Moving discrete breathers in a β-FPU lattice revisited

Moving discrete breathers in a β-FPU lattice revisited Henry Duran1, Jes´ us Cuevas–Maraver2,3, Panayotis G. Kevrekidis4, and Anna Vainchtein1 1Department of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, USA 2Grupo de F´ ısica No Lineal. Departamento de F´ ısica Aplicada I, Escuela Polit´ ecnica Superior, Universidad de Sevilla. C/Virgen de ´ Africa, 7, Sevilla 41011, Spain 3Instituto de Matem´ aticas de la Universidad de Sevilla (IMUS). Edificio Celestino Mutis, Avda. Reina Mercedes s/n, 41012-Sevilla, Spain 4Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA 01003-9305, USA November 18, 2021 Abstract In the present work we revisit the existence, stability and dynamical properties of moving discrete breathers in β-FPU lattices. On the existence side, we propose a numerical procedure, based on a continuation along a sequence of velocities, that allows to systematically construct breathers traveling more than one lattice site per period. On the stability side, we explore the stability spectrum of the obtained waveforms via Floquet analysis and connect it to the energy-frequency bifurcation diagrams. We illustrate in this context examples of the energy being a multivalued function of the frequency, showcasing the coexistence of different moving breathers at the same frequency. Finally, we probe the moving breather dynamics and observe how the associated instabilities change their speed, typically slowing them down over longtime simulations. 1 Introduction Discrete breathers are time-periodic nonlinear modes that arise in lattices due to the interplay of dispersion and nonlinearity [2, 3, 18]. The most common form of such excitations are stationary bright breathers, originally called intrinsic localized modes [26, 31, 35] due to their spatial localization. Breathers were found to exist in Hamiltonian and damped-driven lattices and were experimentally observed in a variety of nonlinear discrete systems, including Josephson junction arrays [5, 36], forced-damped arrays of coupled pendula [13], electrical lattices [17, 27, 30], micromechanical systems [32–34], the denaturation of the DNA double strand [28] and granular chains [6,8,11,38]. In the years since the breathers were first discovered [25], there has been much progress in understanding their existence, spectral stability and dynamical properties [3,18,20]. For instance, explicit criteria for linear [21] and nonlinear [14] stability of stationary breathers have been put 1 arXiv:2111.09167v1 [nlin.PS] 17 Nov 2021 forth. Additionally, it was observed that instability of stationary breathers sets them in motion, and long-lived traveling breathers have been found numerically in various nonlinear lattices [7,10, 12, 19,31]. Breather mobility is of considerable interest because it is associated with energy transport in the lattice; indeed such coherent structures have been proposed as a prototypical means for achieving targeted energy transfer in discrete nonlinear systems [22]. An exact moving breather is time-periodic modulo a shift by one or more lattice spaces. The period is an integer multiple of the period of internal vibrations. Such solutions have been constructed using the Newton iterative method, e.g., for Klein-Gordon [1, 4, 12] and β-FPU [12, 37] lattices. For generic interaction potentials that do not possess a certain symmetry [15,16], moving breathers are no longer spatially localized: instead, they possess oscillatory wings whose amplitude depends on the internal breather frequency and its propagation velocity. The first detailed analysis of this dependence for a β-FPU lattice was performed in [37]. The authors constructed numerically exact moving breathers for several different rational values of the period-wise velocity V1=r/s, where ris the number of lattice sites the breather travels over s periods of the internal vibration. Performing a continuation in internal frequency ωat fixed V1, they investigated how the wing energy (or, equivalently, amplitude) of these breathers depends on their internal frequency. In particular, they studied the mechanism for resonances in the wing amplitude and derived an approximate formula for the resonant frequencies. They also briefly summarized the results of linear stability