Moving discrete breathers in a Klein–Gordon chain with an impurity

Moving discrete breathers in a Klein–Gordon chain with an impurity J Cuevas†, F Palmero†, JFR Archilla†and FR Romero‡ †ETS Ingenier´ıa Inform´atica. Universidad de Sevilla. Avda Reina Mercedes s/n, 41012-Sevilla, Spain ‡Facultad de F´ısica. Universidad de Sevilla. Avda Reina Mercedes s/n, 41012-Sevilla, Spain Abstract. We analyze the influence of an impurity in the evolution of moving discrete breathers in a Klein–Gordon chain with non-weak nonlinearity. Three different behaviours can be observed when moving breathers interact with the impurity: they pass through the impurity continuing their direction of movement; they are reflected by the impurity; they are trapped by the impurity, giving rise to chaotic breathers, as their Fourier power spectra show. Resonance with a breather centred at the impurity site is conjectured to be a necessary condition for the appearance of the trapping phenomenon. This paper establishes a difference between the resonance condition of the non-weak nonlinearity approach and the resonance condition with the linear impurity mode in the case of weak nonlinearity. PACS numbers: 63.20.Pw 63.20.Ry 63.50.+x 66.90.+r Submitted to: J. Phys. A: Math. Gen. E-mail: [email protected] 1. Introduction Intrinsic localized modes or discrete breathers results from the combination of nonlinearity with spatial discreteness. They can be obtained in Klein–Gordon lattices as exact solutions of dynamical equations [1, 2, 3]. In addition, these localized oscillations, under certain conditions, can move and they are usually called moving breathers [4, 5, 6, 7, 8]. The interaction of nonlinear localized oscillations with impurities in a system can play an important role in its transport properties. This problem has been studied during the last decades within different frameworks. The scattering of kinks and envelope solitons with impurities has been studied in one–dimensional atomic lattices with nonlinear interactions [9]. Another approach is concerned with the interaction between high–frequency continuous breathers and impurities in the sine-Gordon model [10, 11]. This approach has been extended to the low–frequency case in [12], and to the case of kinks in the continuous sine-Gordon and ϕ4models [13, 14]. The scattering of a kink by an impurity in the Frenkel–Kontorova model has been considered in [15, 16]. References [9, 15] describe solitons that can be reflected by the impurity or pass through it, depending on the velocity. However, in [10, 11, 13, 14], it is observed solitons can also be trapped for intermediate velocities. Moving discrete breathers in a Klein–Gordon chain with an impurity 2 The interaction of a moving discrete breather with an impurity in a Klein–Gordon chain has been considered by Forinash et al [17]. In this case, it is assumed that the system has weak nonlinearity and three different behaviours can be observed: (a) the moving breather passes through the impurity, (b) it is reflected, or (c) it is trapped by the impurity, originating a depository of energy. All these effects are related to resonances with the impurity modes. In this paper, we are interested in the last approach, i.e., to study the features of the interaction of moving discrete breathers with an impurity at rest in a Klein– Gordon chain of oscillators with non-weak nonlinearity. Although the behaviour in all of these approaches is qualitatively similar, there are some significative differences concerning to the appearance of the trapping phenomenon. In the case of solitons, the different phenomena are velocity–dependent. In discrete lattices, the breather and the impurity mode are related entities [18, 19], i.e., they can be connected through a continuous path. Therefore, the interplay between a breather and an impurity mode can be much stronger than between a soliton and an impurity. In Forinash’s approach, the necessary condition for the appearance of the trapping phenomenon is that the breather frequency resonates with the linear impurity mode one. In our case, that is, a Klein–Gordon chain with non-weak nonlinearity, the necessary condition for the appearance of trapping is that there must exists a breather at the impurity with a frequency close to that of the moving breather. This fact guarantees the existence of a wide range of parameters for which trapping is possible. Nevertheless, this condition is not sufficient, as the trapping phenomenon does not occur when the tails of the a breather centred at the impurity site and the linear impurity mode have different vibration patterns. We propose the hypothesis that both tails must have the same vibration pattern in order that the trapping occurs. 