Energy thresholds for the existence of breather solutions and traveling waves on lattices

Energy thresholds for the existence of breather solutions and traveling waves on lattices. ∗ J. Cuevas, Departamento de Fisica Aplicada I, Escuela Universitaria Polit´enica, C/ Virgen de Africa, 7, University of Sevilla, 41011 Sevilla, Spain N. I. Karachalios, Department of Mathematics, University of the Aegean, Karlovassi, 83200 Samos, Greece F. Palmero, Departamento de Fisica Aplicada I, ETSI Inform´atica, Avd. Reina Mercedes s/n, University of Sevilla, 41012 Sevilla, Spain Abstract We discuss the existence of breathers and of energy thresholds for their formation in DNLS lattices with linear and nonlinear impurities. In the case of linear impurities we present some new results concerning important differences between the attractive and repulsive impurity which is interplaying with a power nonlinearity. These differences concern the coexistence or the existence of staggered and unstaggered breather profile patterns. We also distinguish between the excitation threshold (the positive minimum of the power observed when the dimension of the lattice is greater or equal to some critical value) and explicit analytical lower bounds on the power (predicting the smallest value of the power a discrete breather one-parameter family), which are valid for any dimension. Extended numerical studies in one, two and three dimensional lattices justify that the theoretical bounds can be considered as thresholds for the existence of the frequency parametrized families. The discussion reviews and extends the issue of the excitation threshold in lattices with nonlinear impurities while lower bounds, with respect to the kinetic energy, are also discussed for traveling waves in FPU periodic lattices. 1 Introduction The energy threshold for the formation of a discrete breather in a nonlinear lattice [7], is defined as the positive lower energy bound possessed by the breather. This definition was given in the remarkable paper of S. Flach, K. Kladko and R. MacKay [6]. They considered a generic class of Hamiltonian systems and showed by heuristic and numerical arguments that the energy of a discrete breather family has a positive lower bound for lattice dimension Ngreater than or equal to some critical dimension Nc. On the other hand, when N < Ncthe energy goes to zero as the amplitude goes to zero. It was further predicted that the critical dimension Ncdepends on details of the system but is typically 2 and never greater than 2. Furthermore, for N > Nc, the minimum in energy should occur at positive amplitude and finite localization length. Particular examples concerning the studies of [6] are the discrete nonlinear Schr¨odinger equation DNLS and the nonlinear Klein-Gordon lattice (DKG). For the DNLS equation [5, 12], i˙ ψn+ǫ(∆dψ)n+|ψn|2σψn= 0, σ > 0, n = (n1, n2,...,nN)∈ZN,(1.1) ∗Keywords and Phrases: DNLS lattices, FPU lattices, discrete breathers, traveling waves, energy thresholds, impurities. AMS Subject Classification: 37L60; 35Q55; 47J30. 1 Energy thresholds on lattices 2 the results of [6] were rigorously justified by M. Weinstein in the key work [22]. In (1.1), ǫ > 0 is a discretization parameter ǫ∼h−2with hbeing the lattice spacing, and (∆dψ)nstands for the N-dimensional discrete Laplacian (∆dψ)n∈ZN=X m∈Nn ψm−2Nψn,(1.2) where Nndenotes the set of 2Nnearest neighbors of the point in ZNwith label n. In [22], the existence of the energy excitation threshold was proved through the consideration of the constrained minimization problem IR= inf {H[φ] : P[φ] = R},(1.3) where H[φ] and P[φ] are the fundamental conserved quantities H[φ] = ǫ(−∆dφ, φ)2−1 σ+ 1 X n∈ZN|φn|2σ+2,(1.4) P[φ] = X n∈ZN|φn|2,(1.5) the Hamiltonian and the power, respectively. For instance, it was proved in [22, Theorem 3.1, pg. 678] that if 0 < σ < 2 N, the variational problem (1.3) has a solution for all R>0 and there is no excitation threshold. On the other hand, when σ≥2 N, there exists an excitation threshold Rthresh such that (a) if R>Rthresh then IR<0, and a solution of (1.3) exists and (b) if R<Rthresh then IR= 0, and there is no solution (i.e a ground state minimizer) of (1.3). Actually, when −∞ <IR<0 the infimum in (1.3) is attained. On the other hand, when IR≥0 it follows that IR= 0 (see [22, Proposition 3.1, pg. 679]) and there is no ground state minimizer (see [22, Proposition 4.2 (d), pg. 680]. In the light of the results of Weinstein, the critical dimension predicted by Flach, Kladko and MacKay could be defined from (1.1) as Nc=2 σ.(1.6) However such a definition has a meaning in DNLS when it gives integer values of Nc≥1. Restricted to integer values of σ, we get that Nc= 2 for σ= 1 and Nc= 1 for σ= 2 in consistence with the predictions of [7]. When σ≥3, Ncfrom (1.6) is not an integer, however it suggests that Nc= 1. This implementation for integer values of σis consistent with the predictions of [7] and in equivalence with the definition of the critical exponent (1.1). It is also verified by the numerical studies of [2, 3] and of the present paper. It is important to mention that the critical dimension Nccan be greater than 2 for a large classes of Hamiltonian systems, as it is reported in [11]. The excitation energy threshold concerns “breathers” as time-periodic solutions ψn(t) = eiωtφn, ω > 0, n ∈ZN, t ∈R,(1.7) φn∈ℓ2, spatially localized since φn→0 as |n| → ∞ (as an element of ℓ2-here |n|= max1≤i≤N|ni|). Actually Rthresh corresponds to a minimum on the power attained by the ground-state breather ψn(t) = eiωctφnfor a certain ωc (as a Lagrange multiplier associated to the ground state minimizer φnof (1.3)). A