Revisiting multi-breathers in the discrete Klein–Gordon equation: a spatial dynamics approach

arXiv:2204.11349v2 [nlin.PS] 17 Oct 2022 REVISITING MULTI-BREATHERS IN THE DISCRETE KLEIN-GORDON EQUATION: A SPATIAL DYNAMICS APPROACH ROSS PARKER, JES´ US CUEVAS-MARAVER, P. G. KEVREKIDIS, AND ALEJANDRO ACEVES Abstract. We consider the existence and spectral stability of multi-breather structures in the discrete Klein-Gordon equation, both for soft and hard symmetric potentials. To obtain analytical results, we project the system onto a finite-dimensional Hilbert space consisting of the first MFourier modes, for arbitrary M. On this approximate system, we then take a spatial dynamics approach and use Lin’s method to construct multi-breathers from a sequence of well-separated copies of the primary, single-site breather. We then locate the eigenmodes in the Floquet spectrum associated with the interaction between the individual breathers of such multi-breather states by reducing the spectral problem to a matrix equation. Expressions for these eigenmodes for the approximate, finite-dimensional system are obtained in terms of the primary breather and its kernel eigenfunctions, and these are found to be in very good agreement with the numerical Floquet spectrum results. This is supplemented with results from numerical timestepping experiments, which are interpreted using the spectral computations. 1. Introduction Dynamical models on a one-dimensional lattice have been a topic of interest for well over 50 years. Perhaps the most famous example is the Fermi–Pasta–Ulam–Tsingou (FPUT) model [1,2], which was one of the first problems to be studied using numerical simulations. In this work, we will examine the discrete Klein-Gordon (DKG) equation [3–6] ¨un=d(∆2u)n−f(un),(1) which describes the dynamics of an infinitely long, one-dimensional lattice of particles. The quantity unrepresents the displacement of the nth particle in the integer lattice as a function of the time t. Each particle is harmonically coupled to its two nearest neighbors via the discrete second difference operator ∆2, and the strength of this coupling is quantified by the parameter d. The particles are subject to an external, nonlinear, on-site potential V(u), such that f(u) = V′(u), which can be of different type depending on the model [7, 8]. Common nonlinearities are shown in Table 1. (See subsection 2.1 for the definition of hard and soft potentials). Equation V(u)f(u) sine-Gordon 1 −cos usin u φ4(soft) 1 2u2−1 4u4u(1 −u2) φ4(hard) 1 2u2+1 4u4u(1 + u2) Morse 1 2(1 −e−u)2e−u(1 −e−u) Table 1. Common nonlinearities for the discrete Klein-Gordon equation. 1 2 ROSS PARKER, JES ´ US CUEVAS-MARAVER, P. G. KEVREKIDIS, AND ALEJANDRO ACEVES The DKG equation is the discrete analogue of the nonlinear Klein-Gordon partial differential equation utt =uxx −f(u), which is a prototypical model in the study of nonlinear waves and solitons. One of the the most well-studied forms of this equation is the sine-Gordon equation [3–6], which has periodic nonlinearity f(u) = sin(u), and is completely integrable. The transition between the continuous and discrete models in the sine-Gordon case has been extensively discussed in [9]. The discrete sine-Gordon model, also known as the Frenkel-Kontorova model, was originally devised as a model for describing the dynamics of a crystal lattice near a dislocation core [3, 10], and has been used since its inception in numerous additional applications (see, for example, [3, Chapter 2]), including a mechanical model for a chain of pendula [11,12], arrays of Josephson junctions [13,14], and DNA dynamics [15–17]. Two major classes of coherent structures in the nonlinear Klein-Gordon equation (both discrete and continuous) are of particular interest: kinks, which are heteroclinic structures resembling “wave fronts” that connect two adjacent minima of the potential V(u), and breathers, which are structures that are spatially localized and oscillatory in time. For the continuum sine-Gordon equation, exact, analytical solutions for both of these structures have been found [18]. For discrete systems, the existence and stability of static kinks have been well-studied (see [9,19,20], as well as [21] for results on multi-kink solutions), and there has also been interest in moving kinks [22–25]. We will concern ourselves herein with discrete breather solutions. Discrete breathers have been studied in Hamiltonian [26] and dissipative systems [27], and have applications in areas such as laser scanning microscopy and coupled optical waveguides [28,29]. Existence and stability of discrete breather solutions in Klein-Gordon lattices were first studied by MacKay and Aubry [30, 31] by considering the system near the anticontinuum (AC) limit (d= 0), in which the individual sites in the lattice are uncoupled. The advantage of this approach is that the solution for a single site is known at the AC limit, and this can be continued to small d > 0 using the implicit function theorem. Some results about asymptotic stability of these breathers can be found in [32]. A similar approach has been used for multi-site solitons in the discrete nonlinear Schr¨odinger equation (DNLS) [33,34], which demonstrated that the only potentially stable multi-solitons are those in which adjacent peaks are excited out-of-phase. Analysis of multi-breathers in DKG, in which a finite number of sites in the integer lattice are excited at the AC limit, was done in [35,36], but this was restricted to the case where the excited sites are adjacent, so that a concrete result could be deduced for the dynamics of the relative phases between oscillators at adjacent sites. Indeed, the two methods (the Aubry band method [31] and the MacKay effective Hamiltonian method [37]) explored in the above two publications were shown to yield the same results near this limit in [38]. In this situation, for small d > 0, out-of-phase multi-breathers are stable for soft potentials, and in-phase multi-breathers are stable for hard potentials [35, Theorem 6]. Results on the existence and spectral stability of multi-breathers were extended in [39] to multi-site breathers where any arbitrary, finite set of lattice points is excited at the AC limit; in particular, this allows the excited sites to be separated in the lattice, i.e., there can be “holes” in the lattice. However, spectral stability