2. Model and solutions generation 2.1. Formulation of the model In order to study the effects of impurities on the movement of breathers, we consider a simple model where moving breathers can be generated, that is, a Klein–Gordon chain with nearest neighbours attractive interactions [4, 5]. Its Hamiltonian is given by: H= N ∑ n=1 (1 2˙u2 n+Vn(un) + 1 2C(un−un−1)2),(1) where unrepresents the displacement of the n-th particle with respect to its equilibrium position, Cis a coupling constant and Vn(un) is the substrate potential at the n-th site. We choose Vas the Morse potential, i.e., Vn(u) = Dn(e−u−1)2, which proves to be very suitable for obtaining moving breathers [5, 6, 8]. Dnrepresents the well depth in the n-th site. In this model, the inhomogeneity is introduced assuming a different well depth in only one site, i.e., Dn=Do(1 + αδn,0), then we refer to the particle located at n= 0 as an impurity. αis a parameter which tunes the magnitude of the inhomogeneity. It takes its values in the interval [−1,∞). Hereafter, we will consider Do= 1/2. The results presented here correspond to free ends boundary conditions, although periodic boundary conditions lead to the same results. Moving discrete breathers in a Klein–Gordon chain with an impurity 3 This model has been used extensively in DNA dynamics; in this context it is usually referred to as the Peyrard-Bishop model [20]. In the framework of this model, the variables unrepresent the stretching of the hydrogen bonds connecting each pair of bases, Dis the dissociation energy, and Cis the stacking coupling constant. The inhomogeneity can also be introduced in a similar way varying the mass of a particle, or the coupling constant. The results in the first case are equivalent to the obtained for the inhomogeneity in the potential well. There are, however, several differences when the inhomogeneity is introduced through the coupling constant, and we will make some comments about them at the end of this paper. The Hamiltonian (1) leads to the dynamical equations F({un})≡¨un+V′ n(un) + C(2un−un+1 −un−1) = 0.(2) which have two kinds of solutions, linear ones, which correspond to oscillations of small amplitude, and nonlinear ones, which correspond to intrinsic localized modes or discrete breathers. 2.2. Linear modes The dynamical equations can be linearized if the amplitudes of the oscillations are small. Thus, the equations (2) are transformed in the system of coupled equations: ¨un+ω2 nun+C(2un−un+1 −un−1) = 0,(3) where ωnis the natural frequency of the n-th oscillator in the harmonic limit. It is given by ωn=√2Dn, which implies that ω2 n=ω2 o(1 + αδn,0), with ωo= 1, as Do has been chosen to be 1/2. These equations has N−1 non-localized solutions (being Nthe number of particles) corresponding to linear extended modes and one localized solution, which corresponds to a linear impurity mode. Hereafter, unless stated otherwise, the term mode will be reserved to linear modes. The frequencies of the extended modes can be calculated supposing that they are plane waves (un(t) = uoexp(iω(q)t−nq)) and that the impurity mode decays in the space following a dependence of the form un(t) = uoexp(iωLt)r|n|[17], where ris a spatial decay parameter. Thus, the frequencies of the extended modes are given by: ω(q, α) = √ω2 o+ 4Csin2q(α) 2,(4) where q(α) is the extended modes wave vector, which depends in a nonstraightforward way on α. The frequency of the impurity mode is given by the relation [17]: ω2 L=ω2 o+ 2C+ sign(α)√α2ω4 o+ 4C2.(5) The sign of rindicates the vibration pattern of the impurity mode. Thus, if r > 0, the particles of the mode vibrate in phase and will have a wave vector q= 0. On the contrary, if r < 0, the mode will have a zigzag vibration pattern and a wave vector q=π. The parameter ris given by: r=−sign(α)αω2 o+√4C2+α2ω4 o 2C.(6) Moving discrete breathers in a Klein–Gordon chain with an impurity 4 −1 −0.5 0 0.5 1 1.5 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2αres αc 2ωb ωb Linear modes frequencies α Figure 1. Frequencies of the linear modes versus the parameter α, for C= 0.13. The dependence is qualitatively similar for every value of C. At α=αres and α=αc, two different bifurcations occur, being the first one due to the resonance between the impurity mode and the breather. Therefore, αand rhas opposite sign. This fact also implies that, for the extended modes, q∈(0, π] if α < 0 and q∈[0, π) if α > 0. Figure 1 shows the dependence of the frequencies of the linear modes with α, where the isolated frequencies corresponds to the impurity modes. As it is explained below, the values of the frequency and wave vector of the impurity mode will be the key to explain the occurrence of the trapping phenomenon. 2.3. Stationary and moving breathers A stationary breather can be obtained by solving the full dynamical equations. It can be achieved using common methods based on the anticontinuous limit [21]. The implementation of these methods basically consists in calculating the orbit of an isolated oscillator at fixed frequency ωb, and using this solution as a seed to solve the complete dynamical equations by means of a Newton–Raphson continuation method. If the oscillator initially chosen is the corresponding to the impurity, a static breather centred at the impurity is obtained. It will be called impurity breather. Once a stationary breather is obtained, it can be moved under certain conditions. There exists a systematic method for calculating moving solutions [4, 5] which consists in adding to the velocities of the stationary breather a perturbation of magnitude λ colinear to the