Multibreathers in Klein-Gordon chains with interactions beyond nearest neighbors

Multibreathers in Klein-Gordon chains with interactions beyond nearest neighbors V. Koukouloyannisa,∗, P.G. Kevrekidisb, J. Cuevasc, V. Rothosd aDepartment of Civil Engineering, Technological Educational Institute of Serres, 62124 Serres, Greece bDepartment of Mathematics and Statistics, University of Massachusetts, Amherst MA 01003-4515 cNonlinear Physics Group. Departamento de F´ısica Aplicada I, Escuela Polit´ecnica Superior. Universidad de Sevilla. C/ Virgen de ´ Africa, 7, 41011 Sevilla, Spain dDepartment of Mathematics, Faculty of Engineering, Aristotle University of Thessaloniki,Thessaloniki GR54124 Greece Abstract We study the existence and stability of multibreathers in Klein-Gordon chains with interactions that are not restricted to nearest neighbors. We provide a general framework where such long range effects can be taken into consideration for arbitrarily varying (as a function of the node distance) linear couplings between arbitrary sets of neighbors in the chain. By examining special case examples such as three-site breathers with next-nearest-neighbors, we find crucial modifications to the nearest-neighbor picture of one-dimensional oscillators being excited either in- or anti-phase. Configurations with nontrivial phase profiles emerge from or collide with the ones with standard (0 or π) phase difference profiles, through supercritical or subcritical bifurcations respectively. Similar bifurcations emerge when examining four-site breathers with either nextnearest-neighbor or even interactions with the three-nearest one-dimensional neighbors. The latter setting can be thought of as a prototype for the two-dimensional building block, namely a square of lattice nodes, which is also examined. 1. Introduction The initial numerical inception of anharmonic modes consisting of a few excited sites in nonlinear lattices [1, 2], and their subsequent placement on a rigorous existence basis (under rather generically satisfied non-resonance conditions) in [3] has triggered a huge growth of interest in the theme of the so-called discrete breathers. These are exponentially localized in space, periodic in time states which have subsequently been theoretically/numerically predicted and experimentally verified to arise in a very diverse host of applications. These include (but are not limited to) DNA double-strand dynamics in biophysics [4], coupled waveguide arrays and photorefractive crystals in nonlinear optics [5, 6, 7], breathing oscillations in micromechanical cantilever arrays [8], Bose-Einstein condensates in optical lattices in atomic physics [9], and granular crystals [10]. The interest in this theme has also been mirrored in a wide array of reviews on methods of identifying and analyzing such intrinsically localized modes [11, 12, 13, 14]. More recently, the stability of the discrete breather configurations, especially in the case of the excitation of multiple sites has been of particular interest. One approach to obtaining relevant results consists of the so-called Aubry band theory [14], used e.g. in [15, 16]. This led to the conclusion that for soft nonlinear potentials multi-breathers with any subset of adjacent sites being excited in-phase are unstable, while ones with all adjacent sites in anti-phase can be stable in the vicinity of the so-called anti-continuum limit of uncoupled anharmonic oscillators. A complementary theory that yields insights on both the existence and the stability of multibreathers has been pioneered by MacKay and collaborators; see e.g., [17, 18, 19]. This is the so-called effective Hamiltonian method which is identified by averaging over the period of the ∗Corresponding author Email address: [email protected] (V. Koukouloyannis) Preprint submitted to Elsevier September 10, 2012 unperturbed solution and developing the proper action-angle variables. The extrema of the resulting effective Hamiltonian determine the relative phases of adjacent excited sites in the multi-site breather solution, while the relevant Hessian is intimately connected to the Floquet multipliers of the associated periodic orbit. Using this methodology, the work of [20] retrieved as well as refined the results of [15] for arbitrary phase relations between the excited oscillators of such multi-breather configurations. The equivalence between these two basic methods and their conclusions was recently established in [21]. We should also note in passing that similar results have been acquired also in configurations where there are “holes” between the excited breather sites [22], through higher order perturbation theory generalizing the above conclusions to the cases with one-site holes. On the other hand, the existence and stability of single/multi-site breathers have been studied in diatomic FPU lattices. The work of [23] was based on a discrete Sturm theorem which necessitated (for the separation of the space nand time tvariables) a potential which was at least purely quartic. In the realm of lattices with longer than the nearest-neighbor interactions a variety of issues have been considered such as, e.g. in [24], the existence and bifurcation of quasi periodic traveling waves in nonlocal lattices with polynomial type potentials. In the present work, we consider the generalization of the above settings, which are principally concerned with the interaction between nearest neighbors, to the case with longer range neighbor interactions for Klein- Gordon chains. Upon