different type of lower bounds on the power of DB for DNLS systems but with the same flavor in applications, was derived in our recent works [2, 3]. Using simple arguments based on variational methods [1] or fixed point theorems [10] to establish the existence of solutions (1.7), we were able to show the existence of explicitly given lower bounds on the power of breathers on either finite or infinite DNLS lattices and for different types of nonlinearities (saturable or power). Although some of them depend explicitly on the dimension, a major difference with the excitation threshold of [6, 22] existing only when N > Nc, is that they are valid for any dimension. Thus they do not predict the energy threhshold of [6, 22] but actually the smallest value of the power a DB can have: no periodic localized solution can have power less than the prescribed estimates. This difference becomes evident in the case of the power nonlinearity and in the case where σ < 2/N, the case where the excitation threshold of [6, 22] do not exists: The numerical studies of [2, 3] established the existence of breathers of small frequencies with power very close to the theoretical estimates, the latter becoming sharp Energy thresholds on lattices 3 especially for small values of σ. On the other hand, when σ > 2/N, the theoretical estimates become sharp for the power of breathers associated with large values of σand large frequencies. Similar findings hold also for the DNLS with saturable nonlinearity. Due to these reasons it is necessary to distinguish these lower bounds from the excitation energy threshold of [6, 22]. The lower bounds are “local” in the sense that they are predicting the smallest value of power a breather can have for any given frequency ω. Thus the explicit lower bounds can be useful as a theoretical estimation from below of the minimal power (or energy) of localized excitations of prescribed frequency ω, for arbitrary values of σ, N. This fact has been justified in the examples considered in [2, 3]: Tracing out the theoretical estimates varying ωit has been justified that the lower bounds are satisfied by the DB families considered (parametrized by frequency) and in a sharp manner in many cases and members of the DB families, especially in the case where the excitation threshold do not appears. Our aim in this work is to discuss possible extensions of our observations on the lower bounds for the energy of DB in two classes of inhomogeneous DNLS systems as well as to other localized structures such as traveling waves in FPU lattices. In the context of the DNLS, an important example is that of the interplay of a focusing power nonlinearity with a linear impurity, discussed in Section 2. Such mechanisms are often responsible for the excitation of impurity modes, which are spatially localized oscillatory states at the impurity sites. Our motivation was initiated from [17] and the numerous physical applications listed (arising from superconductors and the dynamics of electron-phonon interactions, to the defect modes in photonic crystals). We consider both the attractive and the repulsive impurity cases. For the existence of nontrivial breathers, we follow two alternative but equivalent approaches. For the focusing case we follow the variational approach via the mountain pass theorem and prove some first lower bounds on the power. Although the defocusing case can be reduced to the focusing one but with the opposite sign of the impurity (under a staggering transformation), the alternative approach of constrained minimization problems revealed new and interesting conditions on the existence of breathers. For instance, a vast difference between the attractive and the repulsive case is verified: while the repulsive case is associated only with the existence of staggered patterns, in the repulsive case we derive sufficient conditions on the the impurity parameter for the coexistence of both staggered and unstaggered patterns (Theorem 2.7-Remark 2.8). We remark that such a scenario was demonstrated in [9] for Klein-Gordon lattices. As in the mountain pass approach, the constrained minimization of the defocusing case comes together with the derivation of the lower bounds on the power of the staggered patterns in the repulsive case and the coexisting staggered and unstaggered patterns in the attractive one. The numerical studies performed, not only verify that the lower bounds are satisfied but also justify that these exact values can be fairly used as a sharp approximation of the smallest possible value of the power in the lattice as the limiting case of small σ, ω indicate. This limiting case is also in accordance with the observations and analysis of [6], on the behavior of DB families in the small amplitude limit. In the light of the numerical findings of Section 2, we continue in Section 3 the study of the DNLS lattice with nonlinear impurities [14, 15, 16, 21], initiated in [3]. We review the existence proof of the excitation threshold by generalizing the concentration compactness arguments of [22], to the most general case of sign-changing nonlinear impurities. To emphasize the differences between the excitation thresholds and the lower bounds on the power, the latter derived in [3] are also reviewed numerically in the case of nonlinearity exponents where the excitation threshold do not appears. This numerical study verifies that the value of the findings of Section 2, are justified for the DNLS with nonlinear impurities. The particular examples we consider are that of a DNLS lattice with a single and a sign changing impurity. The thresholds of the nature discussed in