Discrete breathers in Φ4 and related models

Discrete breathers in φ4and related models Jes´ us Cuevas–Maraver and Panayotis G. Kevrekidis Abstract In this Chapter, we touch upon the wide topic of discrete breather (DB) formation with a special emphasis on the prototypical system of interest, namely the φ4model. We start by introducing the model and discussing some of the application areas/motivational aspects of exploring time periodic, spatially localized structures, such as the DBs. Our main emphasis is on the existence, and especially on the stability features of such solutions. We explore their spectral stability numerically, as well as in special limits (such as the vicinity of the so-called anti-continuum limit of vanishing coupling) analytically. We also provide and explore a simple, yet powerful stability criterion involving the sign of the derivative of the energy vs. frequency dependence of such solutions. We then turn our attention to nonlinear stability, bringing forth the importance of a topological notion, namely the Krein signature. Furthermore, we briefly touch upon linearly and nonlinearly unstable dynamics of such states. Some special aspects/extensions of such structures are only touched upon, including moving breathers and dissipative variations of the model and some possibilities for future work are highlighted. While this Chapter by no means aspires to be comprehensive, we hope that it provides some recent developments (a large fraction of which is not included in time-honored DB reviews) and associated future possibilities. Jes´ us Cuevas–Maraver Grupo de F´ ısica No Lineal, Universidad de Sevilla, Departamento de F´ ısica Aplicada I, Escuela Polit´ ecnica Superior. C/ Virgen de ´ Africa, 7, 41011-Sevilla, Spain, Jes´ us Cuevas–Maraver Instituto de Matem´ aticas de la Universidad de Sevilla (IMUS). Edificio Celestino Mutis. Avda. Reina Mercedes s/n, 41012-Sevilla, Spain e-mail: jcuev[email protected] Panayotis G. Kevrekidis Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA 01003- 4515, USA e-mail: ke[email protected] 1 arXiv:1901.00545v2 [nlin.PS] 28 May 2019 2 Jes´ us Cuevas–Maraver and Panayotis G. Kevrekidis 1 A brief description of discrete breathers: definition, historical perspective and applications Dynamics of localized excitations is, undoubtedly, one of the most important topics within the realm of nonlinear science. In discrete systems, the localized excitations that can be argued to be most generic [1, 2, 3] are the so-called discrete breathers (DBs). These can be defined as time-periodic spatially-localized coherent structures emerging at coupled nonlinear oscillator lattices. This term was coined by Campbell and Peyrard [4] in order to distinguish them from the “continuous” breathers found as exact solutions through the inverse scattering machinery in the sine-Gordon PDE [5]. [It is worthwhile to mention in passing that continuous breathers have a particularly interesting history associated with them in φ4models [6, 7], which is described in detail in a different Chapter of this special volume.] Such DBs are also known as intrinsic localized modes; this name was introduced in order to emphasize that their origin was in the intrinsic nonlinearity of the system, contrary to the case of Anderson modes, where localization stems from lattice disorder [8] and can exist even in the linear limit. A major development that springboarded the study of DBs took place in 1988, through the numerical study of DBs in some prototypical lattice models as reported in two pioneering works [10, 11] by Takeno, Sievers and Kisoda. In these papers DBs are calculated for the first time in the two basic kinds of lattices where they can emerge, namely Fermi–Pasta–Ulam–Tsingou (FPUT)1[10] and Klein–Gordon (KG) [11]. Additionally, some of their stability properties were also determined. FPUT lattices are characterized by the absence of substrate (on-site) potential and the nonlinearity of the inter-site forces, whereas KG lattices possess nonlinear onsite potential and (typically) linear inter-site forces. Interestingly, in those references, potentials of the φ4form and related models were considered. There are also mixed cases of lattices where nonlinearities are present in both the substrate and intersite potentials (see e.g. [12]); these lattices are sometimes denoted as KG/FPUT lattices. While these early works provided credible numerical evidence and planted the seed for studying DBs, they did not rigorously prove their existence. The latter came in a celebrated 1994 paper by MacKay and Aubry [13] and was based on the concept of the so-called anti-continuous (AC) limit. They basically demonstrate in a rigorous fashion, by making use of the implicit function theorem, that DBs can generically exist in nonlinear KG lattices as a result of unobstructed (when some suitable resonance restrictions are avoided) continuation of periodic orbits of individual (uncoupled) oscillators. Mackay–Aubry’s theorem, together with the rigorous proof of stability by Aubry in 1997 [14] and the numerical methods developed by Mar´ ın among others [15] led to an intense interest on DBs in the late 1990’s and the beginning of the 21st Century, not only from the theoretical but also from the 1These lattices have been traditionally denoted as Fermi–Pasta–Ulam, forgetting the outstanding role of Mary Tsingou who was the person responsible for all the numerical simulations in these first computations of nonlinear lattice dynamics [9] Discrete breathers in φ4and related models 3 experimental point of view. The relevant activity has been very well summarized by now in a series of reviews; see, e.g., [1, 2, 3, 16, 17]. DBs have been sought for in many fields of Physics. They have been experimentally generated in arrays of Josephson junctions [18, 19], mechanical [20] and magnetic pendula [21] and microcantilevers [22], nonlinear electrical lattices [23, 24] and granular media [25, 26]. They have been observed in molecular and ionic crystals like the so-called PtCl [27] and antiferromagnets [28], and have been argued as plausible explanations of