revisiting the nearest neighbor case and presenting the effective Hamiltonian formalism (of MacKay and collaborators) there (section II) for existence and stability of multibreathers, in section III, we generalize this formalism to the case of an arbitrary number of neighbors (denoted by r) interacting with each other. By specializing to the case of nearest and next-nearest neighbor interactions (and three-site breathers) as our first case example of the application of the results in section IV, we already infer the fundamental modifications to the standard picture that ensue due to interactions beyond nearest neighbors. These include configurations that have non-standard relative phases between adjacent oscillators, a feature which is absent in the nearest-neighbor interaction case [25] and also symmetry breaking bifurcations that arise due to the “collision” of branches of solutions with such non-trivial phase relations, with more standard ones with relative phases of 0 or πbetween adjacent oscillators. The generic nature of these conclusions is confirmed by considering the case examples of four-site breathers with next-nearest-neighbor interactions in section V and such breathers with interaction ranges of r= 3 in section VI. The latter setting is very close to genuinely two-dimensional setting in a square lattice plaquette, which constitutes our final example in section VII. We close our presentation by some remarks on the parallels of our results with the simpler case of the discrete nonlinear Schr¨odinger lattices [26] (section VIII), which has been examined earlier in [27, 28], as well as a summary of our conclusions and some future directions (section IX). 2. Background: The Classical Klein-Gordon chain εεε εε εε Figure 1: The classical nearest-neighbor Klein-Gordon chain The Hamiltonian of a Klein-Gordon chain with nearest-neighbor (NN) interactions is the following H=H0+εH1= ∞ X i=−∞ 1 2p2 i+V(xi)+ε 2 ∞ X i=−∞ (xi−xi−1)2,(1) which leads to the equations of motion ¨xi=−V′(xi) + ε(xi−1−2xi+xi+1). It is well known that this system supports discrete breather, as well as, multibreather solutions. As indicated above, there are several papers dealing with the existence and stability of these motions; see e.g. [3, 29, 30, 31, 17, 15, 20, 22]. 2 2.1. Persistence of mutibreathers The derivation of the persistence conditions will be based on the notion of the anticontinuum limit and the results of [17]. In the anti-continuum limit ε= 0, we consider all the oscillators of the chain at rest except for n+ 1 “central” ones which move in periodic orbits of frequency ω, but arbitrary phases. This trivially space localized and time-periodic motion is denoted by z0(t) and let S={0,1,...,n}the set of indices corresponding to the central oscillators. To these oscillators, we perform the action-angle (x, p)7→ (w, J) canonical transformation. The Hamiltonian then becomes H=H(wi, Ji, xj, yj) with i∈Sand j∈Z\S. After that, we perform a second canonical transformation ϑ=w0A= n X j=0 Jj φi=wi+1 −wiIi= n X j=i Jji= 1 . . . n where φidenote the nphase differences between the n+ 1 successive oscillators and Iiare the conjugate generalized momenta. Using these variables, the Hamiltonian becomes H=H(φj, Ij, ϑ, A, xj, pj). We define then Heff(φi, Ii, A) = IH◦z(t)dt, where z(t) is periodic orbit obtained by a continuation procedure using constant symplectic “area” A. Since in Heff the variable ϑis ignorable by construction, the conjugate variable Awill be a constant of motion. We expand all the variables involving in power series of εand keep the leading order terms. In this level of approximation, z(t) can be taken equal to z0(t). So, Heff can be written as Heff =H0(Ii) + εhH1i(φi, Ii) (2) where we have omitted constant and higher order terms. The average value of the coupling part of the Hamiltonian is hH1i(φi, Ii) = 1 TIH1(ϑ, φi, Ii)dt where all the calculations have been made along the unperturbed periodic orbit z0. In [17] it was proven that the critical points of the dynamical system associated with Heff are in one-to- one correspondence with the periodic orbits of the original H-system which will be continued for εnonzero but small enough to provide multibreathers. So, by using the form of Heff of (2), we obtain the persistence conditions for the existence of n+ 1-site multibreathers as ∂hH1i ∂φi = 0, i = 1 . . . n, (3) Note that the persistence conditions are the same for every lattice case where the Hamiltonian can be written in the for H=H0+εH1with ∂hH1i ∂φi6≡ 0. In order to obtain specific conditions about the configurations which persist for ε6= 0, to provide multibreathers, we need to calculate hH1i. The motion of the central oscillators for ε= 0 can be described by xi(wi) = ∞ X m=0 Am(Ji) cos(mwi).(4) Since the action Jiremains constant along an orbit in the anticontinuum limit, xidepends only on wi. So, the average value of H1becomes ([20] appendix A) hH1i=−1 2 ∞ X m=1 n X s=1 A2 mcos(mφs) 3 and the persistence conditions (3) become in the case of Klein-Gordon chains with nearest neighbor interactions, ∂hH1i ∂φi = 0 ⇒M(φ)≡ ∞ X m=1 mA2 msin(mφi) = 0, i = 1 . . . n. (5) The function M(φ) possesses the obvious roots φi= 0, π, while it has no others, as it is shown in [25]1. Additionally, the resulting breather states are exponentially localized in space as it can be shown in the framework of [33]. 2.2. Stability of multibreathers The spectral stability of the above mentioned multibreather solutions or, equivalently, the linear stability of the corresponding periodic orbits is determined through its characteristics exponents σi. These exponents are connected with the corresponding Floquet multipliers by the relation λi=eσiT, where T= 2π/ω is the period of the multibreather. Due to the Hamiltonian character of the system there is a pair of exponents identically equal to zero. The non-zero characteristic exponents of the central oscillators correspond to the eigenvalues of the (2n×2n)stability matrix [17] E=ΩD2Heff, where Ω is the matrix of the symplectic form Ω=O−I I O and Iis the n×nidentity matrix. The effective Hamiltonian Heff , as it has already been mentioned, in first order of approximation is given by Heff =H0+εhH1i. So, the stability matrix E, to leading order of approximation and by taking into consideration the form of Heff, becomes E=A B C D =εA1εB1 C0+εC1εD1=     −ε∂2hH1i ∂φi∂Ij−ε∂2hH1i ∂φi∂φj ∂2H0 ∂IiIj +ε∂2hH1i ∂Ii∂Ij ε∂2hH1i ∂φj∂Ii      .(6) Since the only possible solutions are the ones with φi= 0, π and we consider central sites oscillating with the same frequency ω, we get that A1=D1=0and so, the nonzero characteristic exponents are given to leading order of approximation by σ±i=±√ε χ1i+O(ε3/2)i= 1 . . . n, (7) where χ1iare the eigenvalues of the matrix B1·C0. Due to the form of the J7→ Itransformation the C0 matrix becomes (see [20] appendix B) C0=∂2H0 ∂Ii∂Ij =−∂ω ∂J ·L=−∂ω ∂J ·        2−1 0 −1 2 −1 0 ......... 0−1 2 −1 0−1 2        . So, (7) becomes, up to leading order terms, σ±i=±r−ε∂ω ∂J χzii= 1 . . . n, (8) 1One could present the case of phonobreathers (see e.g. [32]) as a counterexample of this statement. But, these motions are substantially different from the multibreathers we study in this work, in the sense that, in the anticontinuoum limit, in the case of phonobreathers all the sites of the lattice are excited, while, in our case there is a specific number of n+ 1 central oscillators. 4 where χziare the eigenvalues of Z=B1·L. For systems of the form (1) we get B1=∂2hH1i ∂φi∂φj =fifor i=j 0 for i6=j, with fi=f(φi) = 1 2 ∞ X m=1 m2A2 mcos(mφi).(9) So, Zcan be written as Z=B1·L=∂2hH1i ∂φi∂φj·        2−1 0 −1 2 −1 0 ......... 0−1 2 −1 0−1 2        =        2f1−f10 −f22f2−f20 ......... 0−fn−12fn−1−fn−1 0−fn2fn        (10) Note that, for linear stability we require all the Floquet multipliers to lie on the unit circle, which is tantamount to all the characteristic exponents being purely imaginary. This depends on the sign of P=ε∂ω ∂J and the sign of χzas it can be seen from (8). Finally, by using some counting theorems [20] for (10), we obtain: Theorem 1. [20] In systems of the form (1), if P≡ε∂ω ∂J <0the only configuration which leads to linearly stable multibreathers, for |ε|small enough, is the one with φi=π∀i= 1 . . . n (anti-phase multibreather), while if P > 0the only linearly stable configuration, for |ε|small enough, is the one with φi= 0 ∀i= 1 . . . n (in-phase multibreather). Moreover, for P < 0(respectively, P > 0), for unstable configurations, their number of unstable eigenvalues will be precisely equal to the number of nearest neighbors which are in- (respectively, in anti-) phase between them. Remark 1: Note that, the form of the matrix C0is such due to the form of the J7→ Itransformation and the fact that in the anti-continuum limit ∂H0 ∂J =ωand ωi=ωfor i= 1 . . . n. So, it is independent of the range of the interaction between the oscillators of the chain and it will remain the same in what follows. On the other hand, the diagonal form of B1will change if longer range interactions are added to the system. So, the theorem will no longer hold but the general methodology will still apply and the characteristic exponents of the multibreather will be given by (8). We will consider this case in what follows. Remark 2: In our previous works we used the term “out-of-phase” for φ=πconfigurations. This was because the only out-of-phase configuration was the φ=πone. In the present work, since, as we will see in the next section, there are out-of phase configurations with φ6=π, we use the term “anti-phase” for the φ=πconfiguration. Remark 3: As indicated in [22], it is possible to generalize the stability considerations to the case where not all of these oscillators are adjacent to each other, however, we will not concern ourselves with this additional complication herein. Remark 4: In the present paper we consider only the linear stability of the multibreathers. One may naturally be concerned about the possibility of a stronger form of a stability result. In that light, it could be expected to acquire exponential stability of these solutions like it is proven in [34] for single-site breathers in Klein-Gordon chains with interactions which decrease algebraically with respect to the the lattice site distances. In addition, in [35] in a different context (in a chain of coupled symplectic maps) it is shown numerically, that there exist areas in the phase space around the periodic orbits of the breathers where, if the motion starts inside it, it will remain there for very long times (in computational terms, practically for ever). These facts imply that one could also acquire exponential stability for this kind of systems. On the other hand, special realizations of 1D chains such as the one considered in [36] can enable the proof of asymptotic stability by taking an appropriate solution limit. 