the present paper can be derived alternatively by using fixed point arguments. This approach has been proved useful for the treatment of the saturable DNLS [2]. Apart from the DNLS lattices, we claim that these methods can be useful in the study of energy thresholds for other interesting localized structures in nonlinear lattices. As a first step to these extensions we derive a lower bound on the energy for traveling waves in Fermi-Pasta-Ulam (FPU) lattices. The lower bound concerns the average kinetic energy of the traveling wave for the standard polynomial anharmonic potential which includes the classical FPU study as a special case. The FPU system is considered in a periodic lattice and the lower bound concerns waves of given speed c > c∗. The value c∗is given explicitly, and as the lower bound, depends also on the number of lattice sites. The condition c > c∗although stronger is in consistency with the restrictions imposed by the analytical results of [8, 23]. The derivation of the lower bound and its discussion is given in Section 4. Energy thresholds on lattices 4 2 Existence of localized modes in finite DNLS lattices with local inhomogeneity. This section is devoted to the DNLS model with a local inhomogeneity studied in [17], supplemented with Dirichlet boundary conditions i˙ ψn+ǫ(∆dψ)n+χnψn+γ|ψn|2σψn= 0,||n|| ≤ K, (2.1) ψn= 0,||n|| > K, (2.2) where ||n|| = max1≤i≤N|ni|for n= (n1, n2,...,nN)∈ZN. Here γ∈Ris the the anharmonicity parameter. Problem (2.1)-(2.2) will be treated by appropriate variatonal methods. For the impurity χ, we consider two possible alternative cases: (A) (Attractive impurity)χn>0,for all n∈ZN K. (R) (Repulsive impurity)χn<0,for all n∈ZN K. The case of a single linear impurity (or single point defect) χn=αδn,n0can be treated with almost identical manner as the attractive and the repulsive impurity. This example will serve for testing numerically the theoretical results which will be presented in this section. Preliminaries. For convenience to the reader, we recall from [2, 3] some preliminary information on various norms and quantities, that will be thoroughly used in the sequel. The finite dimensional problem (2.11)–(2.12) is formulated in the finite dimensional subspaces of the sequence spaces ℓp, 1 ≤p≤ ∞, ℓp(ZN K) = {φ∈ℓp:φn= 0 for ||n|| > K}.(2.3) Note that in the case of the infinite lattice ZN ||φ||q≤ ||φ||p,1≤p≤q≤ ∞ (2.4) 0≤ǫ(−∆dφ, φ)2≤4ǫN X n∈ZN|φn|2.(2.5) For the finite dimensional subspaces it is needless to say that ℓp(ZN K)≡C(2K+1)N, endowed with the norm ||φ||p= X ||n||≤K|φn|p  1 p , and that the well known equivalence of norms , ||φ||q≤ ||φ||p≤(2K+ 1)N(q−p) qp ||φ||q,1≤p≤q < ∞,(2.6) holds. For an 1D-lattice n= 1,...,K, the eigenvalues of the discrete Dirichlet Laplacian −∆dφ=λφ, with φ real, are given explicitly by λn= 4 sin2nπ 4(K+ 1), n = 1,...,K. In the case of the N-dimensional discrete Laplacian, the eigenvalues are: λ(n1,n2,...,nN)= 4 sin2n1π 4(K+ 1)+ sin2n2π 4(K+ 1)+...+ sin2nNπ 4(K+ 1), nj= 1,...,K j = 1,...,N. Clearly, the principal eigenvalue of the discrete Dirichlet problem −∆dφ=λφ, with φreal is given by λ1≡λ(1,1,...,1) = 4Nsin2π 4(K+ 1). Energy thresholds on lattices 5 Restricting the variational characterization of the eigenvalues of the discrete Laplacian in the finite dimensional subspaces ℓ2(ZN K), it follows that λ1>0, can be characterized as λ1= inf φ∈ℓ2(ZN K) φ6= 0 (−∆dφ, φ)2 P||n||≤K|φn|2.(2.7) Then (2.7) implies the inequality ǫλ1X ||n||≤K|φn|2≤ǫ(−∆dφ, φ)2≤4ǫN X ||n||≤K|φn|2.(2.8) From (2.8), we find directly that λ1satisfies the bound λ1≤4N. (2.9) Of course, the upper bound (2.9) follows also from the explicit formula for λ1given above. 2.1 Mountain pass approach in the focusing case γ > 0. In the focusing case γ > 0, substituting the time-periodic solution ψn(t) = eiΩtφn,Ω∈R,(2.10) to (2.1), the stationary analogue of (2.1) reads as −ǫ(∆dφ)n+ Ωφn−χnφn=γ|φn|2σφn,||n|| ≤ K, (2.11) φn= 0,||n|| > K. (2.12) Attractive impurity. Motivated by [17, page 3] and considering the case of the anticontinuum limit ǫ= 0, it follows directly that the attractive impurity (A)only supports time-periodic solutions (2.10) with Ω>0.We have the following Theorem 2.1 (Unstaggered patterns) We consider the DNLS equation (2.1) assuming that (A)is satisfied. If max n∈ZN K χn<Ω,(2.13) there exists a nontrivial φ∈ℓ2(ZN K)such that ψn(t) = eiΩtφn,Ω>0is a solution of the DNLS equation (2.1). Moreover the power of the nontrivial unstaggered periodic solution satisfies the lower bound ǫλ1+ Ω −maxn∈ZN Kχn γ1 σ < R2.(2.14) Proof: We shall follow the variational (minimax) approach of [3, 10]. For instance we seek non-trivial breathers as critical points of C1-functional E:ℓ2→Rdefined as E(φ) = ǫ 2(−∆dφ, φ)2+Ω 2X ||n||≤K|φn|2−1 2X ||n||≤K χn|φn|2−γ 2σ+ 2 X ||n||≤K|φn|2σ+2,(2.15) showing their existence by the mountain pass Theorem (MPT). It can be verified as in [2, 3, 10] that Eis a C1(ℓ2(ZN K),R)-functional. Since we are in a discrete setting it follows directly that any critical point of Eis a solution of (2.1). We proceed by verifying that Epossesses the mountain pass geometry. Since E(0) = 0, we seek next for the existence of z∈ℓ2(ZN K) with ||z||2 2=θ2, satisfying E(z)>0. Now from (2.6), (2.8) and (2.13) we have that E(z)≥ǫ 2(−∆dz, z)2+Ω 2X ||n||≤K|zn|2−maxn∈ZN Kχn 2X ||n||≤K|zn|2−γ 2σ+ 2 X ||n||≤K|zn|2σ+2 ≥ ǫλ1+ Ω −max n∈ZN K χn!θ2 2−γθ2σ+2 2σ+ 2 .(2.16) Energy thresholds on lattices 6 Then, it follows from (2.16) that if condition (2.13) holds, there exists z∈ℓ2(ZN K) with norm ||z||2 2=θ2 satisfying "σ+ 1 γ(ǫλ1+ Ω −max n∈ZN K χn)#1 σ > θ2, for which E(z)>0. We consider next the element ζ=tη ∈ℓ2(ZN K), for some t > 0 and η∈ℓ2(ZN K) with ||η||2= 1 and we observe that E(ζ) = t2 2ǫ(−∆dη, η)2+Ωt2 2−t2 2X ||n||≤K χn|ηn|2−γt2σ+2 2σ+ 2 X ||n||≤K|ηn|2σ+2.