observational findings in systems such as α-Uranium [29] or NaI [30]. They have also been speculated to play an important role in DNA denaturation in an extensive literature reviewed, e.g., in [31], as well as to arise in reconstructive chemical reactions [32] and in the slow decay of luminiscence in Pb-doped alkali-halide crystals as KBr [33], and predicted to exist in crystals of Niobium and Nickel [34] or carbon materials like graphene, carbon nanotubes, fullerenes or hydrocarbons (see [35] for a review). Under suitable conditions, DBs can move along the lattice and are known in that setting as moving breathers. As such, they have been argued to be responsible for the observed tracks in muscovite mica sheets [16, 36] and of the infinite charge mobility (also dubbed as hyperconductivity) experimentally evidenced in such crystals [37, 38]. Admittedly, these are only some examples of an ever increasing list of applications which is by necessity, due to the limited scope of this Chapter, rather incomplete. Nevertheless, it serves to illustrate the generic nature and wide impact of such structures and their broad relevance of study. The present Chapter is devoted to reviewing more concretely some results on the existence, stability and dynamics of DBs in KG lattices with φ4on-site potential, and some other miscellaneous topics related to DBs in such lattices; many of these results have been found for generic KG lattices, but we will focus here on the principal theme of this special volume, namely the φ4potential. Moreover, by choice, the emphasis will be on some recent results, not only due to their connections to the research interests and recent work of the authors, but also because, to the best of our knowledge, the associated findings and the resulting over-arching stability perspective have not been collected in such a summarizing body of work to date elsewhere. 2 The Klein-Gordon lattice and the anti-continuous limit After the brief presentation of the DB concept, in this Section we will introduce some definitions. First of all, we need to address the concept of Klein-Gordon nonlinear dynamical lattice; from a mathematical point of view, it can be defined as a system of coupled second-order ordinary differential equations of the form: Fn(u)≡¨un+V0(un) + X m Cm(un−un+m−un−m)=0 (1) 4 Jes´ us Cuevas–Maraver and Panayotis G. Kevrekidis where nand mare D-dimensional indices, V(un)is the on-site potential, which is not necessarily homogeneous, and Cmis the coupling constant, depending on the distance from the neighbors. In most case examples, the inter-site force is nearestneighbour with Cm=Cδm,1, in which case the previous equation can be written e.g. for the one-dimensional lattice as Fn(u)≡¨un+V0(un) + C(2un−un+1 −un−1) = 0.(2) This dynamical equation derives from the following Hamiltonian: H=X n hn=X n 1 2˙u2 n+V(un) + C 4(un−un−1)2+ (un−un+1)2(3) with hnbeing the energy density. As mentioned above, the anti-continuous (AC) limit introduced by MacKay–Aubry’s theorem [13] is of paramount importance in order not only to prove the existence of discrete breathers in nonlinear KG lattices, but also to give a hint on how to obtain them numerically. The AC limit is that of all the lattice oscillators being uncoupled (i.e. C= 0) and either oscillating with the same frequency ωbor remaining at rest. By virtue of the implicit function theorem, MacKay and Aubry established that the solution can be continued from the AC limit to a finite value of C→0whenever two conditions are fulfilled: (1) the potential of the isolated oscillators is anharmonic and (2) no integer multiples of the DB frequency ωbresonate with the linear modes frequency (i.e. the so-called phonon band which we will quantify further below). If these conditions are fulfilled, a coherent (in the sense that all the sites oscillates with the same frequency) and exponentially localized structure of a DB form will exist. These discrete breathers are exact periodic orbit solutions (up to machine precision) of the nonlinear KG equation. This theorem is not only of theoretical value, but also of practical usefulness as it provides a strategy on how to numerically calculate DBs that was exploited by Mar´ ın and collaborators [15]. Methods based on the AC limit are quite simple: as at the AC limit the oscillators are uncoupled, it suffices to get a periodic orbit of single oscillators subjected to potential V(u)and continue this solution by means of fixed-point methods (like Newton-Raphson) up to the desired coupling. Among the numerical methods used for attaining DBs, we can highlight two: (1) Fourier space methods in which DBs are represented by a Galerkin truncation up to index kmin a Fourier series expansion of the form: un= km X k=−km zkeikωbt,(4) transforming the coupled ODE system (1) into a set of nonlinear algebraic equations; and (2) shooting methods, where DBs are fixed points of the map: ({un(0)},{˙un(0)})→({un(T)},{˙un(T)}),(5) Discrete breathers in φ4and related models 5 with T= 2π/ωbbeing the breather period. Fourier methods have the advantage of using an analytical Jacobian but one has to pay the price of handling with a larger number of equations. In addition, there are pathological potentials [23, 39] for which the convergence of Fourier series is very slow and this method cannot be used. There is a great number of potentials that can be found in the DB literature. They can be classified as soft or hard, if the energy decreases or increases, respectively, with the frequency; a simple way to discern if a potential is soft or hard is by making use of the hardening coefficient h= 3V0000(0) −5V000(0) [40] so that the potential is soft (hard) when h < 0(h > 0). Moreover, a soft (hard) oscillator vibrates with a frequency ωbwhich is smaller (greater) than its natural frequency ωo=pV00(0) (notice that in most cases, ωo= 1). Typical examples of soft potentials include the Morse, Lennard-Jones, cubic (φ3), sine-Gordon and double-well potentials, many of which arise in applications [1, 2]. Hard potentials are usually particular cases of polynomials potential which, depending of