of multi-breathers does not necessarily imply nonlinear stability; even if the multi-breather is spectrally stable, nonlinear instabilities can result from resonance between internal eigenmodes and the continuous spectrum band, as long as the two have opposite Krein signatures [40] (see also the REVISITING MULTI-BREATHERS IN THE DISCRETE KLEIN-GORDON EQUATION 3 simpler example of the discrete nonlinear Schr¨odinger model in [41]). A recent result uses the Schauder fixed point theorem to prove the existence of discrete Klein-Gordon breathers in the setting of a convex on-site potential [42]. Other related work concerns breather solutions in infinite Fermi-Pasta-Ulam-Tsingou (FPUT) lattices, in which the nonlinear potential involves inter-site terms (see [43] and the references therein). Results on the existence of time periodic solutions in FPUT lattices can be found in [43, Chapter 2], as well as the more recent work in [44,45]. In this paper, we take a different approach to multi-breathers. We start with a single-site breather, which we call the primary breather. We take the existence of such a breather for a particular d > 0 as a hypothesis. We then construct a multi-breather by joining together multiple, well-separated copies of this primary breather. Consecutive copies of the primary breather can be either in-phase or out-of-phase. In effect, we replace the condition that the coupling parameter d > 0 is small with the condition that the individual copies of the primary breather are well separated. The construction of complex coherent structures from simple building blocks has a rich mathematical history (see [46], and the references therein). The mathematical technique that we will use to accomplish this is an implementation of the Lyapunov-Schmidt reduction known as Lin’s method. This method has been successfully employed in many systems, including semilinear parabolic PDEs [46, 47] and discrete dynamical systems [48]. We used a similar method to construct multi-solitons in DNLS [49] and multi-kinks in DKG [21]. Heuristically, we use Lin’s method to construct a double breather from the primary breather as follows (see [49, Section 1] for a similar construction of double pulses in DNLS). Let qn(t) be the primary breather solution to (1) with period T, i.e. qn(t+T) = qn(t) for all integers n. Then a T-periodic solution un(t) is a double breather if there is an integer N≫1 such that sup n≤0, t∈[0,T]|un(t)−qn+N(t)|+ sup n≥0, t∈[0,T]|un(t)−qn−N(t)| is small, i.e. un(t) resembles (to leading order) the sum of two copies of the primary breather qn(t) translated by Nunits to the left and to the right, respectively. We note that un(t) must be close to the sum of the two translates of the primary breather for all t∈[0, T]. See the top panel of Figure 1 for an illustration. The double breather (Figure 1b) resembles two sequential copies of the primary breather (Figure 1a). Importantly, however, it is not identical to two adjoined translates of the primary breather; compare the center node (n= 0) of the double breather in Figure 1b to the corresponding node (n= 2) of the primary breather in Figure 1a. For each N≫1, Lin’s method yields a piecewise double breather that comprises three pieces      u1 n(t)n∈(−∞,−N] u2 n(t)n∈[−N, N] u3 n(t)n∈[N, ∞), (2) as shown in Figure 1. This piecewise function will be a genuine double breather if and only if these pieces coincide at n=±N, i.e. u1 −N(t) = u2 −N(t) and u2 N(t) = u3 N(t) for t∈[0, T] (see schematic in Figure 1c). This approach reduces the existence problem for double breathers to solving these two jump conditions. Lin’s method can also be used to determine spectral stability. For a multi-pulse, multikink, or multi-breather, there are specific elements in the point spectrum, which we will 4 ROSS PARKER, JES ´ US CUEVAS-MARAVER, P. G. KEVREKIDIS, AND ALEJANDRO ACEVES (a) (b) (c) Figure 1. Top panel plots ˙un(t) vs. un(t) at each lattice site nfor a primary breather (a) and an in-phase double breather (b) for discrete sine-Gordon. Lattice site n= 0 is located where the axes meet. Period T= 9.8944, coupling parameter d= 0.25. (c) is a schematic showing the piecewise construction of a double breather (2) with Lin’s method, illustrated at t= 0. The piecewise double breather uncomprises three pieces u1 n,u2 n, and u3 n(red stars, blue filled circles, and yellow squares, from left to right). Two copies of the primary breather qnare placed at n=−Nand n=N, respectively (black open circles and purple open circles, from left to right). This is a genuine double breather if and only if the two jumps at n=±Nare zero for all t∈[0, T ]. hereafter term interaction eigenmodes, that result from nonlinear interactions between neighboring copies of the primary coherent structure. For breathers, which are periodic in time, these interaction eigenmodes assume the form of Floquet multipliers or Floquet exponents. (We note that these are called internal modes in [40]; since we find from numerical simulations that other internal modes split off from the continuous spectrum bands as the coupling parameter dis increased, we use the term interaction eigenmodes to refer to this specific set of internal modes). In a similar fashion to how we constructed the multi-breather, we can use Lin’s method to find these interaction eigenmodes by constructing the corresponding eigenfunction. To leading order, this eigenfunction is a piecewise linear combination of translates of the kernel eigenfunction associated with the primary breather. This reduces the spectral problem to a finite-dimensional matrix eigenvalue problem, which can then be REVISITING MULTI-BREATHERS IN THE DISCRETE KLEIN-GORDON EQUATION 5 solved. While the matrix obtained this way has the same form as that in [39, 40], its elements are computed using the primary breather solution for a specific drather than at the AC limit, thus the interaction eigenmodes can be computed more accurately for a greater range of d, i.e. further from the AC limit. For the DKG equation, we follow a similar approach to that which we used in [49]. We reformulate the equation using a spatial dynamics approach by recasting it as a lattice dynamical system on an appropriate function space, and then we use Lin’s method to both construct multi-breathers and determine the interaction eigenmodes associated with these solutions. The major limitation with