(2.17) Taking the limit as t→+∞we get from condition (A), that E(tη)→ −∞. Thus, we derive the existence of ζ∈ℓ2(ZN K) such that E[ζ]<0, by setting tsufficiently large. The final step is to verify that the functional Esatisfies the Palais-Smale (PS) condition. That is, to show that for any sequence {φm}m∈N∈ℓ2(ZN K) such that |E(φm)|is bounded and E′(φm)→0 as n→ ∞, there exists a convergent subsequence. In fact, since ℓ2(ZN K) is finite dimensional it suffices to verify that such a sequence is bounded. Considering a sequence φmof ℓ2(ZN K) be such that |E(φm)|< M′for some M′>0 and E′(φm)→0 as m→ ∞, we get similarly to (2.16), that for msufficiently large M′≥ E(φm)−1 2σ+ 2 hE′(φm), φmi =1 2−1 2σ+ 2ǫ(−∆dφm, φm)2+ Ω||φm||2 2−1 2−1 2σ+ 2X ||n||≤K χn|(φm)n|2 ≥2σ 2(2σ+ 2)(ǫλ1+ Ω −max n∈ZN K χn)||φm||2 2.(2.18) Condition (2.13) is required once again, to get from (2.18) that the sequence {φm}m∈Nis bounded. Therefore the functional Epossesses the geometry required by the MPT and satisfies condition (PS), proving the existence of a nontrivial breather solution (2.10). From (2.1) we get that ǫ(−∆dφ, φ)2+ Ω X ||n||≤K|φn|2−X ||n||≤K χn|φn|2−γX ||n||≤K|φn|2σ+2 = 0.(2.19) Working similarly to (2.16), we have from (2.19) the inequality γ X ||n||≤K|φn|2  σ+1 ≥γX ||n||≤K|φn|2σ+2 ≥ǫ(−∆dφ, φ)2+ Ω X ||n||≤K|φn|2−max n∈ZN K χnX ||n||≤K|φn|2, implying that the power R2=P||n||≤K|φn|2of the critical point satisfies γR2σ+2 ≥(ǫλ1+ Ω −max n∈ZN K χn)R2, from which the estimate (2.14), readily follows.  The results of Theorem 2.1 can be extended to the particular example of the single point defect by similar arguments. Corollary 2.2 (Unstaggered patterns) For the DNLS system (2.1)-(2.2) with a single point defect χn=αδn,n0, α > 0(attractive impurity), the power of the unstaggered periodic solution ψn(t) = eiΩtφn,Ω>0with Ω> α, satisfies the lower bound ǫλ1+ Ω −α γ1 σ < R2, σ > 0.(2.20) Energy thresholds on lattices 7 Remark 2.3 Note that without considering the anticontinuum limit ǫ= 0, the restriction α < Ω + ǫλ1, is derived for the existence of a nontrivial breather solution. However, since the condition holds for any arbitrary ǫ > 0, this fact clearly implies that α < Ω. Condition α < Ωcan be also derived if one considers initially the anticontinuum limit case ǫ= 0. Repulsive impurity. It is interesting that when the impurity is repulsive (R) (cf. [17, page 3]), the case of the anticontinuum limit ǫ= 0 implies the existence of both staggered patterns and unstaggered patterns. Staggered patterns are given in the ansatz ψn(t) = e−iωtφn, ω > 0,(2.21) while the unstaggered ones are (2.10) with Ω >0 (or (2.21) with ω=−Ω<0). It should be pointed out that distinguishing staggered and unstaggered patterns is of physical significance [13]. In the framework of waveguide arrays, the staggered solutions display out-of-phase fields between the neighbor noncentral waveguides (oscillators) whereas the unstaggered ones display in-phase fields in these noncentral waveguides. Working in a very similar manner as for the proof of Theorem 2.1, we have Theorem 2.4 We consider the DNLS equation (2.1) assuming that condition (R)is satisfied. (i) (Staggered patterns) For any ω < −maxn∈ZN Kχnthere exists a nontrivial φ∈ℓ2(ZN K)such that ψn(t) = e−iωtφn,ω > 0is a solution of the DNLS equation (2.1). Moreover the power of the nontrivial staggered periodic solution satisfies the lower bound ǫλ1−ω−maxn∈ZN Kχn Λ1 σ < R2.(2.22) (ii) (Unstaggered patterns). For any Ω>0there exists a nontrivial φ∈ℓ2(ZN K)such that ψn(t) = eiΩtφn,Ω>0 is a solution of the DNLS equation (2.1). Moreover the power of the nontrivial unstaggered periodic solution satisfies the lower bound ǫλ1+ Ω −maxn∈ZN Kχn Λ1 σ < R2.(2.23) In the case of a single point defect with a repulsive impurity we have Corollary 2.5 (i)(Staggered patterns) For the DNLS system (2.1)-(2.2) with a single point defect χn=αδn,n0, α < 0(repulsive impurity), the power of a staggered periodic solution ψn(t) = e−iωtφnwith 0< ω < −α, satisfies the lower bound ǫλ1−ω−α Λ1 σ < R2,−α > 0.(2.24) (ii) (Unstaggered patterns) Any unstaggered solution ψn(t) = eiΩtφnwith Ω>0satisfies the lower bound ǫλ1+ Ω −α Λ1 σ < R2,−α > 0.(2.25) 2.2 Constrained minimization problems for the defocusing case γ < 0 The defocusing case γ=−Λ<0 can be reduced to the focusing one but with the opposite sign of the impurity under the staggering transformation. This transformation is defined as ψn→(−1)|n|ψn,|n|= N X i=1 ni,(2.26) Energy thresholds on lattices 8 (see [13, pg. 066606-7]). We follow this approach in this section which means that we shall consider the defocusing DNLS i˙ ψn+ǫ(∆dψ)n+χnψn−Λ|ψn|2σψn,Λ>0||n|| ≤ K, (2.27) ψn= 0,||n|| > K. The reason for considering alternatively the defocusing problem is that this approach allows for a derivation of new conditions on the impurity χnfor the proof of a coexistence result of staggered an unstaggered solutions. We remark that it was not possible to observe these new conditions by the mountain pass approach of the previous section for the focusing case. On the contrary, the derivation of these new conditions was possible by the consideration constrained minimization problems for the defocusing case, that will be used in the sequel. We are seeking breathers again in the ansatz ψn(t) = e−iΩtφn,Ω∈R.