parameters, can be either soft or hard, namely cubic-quartic or purely quartic (φ4) anharmonicities, where the quartic term arises with a positive sign; see below. This latter potential, on which we will focus in the present Chapter, is given by V(u) = 1 2u2+1 4su4(6) When s= 1 (s=−1), the potential is hard (soft). The orbits u(t)of an isolated oscillator can be expressed in terms of Jacobi elliptic functions and the modulus m is related to the oscillation frequency ωbthrough a transcendental equation. Thus, if s= 1, u(t) = r2m 1−2mcn 2K(m)ωbt π, m, ωb=π 2√1−2mK(m)(7) and, if s=−1 u(t) = r2m 1 + m2cd 2K(m)ωbt π, m, ωb=π 2√1 + m2K(m)(8) Notice that in the soft case, there are heteroclinic orbits separating oscillating states from unbounded ones representing escape from the potential [41]. Another interesting feature that distinguishes soft and hard potentials is related to the oscillation pattern of the tails. When the potential is hard, the tails are staggered and they are unstaggered in the soft case [42]. An example of discrete breathers in soft and hard potentials is shown in Fig. 1 where the profile and the time-evolution is displayed. At the AC limit, it is possible to construct DB-like solutions with more than one excited sites, dubbed as multibreathers. When suitably constructed (typically with the “excited” oscillators in- or out-of-phase as summarized, e.g., in [43]), these are also found to persist when the coupling is switched on. Over the years, some of the DB structures have been endowed with distinguishing names. The Sievers-Takeno 6 Jes´ us Cuevas–Maraver and Panayotis G. Kevrekidis Fig. 1 Profile (top panels), time-dependence of the central sites (middle panels) and Floquet multiplier spectrum (bottom panels) of a 1-site breather in a soft (left panels) and in a hard (right panels) potential. In the former case, parameters are ωb= 0.85 and C= 0.3, whereas in the latter case, ωb= 2.5and C= 1. In the right panels, multipliers with positive (negative) Krein signatures are depicted with red crosses ×(blue pluses +); see the associated discussion around Eq. (15) below. Discrete breathers in φ4and related models 7 mode corresponds to a DB with only one excited site (it can also be denoted as single site or 1-site breather). Also, the Page mode is a multibreather where two adjacent sites are excited; notice that the former is a site-centered breather whereas the latter corresponds to a bond-centered (inter-site-centered) one. When all the sites at the AC limit are excited, we are dealing with a nonlinear phonon or phonobreather; a dark breather [44] is a phonobreather with one (or a few) non-excited site(s), resembling the functional form of a dark soliton of the nonlinear Schr¨ odinger equation. Multibreathers are usually formed by time-reversible oscillators. In these cases, they can be characterized by a coding sequence σ≡ {σn}indicating the phase and the excitation state of the relevant sites at the AC limit. This code is σn= 0 if the n-th oscillator is at rest, σn= 1 if it oscillates with a frequency ωband phase 0 (i.e. un(0) >0) and σn=−1if oscillates with initial phase π(i.e. un(0) <0). For instance, the Sievers-Takeno and Page modes are represented by σ={1}and σ={1,1}(for soft potentials) or σ={1,−1}(for hard potentials), respectively. There are some cases where the code can be more complex as in sine-Gordon potentials, where rotors can coexist with oscillators; in such cases, structures called rotobreathers (see [14, 45]) can emerge if the excited site corresponds to a rotor; they have been experimentally generated in Josephson junction arrays [18, 19]. There are also, however, some special cases where multibreathers are constituted by non-time-reversible oscillators, as demonstrated in [14]. One can find such kind of solutions in 1D chains with periodic boundary conditions in the form of phonobreathers with phase torsion [46, 47], or in 2D percolating clusters or so-called discrete vortices [48, 49, 43]. They can also emerge in 1D lattices with long-range interactions [50, 51]. Additionally, they constitute a potential attractor in periodically forced and damped oscillator networks [52]. The non-resonance condition of MacKay-Aubry’s theorem clearly establishes that the breather frequency ωbmust lie in the gaps between the linear modes (phonons) band; additionally harmonics of this frequency must avoid resonances with the band to ensure the absence of energy dispersing mechanisms affecting the DB. In the case of 1D KG lattices, there is an optical band of phonons given by ω2 ph =ω2 o+ 4Csin2q 2(9) where qis the phonon wavenumber and ωph the frequency of the associated effectively plane wave excitations ∼ei(qn−ωpht). In finite lattices (which are needed for numerical computations), the value of qis quantized with the quantization being determined by the nature of the imposed boundary conditions and the number of lattice nodes N. Consequently, breathers in hard potentials, can be continued until ωbcollides with the upper edge of the phonon band (q≈π), an event occurring when C≈(ω2 b−ω2 o)/4(the approximation symbol is used to take into account the fact that, depending on the quantization of q, it could happen that q=πis not in the band). Resonances in soft potentials are caused by integer multiples of the breather frequency colliding with the frequencies of the phonon band. The relevant critical point emerges when the second harmonic in asymmetric potentials and the third one in symmetric ones (like the purely quartic φ4analyzed in this Chapter) collides 8 Jes´ us Cuevas–Maraver and Panayotis G. Kevrekidis once again with the upper edge of the phonon band; that is, when C≈ω2 b−ω2 o/4 or C≈(9ω2 b−ω2 o)/4, respectively. In addition, in a finite lattice and soft potentials, there are gaps in the phonon band and breathers can “bypass” the resonance frequency, by existing within these finite gaps. In this case, the breather hybridizes with the bifurcating phonon creating a structure called phantom breathers; strictly speaking, they are non-exponentially-localized breathers as they have a tail oscillating with a frequency nωbwith