this method, however, is that for a periodic breather on [0, T], the most appropriate function space is C∞ per([0, T]), which is not a closed subspace of L2 per([0, T ]). It is not straightforward to adapt the necessary mathematical tools, such as the stable manifold theorem and exponential dichotomies, to this space. As an alternative, since we are interested in smooth solutions, we project the system onto the finite-dimensional subspace XMof L2 per([0, T ]) consisting of the first MFourier modes of the standard orthonormal basis, and consider the problem restricted to this finite-dimensional Hilbert space. Although this means that the results we obtain are only approximate, we believe this is a reasonable approximation, since the Fourier coefficients for smooth functions decay exponentially, and this method is valid for arbitrary, finite M(see Figure 4 below). That being said, it is not easy to quantify how good this approximation actually is. For example, as in many other instances, if resonances in the system occur, they may be affected by this truncation. Since numerical discretization of periodic solutions is often done using a Fourier spectral method, this approximation applies directly to this discretization. Finally, the results from this method are in very good agreement with the results obtained by directly computing the Floquet spectrum of the corresponding multi-breather, both for the soft sine-Gordon and the hard φ4potentials. In addition, the spectral pattern agrees qualitatively with that found using the approach in [39, 40] for small d, and the matrix reduction found by using Lin’s method and a spatial dynamics approach has the same form as that in those references. For both soft and hard potentials, the eigenvalue pattern is determined by the phase differences (in-phase vs. out-of-phase) between adjacent copies of the primary breather. For the hard potential, it also depends on whether the distances between copies of the primary breather are an even or odd number of lattice points. This paper is organized as follows. In section 2, we present the mathematical background for the discrete Klein-Gordon equation, including breather solutions, the continuous spectrum, and the reformulation using spatial dynamics. In section 3, we formulate our finitedimensional approximation, which we then use in section 4 to prove results about the existence and spectrum of multi-breathers in the approximate system. The proof of the spectral results is deferred to Appendix A. Numerical results are presented in section 5, which are in very good agreement with the predictions from the main theorems. Results are presented for both the soft sine-Gordon potential and the hard φ4potential. In addition, we present results from timestepping simulations to illustrate the effect of the spectrum on the dynamical evolution of the system. We end with a brief concluding section, which suggests avenues for future research. 2. Mathematical background We will consider the discrete Klein-Gordon (DKG) equation with on-site nonlinearity f(u) ¨un=d(∆2u)n−f(un),(3) 6 ROSS PARKER, JES ´ US CUEVAS-MARAVER, P. G. KEVREKIDIS, AND ALEJANDRO ACEVES on the integer lattice Z, where t∈Ris the evolution time, un(t)∈Ris the displacement of the nth particle in the lattice, (∆2u)n=un+1 −2un+un−1is the discrete second difference operator, and f(u) = V′(u) for a smooth, on-site potential function V(u). We use the following assumptions for the potential V: (i) Vis an even function, and V(0) = 0. This implies V′(0) = 0. (ii) V′′(0) >0. We note that the Morse potential, which is considered in [40], does not satisfy the first assumption, since it is not an even function. For any time t, we take the displacements {un(t)}n∈Z∈ℓ2(Z), and we denote this sequence by u(t). Existence and uniqueness of solutions to (3) is discussed in [40]. Since f(u) is an odd function, if u(t) is a solution to (3), then −u(t) is as well. Equation (3) is Hamiltonian [20,40], and can be written as d dt un vn=0 1 −1 0∂H/∂un ∂H/∂vn,(4) where vn= ˙unis the velocity of the nth particle in the lattice, and His the conserved energy H(u) = ∞ X n=−∞ 1 2v2 n+d 2(un+1 −un)2+V(un).(5) 2.1. Breathers. We are interested in breather solutions to (3), which are periodic in time and exponentially localized in their spatial profile over the lattice. Specifically, a breather solution is a function u∈ℓ2(Z, H2 per[0, T]), where H2 per[0, T] is the Hilbert-Sobolev space of periodic, real-valued functions on [0, T]. The fundamental period Tis the smallest positive real number for which u(t+T) = u(t) for all t. At the AC limit (d= 0), the individual sites in the lattice are decoupled. At each site, un(t) is a T-periodic solution to the nonlinear oscillator equation ¨ φ+V′(φ) = 0,(6) which has conserved energy E=1 2˙ φ2+V(φ). For fixed energy E, equation (6) has a unique, even solution φ(t), which we will call the fundamental AC breather [39]. This solution satisfies the initial conditions φ(0) = aand ˙ φ(0) = 0, where ais the smallest, positive root of V(a) = E. The fundamental period Tof φ(t) is a function of the energy E, and is given by T(E) = √2Za −a du pE−V(u).(7) The potential V(u) is a hard potential if the period Tdecreases as the energy Eincreases, and a soft potential if Tincreases as Eincreases [39, 40]. Pelinovsky and Sakovich prove the existence of multi-site breathers close to the AC limit, i.e. for dsmall, which are even functions of t[39]. Specifically, for a finite set of lattice sites S={k1,...,kN}, with ki< ki+1, they start with a solution u(0)(t) = N X i=1 σiφ(t)eki(8) at the AC limit, where φ(t) is the fundamental AC breather, ekiis the unit vector for site ki in the integer lattice, and σi=±1 is the phase factor for the oscillator at site ki. Adjacent oscillators are in-phase if σiσi+1 = 1, and out-of-phase if σiσi+1 =−1. They then use the implicit function theorem to prove the existence of a multi-site breather u(d)(t) to (3) for REVISITING MULTI-BREATHERS IN THE DISCRETE KLEIN-GORDON EQUATION 7 sufficiently small d[39, Theorem 1]. The fact that in-phase and out-of-phase structures are the only ones available in discrete Klein-Gordon lattices with nearest-neighbor interactions has been shown in [50]. In that light, we will restrict our considerations to such configurations in what follows. Nevertheless, it is relevant to mention in passing that the examination of so-called phase-shift multi-breathers (with relative phases different than 0 or π) in lattices with interactions beyond nearest-neighbor ones remains an active topic of investigation in Klein-Gordon (and DNLS) settings [51]. 