(2.28) Substituting (2.28) in (2.27) we get that φnsatisfies the −ǫ(∆dφ)n−Ωφn−χnφn=−Λ|φn|2σφn,||n|| ≤ K, (2.29) φn= 0,||n|| > K. Attractive case. As it was already mentioned, considering the attractive case with the condition (A) for the impurity is equivalent with the case of the repulsive case for the focusing DNLS (2.1), due to (2.26). However it should be emphasized that the corresponding energy functional do not possesses the mountain pass geometry. Thus we shall consider a constrained minimization problem for the linear energy functional. We start by seeking for staggered patterns (2.28) with Ω >0. The existence of solutions (2.28) with Ω >0, will be given by minimizing the linear energy EΩ[φ] = ǫ(−∆dφ, φ)2−ΩX ||n||≤K|φn|2−X ||n||≤K χn|φn|2,Ω>0.(2.30) More precisely, we consider the minimization problem on ℓ2(ZN K) inf   EΩ[φ] : X |n|≤K|φn|2σ+2 =M > 0   .(2.31) The functional EΩis bounded from below: Consider the ball B=   φ∈ℓ2(ZN K) : X |n|≤K|φn|2σ+2 =M   .(2.32) Then, we have that EΩ[φ]≥ −ΩX |n|≤K|φn|2−X ||n||≤K χn|φn|2 ≥(−Ω−max n∈ZN K χn)C2 2 X ||n||≤K|φn|2σ+2  1 σ+1 = (−Ω−max n∈ZN K χn)C2 2M1 σ+1 . As we are restricted to the finite dimensional space ℓ2(ZN K), it follows that any minimizing sequence associated with the variational problem (2.31) is precompact: any minimizing sequence has a subsequence, converging to a minimizer. Thus Eωattains its infimum at a point ˆ φin B. To derive the variational equation for Eω, we set LR[φ] = X |n|≤K|φn|2σ+2, Energy thresholds on lattices 9 recalling that hL′ R[φ], ψi= (2σ+ 2)Re X |n|≤K|φn|2σφnψ, for all ψ∈ℓ2(ZN K). Then by the Lagrange multiplier rule, DE′ Ω[ˆ φ]−λL′ R[ˆ φ], ψE= 2(−∆dˆ φ, ψ)2−2ΩRe X ||n||≤K ˆ φnψn−2Re X |n|≤K χnˆ φnψn −µ(M)Re X ||n||≤K|ˆ φn|2σˆ φnψn= 0,for all ψ∈ℓ2(ZN K).(2.33) Setting ψ=ˆ φin (2.33), we obtain EΩ[ˆ φ] = (−∆dˆ φ, ˆ φ)2−ΩX |n|≤K|ˆ φn|2−X |n|≤K χn|ˆ φn|2=µ(M) 2X |n|≤K|ˆ φ|2σ+2.(2.34) Furthermore, since EΩ[ˆ φ]≤4ǫN X |n|≤K|ˆ φn|2−ΩX |n|≤K|ˆ φn|2−X |n|≤K χn|ˆ φn|2,(2.35) we get that EΩ[ˆ φ]<0 if Ω>4ǫN −min n∈ZN K χn,(2.36) is satisfied. Note that condition (2.36) is fullfiled for all Ω >0 if 4ǫN < minn∈ZN Kχn. On the other hand, it is required that Ω >4ǫN −minn∈ZN Kχn>0, if 4ǫN > minn∈ZN Kχn. Then assuming (2.36), we find that µ(M)<0. The constant µ(M)/2 can be scaled out from (2.34) by setting ˆ φ= (−µ(M)/2)−1/2σ˜ φ. Scaling further with ˜ φ= (1/Λ))−1/2σφwe get that the non trivial solution of (2.29) satisfies the energy equation (−∆dφ, φ)2−ΩX |n|≤K|φn|2−X |n|≤K χn|φn|2=−ΛX |n|≤K|φ|2σ+2.(2.37) From (2.37), (2.6) and (2.8) it follows that Λ X ||n||≤K|φn|2  σ+1 ≥ΛX ||n||≤K|φn|2σ+2 ≥ −ǫ(−∆dφ, φ)2+ Ω X ||n||≤K|φn|2+ min n∈ZN K χnX ||n||≤K|φn|2 ≥ Ω−4Nǫ + min n∈ZN K χn!X ||n||≤K|φn|2, implying that the power R2=P||n||≤K|φn|2of the critical point satisfies Ω−4Nǫ + minn∈ZN Kχn Λ1 σ < R2, σ > 0.(2.38) The attractive impurity-defocusing case supports also unstaggered patterns. This is compatible with the approach of subsection (2.1). For convenience we set in (2.29) Ω = −ω < 0. The anticontinuum limit suggests that an unstaggered pattern ψn(t) = eiωtφnexists if 0 < ω < minn∈ZN Kχn. Minimizing again the linear energy, instead of (2.36), we require that 0< ω < min n∈ZN K χn−4ǫN, min n∈ZN K χn>4ǫN. (2.39) Energy thresholds on lattices 16 Figure 5: Numerical power against the theoretical estimates (2.43)-(2.44)-(2.45) for 1D and 2D lattices as a function of frequency ωand nonlinearity exponent 0 < σ < 1. Defocusing lattice γ=−1, attractive single point defect α= 0.5. (a) N= 1, ǫ= 0.1 (b) N= 2, ǫ= 0.01. (SC∞) (Sign-changing case) Λn∈ℓ∞and in some S+⊂ZN,{Λn}n∈S+≥0 and in S−:= ZN K\ S+⊆ZN, {Λn}n∈S−≤0, where {Λn}n∈S−6= 0 (not identically the zero sequence in S−). For convenience we write Λnas Λn=−αn, αn≥0n∈ S−⊆ZN, αn6= 0, βn, βn≥0n∈ S+=ZN\S− The existence of unstaggered breathers has been proved via the mountain pass theorem under the same condition to (SC∞) for the sign of Λnbut with the difference that |Λn| ∈ ℓρ, ρ = (q−1)/(q−2) for some q > 2 [10, Theorem 2.6, pg. 125]. This restriction on the summability of Λninduced compactness (by considering auxiliary weighted sequence spaces). Under the relaxed condition (SC∞), the necessary compactness will be recovered by the concentration compactness arguments of [22, Appendix,pg. 689]. Another approach concerning compactness arguments (envelope technique) has been developed in [20]. Under condition (SC∞) we consider the functional H[φ] = ǫ(−∆dφ, φ)2+1 σ+ 1 X n∈ZN Λn|φn|2σ+2,(3.2) and the variational problem on ℓ2 IR= inf (H[φ] : X n∈ZN|φn|2=R > 0).(3.3) We have the following proposition, in analogy with [22]: Lemma 3.1 IR≥0if and only if Rsatisfies the inequality −X n∈ZN Λn|φn|2σ+2 ≤(σ+ 1)ǫR−σ X n∈ZN|φn|2!σ (−∆dφ, φ)2,(3.4) for all φ∈ℓ2. Proof: Let ψ∈ℓ26= 0 be arbitrary. Then z=√R||ψ||−1 ℓ2ψis such that Pn∈ZN|zn|2=R. Since IR≥0 if and only if ǫ(−∆dφ, φ)2≥ − 1 σ+ 1 X n∈ZN Λn|φn|2σ+2,(3.5) Energy thresholds on lattices 17 inserting zin (3.5), we derive as in [22, Proposition 3.1, pg. 679] the inequality (3.4). ⋄ Using (SC∞), we remark that (3.5) is satisfied if Ris such that X n∈S− αn|φn|2σ+2 ≤(σ+ 1)ǫR−σ X n∈ZN|φn|2!σ (−∆dφ, φ)2,for all φ∈ℓ2.(3.6) The next Proposition is a consequence of the condition (SC∞) and [22, Theorem 4.1,pg. 682]: Lemma 3.2 Assuming condition (SC∞), there exists a constant K > 0such that if σ≥2 N, the inequality X n∈S− αn|φn|2σ+2 ≤K X n∈ZN|φn|2!σ (−∆dφ, φ)2,(3.7) holds for all for all φ∈ℓ2. Proof:. By condition (SC∞) and the discrete analogue of the Sobolev-Gagliardo-Nirenberg inequality X n∈ZN|φn|2σ+2 ≤C X n∈ZN|φn|2!σ (−∆dφ, φ)2,(3.8) of [22, Theorem 4.1,pg. 682], we have X n∈S− αn|φn|2σ+2 ≤sup n∈ZN{αn}X n∈ZN|φn|2σ+2 ≤sup n∈ZN{αn}C X n∈ZN|φn|2!σ (−∆dφ, φ)2. Thus K= supn∈ZN{αn}C.