5 3. Klein-Gordon chain with long range interactions The picture radically changes when the chain involves interactions with range longer than mere nearest neighbors. The range parameter rwill be used to indicate the interaction length between the oscillators of the chain. So, for the classic nearest neighbor chain the range is r= 1 as shown in fig. 1 while for the next nearest neighbor (NNN) chain the range is r= 2 as illustrated in fig. 2 etc. The coupling force between the oscillators of the chain is linear and the coupling constants εi, i = 1 . . . r are not, in general, equal. The Hamiltonian of a 1D KG chain with long range interactions is: H= ∞ X i=−∞ [p2 i 2+V(xi)] + 1 2 ∞ X i=−∞ r X j=1 εj(xi−xi+j)2(11) which leads to the equations of motion ¨xi=−V′(xi) + r X j=1 εj(xi−j−2xi+xi+j) 3.1. Persistence of multibreathers Let εj=kjε, with k1= 1, then the Hamiltonian (11) becomes H=H0+εH1= ∞ X i=−∞ [p2 i 2+V(xi)] + ε 2 ∞ X i=−∞ r X j=1 kj(xi−xi+j)2(12) Now, since the Hamiltonian is written in the form H=H0+εH1the persistence conditions (3) can be used. If we consider again n+ 1 “central” oscillators and xi=P∞ m=0 Amcos(mwi), we get for this case hH1i=−1 2 ∞ X m=1 r X j=1 n−j+1 X s=1 A2 mkjcos(m j−1 X l=0 φs+l).(13) Note that, in the above we considered r⩽nsince any interaction of oscillators with r > n does not affect the calculations, which are performed in the anti-continuum limit. So, if one considers r > n then for the calculations in this section it would be equivalent to the choice of r=n. By differentiating Eq. 13 with respect to φiwe get ∂hH1i ∂φi = 0 ⇒ ∞ X m=1 r X p=1 z2 X s=z1 mA2 mkpsin(m p−1 X l=0 φs+l) = 0,(14) or, by taking into consideration the definition of (5), r X p=1 z2 X s=z1 kpM( p−1 X l=0 φs+l) = 0,(15) where z1= max(1, i −p+ 1) and z2=ifor i+p−1⩽n n−p+ 1 for i+p−1> n . Eqs. 13 and 14 (or 15) may be seem cumbersome to handle, they are much easier to use in the concrete examples that will follow in the next sections. Remark: Note that, in order for the system to “see” the long range interaction, and reveal all the relevant phenomena, one has to consider n+ 1-site breathers, with n⩾r. If the case of n < r is considered, only the phenomena which correspond to r=nwill appear. 6 3.2. Stability of multibreathers Since H=H0+εH1the methodology described in section 2 can be used in order to determine the linear stability of multibreathers in this case as well. As we have already mentioned the characteristic exponents of the multibreather provided by the persistence conditions (14) are given, to leading order of approximation, by (8), i.e. σ±i=±r−ε∂ω ∂J χzi, i = 1 . . . n, where χziare the eigenvalues of Zwith Z=B1·L=∂2hH1i ∂φi∂φj·        2−1 0 −1 2 −1 0 ......... 0−1 2 −1 0−1 2        , i, j = 1 . . . n. (16) For linear stability we need all the characteristic exponents to be purely imaginary. So, if P=ε∂ω ∂J <0 we need all the eigenvalues of Zto be negative, while if P=ε∂ω ∂J >0 we need all the eigenvalues of Zto be positive. As it has been already mentioned the form of the matrix Lremains the same as in (10) but the form of B1varies for different values of the range r. Without loss of generality we can consider i⩽j, since the B1matrix is symmetric. Let d=j−i+ 1, then the general form of ∂2hH1i ∂φi∂φjis ∂2hH1i ∂φi∂φj =       0 if d > r 1 2 ∞ X m=1 r X p=d z2 X s=z1 m2A2 mkpcos(m p−1 X l=0 φs+l) if d⩽r,(17) or, by taking under consideration the definition of (9), ∂2hH1i ∂φi∂φj =       0 if d > r r X p=d z2 X s=z1 kpf( p−1 X l=0 φs+l) if d⩽r.(18) Now z1is given by z1= max(1, i−p+d), while z2is still given by z2=ifor i+p−1⩽n n−p+ 1 for i+p−1> n . We cannot formulate a general theorem like Theorem 1 in the case on lattices with r > 1 since the sign of the eigenvalues of Zdepends on the number of the roots the persistence conditions have. As long as there exist only standard multibreathers the situation is described by Theorem 1. But, when phase-shift breathers emerge the situation changes and we cannot a priori determine the sign of χz’s. In that light a theorem analogous to Theorem 1 appears unlikely in a general setting. In order to demonstrate the use of the results of this section, in what follows, we will examine some particular cases. 4. 3-site breathers with r= 2 4.1. The ε1=ε2=εcase 4.1.1. Persistence of multibreathers The simplest case to check the effect of long range interactions is the one of 3 central oscilators (i.e. n= 2) and range r= 2. As we have already mentioned, any range r > 2 would not affect our calculations. First we will check the case k1=k2= 1 ⇒ε1=ε2=ε. 7 ε εε εε εε 1111εε1 εε2εε2 εε2 εε2 Figure 2: The r= 2 Klein-Gordon chain with next nearest neighbor interactions In this case the Hamiltonian (12) reads H=H0+εH1= ∞ X i=−∞ 1 2p2 i+V(xi) + ε 2 ∞ X i=−∞ (xi−xi+1)2+ (xi−xi+2)2. Since we consider a 3-site breather, H1becomes at the anti-continuum limit H1=x2 1+x2 2+x2 3+ (x1−x2)2+ (x1−x3)2+ (x2−x3)2 and hH1i=−1 2 ∞ X m=1 A2 m{cos(mφ1) + cos(mφ2) + cos[m(φ1+φ2)]}, according also to (13), for n= 2 and r= 2. The persistence conditions (14) become ∂hH1i ∂φi = 0 ⇒ ∞ X m=1 mA2 m{sin(mφi) + sin[m(φ1+φ2)]}= 0,for i= 1,2 (19) or, by taking into consideration the definition of M(φ) in (5), M(φi) + M(φ1+φ2) = 0,for i= 1,2.(20) This equation, in addition to the standard solutions φi= 0, π, provides also the solutions φ1=φ2= 2π/3, φ1=φ2= 4π/3. The multibreather solutions with φi6= 0 are called phase-shift multibreathers or phase-shift breathers. The anti-phase and phase-shift configurations are depicted in figs. 3 and 4. For a better visualization one can also refer to videos 1 and 2 [supplementary files 1 and 2 here] (video 3 shows an in-phase configuration [supplementary file 3 here]). In order to produce these figures (and videos) we used the on-site potential V(x) = x2 2−0.15x3 3−0.05x4 4 and initial conditions which correspond to motion with period T= 7 and frequency ω= 2π/T = 2π/7≃ 0.8976. The same potential is used for every numerical calculation throughout this work, although it is straightforward to apply the relevant notions to arbitrary potentials of the Klein-Gordon type. Remark 1: The persistence conditions (20) provide 2 equations. M(φ1) + M(φ1+φ2) = 0 M(φ2) + M(φ1+φ2) = 0.