investigation (without providing a systematic analysis thereof) for the computed solutions. Motivated by these earlier studies, we revisit the problem and conduct a more detailed investigation of the breather existence, stability, dynamics and resonance features. Our analysis extends the results of the earlier work in several ways. Importantly, we consider moving breathers propagating by more than one lattice space (r > 1) over its period, extending the earlier work that had focused chiefly on the r= 1 case. To compute such solutions, we developed a numerical procedure based on a continuation along a sequence of rational values of V1. We show that the total breather energy (the Hamiltonian) and the wing energy are in fact multivalued functions of the internal frequency ω, so that there are several moving breathers with the same ωand different energies. Moreover, our results reveal the truly nonlinear form of the resonances: a rapid increase in wing energy is followed by a more gradual one. Subsequently, we provide a detailed analysis of the linear stability of the obtained solutions focusing on the consequences of the instability associated with real Floquet multipliers µ > 1. In particular, we investigate the dynamics of the breathers perturbed along the corresponding unstable eigenmodes and show that after repeated interactions with the wing oscillations due to the periodic boundary conditions the breather gradually decelerates and eventually becomes nearly stationary, with its velocity oscillating around zero. The paper is organized as follows. We formulate the problem in Sec. 2 and describe our numerical procedures in Sec. 3. In Sec. 4 we examine the dependence of the moving breather with different period-wise velocities on the internal frequency and discuss the multivalued nature of the obtained energies, resonances and stability. Consequences of the observed real instabilities in the breather dynamics are explored in Sec. 5. Concluding remarks can be found in Sec. 6, along with some suggestions for future work. In the Appendix, we discuss additional solutions that coexist with the ones described in the main text but have different linear spectra. 2 2 Problem formulation We consider a lattice of Nparticles with nearest-neighbor interactions governed by a β-FPU potential. In dimensionless variables the Hamiltonian of the system is given by H=1 2 N X n=1 p2 n+ N X n=1 1 2(qn+1 −qn)2+β 4(qn+1 −qn)4= N X n=1 en,(1) where qndenotes the displacement of the nth particle, pn= ˙qn=dqn/dt is its momentum (the mass is rescaled to unity), βmeasures the strength of the nonlinear coupling, and en=1 2p2 n+1 4(qn+1 −qn)2+ (qn−qn−1)2+β 8(qn+1 −qn)4+ (qn−qn−1)4(2) is the site energy density. The equations of motion are ¨qn=qn+1 +qn−1−2qn+β(qn+1 −qn)3−(qn−qn−1)3.(3) In what follows, we assume that Nis even and prescribe periodic boundary conditions: qn+N=qn, pn+N=pn. In the numerical results presented in this work we set β= 1. The β-FPU problem (3) is known to have two types of stationary discrete breather solutions qn(t) = xn(t)that are time-periodic, xn(T) = xn(0), and spatially localized in terms of the relative displacements xn−xn−1. Here, T= 2π/ω is the period of internal oscillations with frequency ω. The first type is the site-centered Sievers-Takeno (ST) mode [35], with displacement that has even symmetry about the center, and the second type is the bond-centered Page (P) mode [26], with odd displacement. The P mode is linearly stable, while the ST mode is unstable [31]. Perturbing an ST mode along an eigenmode corresponding to the instability sets the breather in motion. Our focus here is on moving discrete breathers that propagate rlattice sites over speriods T= 2π/ω of internal oscillations and satisfy [37] {qn(sT)}N i=1 {pn(sT)}N n=1 −(−1)r{qn−r(0)}N n=1 {pn−r(0)}N n=1 =0,(4) where the indices are mod Ndue to periodic boundary conditions. Here sand rare integers, and V1=r s(5) denotes the period-wise velocity of the breather (the number of lattice sites transversed over the period of one internal oscillation), while its translational velocity is given by V2=V1 T=r sT .