⋄ Let us note that under condition (SC∞) the sign of Pn∈ZNΛn|φn|2σ+2 is indefinite. However as it is expected and shown in [3], the existence of unstaggered breathers should be supported mainly by the focusing part of the nonlinearity. Thus it is natural to claim that the existence and the global threshold for unstaggered breathers will be defined from inequality (3.7). Note that (3.7) is a stronger inequality than (3.4), since if φ∈ℓ2 satisfies (3.7), then it satisfies (3.4). In analogy with [22, Eq. 4.2, pg. 680], if K∗is the infimum over all such constants, the excitation threshold denoted by Rsc in the case will be defined for the case σ≥2 Nby the equation (σ+ 1)ǫ(Rsc)−σ=K∗.(3.9) This claim will be justified by a simple treating the defocusing part of the nonlinearity, as well as of the inequality (3.7), on the way to extend the results of [22]. Proposition 3.3 Assume condition (SC∞)and let σ≥2 N. We define Rsc =(σ+ 1)ǫKσ,N 1 σ(3.10) where Kσ,N = inf Pn∈ZN|φn|2σ(−∆dφ, φ)2 Pn∈S− αn|φn|2σ+2 =1 K∗ .(3.11) A. Assume that ||φ||2 ℓ2=R. Then H[φ]≥ǫ(−∆dφ, φ)21−R Rsc σ.(3.12) B. If R < Rsc then IR= 0 and there is no ground state minimizer of (3.3). Proof: A. Using (SC∞) and inequality (3.7) with its best constant K∗we derive that H[φ] = ǫ(−∆dφ, φ)2+1 σ+ 1 X n∈S+ βn|φn|2σ+2 −1 σ+ 1 X n∈S− αn|φn|2σ+2 ≥ǫ(−∆dφ, φ)2−1 σ+ 1 X n∈S− αn|φn|2σ+2 ≥ǫ(−∆dφ, φ)2−ǫ(Rsc)−σRσ(−∆dφ, φ)2, Energy thresholds on lattices 18 thus (3.12). B. Assuming that R < Rthresh, it follows from (3.12) that IR≥0. We consider next some ˜ ψ∈ℓ2such that {˜ ψn}n∈ZN K={˜ ψn}n∈S++{˜ ψn}n∈S−,such that {˜ ψn}n∈S+= 0, {˜ ψn}n∈S−6= 0 ||˜ ψ||ℓ2=√R λ,where λ > 0 arbitrary. Then considering zλ=√R||˜ ψ||−1 ℓ2˜ ψwe observe that ||zλ||2 ℓ2=R and H[zλ] = ǫ(−∆d˜ ψ, ˜ ψ)2−λ2σ+2 σ+ 1 X n∈S− αn|˜ ψn|2σ+2. For λsufficiently large, we get that H[zλ]<0. Therefore if R < Rsc we should have IR= 0. We assume that the infimum is attained at a state ˆ φfor which ǫ(−∆dˆ φ, ˆ φ)2=−1 σ+ 1 X n∈S+ βn|ˆ φn|2σ+2 +1 σ+ 1 X n∈S− αn|ˆ φn|2σ+2,(3.13) X n∈ZN|ˆ φn|2=R. Inserting again inequality (3.7) with its best constant K∗into (3.13) we deduce that ǫ(−∆dˆ φ, ˆ φ)2≤1 σ+ 1 X n∈S− αn|ˆ φn|2σ+2 ≤ǫR Rsc σ < ǫ(−∆dˆ φ, ˆ φ)2.⋄ The last theorem concludes that Rsc is an excitation threshold for σ≥2 N. Theorem 3.4 Assume condition (SC∞)and let σ≥2 N. If R > Rsc then IR<0and there exists a minimizer of the variational problem (3.3). Proof: We shall justify first that if R > Rsc then IR<0: By the definitions (3.10-(3.11) of Rsc and Kσ,N it follows that a φ∗∈ℓ2must exist which do not satisfy inequality (3.7), hence X n∈S− αn|φ∗ n|2σ+2 ≥(σ+ 1)ǫR−σ X n∈ZN|φ∗ n|2!σ (−∆dφ∗, φ∗)2, R > Rsc.(3.14) Then (3.14) clearly implies that X n∈S− αn|φ∗ n|2σ+2 ≥(σ+ 1)ǫR−σ X n∈S− |φ∗ n|2  σ (−∆dφ∗, φ∗)S−, R > Rsc,(3.15) where (·,·)S−, denotes the piece of (·,·)2in the part S−of the lattice. Now we consider the element ζ∈ℓ2, defined as ζ=√R||θ||−1 ℓ2θ, where θn=φ∗ n, n ∈ S− 0, n ∈ S+. Then we observe that H[ζ] = ǫR||θ||−2 ℓ2(−∆dφ∗, φ∗)S−−Rσ+1 σ+ 1||θ||−2σ−2 ℓ2X n∈S− αn|φ∗ n|2σ+2 < ǫR||θ||−2 ℓ2(−∆dφ∗, φ∗)S−−Rσ+1||θ||−2σ−2 ℓ2ǫR−σ X n∈S− |φ∗ n|2  σ (−∆dφ∗, φ∗)S−= 0 Energy thresholds on lattices 19 since ||θ||2 ℓ2=Pn∈S−|φ∗ n|2. Hence IR<0. We proceed by showing the existence of a minimizer. Clearly, IRis bounded form below since H[φ]≥ − 1 σ+ 1 X n∈S− αn|φn|2σ+2 ≥ − 1 σ+ 1 X n∈ZN αn|φn|2σ+2 ≥ − 1 σ+ 1 sup n∈ZN{αn}sup n∈ZN|φn|2σX n∈ZN αn|φn|2 ≥ − 1 σ+ 1Rσ+1. The final step is to prove that every minimizing sequence associated with the variational problem (3.3) is precompact modulo phase translations (cf. [22, Theorem 2.1, pg. 677]. Let {ψm}m∈Na minimizing sequence. Then by the fact that IR<0, we have that H[ψm]≤IR 2,for mlarge enough.(3.16) Then it follows from (3.16) that ||φm||2σ+2 ℓ2sup n∈ZN|Λn| ≥ − X n∈ZN Λn|φm n|2σ+2 ≥ −σ+ 1 2IR>0.(3.17) The lower bound (3.17) shows that the vanishing scenario [22, Theorem 7.1 (2), pg. 689] is excluded for a minimizing sequence. Now we consider a sequence {φm}m∈Nin ℓ2such that ||φm||2 ℓ2→ρ, and H[φm]→ IR.(3.18) The sequence ψm=√R||φm||−1 ℓ2φm, is a minimizing sequence and satisfies (3.17), thus ρ > 0. We prove that ρ=R. (3.19) By contradiction, we suppose that 0< ρ < R. (3.20) We consider the sequences {ζm}m∈N,{ηm}m∈Nsatisfying the dichotomy scenario [22, Theorem 7.1 (3), pg. 689]: This means that for all ǫ > 0, there exists m0≥1 such that for all m > m0 ||ψmk−(ζk+ηk)||ℓ2≤ǫ, ||ζk||2 ℓ2−ρ≤ǫ, (3.21) ||ηk||2 ℓ2−(R−ρ)≤ǫ, with disjoint supports satisfying distance(supp{ζk},supp{ηk})→ ∞. Note that there exists c > 0 such that X n∈ZN Λn|ψmk n|2σ+2 −X n∈ZN Λn|ζk n|2σ+2 −X n∈ZN Λn|ζk n|2σ+2 ≤csup n∈Z|ψmk n−ζk n−ηk n|X n∈ZN|ψmk n|2σ+1 →0,as k→ ∞,(3.22) due to (3.21) and (2.4). Similarly, by (3.21) and (2.5) we can verify that (−∆dψmk, ψmk)2−(−∆dζk, ζk)2−(−∆dηk, ηk)2→0,as k→ ∞.(3.23) Energy thresholds on lattices 20 Using both (3.22) and (3.23), one yields lim k→∞H[ψmk] = lim k→∞H[ζk] + lim k→∞H[ηk],(3.24) lim k→∞H[ζk] + lim k→∞H[ηk] = IR.(3.25) We can easily see that for any φ∈ℓ2and λ > 0, H[φ] = 1 λ2H[λφ]−λ2σ−1 σ+ 1 X n∈S+ βn|φn|2σ+2 +λ2σ−1 σ+ 1 X n∈S− αn|φn|2σ+2 =1 λ2H[λφ]−λ2σ−1 σ+ 1 X n∈ZN Λn|φn|2σ+2. Setting λk=√R ||ζk||ℓ2,µk=√R ||ηk||ℓ2we have ||λkζk||2 ℓ2=Rand ||µkηk||2 ℓ2=Rand H[ζk] = 1 λ2 kH[λkζk]−λ2σ k−1 σ+ 1 X n∈ZN Λn|ζk n|2σ+2 ≥IR λ2 k−λ2σ k−1 σ+ 1 X n∈ZN Λn|ζk n|2σ+2, H[ηk] = 1 µ2 kH[µkζk]−µ2σ k−1 σ+ 1 X n∈ZN Λn|ηk n|2σ+2 ≥IR µ2 k−µ2σ k−1 σ+ 1 X n∈ZN Λn|ηk n|2σ+2. From these inequalities and the definition of IRit follows that H[ζk] + H[ηk]≥ IR1 λ2 k +1 µ2 k−λ2σ k−1 σ+ 1 X n∈ZN Λn|ζk n|2σ+2 −µ2σ k−1 σ+ 1 X n∈ZN Λn|ηk n|2σ+2.