(21) By substraction of equations (21) we get M(φ1) = M(φ2) (22) 8 -20 -10 0 10 20 -2 -1 0 1 2 -20 -10 0 10 20 -2 -1 0 1 2 Figure 3: [Color online] Two snapshots of a 3-site (n= 2), anti-phase (φ1=φ2=π) multibreather in a range r= 2 Klein-Gordon chain with ε1=ε2= 0.02 and frequency ω= 2π/7. See also video 1 [supplementary file 1]. -20 -10 0 10 20 -2 -1 0 1 2 -20 -10 0 10 20 -2 -1 0 1 2 -20 -10 0 10 20 -2 -1 0 1 2 Figure 4: [Color online] Three snapshots of a 3-site (n= 2), phase-shift (φ1=φ26= 0, π) multibreather in a range r= 2 Klein-Gordon chain with ε1=ε2= 0.02 and frequency ω= 2π/7. See also video 2 [supplementary file 2 here]. which has, besides the trivial solutions φi= 0, π, two other obvious solutions: φ1=φ2and φ1+φ2=π for 0 ⩽φi⩽2π. The last solution does not provide any new information because by substituting this into equations (21) we get M(φ1) = M(φ2) = 0, which , as it is shown in [25], only possesses the φi= 0, π solutions. But the φ1=φ2=φsolution can reduce the two equations (21) into equation (23) M(φ) + M(2φ) = 0.(23) Remark 2: Our numerical computations strongly suggest that for all the phase-shift solutions it is φ1=φ2, yet a rigorous proof of this fact is still an open problem. So, equation (23) can be used in order to calculate all the solutions of the persistence conditions (20), except for the mixed one {φ1= 0, φ2=π}(or equivalently {φ1=π, φ2= 0}). Remark 3: In the case under consideration (n= 2, r= 2, ki= 1), all the available solutions correspond to φi’s which make each of the terms of the sum vanish in (19) which obviously provides a zero total. Remark 4: The case under consideration is equivalent to the 3-site breathers on a hexagonal lattice which has already been studied in [31, 37]. It can be effectively considered as a one-dimensional realization of such a lattice. In that context, the phase-shift multibreathers can be alternatively thought as “discrete vortices”, as they are solutions which complete a phase rotation by 2π, as one traverses a discrete contour (which consists of the relevant triangle of sites). Remark 5: As an aside, it should be mentioned that an additional motivation for the consideration of such next-nearest neighbor interactions stems from the consideration of zigzag arrays, similar to the waveguide arrays proposed theoretically in the context of nonlinear optics (and hence in the realm of the DNLS equation) in [38]. Remark 6: The stability of the above mentioned breathers will be discussed at the end of the next section as a special case of the more general unequal coupling one. 9 (a) (b) (c) (d) Figure 10: [Color online] The various families that appear in Fig. 9 are depicted. Portraits of the configuration of the central oscillators for the different multibreather families are shown as insets in the corresponding diagrams. The line (or curve) type in the figures depends on the number of positive χzthat the corresponding family possesses: no positive χzcorresponds to a solid line, 1 positive χzcorresponds to a dashed line, 2 to a dash-dotted and 3 to a dotted line. In (a) two multibreather families are depicted. The first is the in-phase one with {φ1=φ2=φ3= 0 (or 2π)}which possesses 3 positive χz. The second one is the anti-phase one with {φ1=φ2=φ3=π}. It has no positive χzuntil k < k(1) cr = 0.3219 while it has 1 positive χzfor k > k(1) cr . At this point the anti-phase family gives rise, through a supercritical pitchfork, to the the phase-shift {φ1=φ26= 0, π}family. In (b) the mixed 1 configuration is shown {φ1=φ2= 0 or 2π, φ3=π}or {φ1=φ2=π, φ3= 0 or 2π}. These families possess 2 positive and 1 negative χzeach. In (c) the mixed 2 family {φ1=φ3=π, φ2= 0 or 2π}is shown. It possesses 1 positive χz for k < k(2) cr = 1.0736 and no positive χzfor k > k(2) cr . At this point the mixed 2 family collides with the phase-shift one. In (d) the phase-shift {φ1=φ26= 0, π}family is shown and it is represented by φ1=φ3= (3) and φ2= (2) or φ1=φ3= (4) and φ2= (5). It exists for k(1) cr < k < k(2) cr . At k=k(1) cr it bifurcates from the anti-phase family, while at k=k(2) cr it emerges with the mixed 2 family, and possesses no positive χz. Fig. 9 contains all of the above families together, where, when two line segments coincide the more dense is shown. Since for k=k(2) cr it is χz2= 0 we get k(2) cr =−f0 2fπ +f0≃1 or, for the specific potential and frequency used in this work k(2) cr = 1.0736. Remarks on the stability diagram of Figs. 9 and 10: In Figs. 9 and 10 all the multibreather families they exist in the present configuration (n= 3, r= 2) are shown. The kind of the line (or curve) used for every segment depends on the number of positive χzas follows: no positive χz→solid, 1 positive χz→ dashed, 2 positive χz→dashed-dotted, 3 positive χz→dotted. All the families shown in Fig.10 are depicted together in Fig.9. Since, if two segments coincide, the more dense is shown, in Fig.9 we can see only solid and dashed segments. The families which are depicted in these two figures are the following: • {φ1=φ2=φ3= 0 (or 2π)}(in-phase). This family