2.2. Linearization. For a specific coupling constant dand fundamental period T, let u(t) be a breather solution to (3). To study the spectral stability of u(t), we linearize equation (3) about u(t) by substituting the perturbation ansatz u(t) + ǫv(t) and keeping terms of order ǫto obtain the linearized equation ¨vn=d(∆2v)n−f′(un)vn,(9) which can be written as the first order linear system d dt vn wn=wn d(∆2v)n−f′(un)vn(10) by letting wn= ˙vn. Since u(t) has period T, it follows from Floquet theory that its spectral stability depends on the Floquet multipliers, which are the spectrum of the monodromy operator M= Φ(0, T), where Φ(s, t) is the evolution operator for (10). If µis a Floquet multiplier, then the corresponding Floquet exponent λ(which is unique modulo 2πi/T) is related to µby µ=eλT . For every Floquet exponent λ, there is a corresponding solution v(t) = eλtw(t) to the linearized equation (9), where w(t) is periodic with period T(see, for example, [52, Lemma 2.1.29]). Substituting this ansatz into (9), we obtain the Floquet eigenvalue problem d(∆2w)n−f′(un)wn−¨wn= 2λ˙wn+λ2wn,(11) where w∈ℓ2(Z, H2 per[0, T]) ⊂ℓ2(Z, L2 per[0, T]), and we use the inner product hu,viℓ2(Z,L2 per[0,T ]) = ∞ X n=−∞ ZT 0 un(s)vn(s)ds (12) on ℓ2(Z, L2 per[0, T]). We can write equation (11) as L(u)w= (2λ∂t+λ2)w,(13) where the linear operator L(u) is defined by the LHS of (11). Since (3) is Hamiltonian, the Floquet exponents must come in quartets λ=±α±βi. It follows that the Floquet multipliers µcan only occur in one of three patterns: a pair {µ, µ}on the unit circle; a pair {µ, µ−1}on the real line; or a quartet {µ, µ, µ−1, µ−1}off of the unit circle. Therefore, the breather solution u(t) is spectrally unstable unless all of its Floquet multipliers lie on the unit circle. There is always a Floquet exponent at 0 (corresponding to the Floquet multiplier µ= 1), since ˙ uis a solution to (11), i.e. L(u)˙ u= 0, which can be verified by differentiating (3) with respect to t. Furthermore, there exists a solution y∈ℓ2(Z, H2 per[0, T]) which solves L(u)y= 2¨ u,(14) and can be chosen to be perpendicular to ˙ uwith respect to the inner product (12) (see [39, Section 3], noting that the corresponding linear operator to L(u) in [39] has the opposite 8 ROSS PARKER, JES ´ US CUEVAS-MARAVER, P. G. KEVREKIDIS, AND ALEJANDRO ACEVES sign). In fact, we can actually compute y(t). Letting ω= 2π/T be the frequency of the breather, and normalizing the period of the breather to 2πby rescaling the time variable to τ=ωT , as in [53], equation (3) becomes ω2∂2 τun=d(∆2u)n−f(un).(15) Differentiating with respect to ω, we obtain d(∆2∂ωu)n−f′(un)∂ωun−ω2∂2 τ(∂ωun) = 2ω∂2 τun.(16) Changing variables back to t, this becomes L(u)∂ωu=2 ω¨ u, thus y=ω∂ωu. 2.3. Continuous spectrum. The continuous spectrum of L(u) is the set of all λfor which the limiting problem d(∆2w)n−f′(0)wn−¨wn= 2λ˙wn+λ2wn(17) has a bounded solution in n. Following the procedure in [40, Section 2.1], the continuous spectrum consists of the bands λ=i±ω(θ)−2πm Tm∈Z ω(θ) = sf′(0) + 4dsin2θ 2θ∈[−π, π] (18) on the imaginary axis. The corresponding Floquet multipliers comprise two bands on the unit circle, which are symmetric about the real axis, and are given by µ= exp ±isf′(0) + 4dsin2θ 2T!θ∈[0, π].(19) Let m0be the largest nonnegative integer such that T−2πm0>0. At the AC limit, the bands consist of the two points µ= exp(±iθ0), where θ0=T−2πm0. For d > 0, the bands stretch from µ= exp(±iθ0) to µ= exp(±iθ1), where θ1=pf′(0) + 4dT −2πm0. Following the analysis in [40, Section 2.2], if T∈(nπ, (n+1)π) for neven, the upper band has positive Krein signature, and the lower band has negative Krein signature. As dis increased, the bands grow towards (−1,0) (see Figure 2, left). The ends of the bands meet when θ1=π, which occurs when d=1 4(1+2m0)2π2 T2−1, at which point they merge into a single band. They meet again when θ1= 0, which occurs when d=1 4(2+2m0)2π2 T2−1, at which point they comprise the entire unit circle. Conversely, if T∈(nπ, (n+ 1)π) for nodd, the upper band has negative Krein signature, the lower band has positive Krein signature, and the bands grow towards (1,0) as dis increased (see Figure 2, right). 2.4. Spatial dynamics. We now reformulate both the DKG equation (3) and the eigenvalue problem (11) using a spatial dynamics approach, as in [21,49]. Let u(t) be a breather solution to (3) with period T, and define U(n) = (u(n),˜u(n)) = (un, un−1). Then equation (3) can be written as the lattice dynamical system U(n+ 1) = F(U(n)),(20) REVISITING MULTI-BREATHERS IN THE DISCRETE KLEIN-GORDON EQUATION 9 (a) (b) Figure 2. Schematic showing continuous spectrum bands on the unit circle for T∈(nπ, (n+ 1)π) with neven (a) and nodd (b). Krein signature of bands is indicated, and bands grow in the direction of the arrow with increasing d. where Fu ˜u= 2u+1 df(u) + 1 d∂2 tu−˜u u!.(21) We note that since fis a odd function, if U(n) is a solution to (20), then −U(n) is as well. The Floquet eigenvalue problem (11) can similarly be written as W(n+ 1) = DF (U(n)) + (2λ∂t+λ2)BW(n),(22) where DF (U(n)) = 2 + f′(u(n)) d+1 d∂2 t−1 1 0 !, B =1 0 0 0.(23) The zero function is an equilibrium solution to (20). At this point, the standard procedure (see, for example, [21, 46, 49]) is to consider U(n) as a homoclinic orbit of the equilibrium at 0. The complication here is that for the difference equation (20) to be well-posed, we require U(n)∈C∞ per([0, T],R2) for all n, rather than just U(n)∈H2 per([0, T],R2), since each application of Finvolves differentiating twice with respect to t, and equation (20) involves applying Fan arbitrary number of times. In essence, the spatial dynamics formulation in equation (20) is more restrictive than the original system. Since C∞ per([0, T]) is not a closed subspace of L2([0, T]), it is not straightforward to adapt the stable manifold theorem and results on exponential dichotomies to this problem, even if DF (0) has the desired spectral properties. As an alternative, we will consider a finite-dimensional approximation, where we project the problem onto a finite-dimensional subspace of L2 per([0, T]). Roughly, this subspace consists of the first MFourier modes of the standard Hilbert basis for L2 per([0, T ]). Although this is a limited result, we note that since we require U(n) to be smooth, its Fourier coefficients will decay exponentially. Since we can take Mas large as we like, this should be a very good approximation. In addition, since many of the numerical simulations are performed using Fourier spectral methods, this theory applies directly to such numerical discretization. Before we formulate our approximation, we prove some important results about the spectrum of DF (0). 