(6) 3 Numerical Methods To obtain moving breathers, we must find fixed points of the map defined by (4) using the Newton iterative method, with an appropriately perturbed unstable ST stationary breather, whose instability 3 induces the breather mobility, as an initial seed. Here and in what follows, we use symplectic and explicit fourth-order Runge-Kutta-Nystr¨ om algorithm [9] to integrate the equations of motion. We found that over the course of the simulations, the maximum absolute relative difference of the total energy when compared to the initial total energy is on the order of 10−10. We start by constructing an ST breather xn(t)with the given internal frequency ω, using the Newton iterative method and numerical continuation from the anticontinuous limit [24]. Linearizing (3) around the ST breather by setting qn(t) = xn(t) + yn(t)and considering O()terms, we obtain ¨yn−(yn+1 +yn−1−2yn)−3(xn+1 −xn)2(yn+1 −yn)−(xn−xn−1)2(yn−yn−1)= 0, which is used to compute the monodromy matrix Fdefined by y(T) ˙ y(T)=Fy(0) ˙ y(0),(7) where the vector functions y(t)and ˙ y(t)have components yn(t)and ˙yn(t), respectively. The Floquet multipliers µare obtained by finding the eigenvalues of F, once the iterative procedure has converged. A Floquet multiplier satisfying |µ|>1indicates instability. An ST mode has an unstable eigenmode corresponding to a real Floquet multiplier µ > 1; naturally, due to the Hamiltonian nature of the problem, there exists a complementary (inverse) one with µ < 1. Following [10], we obtain the initial seed for a moving breather by applying a kinetic perturbation of the ST breather. Specifically, we use the momentum part δpof the eigenvector associated with the instability, so that our initial guess for the moving breather is given by q p=x(0) 0+λ0 δp,(8) where λis the strength of the perturbation. To construct moving breathers with V1= 1/s for some integer s≥1, we use the Newton iterative method to find fixed points of (4) with initial guess (8) as the values of λare being incremented within some interval. We typically start with λ=−1and increase it by 10−2up to λ= 1. Once this has been completed, we look at the solutions for which the square of the `2norm of the objective function of the Newton iteration, defined by the left hand side of (4), is below some threshold. Doing this allows us to obtain moving breathers on different branches in the (ω,H) plane near the resonance values of ω, as described below. Solutions with other frequency values are then found using parameter continuation along each branch. Typically, this continuation was done in ω, but near the turning points for ωwe used Has a continuation parameter. We found that this method successfully generates moving breathers with r= 1 but has not worked in the examples we considered for velocities with r > 1. To compute moving breathers with period-wise velocity V1=r/s, where r > 1, we have developed the following numerical procedure. We use one of the moving breathers with V1= 1/s0for some integer s0as an initial guess and construct a monotone sequence v1, v2, ..., vkof rational values of the period-wise velocity that are close enough together and satisfy v1= 1/s0 and vk=r/s. These values are chosen in a way that minimizes swhile staying within a prescribed step difference, empirically selected to be between 0.018 and 0.022. Depending on the value of ω chosen, it is possible that for one of the chosen vi, the moving breather solution will be close to 4 a resonance; in this case, a larger step in viis needed to bypass the resonance. For example, to compute a moving breather with V1= 5/7we used the sequence {vi}={1/2,12/23,13/24,9/16,7/12,23/38,5/8,20/31,2/3,11/16,5/7}. We then use a continuation procedure that involves obtaining the moving breather with velocity viusing Newton’s iterative method and the breather with velocity vi−1as the initial guess. An example of a moving breather with V1= 14/23 and ω= 2.5obtained using this method is shown in Fig. 1(a,b). 0 10 20 30 40 50 60 n -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 xn 0 50 0 2 4 610 -9 (a) (b) -1 -0.5 0 0.5 1 Re -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Im (c) Figure 1: (a) Verification of the relation described in (4) for the moving breather with period-wise velocity V1= 14/23 and internal frequency ω= 2.5. The blue circles are the displacements at time t= 23T, while the solid red line is the displacement at time t= 0 shifted to the right by 14 lattice sites. The inset shows the absolute difference between the two sets of displacements. (b) Space-time evolution of the site energy en(t). (c) Floquet multipliers µ associated with the linearization around the solution of panels (a)-(b). The absence of multipliers lying off of the unit circle suggests the spectral stability of the relevant waveform. 