is shown in Fig.10(a) and has 3 positive χz. • {φ1=φ2=φ3=π}(anti-phase). It is shown in Fig.10(a) and has no positive χzfor k < k(1) cr = 0.3219 while it possesses one positive χzfor k > k(1) cr . At this point the anti-phase family bifurcates to provide 16 the phase-shift breather family. • {φ1=φ2= 0, φ3=π}or {φ1=φ2=π, φ3= 0}(mixed 1). These families are depicted in Fig.10(b) and they both possess 2 positive χz • {φ1=φ3=π, φ2= 0 (or 2π)}(mixed 2). It is shown in Fig.10(c). For k < k(2) cr it has 1 positive χz, while for k > k(2) cr it has no positive χz. At k=k(2) cr = 1.0736, this family collides with the phase-shift family. •φ1=φ3, φ26= 0, π (phase shift). This family is represented in Fig.10(d) by φ1=φ3=(3) and φ2=(2) (or φ1=φ3=(4) and φ2=(5)) and has no positive χz. It begins to exist at k=k(1) cr where it bifurcates from the anti-phase family and cease to exist at k=k(2) cr where it collides with the mixed 1 family. Since the stability of the multibreathers is also determined by the sign of P≡ε∂ω ∂J the above are summarized, in terms of the linear stability of the corresponding configurations, in Table 2. P k In-phase φ1=φ2=φ3= 0 Out-of-phase φ1=φ2=φ3=π Phase-shift φ1=φ3, φ26= 0, π Mixed φ1=φ3=π, φ2= 0 Linear Stability P < 0k < k(1) cr unstable stable – unstable P < 0k(1) cr < k < k(2) cr unstable unstable stable unstable P < 0k > k(2) cr unstable unstable – stable P > 0k < k(1) cr stable unstable – unstable P > 0k(1) cr < k < k(2) cr stable unstable unstable unstable P > 0k > k(2) cr stable unstable – unstable Table 2: [Color online] Stability of the various 4-site (n= 3) and range r= 2, breather configurations depending on the values of P≡∂ω ∂J and k. With the dash we denote that this particular family does not exist for this range of values of k. 6. 4-site breathers with r= 3 The natural way to extend our study in 4-site breathers is to consider range of interaction r= 3 (i.e., involving interactions with the 3 closest neighbors on each side of the chain), in order for all the central oscillators to interact with each other. 6.1. Persistence of multibreathers Bearing in mind that ǫi=ǫkiand that k1= 1, hH1iin this case becomes hH1i=−1 2 ∞ X m=1 A2 m{cos(mφ1) + cos(mφ2) + cos(mφ3)+ +k2{cos[m(φ1+φ2)] + cos[m(φ2+φ3)]}+k3cos[m(φ1+φ2+φ3)]}, while the corresponding persistence conditions become M(φ1) + k2M(φ1+φ2) + k3M(φ1+φ2+φ3) = 0 M(φ2) + k2[M(φ1+φ2) + M(φ2+φ3)] + k3M(φ1+φ2+φ3) = 0 M(φ3) + k2M(φ2+φ3) + k3M(φ1+φ2+φ3) = 0. 17 For every ki, there exist the usual φi= 0, π solutions, as well as others as it can be seen in fig. 11. By keeping k3constant, we get various mono-parametric bifurcation diagrams with k2as the parameter. Again, for all the phase shift configurations it is φ1=φ3. In fig. 11, the bifurcation diagrams for two values of k3 are depicted, k3= 0.2 and k3= 0.4. We see that the value of k2(1) cr where the supercritical bifurcation occurs depends strongly on the value of k3, while the value of k2(2) cr remains almost constant at k2(2) cr ≃1.075. The dependence of k2cr with respect to k3is shown in fig. 12. Note that for k3→0 this case coincides with the r= 2 case (i.e., the latter is a special case example) and we retrieve the diagram of Fig. 9. 6.2. Stability As it has already mentioned the stability of the multibreathers is determined by the sign of the eigenvalues χzi, of matrix Z(16). By (17) we get ∂2hH1i ∂φi∂φj =   f1+k2f1+2 +k3f1+2+3 k2f1+2 +k3f1+2+3 k3f1+2+3 k2f1+2 +k3f1+2+3 f2+k2(f1+2 +f2+3) + k3f1+2+3 k2f2+3 +k3f1+2+3 k3f1+2+3 k2f2+3 +k3f1+2+3 f3+k2f2+3 +k3f1+2+3   , and since L=   2−1 0 −1 2 −1 0−1 2    we finally get Z=∂2hH1i ∂φi∂φj·L=   2f1+k2f1+2 +k3f1+2+3 k2f1+2 −f1k3f1+2+3 −k2f1+2 k2f1+2 +k3f1+2+3 −f2−k2f2+3 2f2+k2(f1+2 +f2+3)k2f2+3 +k3f1+2+3 −f2−k2f1+2 k3f1+2+3 −k2f2+3 k2f2+3 −f32f3+k2f2+3 +k3f1+2+3   . Its eigenvalues are, for φ1=φ3and non-specific values of φi, χz1= 2(f1+k2f1+2) χz2,3=f1+f2+k2f1+2 +k3f2φ1+φ2±qf2 1+f2 2−2k2f1f1+2 +k2 2f2 1+2 −2k3f2f2φ1+φ2+k2 3f2 2φ1+φ2. Although such analytical formulas exist and accurately predict the stability and bifurcations of the system, a clearer understanding emerges from the observation of the associated bifurcation diagrams (fig. 11). The diagrams present exactly the same solution families as in Fig. 9, but in this case the value of k2(1) cr , where the supercritical bifurcation occurs, is strongly affected by the value of k3, while, the value of k2(2) cr , where the subcritical bifurcation occur,s remains almost constant with k2(2) cr ≃1.075 (Fig. 12). This indicates that, for the range of values of k3considered in this figure, the parametric interval of k2(the strength of next-nearest neighbor interactions) over which phase-shift solutions exist narrows as k3(the strength of interaction with the third-nearest-neighbors) is increased. 7. 4-site breathers in a 2D square lattice with r= 2 We now turn our considerations to the case of a square lattice, as the one in Fig. 13, with nearestneighbor interactions, not only with the horizontal and vertical neighbors, but with the diagonal as well. The latter interaction is assumed to have a strength ǫ2=kǫ1(where ǫ1≡ǫwill be taken to denote the coupling strength of adjacent nodes along the lattice axes). The Hamiltonian for this system is 18 Figure 11: [Color online] In these panels the various 4-site (n= 3) multibreathers families in a chain with interaction range r= 3 are shown. In the diagram we have considered k2variable, while k3= 0.2 in the left panel and k3= 0.4 in the right panel. The families depicted are qualitatively the same as the ones in the n= 3, r= 2 case. The only difference is that in the present case the value of k2(1) cr where the supercritical bifurcation occurs depends strongly on k3while the value of k2(2) cr where the subcritical bifurcation occurs remains almost fixed. Figure 12: [Color online] The values of the two k2cr, where the bifurcations occur, with respect to k3. Although k2(1) cr depends strongly on k3,k2(2) cr remains almost constant at k2(2) cr ≃1.075. 