16 ROSS PARKER, JES ´ US CUEVAS-MARAVER, P. G. KEVREKIDIS, AND ALEJANDRO ACEVES Remark 1. Substituting yn=ω∂ωqn, we can write (55) as K= ∞ X n=−∞ ZT 0 ˙qn(2ω∂ω˙qn+ ˙qn)dt, (57) which is the solvability condition from [53]. Integrating by parts, we can also write (55) as K= ∞ X n=−∞ ZT 0˙q2 n−2yn¨qndt = ∞ X n=−∞ ZT 0˙q2 n−2ω(∂ωqn)¨qndt. (58) In the AC limit, since the primary breather comprises a single excited site φ(t) at one of the lattice points, equation (58) becomes K=ZT 0˙ φ2−2ω(∂ωφ)¨ φdt =−T2(E) T′(E),(59) by the proof of [39, Lemma 2], where T(E) is defined in (7), noting that −ω∂ωφcorresponds to vin that reference. Corollary 1. If the primary breather QM= (qn, qn−1)has the even spatial symmetry q−n= qn, the matrix Ain Theorem 2simplifies to A=      −a1a1 a1−a1−a2a2 a2−a2−a3a3 ...... am−1−am−1       ,(60) with ai=σiσi+1biand bi=  RT 0˙qNi 2˙qNi 2+1 −˙qNi 2−1dt Nieven RT 0˙qNi−1 2 ˙qNi+3 2−˙qNi+1 2 ˙qNi−3 2dt Niodd. The matrix Ahas one eigenvalue at 0, and the remaining (m−1) eigenvalues {µ1,...,µm−1} are real and distinct. As long as the continuous spectrum does not interfere, there are (m−1) pairs of interaction eigenmodes, which are given by λj=±r−dµj K+O(r−2N)j= 1,...,m−1.(61) These are either real or imaginary by Hamiltonian symmetry. Proof. This follows from Theorem 2 using same argument as in [49, Theorem 5].  Two special cases to consider are the double breather (m= 2), and the multi-breather where all copies of the primary breather are equally spaced. For a double breather, the pair of interaction eigenmodes is given by λ=r2σ1σ2b1d K+O(r−2N).(62) If b1and Khave the same sign, the in-phase double breather is spectrally unstable, and the out-of-phase double breather is spectrally neutrally stable. The reverse is true if b1and K REVISITING MULTI-BREATHERS IN THE DISCRETE KLEIN-GORDON EQUATION 17 have opposite signs. For an m-site multi-breather, if Ni=N1for all i, i.e., the excited sites are equally spaced, then bi=b1for all i. The matrix Afrom (60) reduces to A=b1S, where S=      −σ1σ2σ1σ2 σ1σ2−σ2(σ1+σ3)σ2σ3 σ2σ3−σ3(σ2+σ4)σ3σ4 ...... σm−1σm−σm−1σm       .(63) The eigenvalues of Sdepend only on the phase differences σiσi+1 between consecutive excited sites, and not on the individual phases σi. We note that this matrix reduction of the eigenvalue problem has the same form as that in equation (35) of [39, Lemma 2]. In particular, the matrix Sin (63) and the matrix in that lemma are similar (with similarity transformation given by P= diag(σ1,...,σm)), and thus have the same eigenvalues. The quantity b1corresponds to KNin that lemma. For i= 1,...,m−1, let pi=σiσi+1 be the phase differences between consecutive copies of the primary breather, where pi= 1 indicates in-phase, and pi=−1 indicates out-of-phase. Let n−be the number of negative pi, and n+ the number of positive pi. By [46, Lemma 4] (see also the proof of [39, Lemma 3]), Shas a single eigenvalue at 0, n+positive eigenvalues, and n−negative eigenvalues. If b1and K have the same sign, there are n+pairs of unstable interaction eigenmodes, and n−pairs of neutrally stable interaction eigenmodes. Thus, the only neutrally stable multi-breather is one in which each pair of consecutive copies of the primary breather is out-of-phase. The reverse is true if b1and Khave opposite signs. For multi-breathers in which the excited sites are not equally spaced, the stability pattern depends on the relative signs of each biand K, following [46, Lemma 4]. 5. Numerical results We construct both the primary breather and multi-breathers by numerical parameter continuation from the AC limit using the software package AUTO [57]. To do this, we first choose an energy level E, from which we compute the period Tusing (7), and then we determine the AC breather solution φ(t) by solving (6) with the initial conditions given in subsection 2.1. For the starting solution for the parameter continuation at d= 0, we take uk(t) = ±φ(t) at a finite set of lattice points, and uk(t) = 0 everywhere else. In general, we will take the initial excited sites to be well separated in the lattice, since the results above only hold for well-separated copies of the primary breather. That being said, this numerical method is effective for any starting configuration at the AC limit (see Figure 7, for example, for a double breather comprising two adjacent excited sites). For the parameter continuation, we use the equation d dt un vn=0 1 −1 0∂H/∂un ∂H/∂vn+ǫ∂H/∂un ∂H/∂vn,(64) where His the Hamiltonian defined by (5), and ǫis a small parameter used to break the Hamiltonian structure. The parameter ǫis necessary since homoclinic orbits and periodic orbits in Hamiltonian systems are generally codimension zero phenomena, i.e. they persist as a system parameter (the coupling parameter din this case) is varied; the additional parameter ǫconverts the problem into a codimension one problem (see [58, 59] for more details on this technique). As the coupling parameter dis varied in (64) by numerical 18 ROSS PARKER, JES ´ US CUEVAS-MARAVER, P. G. KEVREKIDIS, AND ALEJANDRO ACEVES parameter continuation, the Hamiltonian-breaking parameter ǫis approximately of order 10−15, thus is negligible. To avoid continuing in the direction of translation symmetry, we add the phase condition h˙uold k, uk−uold ki=ZT 0 vold k(uk−uold k)dt = 0, where uold krefers to the function at the previous continuation step. The index kis chosen to be one of the lattice points at which the initial condition is φ(t). We approximate the integer lattice with a finite lattice of length 2L+ 1, numbered from −Lto L, and we take Dirichlet boundary conditions at the ends of the lattice, i.e. uL+1 =u−L−1= 0. Thus equation (4) becomes an ODE in 4L+ 2 dimensions, with periodic boundary conditions imposed on each variable. Once a breather solution has been found, its Floquet spectrum is found numerically by using Matlab’s eigenvalue solver eig to compute the eigenvalues of the monodromy operator associated with the linearized system (10). Since (10) results from writing a second order linear ODE in 2L+1 dimensions as first order linear system in 4L+2 dimensions, the corresponding eigenvectors are of the form (vn, wn), where vnand wnare (2L+ 1)-dimensional. 