ndepending on the multiple of the breather frequency that resonates [53]. These structures are the discrete analogue of the nanoptera observed at the continuum limit of KG lattices with the φ4double well potential [7]. As shown in [54] for sine-Gordon and φ4double well potentials, continuation up to the continuum limit is similar to a Wannier-Stark ladder. This phenomenon (i.e. breather-phonon hybridization) takes place because the staggering character of the phonons is different than that of the breather; if the breather tails had the same staggering character of the bifurcating phonon, the breather would be smoothly continued from the phonon and its amplitude would be zero at the phonon frequency. Notice that gaps in phonon band appears naturally in disordered lattices because of Anderson localization; breathers in such systems can delocalize [55] or remain localized [56, 57]. The above scenario is typical of Sievers-Takeno and Page modes. When other multibreathers are considered, breathers cannot reach the phonon band and the bifurcation scenario is more complex (see e.g. [54]). Let us also mention that decay of breathers is not necessarily exponential. In lattices with long-range intersite interactions, the decay can be algebraic [58], possibly featuring a transition from exponential to algebraic; for a recent experimental realization of this transition in system based on magnets, see, e.g., [59]. In addition, in some KG/FPUT lattices with φ4intersite potential, breathers can decay superexponentially being dubbed as (nearly) compact DBs [60, 61]. Finally, we must remark that there are several more existence proofs. For instance, a variational proof was introduced in [62]; this is valid only for hard potentials. A proof based on the center manifold theorem was introduced in [40] and is valid for DBs whose frequency is close to the phonon band edge. 3 Stability of discrete breathers This Section can be considered as the core of the present Chapter. The treatment of the topic will be as follows: first of all, we will present an introduction to the Floquet theory applied to the linear stability of discrete breathers; then, we will show different approaches to the linear stability of multibreathers in the vicinity of the AC limit and introduce the energy-vs-frequency monotonicity criterion for the linear stability of breathers and multibreathers at arbitrary coupling; afterwards, nonlinear stability criteria will be summarized together with the dynamical evolution of some examples of unstable solutions. Discrete breathers in φ4and related models 9 3.1 Floquet analysis In order to assess the dynamical robustness of the identified DB solutions and their potential accessibility in physical experiments, a key step is the determination of their stability. In the paper where MacKay and Aubry demonstrate the existence of breathers [13], it is speculated that 1-site breathers are likely to be stable. This was finally proven by Aubry in [14] for finite lattices and MacKay and Sepulchre in [63] for infinite lattices. As breathers are time-periodic solutions of the equation system (1), their spectral stability can be determined by means of a Floquet analysis. To this aim, we need to evaluate the evolution of a perturbation ξn(t)to a solution vn(t). We thus introduce —in e.g. the 1D equation (2)— the solution un(t) = vn(t) + ξn(t)with being a small constant. Then, the equation that the perturbation satisfies to O()is ¨ ξn+V00(vn)ξn+C(2ξn−ξn+1 −ξn−1)=0.(10) This equation can be written in a more compact form as N(v(t))ξ= 0 (11) where ξ≡ {ξn(t)}and v(t)≡ {vn(t)}.Nis known as the linearization operator. If ξ∈ C2, the study of this operator spectrum offers information about stability. If, moreover, ξ∈ E2 s(ωb)(i.e. ξbelongs to the space of time-reversible solutions with frequency ωb), this operator can be identified as the Jacobian (Fr´ echet derivative) of the dynamical equations, i.e. ∂vF(v(t)) = N(v(t)). Equation (11) can be viewed as a the particular case E= 0 of the eigenvalues equation for the Newton operator N(v(t))ξ=Eξ (12) The study of the spectrum of Nis related to Aubry’s band theory [14] as we will see further in the present section. In Hamiltonian dynamical systems, this linearization operator is time-symmetric, real, symplectic and Hermitian. In addition, it is invariant on time translations of period T, so by virtue of Bloch’s theorem, the corresponding eigenfunctions ξ(t)can be expressed as Bloch functions: ξ(t) = eiθt/T υ(t).(13) To perform Floquet analysis, we need to study the spectrum of the Floquet operator F, defined from the following map: Ω(T) = FoΩ(0),with Ω(t)=[ξ(t),˙ ξ(t)] (14) The representation of Foin R2Nis denoted as the so-called monodromy matrix. The eigenfunctions of Foare those of the Newton operator with E= 0. Thus, as a consequence of Bloch’s theorem (13), Ω(T) = exp(iθ)Ω(0). In other words, monodromy eigenvalues (also known as Floquet multipliers) are of the form λ= exp(iθ)with θ∈C.θis dubbed as Floquet argument. 16 Jes´ us Cuevas–Maraver and Panayotis G. Kevrekidis Fig. 3 Dependence with respect to the coupling constant Cof the argument (left panels) and modulus (right panels) of the Floquet multipliers corresponding to the following configurations: (top panels) σ={1,1}multibreather with ωb= 0.8in soft potentials; (middle panels) σ= {1,−1}multibreather with ωb= 2.5in hard potentials; (bottom panels) σ={1,1}multibreather with ωb= 1.5in hard potentials. Notice that in the right panels, the mode with positive (negative) Krein signature is depicted in red (blue). Discrete breathers in φ4and related models 17 operation has a fixed point (if the original lattice has a genuine traveling wave). The period of the associated periodic orbit is directly associated with the speed of the traveling wave (TW) since ωb= 2πs/h where sis the speed of the wave and hthe spacing of the lattice. We thus give a very short proof of the theorem for the case of traveling waves and, by direct analogy, for DBs. Consider the dynamical system: Ut=u pt =J∇H(U)where J=0I −I0. We seek a TW in the form: u=u0(n−st) = u0(ξ)[solving −sU0,ξ =J∇H(U0)] and linearize around it according to: u(ξ, t) = u0(ξ) + W(ξ, t)and p(ξ, t) = p0(ξ) + P(ξ, t). Then, the linearization operator and its adjoint can be found explicitly as: M:= s∂ξ+J∇2H(U0),M∗= (−∇2H(U0)J−s∂ξ) = JMJ. Similarly to the breather problem, the time translation invariance induces a eigenvector and a generalized eigenvector with 0eigenvalue of the form: ∂ξU0(the eigenvector) and M(−∂sU0) = ∂ξU0(the equation for the generalized eigenvector). Then the straightforward proof of the stability theorem is as follows: If an extra eigenvector ˜ Ywith λ= 0 exists, then it must satisfy M˜ Y=∂sU0(yielding an additional generalized eigenvector). This, however, imposes the following symplectic orthogonality condition: 0 = hJ∂ξU0,M˜ Yi=Z(J∂ξU0)·(∂sU0)dξ =Z1 s∇H(U0)·∂U0 ∂s dξ =1 sZ∂H(U0) ∂s dξ =1 sH0(s). Consequently, at this critical point the derivative H0(s)must vanish, and similarly for DBs H0(ωb)=0. As explained in [76] for DBs and [78, 79] for TWs, one can take the calculation further providing estimates of the bifurcating instabilityinducing multipliers on the two sides of the critical point. 3.4 Nonlinear stability As illustrated in [80] for a wide range of NLS type models, linear stability is not the last word, as there are cases of e.g. spatially-antisymmetric solitons, which are linearly stable but the dynamics is can be nonlinearly unstable, if it is evolved over sufficiently long time scales. In both lattice and continuum NLS modes, it was shown in this work that spectral stability may be inconclusive if modes of negative Krein signature exist in the spectrum and if their (nonlinearity induced) harmonics are in resonance with the continuous spectrum. In particular, an ordinary differential equation (ODE) was derived suggesting that for positive energy modes, their energetic content is depleted due to dispersive wave radiation. However, for negative Krein signature modes, the reverse process occurs, eventually pumping (over a slow, power-law in time procedure) the internal mode and ultimately leading to the demise of the coherent structure due to this genuinely nonlinear mechanism. 18 Jes´ us Cuevas–Maraver and Panayotis G. Kevrekidis Fig. 4 The top panels show the energy-frequency dependence of discrete breathers in 2D lattices with a soft potential and C= 0.1(left panels) and a hard potential and C= 0.3(right panels). Bottom panels show the imaginary part of the argument of the Floquet multipliers of these solution families. Based in such findings, in [73] we explored the nonlinear instability of linearly stable multibreathers. We found that nonlinear instability is possible when an integer multiple of the frequency of an internal (i.e. localized) eigenmode resonates with the phonon band and the Krein signature of such a mode and the phonon arc are opposite to each other. Notice that the frequency Ωof the internal eigenmode and its Floquet argument follows the relation θ=±ΩT mod 2π; consequently, if the coupling constant is small enough, the internal mode has not collided with the phonon arc (in fact, this condition is necessary in order for the breather to be stable against potential Hamiltonian Hopf bifurcations), then Ω=θ/T =ωbθ/(2π). In this case, however, a harmonic of such a mode can be located inside the arcs. If it is the 2nd harmonic, then the slow growth follows a t−1/2law, if the 3rd, a t−1/4law etc., as derived by the corresponding ODE [80, 73]. Having in mind the stability properties of multibreathers near the AC limit, the only stable solutions among them are those whose codes σare in phase if the on-site potential is hard and in anti-phase if the potential is soft. As demonstrated in [73], the internal modes that detaches from θ= 0 have, in the upper half-circle (i.e. for Discrete breathers in φ4and related models 19 θ∈[0, π], Krein signature κ= 1 if the potential is soft and κ=−1if it is hard. On the other hand, in order to have nonlinear instability, it is needed that the Krein signature of the phonon arcs in the upper half-circle are the opposite to the internal modes. Because of this, as explained at the beginning of Subsection 3.2, nonlinear instability can only be possible if ωo< ωb≤2ωoif the potential is hard and 2 2k+1 ωo< ωb<1 kωowith k∈Nif the potential is soft. These conditions were corroborated in the detailed numerical computations of [73]. In the simpler case of the NLS and DNLS models, no such conditions need to be imposed, however numerical computations verified the existence of the instability in [80]. Here, we only provide a prototypical case example of the associated phenomenology in the right panel in Fig. 3. This shows the particular case of a the σ={1,1}multibreather in a hard potential. This solution is linearly stable before the Hopf bifurcation takes place, although presents nonlinear instability as the conditions of the paragraph above are fulfilled. The dynamics eventually manifests this instability although the latter only arises over extremely long times, much longer than the linearly unstable cases of the left and middle panels. 3.5 Dynamics In this subsection we will further discuss some prototypical examples of unstable dynamics stemming from linear and nonlinear instabilities of the (2-site) multibreathers shown previously (especially in Fig. 3). First of all, we consider linearly unstable multibreathers in the soft φ4potential. In that case, the main behavior is a blow-up caused by a phenomenon similar to the escape observed in [41]. Left panels of Fig. 5 illustrate the blowing-up dynamics observed in an unstable σ={1,1}multibreather. Notice that blow-up is caused by both exponential and oscillatory instabilities. I.e., the linearly unstable growth leads the oscillators to exit the finite height potential barrier and hence tend to ±∞. In the case of hard potentials, the main dynamical features consist of the transformation of the multibreather into a less energetic 1-site breather. In this transformation, some energy is shed from the breather and, in some cases, it can be localized. The central panels of Fig. 5 shows the dynamics of the σ={1,−1}multibreather. It is natural for the configuration to tend to the site-centered variant as that is generically stable, as we discussed above. Finally, the right panels of the figure consider a linearly stable but nonlinearly unstable multibreather with σ={1,1}displaying a similar behaviour. The nonlinearly unstable dynamics fulfills the conditions of the theorem of [73] as the frequency of the internal mode is Ω= 0.2073 and the spectral bands expands in [0.32,0.5]. As a result, the second harmonic of the internal mode lies in the phonon arc, ultimately (at very long times) slowly feeding the nonlinear growth and eventually leading to the destruction of the two-site configuration. Notice, however, as also indicated above, the characteristically longer (by at least an order of magnitude) time needed for the manifestation of this (nonlinear) instability. 