5 To determine the stability of a computed moving breather, we linearize (3) about it and construct the monodromy matrix Fdefined by {yn+˜r(˜sT)}N n=1 {˙yn+˜r(˜sT)}N n=1=F{yn(0)}N n=1 {˙yn(0)}N n=1,(9) where ˜r=r,˜s=sif ris even and ˜r= 2r,˜s= 2sif ris odd. Fig. 1(c) shows the Floquet multipliers for the breather with V1= 14/23 and internal frequency ω= 2.5. This breather appears to be linearly stable. To explore the consequences of an instability associated with a real Floquet multiplier µ > 1 for a moving breather, we perturb it along the corresponding eigenmode and use the following method to approximate the translational velocity V2of the ensuing waveform as a function of time. The procedure involves computing the location of the center of the energy density of the moving breather. We divide the time interval [ti, tf], where tiis the initial and tfis the final time, into subintervals of length ∆t, thus selecting sample times tisuch that tj+1 −tj= ∆t. Typically, we set ∆t=sT, where Tis the internal period and sis the number of periods the unperturbed breather needs to advance rsites. At each time tj, we compute the energy density en,j and use it to obtain an approximation for the center Xjof the wave Xj=Pn∈{core}nen,j Pn∈{core}en,j .(10) In order to improve the accuracy of this approximation, we use a spline interpolation of the energy density. We then compute (10) including the interpolated points in the core of the moving breather. To determine the width of the core, we start from the maximum of the energy density. We then traverse the chain until the absolute difference between the energy density and wing energy, which is determined by averaging the ten particles that make up the wings, is on the order of 10−4. The distance between the particle where the maximum occurs and the cutoff particle is half of the core width. We choose as a center point the maximum of the interpolated energy density. Once the weighted energy center has been found, we repeat the above procedure using the weighted energy center as the center point. This has little effect for waveforms with small-amplitude wings, but when the wings have larger amplitude, the recalculation is necessary to compensate for the effect they have on the energy density as the center crosses a boundary. The translational velocity V2(t) of the wave is then approximated by V2(tj)≈Xj+1 −Xj tj+1 −tj .(11) 4 Frequency dependence, resonances and stability We now investigate the dependence of the moving breather solutions on the internal frequency ω at fixed period-wise velocity V1and the lattice size N. The results for V1= 1/3and N= 60 are shown in Fig. 2. Panel (a) shows the total energy (Hamiltonian) Has a function of ω. One can see that there is a number of resonances at certain frequency values. At these values, the amplitude of the wing oscillations rapidly increases. Near the resonance frequencies, the breather energy is a multivalued function of ω. Indeed, near each resonance frequency, the curve can be 6 2.1 2.15 2.2 2.25 2.3 2.35 2.4 2.45 2.5 0.5 1 1.5 2 H (a) 2.1 2.15 2.2 2.25 2.3 2.35 2.4 2.45 2.5 2.55 10 -10 10 -8 10 -6 10 -4 10 -2 10 0 ewing /e max 13 -17 12 11 -16 10 -15 (b) 2.23 2.24 2.25 2.26 2.27 2.28 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 max(| |) 1.23 1.24 Re( ) -0.01 0 0.01 Im( ) 1.23 1.24 Re( ) -0.04 -0.02 0 0.02 0.04 Im( ) (c) Figure 2: (a) Energy H, (b) the normalized average site energy ewing/emax of the wings and (c) the maximum moduli of the Floquet multipliers µalong different branches as functions of ωat V1= 1/3and N= 60 near the resonance ω= 2.237. The Floquet multiplier with the maximum modulus has nonzero real and imaginary parts along dashed portions the curve and is real along the solid one. Insets illustrate that this transition occurs due to the collision of a pair of real Floquet multipliers and subsequent emergence of a quadruplet of complex-valued multipliers symmetric about the unit circle (only the pair of such multipliers outside the unit circle is shown in the second inset). Different colors correspond to different branches in (a). The numbers in (b) are the values of mfor each resonance (see the text for detail). split into three pieces: the top branch, the middle branch, and the bottom branch, where the middle and bottom branches are connected to each other by a turning point. We distinguish between the bottom and top branches by alternating colors between green and blue at each resonance. Note that the bottom branch corresponding to one resonance frequency eventually merges with the top branch near another resonance. Along the three branches near each resonance, there are distinct moving breathers with the same internal frequency, as illustrated in Fig. 3 for ω= 2.126. Panel (b) of Fig. 2 shows the corresponding average site energy in a wing portion of the breather, normalized by the maximum site energy. One can clearly see the nonlinear character of the resonances, with rapid increase in wing energy followed by a more gradual one. Both the 7 2.1255 2.126 2.1265 0.57 0.58 0.59 0.6 0.61 H 0 50 n -0.2 -0.1 0 0.1 0.2 qn 0 50 -0.2 -0.1 0 0.1 0.2 0 50 -0.2 -0.1 0 0.1 0.2 Figure 3: The top panel shows a zoomed-in view of the resonance near ω= 2.126. The black vertical line marks ω= 2.126 at which three different moving breathers coexist. These breathers are shown in the three bottom panels, where colors match the respective branches depicted in the top panel. Here V1= 1/3and N= 60. nonlinear form of the resonances and the multivalued nature of the frequency dependence were, apparently, missed in the earlier computations [12, 37]. Yoshimura and Doi in [37] used a normal mode analysis to approximate resonance frequencies. For completeness, we briefly describe the main steps of their derivation. The normal mode coordinates Qm(t),m=−N/2−1, . . . , N/2, are defined by [29] qn(t) = (−1)n √N N/2 X m=−(N/2−1) Qm(t)cos 2π Nmn−sin 2π Nmn, n = 1,2, . . . , N and have the natural frequency associated with the dispersion relation: Ωm= 2 cos πm N.(12) For stationary breathers these modes are approximated in [37] using the method in [23] with frequency ω, which yields Qm(t)≈Amcos(ωt), where Am=π 2√6βN sech π2m N√ω2−4.(13) Complex normal modes Um(t) = 1 2(Qm+Q−m)+ i 2(Qm−Q−m)are then used to construct moving breather solutions with V1=r/s in the form Um(t) = ψm(t)e−imr N s ωt, where ψm(t)are complex-valued functions satisfying d2ψm dt2−i2mrω Ns dψm dt +Ω2 m−mrω Ns 2ψm =−β N Nh X i,j,k=−Nh ΩmΩiΩjΩkψiψjψk·ei[{m−(i+j+k)}r/Ns]ωt∆(m−(i+j+k)), (14) 8 where ∆(m) = (−1)mif r=mN for m∈Zand zero otherwise, Nh=N/2−1, and Ωmis defined in (12). The solution of (14) is then sought in the form ψm(t) = ψ0 m(t) + um(t), ψ0 m(t) = X `=±1 Amei`ωt, um(t) = ∞ X n=−∞ am,nein(ω/s)t,(15) where um(t)is the deviation from the stationary breather ψ0 m(t)with Amgiven by (13), and both components are periodic functions with period sT that are expanded in Fourier series, with coefficients am,n for um(t). Here, only the dominant fundamental frequency components are kept in the expansion for ψ0 m(t) = Amcos(ωt). Substituting (15) into (14) and considering the leading-order approximation in terms of um(t)results in a linear system for am,n. Analysis of this system shows that |am,n|becomes large when its coefficient is close to zero. Setting these coefficients to zero thus yields an approximation for the resonance frequency values ωm,|m|< N/2, at which the mth normal mode is excited. The approximate resonance condition [37] is given by  n s−mr Nsωm= Ωmr1 + 2 Npω2 m−4,(16) where nmay take values n=±sor n=±(s±r), depending on the frequency interval and the value of V1=r/s, and we also recall (12). Using (16), we computed the values of mand ωmfor each resonance shown in Fig. 2(b); the corresponding values of mare shown in the plot. Table 1 compares the predicted values of resonance frequencies with the numerical ones. numer ωmapprox ωmm 2.126 2.129 13 2.237 2.244 12 2.342 2.352 11 2.440 2.454 10 2.464 2.483 −15 2.352 2.364 −16 2.229 2.236 −17 Table 1: Comparison of numerical and approximate resonance values ωmfor V1= 1/3and N= 60. The approximate values are computed using (16). The numerical values were computed by using the wing energy plots, such as Fig. 2(b), and estimating