5.1. Discrete sine-Gordon equation. We first consider the discrete sine-Gordon equation, where the nonlinearity is given by f(u) = sin u. For all of the plots in the section, unless otherwise noted, we use a lattice size L= 15, and the energy at the AC limit is E= 0.25, which corresponds to a period of T= 7.91. We note that 2π < T < 3π, thus the upper continuous spectrum band has positive Krein signature, and the lower band has negative Krein signature (see Figure 2, left). Figure 3a shows the primary breather solution for d= 0.25 at t= 0, which is symmetric on the spatial lattice. The corresponding Floquet spectrum is shown in Figure 3b. There is a single Floquet multiplier (with multiplicity 2) at 1, which comes from translation symmetry in t. The continuous spectrum bands are a discrete set of points, which is a result of the finiteness of the lattice and of the spatial discretization, and have effectively merged into a single band by this value of d. Figure 3c shows the exponential decay of the L2norm on [0, T] of the primary breather solution un(t) at lattice site n, as nis increased. Figure 3d plots the relative error of this decay rate compared to rM, which is defined in (41). The exponential decay of the coefficients of the Fourier series expansion (in t) of the primary breather is shown in Figure 4a. As expected, since the breather is even in t, only even frequencies are represented in the frequency spectrum. Furthermore, the scaled amplitudes for all frequencies k≥20 are below 10−10, suggesting that the finite dimensional approximation is reasonable. Figure 4b shows the exponential decay of the L2norm difference kQM(n)−Q512(n)kℓ2(Z,L2 per[0,T ]) between the solution QM(n) truncated after MFourier nodes and the solution Q512(n) with 512 Fourier nodes, suggesting that an approximation with 32 Fourier nodes is more than adequate for this set of parameters. Figure 5 shows the initial conditions (solution at t= 0) for the out-of-phase (left) and in-phase (right) double breather for N1= 6 at d= 0.25, together with their Floquet spectra and the eigenfunction corresponding to the interaction eigenmode. The out-of-phase double breather has a pair of interaction eigenmodes on the unit circle, thus is spectrally neutrally stable, while the in-phase double breather has a pair of real interaction eigenmodes off of the unit circle and on the positive real axis, thus is spectrally unstable. This same pattern holds for all N1, both even and odd. We note that for the solvability condition, we have K < 0 for the sine-Gordon potential, and that the quantity b1<0 in (62) for both even and odd REVISITING MULTI-BREATHERS IN THE DISCRETE KLEIN-GORDON EQUATION 19 (a) (b) (c) (d) Figure 3. (a) Initial condition un(0), and (b) Floquet spectrum, for primary breather for d= 0.25 for discrete sine-Gordon. (c) Semilog plot of L2norm of solution un(t) over one period [0, T ] vs. the lattice index nfor varying coupling parameter d. (d) Relative error of exponential decay rate seen in (c) versus that predicted by equation (42). N1. Since Kand b1have the same sign for all N1, this agrees with the stability predictions following equation (62). Figure 6a shows the exponential decay of the Floquet exponents λ(as computed from the monodromy matrix) for in-phase double breathers as the separation distance N1is increased. (Recall that the Floquet exponents λand Floquet multipliers µare related by µ=eλT ). This is similar for out-of-phase double breathers (not shown). The remaining panels in Figure 6 show the relative error between the computation of these Floquet eigenmodes using the formula (62) and the computation of these modes from the monodromy matrix. As expected, the relative error increases with the coupling parameter d, and decreases with the separation distance N1. Figure 7 shows the initial condition un(0) and Floquet spectra for two special two-site breathers, for which the two excited sites at the AC limit are in adjacent lattice nodes. We will call these inter-site centered breathers, in analogy to the inter-site centered soliton in DNLS [60]. As predicted by [40], the in-phase inter-site centered breather 20 ROSS PARKER, JES ´ US CUEVAS-MARAVER, P. G. KEVREKIDIS, AND ALEJANDRO ACEVES (a) (b) Figure 4. (a) Semilog plot showing the decay of the coefficients of the Fourier series expansion for a single-site breather at the center site u0(t) and neighboring site u1(t) for discrete sine-Gordon. The vertical axis is rescaled so that the fundamental frequency has amplitude of 1. (b) Semilog plot showing the L2norm difference between the solution QM(n) truncated after MFourier nodes and the solution Q512(n) with 512 Fourier nodes. Coupling parameter d= 0.25 in both cases. is unstable, and the out-of-phase inter-site centered breather is spectrally stable, at least for small d. Since the two copies of the primary breather in the inter-site centered breather are not well separated, the eigenvalue results of section 4 do not apply. This is a distinguishing feature between the theory presented herein and the earlier results, e.g., of [35, 36]. The latter are well-tailored to this case, while the theory presented herein is suitable for the case of well-separated breathers, as described above. For completeness here, and also per its intrinsic interest, we show in Figure 8 the bifurcation diagram for an out-of-phase double breather, starting from the AC limit. The parameter continuation starts on the lower branch, where the double breather has a pair of interaction eigenmodes on the unit circle (label 1). The upper branch is a double breather whose constituent breathers are out-of-phase, yet each encompasses two in-phase adjacent excited sites (label 7), i.e. two inter-site centered breathers, which we recall leads to instability (Figure 7b); this breather has two pairs of real interaction eigenmodes, in addition to a pair of interaction eigenmodes on the unit circle. The middle branch is an asymmetric double breather, comprising one