20 Jes´ us Cuevas–Maraver and Panayotis G. Kevrekidis Fig. 5 Dynamical evolution of the unstable configurations displayed in Fig. 3. Top panels show the space-time dependence of the energy density, middle panels depict the energy density evolution of the central sites and bottom panels compare to the profiles of the perturbed breather at the beginning and at the end of the simulation. The multibreathers considered at the figures are: (left panels) σ={1,1}in a soft potential with ωb= 0.8and C= 0.2; (central panels) σ={1,−1} in a hard potential with ωb= 2.5and C= 0.5; (right panels) σ={1,1}in a hard potential with ωb= 1.5and C= 0.1. 4 Some glimpses on other breather features Although the core of the present chapter focuses on the stability properties of discrete breathers in φ4lattices, we mention in passing some developments related to a judicious selection of aside topics, namely moving breathers and the generation of discrete breathers in lattices with dissipation. Discrete breathers in φ4and related models 21 4.1 Moving breathers As mentioned in the previous section, when an an anti-symmetric eigenmode detaches from the phonon arc, it can reach θ= 0 bringing about a tangent bifurcation. Just at θ= 0, the mode is marginal and resembles a translational mode, so that a perturbation along it can set the breather into motion. Contrary to continuous breathers, moving discrete breathers generically radiate phonons (see e.g. [72]) and they eventually stop. Due to the special features of moving breathers, there does not exist a systematic underlying mathematical theory that can clearly characterize them. Some attempts of defining a Peierls-Nabarro barrier similar to kinks have been performed, but they only seem to work in FPUT lattices close to the continuum limit [81]. In any case, there must be a mechanism alike to Peierls-Nabarro barrier which is related to the existence of tangent bifurcations of the translational mode that detaches from the phonon arc. Because of this, moving breathers can only be observed in lattices for which the breathers experience such bifurcations. There are only a few reported cases of one-dimensional KG lattices, namely, with Morse, sine-Gordon and doublewell potentials [42]. We have also been able to generate moving breathers in twodimensional KG lattices with Morse potential, a result that have not been published yet. Moving breathers (with high mobility) has been observed in two-dimensional lattices with in-plane degrees of freedom modeling e.g. muscovite mica [16] which are also known as quodons and, as mentioned in the Introduction, are speculated to play an important role in charge transport properties in such materials. Moving breathers do not exist in KG lattices with the φ4potential, as the tangent bifurcation of the translational mode does not take place. However, such a bifurcation was observed for KG/FPUT lattices with on-site and interaction potentials of the hard φ4form [64]. Such a lattice has been used for modeling micromechanical cantilever arrays [22]. In [82] we generated moving breathers in such a model and analyze their interaction with geometrical defects. 4.2 Dissipative lattices Most of the experimental findings of discrete breathers have been achieved on lattices with dissipation and external driving, such as micromechanical [22], pendula [20] and Josephson junctions [18, 19] arrays, nonlinear electrical lattices [23, 24] or granular media [25, 26]. Such classes of systems remain quite popular to this day with numerous variations continuously arising including, e.g., piecewise-linear systems emulating the β-form of the celebrated FPUT lattice [83], or electrical systems involving beyond-nearest-neighbor interactions [84]. As demonstrated in [85], discrete breathers in dissipative lattices can also exist away from the anticontinuum limit. Contrary to Hamiltonian lattices, there are no resonances with phonons (the spectrum of plane wave excitations is pushed to the left half of the complex spectral plane); as a result such states are now poten- 22 Jes´ us Cuevas–Maraver and Panayotis G. Kevrekidis tial attractors of the system. The work of [52] shows the complex phenomenology that is observed in driven and damped Frenkel–Kontorova lattices. If the system is driven with a frequency ωb, discrete breather solutions acquire the same frequency as the driving force. In general, all the lattice sites oscillate with the same frequency; however, we have found an electrical lattice where subharmonic resonance emerges (that is, the excited sites of the breather oscillate with half of the frequency of the low amplitude sites) [86]. Recently, multistable variations of pendula, potentially applicable to SQUID metamaterials, have been manifested as potential sources of more complex breathing patterns such as the celebrated chimera states [87]. Discrete breathers have been studied in driven and damped one-dimensional KG lattices with hard φ4potentials in [88]. Such a lattice is defined by equation: ¨un+α˙un+V0(un) + C(2un−un+1 −un−1) = Fn(t) + ηn(t)(21) with Fn(t)being a periodic function of frequency ωband ηn(t)is a Gaussian white noise with zero mean and autocorrelation < ηn(t)ηm(t0)>= 2Dδnmδ(t−t0). In the deterministic case (D= 0) and for a staggered driving of the form Fn(t) = (−1)nfsin(ωbt), the phenomenology is quite simple: at given damping and frequency, discrete breathers only exist above a threshold fth. However, if the noise is introduced in the lattice, there are two interesting phenomena: if the driving amplitude is suprathreshold (f > fth), breathers with frequency ωbcan be generated even if the initial condition is uniform; if the driving amplitude is subthreshold (f < fth), breathers are still produced by the concerted action of noise and the driving force, in a way that noise, on the one hand, enables system transitions between the uniform and the coherent localized states and, on the other hand, destroys any degree of order of the system if its amplitude is large enough: in other words, we are dealing with a stochastic resonance phenomenon. 