the frequency at the center of the gap that separates branches corresponding to each resonance. Panel (c) of Fig. 2 shows the maximal moduli of the Floquet multipliers associated with the computed breathers near the resonance ω= 2.237 as a representative example. As the top branch nears a resonance, a real instability, which corresponds to a real Floquet multiplier µ > 1, manifests itself. As ωcontinues to increase along the branch, and the wings of the moving breathers become more pronounced, this real instability is accompanied by the emergence of complex instability modes associated with Floquet multipliers µthat have nonzero imaginary part and satisfy |µ|>1. As can be seen in the insets, the largest real multiplier is accompanied by a smaller real one that eventually collides with it. This collision results in the formation of a symmetric quadruplet of complex-valued multipliers. Meanwhile, both the bottom branch and the middle branch are 9 2.35286 2.35288 2.3529 2.35292 2.35294 2.35296 1.472 1.473 1.474 1.475 1.476 1.477 1.478 1.479 H (a) 2.35286 2.35288 2.3529 2.35292 2.35294 2.35296 1 1.00005 1.0001 1.00015 1.0002 1.00025 max(Re( )) (b) Figure 9: (a) Energy Hversus frequency ωand (b) the maximum real Floquet multipliers µalong the two solutions near the resonance at ω= 2.352. The colors correspond to those used in Fig. 8. Here V1= 1/3and N= 60. In the left panel the blue and red branches cannot be distinguished over the scale of the figure (see also the magnified pictures in the left panels of Fig. 8). 0 1000 2000 3000 4000 5000 6000 7000 8000 t -29 -28 -27 -26 -25 -24 log(|c(t) - V2|) (a) 0.6 0.8 1 1.2 1.4 1.6 1.8 t10 4 -32 -30 -28 -26 -24 -22 -20 -18 -16 log(|c(t) - V2|) (b) 1.8 1.9 2 2.1 2.2 2.3 t10 4 -21 -20.5 -20 -19.5 -19 -18.5 -18 log(|c(t) - V2|) (c) Figure 10: Time evolution of the absolute difference between the computed speed cand the initial initial translational velocity V2= 1/(3T)for the moving breather with the largest real Floquet multiplier µ= 1.0023 perturbed along the corresponding unstable eigenmode. Panels (a) and (c) show the early and late stages of the evolution, while panel(b) depicts over the entire time span. The maximum modulus of the Floquet multipliers is |µ|= 1.0104. In panels (a) and (c), the red lines correspond to the best linear fit measuring the growth rate of the wave. The line in panel (a) measures the initial growth due to the real instability, and the second line measures the growth due to the complex instability. In panel (b), the darker region corresponds to the emergence of the complex instability as the main factor in the growth of the perturbed moving breather. Here N= 60,ω= 2.355,V1= 1/3, and the strength of the perturbation is = 10−7. associated with an eigenmode along which the dynamics was initially perturbed. The second, depicted in Fig. 10(c), is determined by the maximal-modulus Floquet multipliers µ= 0.5034 ± 0.8761iwith |µ|= 1.0104. The middle panel of Fig. 10(b) captures the transition from the former to the latter. We note that this is different from the example shown in Sec. 5, where the real Floquet multiplier corresponding to the eigenmode along which the moving breather was perturbed also had the largest modulus among the Floquet multipliers. The red lines in panels (a) and (c) measure the growth rate and have the slope ln(|µ|)/(6T), where µis the corresponding multiplier and we 16 have used the fact that V1= 1/3for the unperturbed breather. Comparing the lines of growth rate for maximum real multiplier and the complex multiplier with maximum modulus to the early and late stages, respectively, of the evolution in the simulation results yields an absolute difference of size O(10−5)in both cases, indicating that the two regimes are indeed dominated by the two distinct types of instability. At later times, the velocity evolution is similar to that for the examples discussed in Sec. 5. In short, this detailed examination of the associated dynamical evolution revealed that the instability growth rates captured by our Floquet analysis, even when very small, accurately reflect the instability features of the associated solutions and hence appear to be real features of the wave dynamics. References [1] J. F. R Archilla, Y. 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