single-site breather and another intersite-centered breather which is out-of-phase with it (label 8); this breather has one pair of real interaction eigenmodes, and one pair of interaction eigenmodes on the unit circle. There is a corresponding branch (not shown) where the order of the two breathers is reversed. As dis increased along the lower branch, a pair of internal modes appears on the unit circle (label 2); these have opposite Krein signature from the nearby interaction eigenmode. These collide and move off of the unit circle (label 3) in the form of a complex quartet (corresponding to an oscillatory instability). At this point, a second pair of internal modes has appeared on the unit circle, which then passes between the first pair of Floquet modes (not shown). By label 4, the pair of eigenmodes which left the unit circle has rejoined the unit circle. Between label 4 and label 5, the second pair of internal modes collides (with the branch traced by label 8 REVISITING MULTI-BREATHERS IN THE DISCRETE KLEIN-GORDON EQUATION 21 (a) (b) (c) (d) Figure 5. Initial condition un(0) with Floquet spectrum in inset (top) and eigenfunction (vn, wn) corresponding to interaction eigenmode (bottom) for out-of-phase double breather (left) and in-phase double breather (right) for discrete sine-Gordon. Orange solid line in Floquet spectrum plot corresponds to unit circle. Coupling parameter d= 0.25, breather distance N1= 6. and label 9) at (1,0) in a subcritical pitchfork bifurcation and moves off of the unit circle. This pitchfork is symmetry-breaking, and produces the asymmetric, middle branches of the bifurcation diagram, indicated by the dashed line. Between label 5 and label 6, there is a turning point (fold bifurcation), in which the first pair of internal modes collides and moves off of the unit circle. This collision represents a saddle-center bifurcation with the branch corresponding to label 7, designated by the dotted line, a branch that bears generically two real multiplier pairs. We find it quite relevant to recall that qualitatively similar pitchfork bifurcations are seen for multi-kinks in DKG ([21, Figure 4]) and for vortex pairs in the 2-dimensional DNLS equation ([61, Figure 4]). Indeed, this seems to represent a generic mode of disappearance in such discrete nonlinear lattice dynamical systems of states that do not persist all the way to the continuum limit. Figure 9 shows the bifurcation diagram for an in-phase double breather, again starting from the AC limit. The diagram is qualitatively similar, in that two bifurcations (a pitchfork 22 ROSS PARKER, JES ´ US CUEVAS-MARAVER, P. G. KEVREKIDIS, AND ALEJANDRO ACEVES (a) (b) (c) (d) Figure 6. (a) Semilog plot of magnitude of Floquet exponent λcorresponding to interaction eigenmode vs. N1for in-phase double breather with d= 0.1,0.15,0.2,0.25 for discrete sine-Gordon. (b) Semilog plot of the relative error of Floquet multipliers of double breathers vs. N1for the case of coupling parameter d= 0.25. Semilog plot of the relative error of interaction eigenmode computation vs. dfor out-of-phase double breathers (c) and in-phase double breathers (d) for breather distance N1= 4,5,6,7. and a saddle-center) result from collisions of internal modes at (1,0). The main difference is that, since the starting in-phase breather has a pair of real interaction eigenmodes, these are not involved in any collisions with the internal modes. In both cases, there is a turning point (fold bifurcation) in the bifurcation diagram at a critical value d0of the coupling parameter d, at which point the parameter continuation reverses direction in d, again marking a saddlecenter bifurcation. For the out-of-phase double breather, this occurs after the pitchfork bifurcation of the site-centered breathers with the asymmetric ones (see inset in Figure 8). On the contrary, for the in-phase double breather, the pitchfork bifurcation occurs between the branch comprising a pair of inter-site centered breathers and the asymmetric branch bearing one on-site and one inter-site centered breather (see inset in Figure 9). A plot of the REVISITING MULTI-BREATHERS IN THE DISCRETE KLEIN-GORDON EQUATION 23 (a) (b) Figure 7. Initial condition un(0) and Floquet spectrum (inset) for out-of-phase inter-site centered breather (a), and in-phase inter-site centered breather (b) for discrete sine-Gordon. Coupling parameter d= 0.1. Figure 8. Bifurcation diagram plotting energy H(u) vs. dfor out-of-phase double breather in the soft sine-Gordon potential with N1= 6. Points of interest are marked with black dots and labeled with circled numbers, which correspond to Floquet spectra (and in select cases also to the configurations at the bottom left). Continuous spectrum bands are not shown for clarity. Bifurcations are indicated in the inset with a black cross. 24 ROSS PARKER, JES ´ US CUEVAS-MARAVER, P. G. KEVREKIDIS, AND ALEJANDRO ACEVES Figure 9. Bifurcation diagram plotting the energy H(u) vs. dfor in-phase double breather in soft sine-Gordon potential with N1= 6. Points of interest are marked with black dots and labeled with circled letters, which correspond to Floquet spectra. Again, see bottom left for select spatial configurations. Continuous spectrum bands are not shown for clarity. Bifurcations are indicated with a black cross. Figure 10. Turning point of parameter continuation d0vs. separation distance N1for out-of-phase and in-phase double breathers for discrete sine-Gordon. turning point d0vs. the separation distance N1suggests that d0increases linearly with N1 for both the in-phase and the out-of-phase double breather (Figure 10). REVISITING MULTI-BREATHERS IN THE DISCRETE KLEIN-GORDON EQUATION 25 Finally, we demonstrate the effect of the interaction eigenmodes on the dynamics of the double breathers by performing long-term numerical timestepping experiments. For a timestepping scheme, we use a symplectic and symmetric implicit Runge-Kutta method [62] to preserve the symplectic structure of the Hamiltonian system. Specifically, we use the Matlab implementation of the irk2 scheme of order 12 from [63]. For each experiment, we perturb the stationary breather by adding the eigenvector (vn, wn) corresponding to the interaction eigenmode µ, multiplied by a small factor δ. (If the eigenvector is complex, we use the real part of the eigenvector for the perturbation). For a double breather