direction of the pinning mode and letting the system evolve in time. The pinning mode is an anti-symmetric linear localized mode, which may appear in the set of linear perturbations of the system provided the coupling is strong enough [6]. Thus, a perturbation in its direction breaks the translational symmetry of the system and make the breather move. The results presented here correspond to a frequency ωb= 0.8 and a coupling Moving discrete breathers in a Klein–Gordon chain with an impurity 5 Figure 2. Different regimes in the interaction of a moving breather with an impurity introduced as an inhomogeneity in the potential well depth. C= 0.13, although other values of the same order give qualitatively similar results. In this way, we obtain moving breathers with low phonon radiation for values of the perturbation λ.0.2. It is worth remaking that the nonlinearity of our system is higher than the considered in [17], as, in that paper, the frequencies of the localized excitations oscillate between 0.922 and 0.980 (in our frequency units, i.e. normalized to the linear frequency at zero coupling ωo= 1), which are very close to the linear ones. We have considered different values of the parameter αin the range −1≤α≤1. 3. Interaction of moving breathers with an impurity 3.1. Numerical observations We have studied the behaviour of moving breathers when they interact with an impurity. The study has been performed varying the value of the inhomogeneity parameter α. Four different regimes, separated by critical values of the parameter α, have been found (see figure 2): (i) Barrier. The impurity acts as a potential barrier. It occurs either with α > 0 or α∈(−1, α1) with α1<0. As the moving breather reaches the impurity, it is generally reflected, leaving the impurity excited during a short time, whose amplitude decreases with |α|. The only exception to this behaviour occurs for α&0. In this case, the breather can pass through the impurity provided the translational velocity is high enough. (ii) Excitation. The impurity is excited and the breather is reflected. It occurs for α∈(α1, α2). The energy of the excited impurity is larger than the energy of the impurity breather. Thus, the excited impurity vibrates with a frequency lower than ωbas the on–site potential is soft. This behavior is shown in figure 3. (iii) Trapping. The breather is trapped by the impurity. It occurs in the interval α∈(α2, α3). When the moving breather is close to the impurity, it becomes trapped while its center oscillates between the neighbouring sites, as figure 4 shows. Furthermore, the trapped breather emits a great amount of phonon radiation and seems to be chaotic, as can be appreciated from its Fourier power spectrum (Figure 5). (iv) Well. The impurity acts as a potential well. It occurs for α∈(α3,0) and consists of an acceleration of the breather as it approaches to the impurity, and a deceleration after the impurity has been passed through. Moving discrete breathers in a Klein–Gordon chain with an impurity 6 −20 −10 0 10 20 0 50 100 150 200 250 0 0.2 0.4 Periods Position Energy 0 50 100 150 200 250 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 Energy density at the impurity site Periods EN Figure 3. Interaction of a breather with an impurity for α=−0.52 and λ= 0.1, which corresponds to the impurity excitation case. Top: Evolution of the moving breather. Bottom: Evolution of the energy density of the impurity. ENis the energy of the impurity breather. The transition between the different regimes is somehow diffuse, that means, the values of the critical values of αcannot be exactly determined. Furthermore, they are slightly dependent on the breather velocity. An estimation of the critical values for ωb= 0.8, C= 0.13 and λ= 0.1 leads to: α1≈ −0.54, α2≈ −0.49 and α3≈ −0.02. As commented above, these regimes have also been found for different values of the breather frequency. For stronger coupling, the phonon radiation is significant and could mask some of the described effects. Moving discrete breathers in a Klein–Gordon chain with an impurity 7 −20 −10 010 20 0 100 200 300 400 500 0 0.2 0.4 Periods Position Energy Figure 4. Evolution of the moving breather for α=−0.3 and λ= 0.1, which corresponds to the trapping case. The moving breather becomes trapped by the impurity; afterwards, the breather emits phonon radiation and its energy centre oscillates between the sites adjacent to the impurity, 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −1 0 1 2 3 log|S(ω)| ω 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 1 2 3 log|S(ω)| ω Figure 5. Fourier power spectra for the trapped breather of figure 4 (top) and a stationary breather with the same values of Cand ωb(bottom). The spectrum on the top panel is characteristic of chaos. Moving discrete breathers in a Klein–Gordon chain with an impurity 8 −1 −0.8 −0.6 −0.4 −0.2 0 −1 −0.5 0 0.5 1 1.5 2 uo(t=0) α Figure 6. Pitchfork bifurcation for ωb= 0.8 and C= 0.13. αres =−0.5328. 