5 Outlook and future directions From the above discussion, it is clear that the theme of discrete breathers is one that is gradually maturing and emerging in a wide range of applications and a diverse array of systems. In particular, we are gradually departing from the simplest nearest-neighbor scenarios of either just KG or just FPUT types and moving on to a new, more elaborate phase where systems can be designed with beyond-nearest [84] and even long-range interactions [59] and also with multiple and potentially competing [12] interactions, or with ones that are progressively more amenable to analytical considerations [83]. This suggests that there is a significant need for further theoretical and computational developments to support the corresponding emerging experimental platforms. While the early stages of development of DBs favored analytical proofs of existence and associated techniques of numerical existence and stability, subsequent ones favored a more systematic exploration of spectral properties and an attempt to Discrete breathers in φ4and related models 23 classify the different multibreathers and offer systematic guidelines about when they may be expected to be dynamically robust. In this Chapter, we summarized some of this systematic effort in the previous decade and some of its crystallized results and convergence of different methods over the past few years. More recently, further tools have arisen in probing spectral and dynamical features of DBs. Among others, we have explored and summarized here the energy-versus-frequency monotonicity criteria and how they relate to linear instabilities and given connections between these and the stability of traveling waves in lattices. Additionally, we have warned the reader against the naive expectation that spectral stability is the full story, presenting case examples where this fails to be true due to the nonlinear instability of internal modes with opposite Krein signature than that of the phonon arcs. The slow, power-law nature of the latter instabilities, as opposed to the exponential growth of linear instabilities was highlighted. Lastly, some possibilities of further developments towards moving breathers or dissipative lattices were briefly touched upon. Clearly, there is need for further theoretical systematics. Many of the relevant points were brought up in parts of our discussion. Understanding the stability of phase-shift multibreathers and vortex breathers in higher-dimensional systems is an important open topic. Carrying out the associated stability computations to higher order is particularly relevant. Recent work, in fact, brings up the possibility that relevant solutions may fail to exist at higher orders in some important case examples [89]. Studying also long-range interactions may bring about surprises and produce gaps in the spectrum where novel DBs may exist, as per the recent work of [90]. Again, this is a topic meriting further exploration. The study of systems with nontrivial tails (nanoptera) and the examination of whether the stability features/criteria presented herein apply to them is also an open topic. The same holds true for systems with external drive and damping: can we offer some guidelines to characterize their stability characteristics, suitably adapting what we know in the more structured Hamiltonian cases or perhaps not ? Plus then there are topics which, while touched upon, still seem fairly poorly understood or wide open for new insights: among them moving breathers, or quasi-periodic solutions and their existence and stability, as well as the role of DBs in asymptotic dynamics and thermalization (see, e.g., [91] for a recent summary in the discrete NLS case) are only some that come to mind. In summary, discrete breathers may have matured but have many more challenges to offer for the years to come both at the fundamental, at the computational and at the experimental level... Acknowledgements This material is based upon work supported by the National Science Foundation under Grant No. DMS-1809074 (P.G.K.). J.C.-M. thanks financial support from MAT2016- 79866-R project (AEI/FEDER, UE). P.G.K. also gratefully acknowledges support from the USAFOSR under Grant No. FA9550-17-1-0114. 24 Jes´ us Cuevas–Maraver and Panayotis G. Kevrekidis References 1. S. Flach, C.R. Willis, Phys. Rep. 295, 181 (1998) 2. S. Flach, A.V. Gorbach, Phys. Rep. 467, 1 (2008) 3. S. Aubry, Physica D 216, 1 (2006) 4. D.K. Campbell, M. Peyrard, in CHAOS/XAOC: Soviet-American Perspectives on Nonlinear Science, ed. by D.K. Campbell (American Institute of Physics, New York, 1990), p. 305 5. M. Ablowitz, D.J. Kaup, A.C. Newell, H. Segur, Stud. App. Math. 53, 249 (1974) 6. H. Segur, M. Kruskal, Phys. Rev. Lett. 58, 747 (1987) 7. J.P. Boyd, Nonlinearity 3, 177 (1990) 8. P.W. Anderson, Phys. Rev. 109, 1492 (1958) 9. T. Dauxois, Phys. Today 61, 55 (2008) 10. A.J. Sievers, S. Takeno, Phys. Rev. Lett. 61, 970 (1988) 11. S. Takeno, K. Kisoda, A.J. Sievers, Prog. Theor. Phys. Suppl. 94, 242 (1988) 12. M. Kimura, T. Hikihara, Chaos 19, 013138 (2009). 