(3.26) We shall pass to the limit in (3.26) as k→ ∞: We observe that lim k→∞1 λ2 k +1 µ2 k=ρ R+R−ρ R= 1, as well as, that due to the assumption (3.20) lim k→∞λ2σ k=Rσ ρσ>1, lim k→∞µ2σ k=Rσ (R−ρ)σ>1. Then by using (3.22) we infer that lim k→∞H[ζk] + H[ηk]≥ IR−ξ−1 σ+ 1 lim k→∞ X n∈ZN Λn|ψmk n|2σ+2, ξ = min Rσ ρσ,Rσ (R−ρ)σ. Application of the lower bound (3.17) to ψmkimplies the inequality IR−ξ−1 σ+ 1 lim k→∞ X n∈ZN Λn|ψmk n|2σ+2 ≥ IR−ξ−1 2IR>IR, since IR<0. In conclusion, lim k→∞H[ζk] + H[ηk]>IR, which contradicts (3.25). Henceforth the dichotomy scenario must be excluded. ⋄ Energy thresholds on lattices 21 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Ω PΩ Nonlinear impurity. σ=2. N=1. ε=0.25 1.1 1.2 1.3 1.4 1 1.01 1.02 1.03 1.04 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 0 0.5 1 1.5 2 2.5 Ω PΩ Nonlinear impurity. σ=10. N=1. ε=0.25 1.5 2 2.5 3 3.5 1.09 1.1 1.11 1.12 1.13 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Ω PΩ Sign−changing nonlinearity. σ=1. N=2. ε=0.15 1.26 1.28 1.3 1.32 1.34 1.36 0.79 0.795 0.8 0.805 0.81 0.815 0.82 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 0 0.5 1 1.5 2 2.5 Ω PΩ Sign−changing nonlinearity. σ=2. N=2. ε=0.15 1.3 1.4 1.5 1.6 1.7 1.04 1.05 1.06 1.07 1.08 1.09 1.1 Figure 6: Numerical power for single site breathers centered at the nonlinear impurity site for N= 1,2. (a) Single nonlinear impurity σ= 2, N= 1 (σ=σcrit), (b) Single nonlinear impurity σ= 10, N= 1 (σ > σcrit), (c) Sign-changing impurity σ= 1, N= 2 (σ=σcrit), (d) Sign-changing impurity σ= 2, N= 2 (σ > σcrit). The insets show magnifications of the region where the power of periodic solutions attains the excitation threshold Rsc of Theorem 3.4. Energy thresholds on lattices 22 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Ω PΩ Nonlinear impurity. σ=1. N=3. ε=0.05 0.65 0.7 0.75 0.8 0.4 0.45 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Ω PΩ Nonlinear impurity. σ=2. N=3. ε=0.05 0.65 0.7 0.75 0.8 0.85 0.7 0.72 0.74 0.76 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Ω PΩ Sign−changing nonlinearity. σ=1. N=3. ε=0.05 0.62 0.64 0.66 0.68 0.7 0.72 0.38 0.39 0.4 0.41 0.42 0.43 0.44 −1 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Ω PΩ Sign−changing nonlinearity. σ=2. N=3. ε=0.05 −0.14 −0.12 −0.1 −0.08 −0.06 0.695 0.7 0.705 0.71 Figure 7: 3D DNLS lattices with nonlinear impurities. (a) Numerical power for single site breathers centered at the nonlinear impurity site for the single nonlinear impurity σ= 1, N= 3 (σ > σcrit), (b) for the single nonlinear impurity σ= 2, N= 3 (σ > σcrit), (c) for the sign-changing impurity σ= 1, N= 3. (d) Numerical power for single site unstaggered breathers far from the sign-changing impurity σ= 2, N= 3. The insets show magnifications of the region where the power of each breather family attains the excitation threshold Rsc of Theorem 3.4 in the 3D case. 3.1 Numerical studies in the case of nonlinear impurities In this section we present numerical studies for the DNLS lattices with nonlinear impurities with a twofold purpose. The first aim is to present the numerical verification on the existence of the excitation threshold. The second is to review the lower bounds derived in [3], emphasizing the differences between the lower bounds and the excitation thresholds. The present study is also extended to 3D-lattices, thus covering all the cases of physical significance with respect to the dimension. The models we consider are those of a single nonlinear impurity, Λn=δn,0[14, 15, 16, 21], and of a sign-changing anharmonic parameter. In the latter, we have considered the simplest case where all sites have Λn= 1 except for one of them which is −1. The panel of pictures in figure 6, demonstrates the results of the numerical study for 1D and 2D lattices. For these cases, the critical exponent σcrit = 2/N for the nonlinearity exponent σis σcrit = 2,for N= 1 σcrit = 1,for N= 2, and the study concerns cases σ≥σcrit where the excitation threshold appears. This numerical study gives the numerical verification of Theorem 3.4 on the generalization of the existence of the excitation threshold in the Energy thresholds on lattices 23 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 0 0.5 1 1.5 2 2.5 Ω PΩ Sign−changing nonlinearity. σ=0.1. N=1. ε=0.25 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 0 0.5 1 1.5 2 2.5 Ω PΩ Sign−changing nonlinearity. σ=0.1. N=2. ε=0.15 −30 −20 −10 0 10 20 30 −0.04 −0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 n φn Sign−changing nonlinearity. σ=0.1. N=1. ε=0.25. Ω=1.28 −10 −5 0 5 10 −10 −5 0 5 10 −0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 n1 Sign−changing nonlinearity. σ=0.1. N=2. ε=0.15. Ω=1.32 n2 φn1,n2 Figure 8: Numerical power for single site staggered breathers centered at the nonlinear impurity site, for the DNLS with sign-changing impurity for σ= 0.1< σcrit. (a) N= 1, (b) N= 2. The lower bound Rlb(Ω) is fulfilled. (c), (d) Breather profiles with power close to the lower bound. Energy thresholds on lattices 24 case of nonlinear impurities. The inset in each picture demonstrates the positive lower energy bound Rsc of Theorem 3.4. The numerical power as well as the excitation threshold Rsc has been calculated for staggered breathers of the defocusing DNLS. The numerical power has been drawn against the theoretical lower bound Rlb := Ω−4Nǫ maxn∈S+{Λn}1 σ < R2,Ω>4ǫN, σ > 0,(3.27) derived in [3, Theorem 3.1, pg. 77], and justifies that Rlb is fulfilled for breathers with frequency Ω >4ǫN. Picture (b) indicates that Rlb(Ω) is becoming a satisfactory estimate for large σand for large values of frequencies Ω, as an Ω-dependent lower bound for existence. The effectiveness and strength of the theoretical explicit lower bounds become even more apparent in the 3D-lattices. In Figure 7, the numerical study of (3.27) for the cases N= 1,2 in [3] is extended to the case N= 3. For the 3D-lattice, the critical exponent is σcrit =2 3,for N= 3. The