un(t), let uL(t) = u−N− 1(t) and uR(t) = uN+ 1(t) be the center sites (i.e. the sites of maximal excitation) of the left and the right breather, respectively. First, we look at the out-of-phase double breather (Figure 11a). We choose the coupling parameter d= 0.2 and separation distance N1= 4 so that we avoid the (nonlinearity induced) 1:2 resonance between the interaction eigenmode and the continuous spectrum bands, which leads to a nonlinear instability (see [40] for details). Although we expect that nonlinear instabilities due to these higher order resonances will occur, they will be manifested at times beyond the time intervals of our simulations. When the out-of-phase double breather is perturbed, the peak amplitudes of uL(t) and uR(t) oscillate in opposite directions with frequency given by the imaginary part of log(µ)/T (see Figure 11), with relative error less than 0.005, where µis the Floquet multiplier on the unit circle corresponding to the interaction eigenmode. When the in-phase double breather is perturbed in the same way, the oscillations in uL(t) initially increase in both amplitude and period, while those in uR(t) decrease in both amplitude and period (Figure 12d). (These are reversed if the perturbation is in the opposite direction). We can interpret this behavior by looking at the form of the corresponding eigenfunction (Figure 5, right). An addition of a small multiple of this eigenfunction to the double breather at t= 0 causes one breather to increase in both amplitude and velocity (thus in energy), and the other to decrease in both amplitude and velocity. Since sine-Gordon is a soft potential, the breather which increases in energy also increases in period, and the breather which decreases in energy also decreases in period. For separation distance N1= 6, as tevolves, we see that the periods of uL(t) and uR(t) vary periodically in opposite directions (Figure 12c), thus the phase differences between the two breathers vary periodically as well (Figure 12a and Figure 12b). The amplitudes of uL(t) and uR(t) also vary periodically in opposite directions (Figure 12e). In effect, the perturbed in-phase double breather appears to be oscillating about the neutrally stable, out-of-phase double breather, a feature that persists robustly over longer time scales. The growth rate of the difference in ℓ2norm between the perturbed and unperturbed solutions is given by µ1/T , with a relative error of less than 0.001, where µ > 1 is the larger of the Floquet multiplier pair (Figure 12f). If the separation distance N1is smaller, the two breathers attract each other, and eventually merge into a localized, single-site breather state (see Figure 13 for separation distance N1= 4). This limiting solution is close to a higher energy, single-site breather, and is located between the initial starting sites for the pair of breathers. The energy density at lattice site n, which is plotted in the bottom panels of Figure 13, is given by hn=1 2v2 n+V(un) + d 4(un−un+1)2+ (un−un−1)2,(65) where vn= ˙un, so that the Hamiltonian (5) is the sum of the energy densities over the entire lattice. A plot of the total energy vs. time (Figure 13b) indicates that energy is conserved by the numerical scheme over the simulation time, providing validation of this result. 32 ROSS PARKER, JES ´ US CUEVAS-MARAVER, P. G. KEVREKIDIS, AND ALEJANDRO ACEVES Appendix A. Proof of Theorem 2 The proof is a straightforward adaptation of the proof of [49, Theorem 2]. Let Y(n) = (y(n),˜y(n)) = ω∂ωU(n). We take as an ansatz for the eigenfunction W(n) the following piecewise linear combination W± i(n) = ci(˙ U± i(n) + λY ± i(n)) + ˜ W± i(n),(66) where ci∈C, and W± i(n) is defined on the same interval as U± i(n) in (48). Substituting (66) into (40), using the relations ˙ U± i(n+ 1) = DFM(U(n)) ˙ U± i(n) Y± i(n+ 1) = DFM(U(n))Y± i(n) + 2∂tB˙ U± i(n),(67) and simplifying, we obtain the equation ˜ W± i(n+ 1) = DFM(σiQM(n)) ˜ W± i(n) + λ2ciB[2∂tY± i(n) + ˙ U± i(n)] + [G± i(n) + (2λ∂t+λ2)B]˜ W± i(n) + ciλ3BY ± i(n),(68) where G± i(n) = DFM(U± i(n)) −DFM(σiQM(n)).(69) In addition to solving (68), the eigenfunction W± imust satisfy matching conditions at n= ±Niand n= 0. As in [21,46,49], this will in general not be possible. Instead, we solve the system ˜ W± i(n+ 1) = DFM(σiQM(n)) ˜ W± i(n) + λ2ciB[2∂tY± i(n) + ˙ U± i(n)] + [G± i(n) + (2λ∂t+λ2)B]˜ W± i(n) + ciλ3BY ± i(n) ˜ W+ i(N+ i)−˜ W− i+1(−N− i) = Cic ˜ W+ i(0) −˜ W− i(0) ∈CZM(n), (70) where Cic= [ ˙ U− i+1(−N− i) + λY − i+1(−N− i)]ci+1 −[˙ U+ i(N+ i) + λY + i(N+ i)]ci.(71) A solution to (70) is an eigenfunction if and only if the mjump conditions ξi=hσiZM(0),˜ W+ i(0) −˜ W− i(0)i= 0 i= 1,...,m (72) are satisfied. We proceed as in the proof of [49, Theorem 2]. Since DFM(0) is hyperbolic, and kQM(n)kL2 per([0,T ]) ≤Cr−|n| M by the stable manifold theorem, we can adapt the results of [65] to decompose the evolution operator Φ(m, n) of the variational equation (43) in exponential dichotomies on Z±. We then rewrite (68) as a fixed point problem using the discrete variation of constants formula (see, for example, [49, Lemma 3]), and project onto the stable and unstable subspaces of the exponential dichotomy. From there, we follow the steps in [49], using the estimates (51), to REVISITING MULTI-BREATHERS IN THE DISCRETE KLEIN-GORDON EQUATION 33 obtain a unique solution to (70). The jump conditions (72) become ξi=σiσi+1hZM(N+ i),˙ QM(−N− i)i(ci+1 −ci) +σiσi−1hZM(−N− i−1),˙ QM(N+ i−1)i(ci−ci−1) −1 dλ2 ∞ X n=−∞ DZM(n+ 1), B[2∂tY(n) + ˙ QM(n)]E+R(λ)i(c), (73) where the remainder term has the uniform bound kR(λ)(c)kX2 M≤Cr−N M+|λ|3. 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KEVREKIDIS, AND ALEJANDRO ACEVES Department of Mathematics, Southern Methodist University, Dallas, TX 75275, USA Email address:[email protected] Grupo de F´ ısica No Lineal, Departamento de F´ ısica Aplicada I, Universidad de Sevilla. Escuela Polit´ ecnica Superior, C/ Virgen de Africa, 7, 41011-Sevilla, Spain Instituto de Matem´ aticas de la Universidad de Sevilla (IMUS). Edificio Celestino Mutis. Avda. Reina Mercedes s/n, 41012-Sevilla, Spain Email address:[email protected] Department of Mathematics and Statistics, University of Massachusetts, Amherst MA 01003, USA Email address:[email protected].edu Department of Mathematics, Southern Methodist University, Dallas, TX 75275, USA Email address:[email protected]