13. R.S. MacKay, S. Aubry, Nonlinearity 7, 1623 (1994) 14. S. Aubry, Physica D 103, 201 (1997) 15. J.L. Mar´ ın, S. Aubry, Nonlinearity 9, 1501 (1996) 16. J.F.R. Archilla, N. Jim´ enez, V.J. S´ anchez-Morcillo, L.M. Garc´ ıa-Raffi (eds.), Quodons in Mica. Nonlinear Travelling Excitations in Crystals (Springer, Cham, 2015) 17. S.V. Dmitriev, E.A. Korznikova, Yu.A. Baimova, M.G. Velarde, Phys. Uspekhi 59, 446 (2016) 18. E. Tr´ ıas, J.J. Mazo, T.P. Orlando, Phys. Rev. Lett. 84, 741 (2000) 19. P. Binder, D. Abraimov, A.V. Ustinov, S. Flach, Y. Zolotaryuk, Phys. Rev. Lett. 84, 745 (2000) 20. J. Cuevas, L.Q. English, P.G. Kevrekidis, M. Anderson, Phys. Rev. Lett. 102, 224101 (2009) 21. F.M. Russell, Y. Zolotaryuk, J.C. Eilbeck, T. Dauxois, Phys. Rev. B 55, 6304 (1997) 22. M. Sato, B.E. Hubbard, A.J. Sievers. Rev. Mod. Phys. 78, 137 (2006) 23. F. Palmero, L.Q. English, J. Cuevas-Maraver, R. Carretero-Gonz´ alez, P.G. Kevrekidis, Phys. Rev. E 82, 026605 (2011) 24. L.Q. English, F. Palmero, J.F. Stormes, J. Cuevas-Maraver, R. Carretero-Gonz´ alez, P.G. Kevrekidis, Phys. Rev. E 88, 022912 (2013) 25. N. Boechler, G. Theocharis, S. Job, P. G. Kevrekidis, Mason A. Porter, and C. Daraio Phys. Rev. Lett. 104, 244302 (2010). 26. C. Chong, F. Li, J. Yang, M.O. Williams, I.G. Kevrekidis, P.G. Kevrekidis, C. Daraio, Phys. Rev. E 89, 032924 (2014) 27. B.I. Swanson, J.A. Brozik, S.P. Love, G.F. Strouse, A.P. Shreve, A.R. Bishop, W.-Z. Wang, M.I. Salkola, Phys. Rev. Lett. 82, 3288 (1999) 28. M. Sato, A.J. Sievers, Nature 432, 486 (2004) 29. M.E. Manley, M. Yethiraj, H. Sinn, H.M. Volz, A. Alatas, J.C. Lashley, W.L. Hults, G.H. Lander, J. L. Smith, Phys. Rev. Lett. 96, 125501 (2006) 30. M.E. Manley, A.J. Sievers, J.W. Lynn, S.A. Kiselev, N.I. Agladze, Y. Chen, A. Llobet, A. Alatas, Phys. Rev. B 79, 134304 (2009) 31. M. Peyrard, Nonlinearity 17, R1 (2004) 32. J.F.R. Archilla, J. Cuevas, M.D. Alba, M. Naranjo, J.M. Trillo, J. Phys. Chem. B 110, 24112 (2006) 33. E. Mih´ okov´ a, L. S. Schulman, J. Phys. A: Math. Theor. 43, 183001 (2010) 34. M. Haas, V. Hizhnyakov, A. Shelkan, M. Klopov, A.J. Sievers, Phys. Rev. B 84, 144303 (2011) 35. J.A. Baimova, E.A. Korznikova, I.P. Lobzenko, S.V. Dmitriev, Rev. Adv. Mater. Sci. 42, 68 (2015) 36. J.F.R. Archilla, F.M. Russell, Lett. Mater. 6, 3 (2016) 37. F.M. Russell, in Nonlinear Systems, Vol. 2, ed. by J.F.R. Archilla, F. Palmero, M.C. Lemos, B. S´ anchez-Rey, J. Casado-Pascual. (Springer, Cham 2018), p. 241 38. F.M. Russell, J.F.R. Archilla, F. Frutos, S. Medina-Carrasco, Europhys. Lett. 120, 46001 (2017) Discrete breathers in φ4and related models 25 39. G. James, P.G. Kevrekidis, J. Cuevas, Physica D 251, 39 (2013) 40. G. James, B. S´ anchez-Rey, J. Cuevas, Rev. Math. Phys. 21, 1 (2009) 41. V. Achilleos, A. ´ Alvarez, J. Cuevas, D.J. Frantzeskakis, N.I. Karachalios, P.G. Kevrekidis, B. S´ anchez-Rey, Physica D 244, 1 (2013) 42. D. Chen, S. Aubry, G. Tsironis, Phys. Rev. Lett. 77, 4776 (1996) 43. J. Cuevas, V. Koukouloyannis, P.G. Kevrekidis, J.F.R. Archilla, Int. J. Bif. Chaos 21, 2161 (2011) 44. A. ´ Alvarez, J.F.R. Archilla, J. Cuevas, F.R. Romero, New J. Phys. 4, 72 (2002) 45. S. Takeno, M. Peyrard, Physica D 92, 140 (1996) 46. T. Cretegny, S. Aubry, Physica D 113, 162 (1998) 47. J. Cuevas, J.F.R. Archilla, F.R. Romero, J. Phys. A: Math. Theor. 44, 035102 (2011) 48. T. Cretegny, S. Aubry, Phys. Rev. B 55, R11929 (1997) 49. V. Koukouloyannis, R.S. MacKay, J. Phys. A: Math. Gen. 38, 1021 (2005) 50. V. Koukouloyannis, P.G. Kevrekidis, J. Cuevas, V. Rothos, Physica D 242, 16 (2013) 51. Z. Rapti, Phys. Lett. A 377, 1543 (2013) 52. J.L. Mar´ ın, F. Falo, P.J. Mart´ ınez, L.M. Flor´ ıa, Phys. Rev. E 63, 066603 (2001) 53. A.M. Morgante, M. Johansson, S. Aubry, G. Kopidakis, J. Phys. A: Math. Gen. 35, 4999 (2002) 54. M. Chirilus-Bruckner, C. Chong, J. Cuevas-Maraver, P.G. Kevrekidis, in The sine-Gordon model and its applications, ed. by J. Cuevas-Maraver, P.G. Kevrekidis and F.L. Williams (Springer-Verlag, Heidelberg, 2014), p. 31 55. G. Kopidakis, S. Aubry, Physica D 130, 155 (1999) 56. G. Kopidakis, S. Aubry, Physica D 139, 247 (2000) 57. J.F.R. Archilla, R.S. MacKay, J.L. Mar´ ın, Physica D 134, 406 (1999) 58. S. Flach, Phys. Rev. E 58, R4116 (1998) 59. M. Moler´ on, C. Chong, Alejandro J. Mart´ ınez, M.A. Porter, P.G. Kevrekidis, C. Daraio, arXiv:1801.09560. 60. B. Dey, M. Eleftheriou, S. Flach, G.P. Tsironis, Phys. Rev. E 65, 017601 (2001) 61. J.C. Comte, Phys. Rev. E 65, 067601 (2002) 62. S. Aubry, G. Kopidakis, V. Kadelburg, Discrete Continuous Dyn. Syst. Ser. B 1, 271 (2001) 63. R.S. MacKay, J.-A. Sepulchre, Physica D 119, 148 (1998) 64. J.L. Mar´ ın, S. Aubry, M. Flor´ ıa, Physica D 113, 283 (1998) 65. J.F.R. Archilla, J. Cuevas, B. S´ anchez-Rey, A. ´ Alvarez, Physica D 180, 235 (2003) 66. V. Koukouloyannis, P.G. Kevrekidis, Nonlinearity 22, 2269 (2009) 67. D.E. Pelinovsky, A. Sakovich, Nonlinearity 25, 3423 (2012) 68. J. Cuevas, J.F.R. Archilla, Yu.B. Gaididei, F.R. Romero, Physica D 163, 106 (2002) 69. J. Cuevas, J.F.R. Archilla, J.F.R. Romero, Nonlinearity 18, 769 (2005) 70. P.G. Kevrekidis, The discrete nonlinear Schr¨ odinger equation: Mathematical Analysis, Numerical Computations and Physical Perspectives (Springer-Verlag, Heidelberg 2009) 71. J.L. Mar´ ın, S. Aubry, Physica D 119, 163 (1998) 72. S. Aubry, T. Cretegny, Physica D 119, 34 (1998) 73. J. Cuevas-Maraver, P.G. Kevrekidis, D.E. Pelinovsky, Stud. Appl. Math. 137, 214 (2016) 74. S. Flach, K. Kladko, R.S. MacKay, Phys. Rev. Lett. 78, 1207 (1997) 75. M. Kastner, Nonlinearity 17, 1923 (2004) 76. P.G. Kevrekidis, J. Cuevas-Maraver, D.E. Pelinovsky, Phys. Rev. Lett. 119, 094101 (2016) 77. N.G. Vakhitov and A.A. Kolokolov, Radiophys. Quantum Electron. 16, 783 (1973). 78. J. Cuevas-Maraver, P.G. Kevrekidis, A. Vainchtein, H. Xu, Phys. Rev. E 90, 042903 (2017) 79. H. Xu, J. Cuevas-Maraver, P.G. Kevrekidis, A. Vainchtein, Phil. Trans. Roy. Soc. A 376, 2017092 (2018) 80. P.G. Kevrekidis, D.E. Pelinovsky, A. Saxena, Phys. Rev. Lett. 114, 214101 (2015) 81. M. Kastner, J.-A. Sepulchre, Discrete Continuous Dyn. Syst. Ser. B 5, 719 (2005) 82. J. Cuevas, P.G. Kevrekidis, Phys. Rev. E 69, 056609 (2004) 83. Y. Watanabe, T. Nishida, Y. Doi, N. Sugimoto, Phys. Lett. A 382, 1957 (2018). 84. X.-L. Chen, S. Abdoulkary, P.G. Kevrekidis, L.Q. English, arXiv:1806.07494. 85. J.-A. Sepulchre, R.S. MacKay, Nonlinearity 10, 679 (1997)