19 i=-2 i=-1 i=0 i=1 i=2 i=3 j=-2 j=-1 j=0 j=1 j=2 j=3 0 1 23 Figure 13: [Color online] In the square lattice under consideration each oscillator is coupled with its neighbour not only in the horizontal and vertical directions but in the diagonal directions as well. So, every lattice site interacts with its 8 neighbours instead of the 4 of the classical KG square lattice configuration. In this setting, we consider 4-site breathers. Let the encircled oscillators in the figure be the central oscillators which are denoted as 0, 1, 2, 3 respectively H= ∞ X i=−∞ ∞ X j=−∞ 1 2p2 ij +V(xij )+ε1 2 ∞ X i=−∞ ∞ X j=−∞ (xij −xi−1,j )2+ (xij −xi,j−1)2 +ε2 2 ∞ X i=−∞ ∞ X j=−∞ (xij −xi−1,j−1)2+ (xij −xi−1,j+1)2 (27) or H=H0+εH1= ∞ X i=−∞ ∞ X j=−∞ 1 2p2 ij +V(xij )+ +ε 2 ∞ X i=−∞ ∞ X j=−∞ (xij −xi−1,j )2+ (xij −xi,j−1)2+k(xij −xi−1,j−1)2+ (xij −xi−1,j+1)2 (28) We consider 4 “central” oscillators in the anti-continuum limit and we denote them by 0, 1,2 ,3 as it is shown in fig.13. We have then φ1=w1−w0,φ2=w2−w1,φ3=w3−w2and φ4=w0−w3. But since, by construction, we have φ4= 2π−φ1−φ2−φ3, we have finally only 3 independent φi’s. The hH1iin this case is hH1i=−1 2 ∞ X m=1 A2 m{cos(mφ1) + cos(mφ2) + cos(mφ3) + cos[m(φ1+φ2+φ3)] +k{cos[m(φ1+φ2)] + cos[m(φ2+φ3)]}} and the corresponding persistence conditions are M(φ1) + M(φ1+φ2+φ3) + k M(φ1+φ2) = 0 M(φ2) + M(φ1+φ2+φ3) + k[M(φ1+φ2) + M(φ2+φ3)] = 0 M(φ3) + M(φ1+φ2+φ3)] + k M(φ2+φ3) = 0 This case coincides with the 1D 4-site r= 3 chain case, with k3=k1= 1 and k2=k. Hence, the results 20 (both for the persistence and the stability of the solutions) are a special case of the previous section. All the existing multibreather families of this configuration are depicted in fig. 14. In this diagram we can observe the appearance of {φ1=φ2=φ3=π/2}family, which is the vortex solution of the classical square Klein-Gordon lattice; see e.g., the relevant discussion in [21]. In addition, there are several phase-shift families, the stability of which will be analyzed below. Interestingly, all these phase-shift breathers cease to exist at a critical value of k=k(2) cr = 1.03549 except of the vortex one. Figure 14: [Color online] In this diagram, the various multibreather families of the square lattice configuration, are depicted. The value of kdetermines the strength of the coupling in the diagonal direction. For a more detailed description of the particular families one can also refer to Fig. 15. The mutibreather families that are supported by the present configuration are described in what follows. • {φ1=φ2=φ3= 0}(in-phase) It is shown in Fig.15(a). It possesses 3 positive χzindependently of the value of k2. • {φ1=φ2=φ3=π/2}(vortex). It is shown in Fig.15(a). It has no positive χzindependently of the value of k2and does not interact (i.e., collide) with any other family of solutions. • {φ1=φ2=φ3=π}(anti-phase). It is shown in Fig.15(a). It has no positive χzfor k < k(1) cr = 0.96572, while for k > k(1) cr it acquires 2 positive χz. • {φ1=φ3=π,φ2=φ}(phase-shift 1). It is represented by in Fig.15(b) φ1=φ3= (1) and φ2= (2) or φ1=φ3= (1) and φ2= (3). It exists for k(1) cr < k2< k(3) cr . It bifurcates from the anti-phase family at k=k(1) cr and possesses 3 negative χzfor k(1) cr < k < k(2) cr = 1.0344. At k2=k(2) cr it collides with the phase-shift 2 family in a bubcritical pitchfork bifurcation. So, for k(2) cr < k < k(3) cr it possesses 2 negative and a positive χz. At k=k(3) cr it collides with the mixed family. • {φ1=φ3=π,φ2= 0}(mixed). It is shown in Fig.15(c). It has 2 positive χzfor k < k(3) cr = 1.0355 and 1 positive χzfor k2> k(3) cr . •(phase-shift 2a) It is represented in Fig.15(d) by φ1=φ3= (4) and φ2=(8) which collides with the φ1=φ3= (5) and φ2=(6) for k2=k(2) cr and have 1 positive χz. 21 •(phase-shift 2b) It is represented in Fig.15(d) by φ1=φ3= (4) and φ2=(9) which collides with the φ1=φ3= (5) and φ2=(7) for k2=k(2) cr and have 1 positive χz. Notes on Figs. 14 and 15: 1. At k=k(2) cr the φ2= (3) branch of the phase-shift 1 family collides with the phase-shift 2a family, while the the φ2= (2) branch of the phase-shift 1 family collides with the symmetric (and equivalent) of branches φ2= (7) and φ2= (9) of the phase-shift 2b family. 2. At k= 0 the phase-shift 2 family approaches very much the vortex family, so it is plausible to expect that at that point they coincide. Yet, there is a small difference in the values of φiof the two families due to the nonlinear character of the oscillators constituting the lattice, i.e., due to the existence of an infinity of terms in the development (4). For a smaller frequency of oscillation, the nonlinear character of the oscillation becomes stronger, hence the terms after the first in (4) become larger and the two families are more clearly separated. 