structure.The insets of Fig. 4(c) show the spatial Fourier transform before ( = 1.6 s) and after ( = 2.2 s) the turning point (more details of the Fourier analysis of the experiment is given in the Supplemental Material).
Finally, we measure the width of the structures using the halfwidth at half maximum (HWHM) metric.The HWHM is given by  max −  half , where  max is the time the maximum is attained and  half is the time where the trajectory first attains half the maximum value.The experimentally measured values are shown as the black triangles in Fig. 4(a).The width of the structure can be predicted theoretically using the real part of the Floquet exponent.In particular, assuming the envelope of the breather follows the form sech() (which we demonstrated previously was a reasonable assumption), we can compute the HWHM as sech −1 (1∕2)∕ 5 .The prediction in the lossless ( = 0) and damped ( = 0.15 Ns/m) cases are shown as red solid squares and open blue squares, respectively, in Fig. 4(a).For sufficiently large values of Δ, there is good agreement between theory and experiment, and both show the structure becomes more narrow as the wavenumber goes deeper into the gap.This is consistent with classic breathers in space-periodic systems.

CONCLUSION
We explored a new class of solutions, phononic wavenumber bandgap breathers and their damped analogues, in a timevarying lattice.Several avenues of inquiry follow naturally from this study, including the possible existence of genuine q-gap breathers (i.e., with both tails decaying to zero), an analytical description of the breather profiles via a multiple-scale analysis, and the exploration of such structures in higher spatial dimensions.

1 FIG. 2 .
FIG. 2. (a) Plot of  = ∕(2) where  is the imaginary part of the Floquet exponent (blue curve) and , the real part of the exponent (red curve) for an infinite lattice.The black dots show the corresponding values for a finite sized lattice with  = 11.The modulation frequency is  mod = 1∕ = 45 Hz.The shaded gray region indicates the region of instability (i.e., the wavenumber bandgap).(b) Plot of the function   from Eq. (7) in the infinite lattice (solid blue line) and in the  = 11 lattice (solid black dots).The plane wave is unstable when |  | > 1, which is highlighted by the gray region.The inset shows the corresponding Floquet multipliers in the complex plane.There are two multipliers lying outside the unit circle (also shown) demonstrating the instability of a general solution.

8 FIG. 3 .
FIG. 3. Wavenumber bandgap breather for  mod = 45 Hz.(a) Intensity plot of velocity after initializing the unstable plane wave with wavenumber  5 with  = 0. (b) Time series of the velocity of node 6 of panel (a).The dashed line shows the best-fit envelope (in the least square sense).The red dots are the solution sampled every 2 seconds.The insets show the spatial Fourier transform before (left) and after (right) the turning point.(c) Plot of the Poincaré map of the solution shown as red dots in panel (b).The yellow squares and blue circles correspond to the same simulation with  = 0.075 Ns/m and  = 0.15 Ns/m, respectively.

1 FIG. 4 .
FIG. 4. (a) Plot of the breather amplitude for the  = 0 (red dots) and  = 0.15 Ns/m (blue circles) simulations vs. distance of the wavenumber to the bandedge, Δ.The lines show the best fit function of the form  1 Δ  2 .The markers with error bars are the measured amplitude from the experiment.The solid red squares and open blue squares show the theoretical HWHM for the  = 0 and  = 0.15 Ns/m cases, respectively.The black triangles are the experimentally measured HWHM.(b) Time series example for the  = 0.15 Ns/m simulation with Δ = 0.147.The dashed line shows the best-fit envelope of the solution up until the maximum.The insets show the spatial Fourier transform before (left) and after (right) the turning point.(c) Same as panel (b), but for the experiment.