examples considered σ= 1,2 correspond to the supercritical case σ > σcrit and the insets in each picture justify numerically the appearance of the excitation threshold Rsc in the three dimensional lattice. The range of frequencies 0.8<Ω<1 in pictures (a), (b), (c), corresponds to the members of the staggered breather family with power approaching the lower bound Rlb(Ω) with increased accuracy and in a quite sharp manner for σ > σcrit. The insets in each picture justify numerically the appearance of the excitation threshold Rsc in the three dimensional lattice. To elucidate further the difference between the excitation threshold Rsc and the lower bound Rlb(Ω), let us remark that the example of frequencies 0.8<Ω<1 corresponds to frequencies Ω>Ωthresh =the frequency of the minimizer on which the excitation threshold Rsc is attained. In the example (c), it is found that Rsc = 0.6968 at Ωthresh = 0.69. Picture (d) in particular, shows the numerical power of the family of unstaggered breathers which are infinitely far from the sign-changing impurity against the theoretical estimate ˆ Rlb(Ω) = ǫλ1−Ω −minn∈S−{Λn}1 σ < R2,Ω∈(−∞, ǫλ1), σ > 0, of [3, Remark 3.2, pg. 80]. Estimate (3.28) is valid for any unstaggered breather ψn(t) = e−iΩtφn, for any Ω <0. The range of frequencies −0.5<Ω<−0.2 in (d), corresponds to the members of the unstaggered breather family far from the impurity with power very close to the lower bound ˆ Rlb(Ω). For this breather family it is found that Rsc = 0.6970 at Ωthresh =−0.09. We remark here that a complete study of ˆ Rlb(Ω) requires also the study of the unstaggered breathers which are centered at the site adjacent to the impurity, since our numerical studies revealed that there is a range of frequency for which the smallest power breather is attained for this kind of breather family. Figure 8 considers the example of DNLS lattices with sign-changing impurity in the case σ= 0.1, a “limiting case” of σ < σcrit where the excitation threshold do not exist. This study shows that although there is no excitation threshold Rsc, breathers with Ω >4ǫN have power fulfilling Rlb(Ω). Breathers with frequency 1<Ω<1.3 and 1.2<Ω<1.4 in the 1D and 2D case respectively, are the real examples with power approaching sharply to the theoretical lower bound Rlb(Ω), justifying that this bound can be considered as a theoretical prediction of the smallest power for any breather family. Pictures (c) and (d) show two members of these families. In Figure 9, where the 3D case is considered, we observe that the range of frequencies for breathers close to the lower bound is increased. The results of the numerical study can be summarized in Figure 10, where the power of staggered breathers has been plotted as a function of Ω and σ. The upper (blue) surface corresponds to the numerical power while the lower (red) surface to the lower bound (3.27). The comparison between both surfaces makes evident the effectiveness of the theoretical estimates as a local prediction with respect to the frequency, of the smallest power for breathers in DNLS lattices with nonlinear impurities. 4 Lower bounds on the kinetic energy of traveling waves of periodic FPU lattices In this section, we prove the existence of a lower bound on the kinetic energy for the existence of traveling waves of prescribed speed, on a finite lattice, supplemented with periodic boundary conditions. For instance, we shall Energy thresholds on lattices 25 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Ω PΩ Sign−changing nonlinearity. σ=0.1. N=3. ε=0.05 0.8 0.82 0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 1 0 0.005 0.01 0.015 0.02 0.025 Ω PΩ Sign−changing nonlinearity. σ=0.1. N=3. ε=0.05 Figure 9: (a) Numerical power for single site staggered breathers centered at the nonlinear impurity site, for the DNLS with sign-changing impurity for σ= 0.1< σcrit for N= 3. (b) Magnification for the range of frequencies 0.8<Ω<1. consider the equation of motion ¨qn=V′(qn+1 −qn)−V′(qn−qn−1),(4.1) where qndenotes the displacement of the nth particle from its equilibrium position, and V(qn+1 −qn) is the anharmonic interaction potential. The Hamiltonian of system (4.1) is H=X n1 2˙q2 n+V(qn+1 −qn).(4.2) A solitary wave of (4.1) is a solution of the form qn(t) = u(n−ct) = u(z),with u(z)→0 as |z| → ∞.(4.3) Substitution of (4.3) in (4.1) results in the forward-backward differential-difference equation c2u′′(z) = V′[u(z+ 1) −u(z)] −V′[u(z)−u(z−1)].(4.4) We refer to the monograph [18] for rigorous results on the analysis of periodic oscillations and traveling waves of FPU lattices. In this first investigation will shall consider the case of a periodic lattice of 2Lparticles, −L≤n≤L. This case is of the same importance for numerical simulations with periodic boundary conditions, since the infinite lattice cannot be modeled numerically. We shall consider the case of the potential V(u) = γ2u2 2+α2up p, γ ≥0, p > 2,(4.5) which covers the case of the classical Fermi, Pasta and Ulam study (FPU). In the FPU study p= 3 or 4, [4]. It is important at this point, to remark some important results on the existence of traveling waves for the system (4.1). In the case of the potential (4.5) the existence of non-trivial traveling waves in infinite lattices, has been resolved through the study of (4.4) by variational methods in [8, 23].The approach of [8] is based on minimizing the average kinetic energy T(u) := 1 2ZR u′(z)2dz, (4.6) subject to the constraint RRV[u(z−1) −u(z)]dz =K(prescribed average potential energy). The speed cof the wave is given by an unknown Lagrange multiplier. The approach of [23] is based on the application of the