8. Discussion - Comparison with the DNLS results It should be noted here that a number of results similar to the ones presented herein have been recently presented in the context of the DNLS equation e.g. in [27, 28]. The setting of the DNLS essentially reflects a special case example of our Klein-Gordon calculation where instead of the existence and stability conditions reflecting a sum over all the harmonics due to the U(1) invariance of the underlying model, only the first harmonic is present. Nevertheless, the latter is sufficient to induce a number of the conclusions that we inferred herein. In particular, next-nearest neighbor interactions create phase-shift multibreathers (which were also parallelized to discrete vortex breathers in hexagonal lattices), as illustrated in [27]. As also shown in the same work, the long range interactions may drastically affect the stability properties of twodimensional discrete vortices (in square lattices). On the other hand, the work of [28] provided a different analytical handle, via variational approximations, on the solutions that arise in settings with long range interactions. Furthermore, it was able to capture phenomena (both analytically and numerically) such as the supercritical or subcritical pitchfork bifurcations for such phase-shift multibreather solutions with NNN interactions. For instance, in the DNLS case the supecritical bifurcation leading to the emergence of such solutions would happen precisely at k= 0.5 (due to the relevance of just the first harmonic) and not at k= 0.48286 as obtained here in section IV B for the Klein-Gordon case. Nevertheless, the basic phenomenology remains intact. It should also be noted here that the DNLS enables a wealth of additional results on the existence of localized modes e.g., based on homoclinic orbits and map type approaches [39], as well as on their asymptotic stability e.g., based on dispersive decay estimates [40]. Nevertheless, there are techniques that are common to both DNLS and Klein-Gordon type lattices, including the ones of continuation from the anti-continuum limit as used here. Another class of such common techniques, but from the “opposite limit”, namely the continuum has been developed for the DNLS, with a recent example being the work of [41]. There, the use of ideas from finite element methods and variational analysis has enabled a proof of convergence to the ground state and a characterization of the localization length. Using, once again, the analogies between the DNLS and Klein-Gordon lattices, the work of [42] enabled the extension of results of existence of DNLS localized modes [43] in the small amplitude, near-continuum limit to the breathers of the Klein-Gordon type of chains. 9. Conclusions Classical Klein-Gordon chain with nearest neighbor interactions support multibreather solutions only with phase differences between successive oscillators of φi= 0, π. There, the stability scenaria are specific and well known. For a KG chain with P=ε∂ω ∂J <0 the anti-phase configuration is the only stable one, while for P=ε∂ω ∂J >0 the in-phase configuration is the only stable multibreather solution. 22 On the other hand, in chains with long range interactions the picture is substantially different. First of all, in such chains, multibreathers with φi6= 0, π (phase-shift multibreathers) can be supported in addition to the standard φi= 0, π ones. The existence of phase-shift multibreathers as well as the specific φi’s of such profiles depend on the various coupling parameters εiwithin the chain. There are critical values of ki=εi/ε1 past which a bifurcation occurs (typically a supercritical pitchfork) and phase shift breathers begin to exist. Past this bifurcation point, the stability properties of the existing multibreathers are significantly modified, although this also depends on the particular (soft or hard) nature of the nonlinearity. As, however, additional parameters are tuned (e.g., higher ranges of neighbor interactions), it is also possible for such phase-shift solutions to terminate in subcritical pitchfork bifurcations. These results are not unique to the realm of one-dimensional lattices with higher range of interactions. They can also be developed for two-dimensional square lattices in which case they may lead to bifurcations or terminations of the families of discrete vortices which arise therein. Such vortices sustained by the twodimensional analogs of the lattice can be of either a symmetric or asymmetric type. In particular, in the case considered herein, the presence of diagonal coupling within the square was critical in inducing the emergence of asymmetric such patterns. This study opens a number of a directions for further investigation. Firstly, it would be very relevant to examine particular functional forms of the decay of the long range interactions (e.g., exponentially or polynomially decaying ones) to identify whether any systematic conclusions can be derived on the basis of such decay laws. As one such example, where the nature of the interactions plays a critical role on the properties of the localized modes, we mention the work of [44] on single-site localized modes of the DNLS with algebraically decaying interactions showing a crossover in their decay properties, and a modification of their existence energy thresholds as a function of the exponent of the algebraic decay of the interactions. Secondly, it would also be very interesting to examine the interplay of the geometry of higher dimensional lattices (and the interactions that they present) with the strength of the long range interactions that can be considered therein and to try to derive some general conclusions about the possible stable/unstable discrete soliton and discrete vortex solutions. 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