3.2. Discussion Some of the results in the last subsection can be explained from the properties of the impurity breather. Particularly, if a continuation of a stationary breather is performed varying the parameter α, a bifurcation appears for α=αc>0 and another one for α=αres <0 (see Figure 1). The first one is originated by a localized Floquet eigenmode which abandons the unit circle and leads to a breather extinction, i.e., the impurity breather does not exist for α > αc. In the second case, the breather bifurcates with the zero solution through a pitchfork bifurcation (see figure 6) in the space of time–reversible solutions of frequency ωb‡. In this bifurcation, the outer branches correspond to impurity breathers either with un(0) >0 or un(0) <0, while the central branch corresponds to the zero solution, i.e. all the oscillators are at rest. For α=αres the frequency of the impurity mode is the same as the frequency of the impurity breather with the moving breather frequency, i.e., ωL=ωb.αres can be calculated from the equation (5) as a function of ωband C: αres =−√(ω2 b−ω2 o)(ω2 b−ω2 o−4C) ω2 o (7) Thus, for C= 0.13 and ωb= 0.8, αres =−0.5628, which is lower than α1. However, the trapped breather does not exist for α > 0. It indicates that the condition α∈(αres,0) might hold in order that the trapped breather exists. The scenario for the trapped breathers when α < 0 is the following: in this case, the impurity mode has q= 0, and also all the particles of the impurity breather vibrates in phase; this vibration pattern indicates that the impurity breather bifurcates from plane waves with q= 0 [22], i.e., the impurity bifurcates from the impurity mode and it will be the only localized mode that exists when the impurity is excited ‡Note that the dynamical equations (2) do not correspond to a standard dynamical system ¨x=f(x, t), where usually pitchfork bifurcations are described Moving discrete breathers in a Klein–Gordon chain with an impurity 9 for α > αres. Thus, when the moving breather reaches the impurity, it can excite the impurity mode. In fact, we have performed a successful continuation from the impurity breather to the impurity mode at constant action and α[18, 19], and varying a parameter swhich tunes the nonlinearity of the system. This parameter is introduced by changing the on–site potential to the expression V∗ n(un) = Dn(u2 n−su2 n)+sVn(un), where Vn(un) is the original potential (1). When αres < α < α2, the impurity breather is unable to create a trapped entity. The energy of the impurity mode decreases with |α|, thus there must be a minimum value of the impurity breather energy for the existence of trapping. The narrow window of impurity excitations observed in the interval (α1, α2) can be due to a resonance of the moving breather with the impurity breather with a frequency slightly smaller than ωb. For α < αres, the trapped breather cannot be generated, and the moving breather is always reflected. In addition, the impurity breather does not exist. Therefore, there might be a connection between both facts, i.e., the existence of the impurity breather seems to be a necessary condition in order to obtain a trapped breather. If α > 0, the scenario is different. In this case, the impurity mode has q=π but the impurity breather sites vibrate again in phase, that is, the impurity breather does not bifurcate from the impurity mode. Thus, there are two different localized excitations for α > 0: the (linear) impurity mode and the (nonlinear) impurity breather. But, actually, the equations that govern the system are nonlinear, so the linear modes can only correspond to low-amplitude oscillations. In the case of the impurity breather, the linear regime corresponds to the tails. Thus, if the moving breather reaches the impurity site, it will excite the impurity breather and the tails of the impurity mode. But the latter vibrates in zigzag. As a consequence, there will be two different linear localized entities: the tails of the impurity mode (vibrating in zigzag) and the tails of the impurity breather (vibrating in phase). Therefore, we conjecture that the existence of both linear localized entities at the same time may be the reason why the impurity is unable to trap the breather when α > 0. It can be thought whether resonances with the harmonics of the breather frequency could have consequences in the interaction of the moving breather with the impurity. In our system, these resonances occur for α > αc, and therefore, the impurity breather does not exist, so that no trapping effects take place, leading only to breather reflections. We summarize here our hypothesis for the existence of trapping: Trapping hypothesis:The existence of an impurity breather for a given value of αis a necessary condition for the existence of trapped breathers. However, if there exists an impurity mode with a different vibration pattern to the impurity breather one, the trapped breather does not to exist. 4. Inhomogeneity in the coupling parameter In order to check whether the hypothesis proposed in the last section holds for different situations, we consider a chain of oscillators for which the imhomogeneity is introduced through the coupling constants. In this case, the Hamiltonian can be written as: H=∑ n(1 2˙u2 n+V(un) + 1 4Cn[(un−un−1)2+ (un−un−1)]),(8) which leads to the following dynamical equations: