Escape dynamics in the discrete repulsive φ4 model

Escape Dynamics in the Discrete Repulsive φ4-Model V. Achilleos1, A. ´ Alvarez2, J. Cuevas3, D. J. Frantzeskakis1, N. I. Karachalios4, P. G. Kevrekidis5, B. S´anchez-Rey3 1Department of Physics, University of Athens, Panepistimiopolis, Zografos, Athens 15784, Greece 2Grupo de F´ısica No Lineal. Universidad de Sevilla. ´ Area de F´ısica Te´orica. Facultad de F´ısica, Avda. Reina Mercedes, s/n, 41012 Sevilla, Spain 3Grupo de F´ısica No Lineal. Universidad de Sevilla, Departamento de F´ısica Aplicada I, Escuela Polit´enica Superior, C/ Virgen de Africa, 7, 41011 Sevilla, Spain 4Department of Mathematics, University of the Aegean, Karlovassi, 83200 Samos, Greece 5Department of Mathematics and Statistics, University of Massachusetts, Amherst MA 01003-4515, USA (Dated: October 23, 2012) We study deterministic escape dynamics of the discrete Klein-Gordon model with a repulsive quartic on-site potential. Using a combination of analytical techniques, based on differential and algebraic inequalities and selected numerical illustrations, we first derive conditions for collapse of an initially excited single-site unit, for both the Hamiltonian and the linearly damped versions of the system and showcase different potential fates of the single-site excitation, such as the possibility to be “pulled back” from outside the well or to “drive over” the barrier some of its neighbors. Next, we study the evolution of a uniform (small) segment of the chain and, in turn, consider the conditions that support its escape and collapse of the chain. Finally, our path from one to the few and finally to the many excited sites is completed by a modulational stability analysis and the exploration of its connection to the escape process for plane wave initial data. This reveals the existence of three distinct regimes, namely modulational stability, modulational instability without escape and, finally, modulational instability accompanied by escape. These are corroborated by direct numerical simulations. In each of the above cases, the variations of the relevant model parameters enable a consideration of the interplay of discreteness and nonlinearity within the observed phenomenology. I. INTRODUCTION a. The escape problem for systems of coupled objects. The escape problem for single particles of coupled degrees of freedom or for small chains of coupled objects, concerns their exit from the domain of attraction of a locally stable state [17, 18]. Its investigation is crucial for the understanding of evolutionary processes in natural sciences, representing the transition from a stable state to another, or to extinction and collapse. Transition state theory deals exactly with this kind of problem by referring to the case where the considered objects, initiating from the domain of attraction of a locally stable state, escape to a neighboring stable state crossing a separating energy barrier. A natural example of this sort involves a bistable potential, where the barrier corresponds to the height of the saddle point separating the two minima of the potential. This mechanism has been widely explored in numerous physical settings, including (but not limited to) phase transitions, chemical kinetics, pattern formation and the kinematics of biological waves – see, e.g., Refs. [14, 31, 32, 38] and references therein. Depending on the geometry of the underlying potential energy landscape which may possess multiple minima, topological excitations with a “kink shape” (which can be thought of as instantons) interpolating between the adjacent minima may exist and play a critical role in the study of transition probabilities. Another example of such an escape process is the one whereby the considered objects, initiating within the basin of attraction of a locally stable state, escape towards infinity. This process is often termed collapse and it is associated with the emergence of finite time singularities for the corresponding dynamical system. Collapse or blow-up in finite time is quite widespread as an instability mechanism in diffusion processes and wave phenomena in nonlinear media [5, 36, 40–42]. Generally, the collapse phenomenon features the domination of nonlinearity when the initial data are sufficiently large; however, the escape process leading to collapse may arise in more subtle cases where the initial data may be contained even deeply within the domain of attraction of the stable state. This process indicates the existence of interesting energy exchange and transfer mechanisms between the interacting objects. We remark that, generally, collapse does not occur in integrable systems related to solitons or intrinsic localized modes [10]. As concerns discrete systems, the most prototypical among the known integrable differential-difference equations, namely the Ablowitz- Ladik and the Toda lattices, also do not exhibit collapse behavior [1, 44]. In both the continuum and the discrete cases this feature can be associated with the presence of an infinity of conservation laws, which seems highly restrictive towards the possibility of collapse-type events. A seminal work examining the underlying energy mechanisms supporting escape is the work of Kramers [23, 30], implying that external, mainly stochastic, energy sources generate energy dissipation and fluctuation enabling the escape process. These mechanisms have been investigated in many variants of damped and driven systems and the theory of thermally activated escape has been generalized to systems with many degrees of freedom, with numerous 2 physical applications [11, 18]. b. Deterministic escape for nonlinear Hamiltonian systems of coupled oscillators. The possibility of an unforced, purely deterministic Hamiltonian system realizing an escape process from the region of phase space sustaining oscillations around a stable fixed point will be our focal theme herein. This nonlinear dynamics scenario has been investigated in Refs. [15, 19, 20] for a system of coupled harmonic oscillators subjected to a nonlinear external potential, namely: ¨ Un−(Un+1 −2Un+Un−1) + W′(Un) = 0.(1.1) The system (1.1), known as the Discrete Klein-Gordon (DKG) equation, is a fundamental model used in different contexts, including crystals and metamaterials, ferroelectric and ferromagnetic domain walls, Josephson junctions, nonlinear optics, complex electromechanical devices, the dynamics of DNA, and so on [10, 12, 13, 21, 27]. In Refs. [19, 20] the case of a cubic potential of the form: W(U) = ω2 d 2U2−β 3U3, β > 0,(1.2) is considered. This is a prototypical example of potentials which may support collapse, since it possesses a metastable equilibrium and becomes negatively unbounded after crossing its maximum. Importantly, it may support escape dynamics associated to collapse. Here, collapse will be taken to mean that one or more of the oscillators acquire an amplitude U→ ∞. For instance, it is established [19, 20] that an initially almost uniformly supplied energy, associated with an almost flat mode, may progressively be redistributed internally, creating unstable growing nonlinear modes, and eventually concentrated within a few units forming a localized mode that may overcome in amplitude the height of the potential well. More precisely, the whole chain is initialized by an appropriate randomization to an almost homogeneous state enabling non-vanishing interactions and the exchange of energy among the units. The total energy of this initial state is greater than the potential energy of the saddle which defines the depth of the well; however, the position of each of the oscillators within this state is close to the bottom of the respective well and thus the energy of each unit is significantly below the energy barrier of the saddle. Due to the emergence of modulational instability [29], localized excitations are formed and the total energy becomes concentrated in confined segments of the chain. This energy exchange mechanism results in the emergence of a critical mode in which one of the lattice units overcomes the saddle point barrier. c. Model and principal aims. In the present work, the collapse instability, as well as the above escape scenario will be examined by both analytical (based mainly on analysis techniques involving differential and algebraic inequalities for suitable norms of the system) methods and selected numerical experiments that will be used to complement/corroborate the analytical results. Our model will be the following Klein-Gordon chain, characterized by a quartic potential and a dissipative term: ¨ Un+γ˙ Un−(Un+1 −2Un+Un−1) + ω2 d(Un−βU3 n) = 0, β > 0, γ ≥0.(1.3) The lattice is assumed to be infinite (n∈Z) and we consider initial conditions: Un(0) and ˙ Un(0) ∈ℓ2,(1.4) thus the results refer to spatially localized solutions (as the definition of the phase space ℓ2directly implies). In some cases, especially in numerical simulations, we will consider Dirichlet or periodic boundary conditions. When the dissipation parameter γ= 0, the DKG system (1.3) describes the equations of motion derived by the Hamiltonian: H(t) = 1 2 +∞ X n=−∞ ˙ U2 n+1 2 +∞ X n=−∞ (Un+1 −Un)2+ω2 d 2 +∞ X n=−∞ U2 n−βω2 d 4 +∞ X n=−∞ U4 n,(1.5) while the case γ > 0 corresponds to the linearly damped analogue of the problem. Equation (1.3) implies that each individual oscillator of unit mass evolves within the quartic on-site potential W(U) = ω2 d 2U2−βω2 d 4U4.(1.6) The repulsive φ4-potential (1.6), illustrated in Fig. 1(a), has a stable minimum at Umin = 0, corresponding to the rest energy Emin =W(0) = 0, and two unstable maxima located at U∓ max =∓1 √β, corresponding to Emax =W∓1 √β= ω2 d 4β. 3 −2 −1 0 1 2 0 5 10 15 20 25 U W(U) ωd=100 ωd=10 −2 −1 0 1 2 −2 −1 0 1 2 U ˙ U Figure 1: (a) The quartic on-site potential (1.6) for β= 1 and two different values of ωd(ω2 d= 10 and ω2 d= 100). (b) The phase plane for the single repulsive Duffing equation ¨ U+ω2 dU−ω2 dβU3= 0 for β= 1 and ω2 d= 3.2. Blow-up occurs only for initial positions outside the potential well (i.e., for all orbits beyond the heteroclinic orbit that is connecting the two saddles). System (1.3) shares similar phenomenological properties with those of simplified models for DNA dynamics [2, 8, 9]. To be more specific, the prototypical similarities are the following. The case β > 0 is the physically relevant case for the study of energy-localization mechanisms in hydrogen-bonded crystals or DNA molecules, since in this type of models the potentials soften for large amplitudes [29]. In particular, the potential (1.6) (which is also known as “soft quartic potential”) may correspond to the hydrogen-bridge bond between nucleotides. As it is typical in the modeling of chemical bonds [9, Section 2.1, pg. 269], (1.6) represents qualitatively a potential with a hard repulsive part and a softer attractive part, and it diverges as U→ ∞. The region [U− max, U+ max], which corresponds to the potential well of depth Emax, describes the region of amplitudes below the breaking of the chemical bonds. Here, the escape problem can simply be described as escaping from the potential well (towards infinite values of the field) by crossing the saddle points U∓ max in the configuration space. In order to have escape of particles over the energy barrier Emax to the region U > Umax, a sufficient amount of energy has to be supplied. Regarding the large-time behavior of solutions, the escape leads to blow-up of solutions in finite time. d. Structure of the presentation and main findings. The paper organization and the main results can be described as follows. Our aim is to progressively examine the potential blow-up of initial conditions involving more oscillators. First, in Section II, we study the collapse problem for the Hamiltonian system (γ= 0), i.e., the formation of finite time singularities, and find the following. •Since the system can be seen to belong in the class of second-order evolution equations of divergent structure, it can be naturally treated, with respect to global non-existence, by the energy-type methods of [16]. In particular, using differential and algebraic inequalities we derive conditions under which suitable norms (such as the ℓ2norm) will diverge in finite time. •Focusing on initial conditions of a single excited unit, with the other units initially located at the minimum Umin = 0, we derive an analytical value for the initial position of the single unit outside the potential well, which serves as a sufficient condition (threshold) for global non-existence. This value depends strongly on the parameter ω2 dcontrolling the interplay of discreteness and nonlinearity. Our numerical simulations demonstrate that, indeed, blow up occurs when the analytical prediction is satisfied and a concerted escape may follow when we are in a regime where the coupling effect is significant. This is the “drive over” phenomenon, where a particular oscillator drives its neighbors over the collapse barrier. A reverse phenomenon that we also identify numerically and explore herein is the “pull back” effect, where the attraction of an initially escaped unit by the neighbors back to the stability domain, leads to solutions which exist globally in time. •A detailed numerical analysis of the single excited site initial data reveals the existence of a “true” threshold for the position of the excited unit outside of the well, which acts as a separatrix for the above two different dynamical behaviors. For this threshold, the ω2 d-dependent analytical prediction that we analytically identify (for collapse) serves as an upper bound; the relevant comparison of the analytical and true (numerically obtained) threshold is analyzed in the different limits of the discreteness parameter. •We also examine the global existence result of [16, Lemma 2.1, pg. 455] in view of the Hamiltonian system (1.3) (γ= 0). We find the following: assuming negative initial time derivative of the ℓ2norm along with a 4 non-positive initial Hamiltonian energy fails to ensure global existence. The numerical results seem to fully support this concern. Next, Section III deals with the collapse problem for the linearly damped system (1.3) for γ > 0. Using the abstract energy methods developed in Ref. [35] (having the advantage that they take into account the geometry of the potential energy), we find the following. •We derive a γ-independent prediction for the position threshold outside the potential well for the single excited unit. The numerical experiments of section III A verify the relevance of this criterion and that, even for significantly large values of γ > 0, the induced friction does not modify the scenarios found in the conservative case. In particular, friction does not affect the “pull back” effect by increasing the threshold value (as might intuitively be expected). •The γ-independence of the threshold criterion, illustrates that only the strength of the binding forces are responsible for the “pull back” or “drive over” effects. To this end, the criterion is also applied successfully to the case where γ= 0, yielding an alternative and improved upper bound to the threshold for collapse or stability. This is also corroborated by our numerical simulations. In Section IV, we continue our analysis by studying the escape dynamics for multi-site excitations of the Hamiltonian system (γ= 0). Here, we extend our arguments for a single-site excitation to the case of a few and ultimately to many site excitations, but this time, positioned inside the potential well. Our results in this setting are as follows. •We derive an analytical threshold for collapse based on the initial “energy” of a short-length lattice-segment, depending on the discreteness parameter ω2 d. The numerical results (based on a three excited units initial configuration) validate the theoretical expectations. The threshold separates the remaining of the segment in the potential well from its escape which, in turn, is leading to collapse. •Combining a violation of a small initial data global existence result (which can be proved by implementing the methods of Ref. [6]), with the modulational instability analysis (based on the discrete nonlinear Schr¨odinger (DNLS) approximation [29]), we reveal the existence of three different regimes for the amplitude of a plane wave initial configuration, distinguishing between the following dynamical behaviors: modulational stability, modulational instability without escape and finally to modulational instability combined with escape. We also examine how the violation of a small data global existence criterion connects to the numerical amplitude value beyond which modulational instability leads to escape. •For the linearly damped DKG chain, we perform numerical simulations which reveal a transient modulation instability regime for small values of damping. Nevertheless, the instability is completely suppressed eventually, in accordance with the predictions of the damped DNLS approximation. Finally, Section V offers a discussion and a summary of our results. The complementary appendices section contains the proofs of the global nonexistence and global existence results which have been used in this paper for the analysis of the escape dynamics. e. Notation. We shall use when convenient, the short-hand notation {Un(t)}n∈Z=U(t)∈ℓ2,{Un(0)}n∈Z= U(0) ∈ℓ2and {˙ Un(0)}n∈Z=˙ U(0) ∈ℓ2, for the solution for t > 0 and the initial conditions at t= 0 respectively. In this notation, ∆2stands for the one-dimensional discrete Laplacian {∆2U}n∈Z=Un+1 −2Un+Un−1, defined on ℓ2. We shall also denote by (U, Y )ℓ2= +∞ X n=−∞ UnYn,||U||2 ℓ2= +∞ X n=−∞ U2 n, the squared-ℓ2inner product and norm respectively. II. THE CASE OF THE HAMILTONIAN SYSTEM We may start discussing the conditions for the existence of finite-time singularities, and their relevance to the problem of escape dynamics. To this end, we have reviewed and implement in the discrete case the method of Ref. [16], on the derivation of a differential inequality for the norm x(t) := ||U(t)||2 ℓ2, and verify that it blows-up in finite time under appropriate sign conditions on the initial Hamiltonian and the time derivative of the norm. 5 Theorem II.1 We assume that the initial data (1.4) for the system (1.3) are such that H(0) ≤0and (U(0),˙ U(0))ℓ2> 0. Then Tmax <∞. More precisely, the solution blows up (in the sense that ||U(t)||2 ℓ2becomes unbounded) on the finite interval (0, Tb), with Tb= 2 ||U(0)||2 (U(0),˙ U(0))ℓ2 ,and Tmax ≤Tb.(2.1) Proof: See Appendix A.  A. Numerical Study 1: Connection of the results of Theorem II.1 with escape dynamics Initiating here our numerical studies, we intend to discuss the possible connections of Theorem II.1 with the problem of escape dynamics. The numerical studies will consider the system (1.3) in a form involving the variable discretization parameter ǫ > 0, namely: ¨ Un−ǫ(Un+1 −2Un+Un−1) + ω2 d(Un−βU3 n) = 0, β > 0, ǫ =1 h2,(2.2) on the interval [−L, L], supplemented with Dirichlet boundary conditions. The solution of (2.2) is described by the vector U∈RK+2 U(t) = (U0(t), U1(t),...UK+1(t)) , U0(t) = UK+1(t) = 0,(2.3) where Un(t) := U(xn, t), xn=−L 2+nh,n= 0,...,K + 1, for all t≥0. With the change of variable t→1 ht, we rewrite the initial-boundary value problem for (2.2) in the form ¨ Un−(Un+1 −2Un+Un−1) + Ω2 d(Un−βU3 n) = 0, t > 0n= 1, . . . , K, Ω2 d=h2ω2 d,(2.4) U(0),˙ U(0) ∈RK+2,(2.5) U0=UK+1 = 0, t ≥0.(2.6) Our rescaling of the lattice parameter out of the problem makes it important that we add here a comment about the nature of the continuous and the so-called anti-continuous limit. In the latter, we need to consider within Eq. (1.3) the limit of ω2 d→ ∞. There, the nonlinearity dominates, and we are essentially in the regime of individual (uncoupled) oscillators. On the other hand, for small values of ω2 d, the linear coupling term becomes important. However, due to the nature of our model, we are reaching the asymptotically linear limit (rather than the continuum limit) as ω2 d→0. We will perform numerical simulations for the simplest case of initial data Un(0) = Aδn,0,(2.7) ˙ Un(0) = Bδn,0,(2.8) where δn,0= 1 at site n= 0 and zero elsewhere (Kronecker δ), for A, B ∈R. For the initial data (2.7)-(2.8), the condition for the initial Hamiltonian H(0) ≤0 reads: B2 2+1 + Ω2 d 2A2≤βΩ2 d 4A4,Ω2 d=h2ω2 d.(2.9) Then, it can readily be observed that the quartic equation βΩ2 d 4A4−1 + Ω2 d 2A2−B2 2= 0,(2.10) possesses the unique positive root (for A2): A2 ∗= 2    1 + Ω2 d 2 βΩ2 d +v u u t1 + Ω2 d 22 β2Ω4 d +B2 2βΩ2 d     ,Ω2 d=h2ω2 d.(2.11) 6 Therefore, for the initial data (2.7)-(2.8), the condition (2.9) is satisfied if A < −A∗or A > A∗.(2.12) It can be easily checked from (2.11) that −A∗< U− max =−1 √βand A∗> U+ max =1 √β,for all ω2 d, h, β > 0, recalling that U∓ max denote the location of the saddle points of the on-site potential. The condition on the sign of the time derivative of the ℓ2norm for the initial data (2.7)-(2.8) reads (Un(0),˙ Un(0))ℓ2=AB > 0.(2.13) First, we remark that condition (2.12) is not directly related to the core question of escape dynamics, which is the escape from the potential-well of the on-site potential for initial configurations of the chain inside the well: assuming initially only one excited unit, the conditions of Theorem II.1 imply that the initial position of this unit should be located outside the interval (U− max, U+ max) defining the well. On the other hand, exciting initially only one unit outside the well, with all the others located at the minimum Umin = 0, is indeed of strong physical relevance in connection to another central question regarding the escape dynamics [19, pg. 041110-4]: does this initially escaped unit continue its excursion beyond the saddle-point barrier or can it, in fact, be pulled back into the bound chain configuration Un∈(U− max, U+ max) by the restoring binding forces exerted by the neighbors? Alternatively, can the unit which is initially located outside the well, drag neighboring ones closer to or, in a more extreme scenario, over the barrier? The answer to these important questions should critically depend on the strength of the interactions between the oscillating units, imposed by the linear coupling. In view of the above questions, the conditions (2.12)-(2.13) seem to be quite relevant, in the sense that they determine the behavior of the whole chain regarding the escape dynamics if the analytical threshold (2.12) is crossed by at least one unit: in the anti-continuum limit ǫ→0 [or Ω2 d→ ∞ since, from our scaling Ω2 d=O1 ǫ=O(h2)], according to Theorem II.1, we expect that the initially escaped unit will continue its motion beyond the saddlepoint barrier, while the other units should remain in the well. This is because in this case, the escaped unit is not interacting with the other units of the chain. The fact that ||U(t)||ℓ2approaches infinity at finite time is due to the unboundedness of the energy of this unit in finite time; while the other units remain in the well, the excursion of the escaped unit implies actually the unboundedness of the sup-norm ||U(t)||∞= supn∈Z|Un(t)|in finite time and, in turn, the unboundedness of the ||U(t)||ℓ2-energy due to the inequality: ||U(t)||∞≤ ||U(t)||ℓ2.(2.14) Note that in the infinite lattice only this side of the inequality holds in contrast with the finite lattice, where the equivalence of norms ||U(t)||∞≤ ||U(t)||ℓ2≤√N||U(t)||∞,(2.15) is valid in the N-dimensional space. In the discrete regime ǫ=O(1) (i.e., for moderate values of Ω2 d, as e.g., Ω2 d= 10 used in Ref. [4]) and in the continuum limit ǫ→ ∞, it is expected from Theorem II.1 and (2.14) that a concerted escape of the whole chain will take place, while the time of escape of the whole chain should depend on the coupling strength. The above analytical expectations and questions have been tested for β= 1, h= 0.5 and varying the parameter ω2 d. The initial speed is taken to be B= 0.001, i.e., a small initial velocity of the one-excited unit is provided in order to satisfy the condition (2.13). The first observation of the numerical studies is the justification of (2.12)-(2.13) as sufficient conditions for blow-up and for the continuation of the excursion of the one-excited unit beyond the saddle point barrier when A > A∗. The second question examined numerically, concerns the dynamics of the chain when the one-excited unit is located in the region 1 < A < A∗, i.e., beyond the location of the saddle point, but below the analytical critical value A∗. There, our numerical simulations revealed that blow-up occurs for initial positions of the unit A < A∗; hence, conditions (2.12)-(2.13), although sufficient, are not strictly necessary for blow-up, which was found to occur for the one-excited unit even when its location is below the analytical value A∗. However, and more importantly, the numerical studies revealed the existence of a new barrier (threshold) Uthresh, located beyond the position of the saddle point U+ max = 1, with an important dynamical property. Although U+ max < Uthresh, when the excited unit is initially located at position Asatisfying U+ max < A < Uthresh ≤A∗, it may be pulled back to the potential well and the solution exists globally. As mentioned above, the position of the barrier Uthresh, 7 -0.8 -0.4 0.0 0.4 0.8 1.2 1.6 n 4 2 -4 -2 0 (a) 3 4 2 1 0 t u n 0 1 2 3 4 5 6 7 8 (b) 2 4 -4 -2 0 1.2 1 0.2 0.6 0.4 0.8 t 0 u n n Figure 2: (a) Evolution of a single-excited unit initially located at A= 1.2, satisfying U+ max = 1 < A < A∗= 2.45; the value of U+ max is indicated by the dashed (red) line. Although the unit is initially beyond the saddle point (but below both the numerical threshold Uthresh ∼1.8 and the analytical value A∗), it is pulled back in the potential well, leading to globally existing dynamics. Lattice parameters are β= 1, Ω2 d= 1, for h= 0.5 and ω2 d= 4. (b) In this case, the single-excited unit, initially at A= 2, is pulling the adjacent neighbors towards escape from the potential well. Here, A∗= 4.24 and Uthresh ∼1.9. The lattice parameters are: β= 1, Ω2 d= 0.25, for h= 0.5 and ω2 d= 1. satisfying U+ max <Uthresh ≤A∗, is determined by the restoring binding forces. This fact is demonstrated in Fig. 2(a), showing the evolution of the chain in strong coupling regime (Ω2 d= 1). A single-excited unit, of initial amplitude A= 1.2 (satisfying U+ max = 1 < A < A∗), is being pulled back inside the well, due to the sufficiently strong coupling with its neighbor units. This verifies our analytical prediction, that restoring forces (depending on the discreteness) can prevent the escape of elements of the chain initialized outside the potential well. This is what we will refer to as the “pull back” effect. After this effect takes place for the single-excited unit and it gets “retracted” inside the potential well (U− max, U+ max), the whole chain performs globally existing wave motions. The fact that blow-up occurs for A > A∗= 2.44, as well as the result of Fig. 2(a), confirm the existence of the threshold Uthresh and motivate a further numerical investigation to determine its exact location. The corresponding numerical value is found to be Uthresh ∼1.8< A∗for the near-continuum case (Ω2 d= 1). Concluding, we have shown that: 1. blow-up occurs when A > Uthresh; 2. a single-excited unit is pulled back to the potential well when U+ max ≤A≤ Uthresh; 3. finally and in connection to our analytical result, A∗provides an upper bound for the exact value of Uthresh ∈ (U+ max, A∗) and blow-up always takes place, as predicted for A > A∗. The dependence of the analytical blow-up threshold prediction A∗on the discreteness parameter ω2 d(for fixed values of β= 1 and h= 0.5), is depicted by the solid (black) line in Fig. 3. The respective plot for the numerically obtained value, Uthresh is also depicted –in the same plot– by (red) circles. Comparing the two results, we conclude that A∗ tends to be more accurate as an upper bound for the “real” barrier Uthresh, for small to moderate values of ω2 d. On the other hand, a constant –but reasonable– discrepancy between the analytical and the numerical value is found in the intermediate regime between moderate and large values of ω2 d(the latter corresponding to the near anti-continuum regime of essentially uncoupled nonlinear sites). This discrepancy can be explained as follows: approaching the anticontinuum limit as ω2 d→ ∞, it is naturally expected that the threshold value Uthresh →U+ max =1 √βsince, due to the increasingly weaker interactions between the oscillating units, the system (2.2) is asymptotically reduced to a single repulsive Duffing equation for the single-excited unit –see Fig. 1(b). On the other hand, it follows from (2.11) that A∗→√2 √βas ω2 d→ ∞. Thus, a constant difference between the threshold values is found in this regime, namely A∗−Uthresh ∼√2−1 √β(for large values of ω2 d), which is observed in Fig. 3. In the asymptotically linear limit of ω2 d→0 (Ω2 d→0), Eq. (2.11) predicts that A∗→ ∞. This prediction can also be justified, given the absence (in that limit) of the nonlinearity that is responsible for the blow-up. In the 8 0 100 200 300 400 0.5 1.0 1.5 2.0 2.5 3.0 A +,* A * u thresh 2 d Figure 3: Theoretical predictions for the blow-up amplitude for a single site excitation, as a function of ω2 d, compared to the actual numerical value Uthresh. The solid (black) curve corresponds to the analytical upper bound A∗[cf. Eq. (2.11)] and the dashed (blue) curve to the analytical value A+,∗[cf. Eq. (3.12)]. Bullets (in red) denote the numerical values Uthresh. An asymptotic discrepancy appears for A∗as ω2 d→ ∞ while the accuracy of A+,∗increases as the anti-continuum limit of uncoupled sites is approached– see discussion in the text. The parameter values are: h= 0.5, β= 1. corresponding limit, we observe numerically that Uthresh → ∞, in accordance with the analytical prediction A∗→ ∞. The limits A∗,Uthresh → ∞ as ω2 d→0 are shown in Fig. 3. Furthermore, as the above limit is approached and due to the increasingly more significant role of the interactions between the units, if the excited amplitude has crossed the barrier Uthresh for blow-up, it will enforce the other units to escape from the potential well. This is “drive over” the barrier scenario, clearly demonstrated in Fig. 2(b), which shows the evolution of the chain for lattice parameters Ω2 d= 0.25, i.e., ω2 d= 1 for h= 0.5. The single-excited unit with an initial amplitude Uthresh ∼1.9< A = 2 < A∗, pulls the adjacent neighbors, enabling them to cross the saddle point towards escape from the potential well. The accuracy of the analytical prediction for the escape threshold will be improved in the next section dealing with the linearly damped version of (1.3). The improvement, also depicted in Fig. 3 with the dashed (blue) line will be discussed there, in detail. B. Discussion on the sign-condition (2.13). We conclude this section with a discussion of [16, Lemma 2.1, pg. 455] for the lattice dynamical system (1.3)- (1.4). According to the abstract results of Ref. [16], keeping the condition H(0) ≤0 and violating the sign condition (U(0),˙ U(0))ℓ2>0 should lead to global existence. In terms of the initial data (2.7)-(2.8), the result [16, Lemma 2.1, pg. 455] implies that the initially escaped unit will be pulled back and return to the potential well. For example, under the assumption ˙x(0) = 2(U(0),˙ U(0))ℓ2<0,(2.16) we will reconsider the differential inequality (5.18) in the form ¨x(t) [ ˙x(t)]2≥3 2x(t).(2.17) Integrating (2.17) in the interval [0, t] for arbitrary t∈[0, T ], the interval of existence, we find that 9 0.00 0.02 0.04 0.06 0.08 1.75 1.80 1.85 1.90 1.95 2.00 0.00 0.02 0.04 0.06 0.08 -22.710 -22.712 -22.714 -22.716 -22.718 -22.720 0.00 0.02 0.04 0.06 0.08 3.0 3.2 3.4 3.6 3.8 4.0 0.00 0.02 0.04 0.06 0.08 -10 0 10 20 30 40 t U 0 (t) H(t) t x(t) t dx/dt t Figure 4: Time evolution of the amplitude U0, Hamiltonian H(denoted by H(t) in the figure), norm xand time derivative of the norm dx/dt for a single-excited unit with initial data (2.7)-(2.8) with A= 1.8 and B=−5.5; the lattice parameters are Ω2 d= 10 (ω2 d= 40, h= 0.5), β= 1, corresponding to a highly nonlinear discrete regime. The top left panel shows the excursion to infinity for the one-excited unit; the top right panel depicts the negative (constant) Hamiltonian H(t) = H(0) ∼22.7; the bottom left panel shows the change of the slope of the norm x(t) and its increase to infinity after time t∗; the bottom right panel depicts the change of sign of ˙x(t) after time t∗. −1 ˙x(t)+1 ˙x(0) ≥3 2Zt 0 1 x(s)ds, which can be rewritten as 1 ˙x(t)≤1 ˙x(0) −3 2Zt 0 1 x(s)ds. (2.18) Then, the assumption ˙x(0) = 2(U(0),˙ U(0)) <0, as well as the positivity of the norm function x(t) = ||U(t)||2 ℓ2≥0 for all t∈[0, T ] in the interval of existence [0, T ] should imply that 1 ˙x(t)≤0, i.e., ˙x(t)≤0,for all t∈[0, T ].(2.19) Integrating (2.18) once more in [0, T ], we see that x(t)≤x(0),for all t∈[0, T].(2.20) Letting t→ ∞ in (2.20) implies that Tmax =∞and lim sup t→∞ ||U(t)||2 ℓ2≤ ||U(0)||2 ℓ2.(2.21) Hence, the solution U(t) = {Un(t)}n∈Zshould be defined in [0,∞) and be uniformly bounded. However, the validity of (2.18) for all t∈[0, T ] –the interval of existence– should be put under question: Although the continuity of ˙x(t) = 2(U(t),˙ U(t))ℓ2<0 in [0, T ], the assumption ˙x(0) = 2(U(0),˙ U(0))ℓ2<0 and (2.18) guarantee 16 Condition (4.15) is not establishing escape by itself, since it is certainly satisfied with t1≡Tmax, when the additional restriction on the initial Hamiltonian energy HI′(Z(0)) on I′, HI′(0) <Φ(ρ∗) = (µ1+ω2 d)2 4βω2 d ,(4.20) holds. For instance, we may repeat the proof of Theorem III.1 for the excited segment on I′and show that the solution Z(t) of (4.9)-(4.11) cannot exist globally in time. With the aim to avoid the restriction on the initial energy (4.20) and see if the condition (4.15) suffices for escape, we consider –in the set-up of (4.9)-(4.11)– the invariant region introduced in [33] (see also Ref. [47]). For instance, we consider the set W(Z) := {Z∈ SI′:I(Z)>0 and J(Z)< d}, where the functional Iis given by I(Z) = ǫ λ+1 X j=0 (Zj+1 −Zj)2+ω2 d λ+1 X j=0 Z2 j−βω2 d λ+1 X j=0 Z4 j, and d= inf J(Z) denotes the infimum over all Z∈ SI′of the functional J(Z) = ǫ 2 λ+1 X j=0 (Zj+1 −Zj)2+ω2 d 2 j=λ+1 X j=0 Z2 j−βω2 d 4 λ+1 X j=0 Z4 j. The analysis of Ref. [33] for the continuum limit suggests that the set W(Z)is invariant under the flow associated to (4.9)-(4.11) and solutions are global in time. More precisely, if the initial data Z(0) ∈ W(Z), i.e., I(Z(0)) >0,(4.21) and HI′(0) < d, (4.22) then the solution Z(t) of (4.9)-(4.11) satisfies Z(t)∈ W(t),for all t∈[0,∞),(4.23) i.e., Tmax =∞. On the other hand, a solution which blows-up in finite time should initiate from initial data violating the condition (4.21). It is interesting to observe that in the discrete case, an initial condition which violates (4.21) satisfies (4.15). Proposition IV.2 If the initial data Z(0) ∈ SI′satisfy I(Z(0)) <0,(4.24) then they satisfy (4.15). Proof: Condition (4.24) reads as βω2 d λ+1 X j=0 Z4 j(0) > ǫ λ+1 X j=0 (Zj+1(0) −Zj(0))2+ω2 d λ+1 X j=0 Z2 j(0). Applying (4.13) and (5.29) for Z(0), we have ǫ λ+1 X j=0 (Zj+1(0) −Zj(0))2+ω2 d λ+1 X j=0 Z2 j(0) ≥(µ1+ω2 d) λ+1 X j=0 Z2 j(0) ≥(µ1+ω2 d)  λ+1 X j=0 Z4 j(0)  1 2 . Thus, we have derived that βω2 d||Z(0)||4 4>(µ1+ω2 d)||Z(0)||2 4, which implies that ||Z(0)||4>pµ1+ω2 d ωd√β, i.e., (4.15) is satisfied. ⋄ 17 B. Numerical study 3: Evolution of the lattice segment Propositions IV.1 and IV.2 indicate that (4.15) may suffice for escape dynamics, taking into account that the connection with this behavior shall be established if (4.15) may describe initial configurations of the segment Z(0) inside the potential well. The numerical investigation of this issue is the purpose of the numerical study of this section. With the change of variable t→1 ht, we rewrite the system (4.9)-(4.11) as ¨ Zj−∆dZj+ Ω2 dZj−βΩ2 dZ3 j= 0, t > 0j= 1,...,λ, Ω2 d=h2ω2 d,(4.25) Z(0),˙ Z(0) ∈Rλ+2,(4.26) Z0=Zλ+1 = 0, t ≥0.(4.27) We will consider a set of parameter values which will span the whole regime, from the asymptotically linear to the anti-continuum limit. For a lattice spacing of h= 0.5 we fall in case A, where the endpoints of the unit interval I are occupied by oscillators. In this case, the unit interval consists of λ= 3 oscillators, with the m= 1 included in I [see also (4.3)] together with the two endpoints. The length of the interval I′, which is composed by Iand the two adjacent neighbors, has length L′= 2 and consists of λ+ 2=5 oscillators [see also (4.4)]. Since the number of points is λ= 3, we will examine the evolution of the 3-point segment, by considering the simplest case of initial data, that is zero velocities ˙ Z(0) = (0,0,0), and initial positions of the form Z(0) = (A, A, A). For these initial data, the ℓ4-norm is ||Z(0)||4=  λ+1 X j=0 Z4 j(0)  1 4 =  λ X j=1 Z4 j(0)  1 4 =  3 X j=1 A4  1 4 = 31 4A. From (4.14), the eigenvalue µ1(I′)≃2.34315, and the condition (4.15) results in a critical value for the displacements A, depending on ωdas follows: A > Acrit := pµ1+ω2 d 31 4ωd .(4.28) As will be justified below, the critical value Acrit is physically relevant for the escape dynamics. This is because in the discrete regime, it yields initial segment configurations inside the well and has the physically expected behavior in the limiting cases of the asymptotically linear and of the anticontinuous limit. In the left panel of Fig. 7 we show the time evolution of the 3-unit segment in the moderately discrete regime (Ω2 d= 10), with an initial condition, Z(0) = (A, A, A) where A= 0.79 > Acrit. As shown in this panel, the segment initially performs one oscillation, but subsequently reorganizes so that at time t∼1.5 the central unit (j= 0) has already crossed the barrier Uthresh ∼1.8 and it will lead the chain to collapse. In particular, it can never return to the potential well, according to Theorem II.1, due to the autonomous nature of the system at hand. The energy transfer from the adjacent units of the segment to the central unit (resulting in growing amplitude oscillations of this unit), is visualized in the right panel of Fig. 7. The potential energy stored in the 3-unit segment (light colored area) is progressively localized within the central unit. The dark area between the patterns visualizes the passing from the bottom of the potential well Umin = 0 (with a maximal kinetic energy), as the units oscillate. The same escape dynamics have also been confirmed numerically, for the set of parameters in the strongly discrete regime (Ω2 d= 100), with the initial condition A= 0.77 > Acrit. The collapse time in this case is t≃6.4, which is larger compared to the respective one in the moderately discrete regime. This is expected due to the weaker interaction between the units, which results in a longer time interval during which the energy is redistributed, the more discrete the lattice becomes. Besides that, in the strongly discrete regime, the depth of the potential well is considerably increased, which has, as a result, a longer time needed for the excited units to climb over the saddle points and finally escape (cf. Fig. 1). The analytically obtained threshold Acrit, for which the segment Z(0) = (A, A, A) will exhibit escape dynamics, is plotted as a function of the coupling parameter ω2 din Fig. 8 –see the continuous (black) line; for comparison, the numerically obtained threshold is also presented in this figure by (red) dots. The observation that limω2 d→0Acrit =∞ as ω2 d→0 reflects the fact that blow-up is not possible in the asymptotically linear regime. The numerical simulations verified the existence of a critical value ω2 d,min, as we approach the asymptotically linear from the discrete regime, below which escape is prevented, and the energy stored in the 3-units segment is dispersed along the chain leading to global existence. For large values of ω2 dwe observe that the numerical threshold converges slowly to the limit limω2 d→∞ Acrit ∼0.75. 18 -0.69 0.00 0.69 1.38 2.07 u n 4 2 -4 -2 0 t 1.5 1.2 0.9 0.6 0.3 0 n j t 0.000 1.000 2.000 3.000 4.000 5.000 6.000 7.000 8.000 9.000 1.41.210.80.60.40.20 -3 -2 3 2 0 -1 1 Figure 7: Left panel: escape dynamics for the 3-units segment Z(0) = (A, A, A) with A= 0.79 > Acrit; clearly, the central unit j= 0 is escaping from the potential well. Right panel: contour plot showing the evolution of the potential energy of the chain in the case of an initially excited 3-units segment; the plot demonstrates the energy transfer from the adjacent units to the central unit and its localization therein. The parameters used are Ω2 d= 10 (ω2 d= 40, h= 0.5) and β= 1. 0 100 200 300 400 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 A crit A esc 2 d Figure 8: The analytically obtained threshold Acrit, as a function of the coupling parameter ω2 d(continuous (black) curve), against the numerical value Aesc ((red) dots) for the escape of the 3-units segment. Other parameters used are h= 0.5, β= 1. C. Modulational instability mechanism In this section, we will briefly review initially the modulational instability mechanism [19], with the subsequent aim to investigate its connection with escape dynamics of plane wave initial data [29]. This will complete our program of examining blow-up for the chain starting from one site excitations and progressively passing to few and finally to many site excitations. The modulation instability mechanism concerns the instability of a plane wave of the form Un= Φ0ei(qn−ωt)+ c.c,(4.29) under small perturbations of different wave numbers. When this instability occurs, the exponential growth of the perturbations creates localized excitations. The question at hand here is if a sizeable growth of the perturbations may 19 create a critical localized excitation which, in turn, may lead to escape dynamics. Let us recall first that in the linear limit β= 0 of (2.2), the existence of plane waves (4.29) is associated with the dispersion relation: ω2= 4ǫsin2q 2+ω2 d.(4.30) For q= 0 and q=π, (4.30) gives ω2=ω2 dand ω2=ω2 d+ 4ǫrespectively. Thus, the linear spectrum has a gap of size ωd, and is bounded by the frequency ω2 max =ω2 d+ 4ǫ. In the nonlinear case β > 0, we seek for solutions of (2.2) of the form Un(t) = φn(t)e−iωdt+ c.c, φn∈C,(4.31) where φn(t) denotes the varying envelope. Substitution of (4.31) into (2.2) implies that the varying envelope φn(t) satisfies the equation ¨ φn−2iωd˙ φn−ω2 dφn=ǫ(φn+1 −2φn+φn−1)−ω2 dφn+βω2 d3|φn|2φn+φ3 ne2iωdt+ c.c.(4.32) Under the assumption of the slow-variation in time of φn(t) with respect to the main oscillation at frequency ωd, i.e., ˙ φn<< ωdφn, in the rotating-wave approximation, we may keep only the terms proportional to e±iωdt; this way, (4.32) results in the DNLS equation: 2iωd˙ φn+ǫ(φn+1 −2φn+φn−1) + 3βω2 d|φn|2φn= 0.(4.33) We proceed by seeking solutions of (4.33) of the form: φn(t) = φ0eiθn, θn(t) = qn −˜ωt, φ0∈R.(4.34) Inserting (4.34) in (4.33), we obtain 2ωd˜ωφ0ei qn +ǫφ0ei q(n+1) −2ei qn +ei q(n−1)+ 3βω2 d|φ0|2φ0ei qn = 0, which leads to the following dispersion relation for the frequency ˜ω: 2ωd˜ω= 4ǫsin2q 2−3βω2 d|φ0|2.(4.35) In summary, the validity of the single-frequency approximation leading to the DNLS equation (4.33) for the varying envelope φn(t) of the solutions (4.31) of the DKG equation (2.2), assumes that the gap frequency ωdis large compared to the other frequencies of the system, 4ǫ << ω2 d,(4.36) and the nonlinearity is weak, in the sense: βφ2 0<< 1.(4.37) Having the restrictions (4.36) and (4.37) in mind, one can study the modulational instability conditions of (4.34) by considering the ansatz φn= (φ0+χn)ei(θn+ψn),(4.38) where χnand ψnare small perturbations of the amplitude and phase, respectively, i.e., |χn(t)|<< φ0,and |ψn(t)|<< |θn(t)|. Then, substituting (4.38) in the DNLS (4.33), it turns out that χnand ψnsatisfy the system of linear equations 2ωd˙χn+ǫ[(χn+1 −χn−1) sin q+φ0(ψn+1 −2ψn+ψn−1) cos q] = 0,(4.39) −2ωdφ0˙ ψn+ǫ[(χn+1 −2χn+χn−1) cos q−φ0(ψn+1 −ψn−1) sin q] + 6βω2 dφ2 0χn= 0.(4.40) Assuming that the perturbations χnand ψnhave the form of plane waves, namely, χn(t) = χ0ei(Qn−Ωt),(4.41) ψn(t) = ψ0ei(Qn−Ωt),(4.42) 20 where χ0and ψ0are constant amplitudes, while Qand Ω denote wavenumber and frequency respectively, we can derive from (4.39)-(4.40) the following system for χ0, ψ0,  −2i(ωdΩ−ǫsin Qsin q)−4ǫφ0sin2Q 2cos q −4ǫsin2Q 2cos q+ 6βω2 dφ2 02iφ0(ωdΩ−ǫsin Qsin q) ·χ0 ψ0=0 0.(4.43) For nontrivial solutions, we require the determinant of the matrix in (4.43) to be zero, which leads to the dispersion relation: (ωdΩ−ǫsin Qsin q)2=ǫsin2Q 2cos q4ǫsin2Q 2cos q−6βω2 dφ2 0.(4.44) From this, the condition for modulation instability, i.e., for Ω with a nonvanishing imaginary part can directly be obtained from (4.44); this condition is expressed (for cos(q)>0; if this sign changes, so does the sign of the inequality below) in terms of the amplitude φ0and the wavenumber Qas: 6βω2 dφ2 0>4ǫsin2Q 2cos q. (4.45) In the case of the mode with q= 0, the condition for φ0leading to modulational instability is: φ0>v u u t 2ǫsin2Q 2 3βω2 d := φ0,MI.(4.46) Condition (4.46) should be combined with restrictions (4.36) and (4.37) which, in our case, are: 1>> βφ2 0and ω2 d>> 4ǫ. (4.47) We also remark that in the case of modes with 0 ≤q < π 2, since 6βω2 dφ2 0−4ǫsin2Q 2cos q≥6βω2 dφ2 0−4ǫ, instability of a plane wave occurs if φ2 0>2 3 ǫ βω2 d =2 3βh2ω2 d =2 3βΩ2 d ,Ω2 d=h2ω2 d,(4.48) where we have used the scaling t→1 htas before. D. Small data global existence conditions and their violation The last scenario which will be considered is based on the violation of small data, global existence conditions, and the investigation of the possible connection of this violation with the generation of instabilities. For the derivation of the conditions for global existence, which shall guarantee non-escape dynamics, we shall consider a discrete variant of the method of Ref. [6], and use the energy functional: E(t) := || ˙ U(t)||2 ℓ2+||U(t)||2 ℓ2 1= +∞ X n=−∞ ˙ U2 n+ +∞ X n=−∞ (Un+1 −Un)2+ω2 d +∞ X n=−∞ U2 n,(4.49) where ||U||2 ℓ2 1denotes the “linear coupling energy”-norm ||U||2 ℓ2 1:= +∞ X n=−∞ (Un+1 −Un)2+ω2 d +∞ X n=−∞ U2 n. 21 We shall also consider the quantities E±= ω2 d1±r1−2β ω2 d E(0) 1 + β 2ω2 d β,(4.50) which are real and positive if E(0) satisfies E(0) <ω2 d 2β1 + β 2ω2 d.(4.51) Then, the small data, global existence result is stated in the following Theorem. Theorem IV.1 We consider the energy function (4.49) and we assume that the initial data (3.2) have energy E(0) <min    1,ω2 d 2β1 + β 2ω2 d   .(4.52) Then the solution of (1.3) exists globally in time (Tmax =∞), and satisfies for all t∈[0,∞)the energy bound E(t)< E−.(4.53) Proof: See Appendix C.  On the other hand, if the smallness condition (4.51) is violated, i.e., E(0) >Ω2 d 2β1 + β 2Ω2 d,(4.54) the inequality (5.86) is valid for E(t)∈[0,∞) and the possibility of unbounded solutions cannot be excluded. With this observation at hand, we may investigate the dynamics of the chain for initial data satisfying (4.54) to see if this violation condition may give an analytical value of amplitudes leading to blow-up and escape. Let us also remark that Ω2 d 2β1 + β 2Ω2 d>1 when Ω2 d> β(1 + √2).(4.55) E. Numerical study 4: Modulational instability and violation of conditions for global existence We conclude this section with the last numerical study, investigating the modulational instability conditions (4.46)- (4.47), and the violation of small data global existence (4.54)-(4.55). We will consider lattice parameters in the strongly discrete regime, namely, h= 0.5 and ω2 d= 400 (Ω2 d=ω2 dh2= 100), while for the nonlinearity parameter we will use the value β= 0.1. We intend to examine (4.48), i.e., the conditions for modulational instability and stability for amplitudes φ0> φ0,MI and φ0< φ0,MI, respectively. For our analysis we will focus on the region of modulation instability in the (q, Q)-plane defined in [29, Fig. 2(b), pg. 3200], and particularly on the line q= 0; in this case, the corresponding domain for the wavenumbers is 0,3π 4, and we fix Q=π 2. For the above set of parameters, the critical value of the amplitude φ0,MI defined by the right-hand side of (4.48) is: φ0,MI = 0.18.(4.56) The violation condition (4.54) will also be tested for plane wave initial data, and we will derive the relevant critical value of the amplitude Φ0,crit, as shown below. The plane wave (4.29) can be rewritten as Un(t) = 2φ0cos(qn −ωt) and, thus, at t= 0: Un(0) = 2φ0cos qn, and ˙ Un(0) = 2ωφ0sin qn, (4.57) 22 In order to test the condition (4.54) numerically we should consider a finite lattice of Kunits. In this case, the initial data (4.57) have initial energy E(0) := Epw(0) given by Epw(0) = 4ω2φ2 0 K X n=1 sin2qn + 4φ2 0 K X n=1 [cos(q(n+ 1)) −cos qn]2+ 4φ2 0Ω2 d K X n=1 cos2qn. It readily follows that the mode with q= 0 has energy Epw(0) = 4Ω2 dφ2 0K, (4.58) and the condition (4.54) will be satisfied for a chain of Kunits if φ0>v u u t 1 8Kβ 1 + β Ω2 d:= φ0,crit.(4.59) On the other hand, to get a K-independent condition for the amplitude, we first observe that the mean value of the energy Epw(0) of the the q= 0 mode denoted by Epw(0) is due to (4.58) Epw(0) = 1 KEpw(0) = 4Ω2 dφ2 0. Since Epw(0) >Epw(0) the condition (4.54) will be satisfied for a chain of arbitrary size if Epw(0) >Ω2 d 2β1 + β 2Ω2 d, implying the inequality φ0>v u u t 1 8β1 + β Ω2 d:= Φ0,crit.(4.60) For the example of lattice parameters considered above, the critical value of the amplitude Φ0,crit is found to be Φ0,crit = 1.11.(4.61) Note that both critical values for the amplitudes (4.56) and (4.61) are physically relevant regarding escape dynamics since in the case β= 0.1 the saddle points are located U∓ max =∓1 √β≃3.16 (and hence the initial excitation places all nodes inside their corresponding wells). It should be also recalled that the analytical condition (4.60) rigorously does not ensure global existence or escape (blow-up), since it simply violates the global existence condition (4.52) of Theorem IV.1 for plane waves. The numerical simulations below will examine whether data satisfying the analytical condition (4.60)] may lead to escape dynamics or not. The initial condition used in the numerical study is of the form [cf. Eq. (4.57) for q= 0]: Un(0) = 2φ0+ ˆǫcos(Qn),ˆǫ << φ0,(4.62) where the amplitude of the small perturbation is ˆǫ= 0.005. Figure 9 demonstrates the transition from modulational stability to escape dynamics for plane wave initial data, by plotting the evolution of the ratio V(t) = max nHn(t) Hn(0),(4.63) where the quantity Hn=1 2˙ U2 n+1 2(Un+1 −Un)2+ω2 d 2U2 n−βω2 d 4U4 n,(4.64) is the energy density of the system, and Hn(0) corresponds to the initial energy density, at t= 0. The figure demonstrates the transition from modulational stability (left) to modulational instability (middle) and finally to self-organized escape (right) upon varying the amplitude φ0, as is explained in more detail below. 23 0 200 400 600 800 100 0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 0 200 400 600 800 1000 1 2 3 4 5 6 7 012345 1 2 3 4 5 t 0 0 t 0 V(t) t 024 0 20 |max (u j )| t Figure 9: Evolution of the maximum energy density, normalized to the initial energy density, V(t) [cf. Eq. (4.63)], of the lattice chain. The initial condition, in all three cases, is: Un(0) = 2φ0+ ˆǫcos(Qn), with ˆǫ= 0.005, Q=π 2. The three panels correspond, respectively, to modulational stability for φ0= 0.07 (left panel), modulational instability but without collapse for φ0= 0.15 (middle panel), and escape dynamics for φ0= 1.25 (right panel). The inset in the right panel shows the evolution of the maximum amplitude among the Un, clearly demonstrating the escape at t≃4.5. The left panel of Fig. 9 shows the evolution of (4.63) for φ0= 0.07 < φ0,MI and justifies (for this choice) the modulational stability of the plane wave; we observe that V(t) shows a small fluctuation around its initial value V(0) = 1, thus depicting the stability of the configuration over the entire time interval of the numerical integration of the system (1000 time units). Note that the same behavior was found for other values of the amplitude φ0, up to φ0<0.15. The situation changes drastically for φ0= 0.15: beyond this value, modulational instability starts manifesting itself, as shown in the middle panel of Fig. 9. The onset of the instability is characterized by a significant increase of V(t) from its initial value V(0) = 1 (an order of magnitude larger than that observed in the previous case), which implies exponential growth of the excited mode. We mention that φ0= 0.15 < φ0,MI = 0.18; this deviation is expected due to the approximate description of the initial equation (2.2) by the DNLS model (4.33). The long-time behavior of the system, in this modulationaly unstable regime, is shown in Fig. 10 (the parameter values, as well as the initial condition, are the same with the ones in the middle panel of Fig. 9). The contour plot of the energy density Hn [cf. Eq. (4.64)] reveals that up to t≃2000 a periodic pattern is formed, due to the modulational-instability-induced exponential growth of the excited mode. This is a so-called breather lattice (see e.g. [28] for a relevant discussion of such waveforms in integrable models where they exist in analytically tractable form). However, after t= 2000, and due to the highly nontrivial dynamical interactions of such modes coming into play, at this stage, we observe the formation of moving breathers, each of which acquiring a small fraction of the total amount of energy. These breathers interact with each other and exchange energy; during this process, it is observed that some “prevailing” breathers (the ones with the larger energy) seem to “absorb” the energy of the breathers that they interact with. This process continues for a long time leading to a gradual coarsening of the chain dynamics. Such processes have been debated at considerable length for Hamiltonian systems (with and without ℓ2norm conservation properties); a recent example of such a discussion containing an account of earlier work can be found in [22]. In our long time dynamics, one can clearly identify three strong “eventual” breathers; these are located far enough from each other, so that no interactions between them are observed (at least until the end of the simulation, at t= 10,000). A similar effect has also been observed in Ref. [7] (see Fig. 7 of this work and also references therein). The relevant evolution also brings forth the potential for connections of case examples such as that of Fig. 10 with chaotic dynamics. In particular, what we can see here, but also in Figs. 2 and 7 of [7], is that the MI manifestation leads not just to a simple unstable wavenumber, but rather to a band of unstable wavenumbers. The dominant one among them leads transiently to the formation of a structure resembling a breather lattice [28], but as shown in both of the above examples, as well as in Figs. such as Fig. 3 of [28], such states are unstable. Thus, the excitation of all the additional unstable modes eventually through an apparently chaotic evolution mixing these modes, leads to a self-organizing end result whereby a few dominant/robust discrete breather structures finally prevail. Let us now discuss the connection of the modulational instability mechanism with the emergence of escape dynamics. Generally, modulational instability gives rise to an exponential growth of the plane wave perturbations and, as 24 t n 2000 4000 6000 8000 10000 −40 −20 0 20 40 20 40 60 80 100 120 140 160 180 Figure 10: Contour plot, showing the long time evolution of the energy density Hn, for the same initial condition as in the middle panel of Fig. 9. Up to t∼2000, the chain follows a periodic (breather lattice) pattern due to the exponential growth of the initially excited mode. Beyond that point, we observe the creation of small breathers which collide and exchange energy. At the end of the simulation (t= 10,000), the total energy density is chiefly distributed among only 3 distinct localized breathers. a result, a large-amplitude localized mode –composed by one or few oscillators– may be formed. However, the modulational instability mechanism is not enough by itself to ensure escape dynamics, and modulationally unstable states corresponding to localized, even chaotic excitations, may exist globally in time. Our numerical simulations have shown that there exists a threshold of the amplitude φ0separating the modulational instability regime from the escape dynamics regime; this threshold was found to be (for the set of parameters used in Fig. 9) φ0= 0.98. The right panel of Fig. 9 depicts the evolution of (4.63) for φ0= 1.25, i.e., beyond the threshold value. It is observed that for a short time interval V(t) remains close to its initial value while, at a later time, a sudden and sharp increase appears; this is also associated with a sharp increase of the maximum amplitude along the chain, leading to the escape from the potential well of the critical unit possessing this amplitude. The inset in the right panel of Fig. 9 shows the evolution of maxn|Un(t)|crossing the saddle point barrier. It is interesting to observe, that the analytical value Φ0,crit [cf. Eq. (4.61)] appears to yield a value proximal to the amplitude threshold separating the modulational instability from the escape dynamics regime. We have confirmed numerically that an initial condition of the form of a randomly perturbed plane wave, namely, Un(0) = 2Φ0,crit + ˜ǫ·noise,˜ǫ= 10−4,(4.65) leads to escape dynamics and blow-up in finite time, and the evolution of V(t) is similar to the one shown in the right panel of Fig. 9. Additionally, it should be noted that our simulations have also confirmed the fact that Φ0,crit is almost independent of the number Kof lattice sites, as seen in Table I: it is clearly observed that the numerically found critical amplitude takes the constant value φ0= 0.90 for chains composed by more than K= 100 sites (and up to K= 1000), while it takes a slightly larger value for smaller chains. The linearly damped system: Transient modulational instability within the exponential decay of solutions. In the case of the linearly damped system (3.5), seeking for solutions (4.31) under the slow-variation in time approximation, we derive the linearly damped analogue of the DNLS (4.33), which is iωd˙ φn+iωd γ 2φn+ǫ(φn+1 −2φn+φn−1) + 3 2βω2 d|φn|2φn= 0.(4.66) Φ0,crit = 1.11 K 20 50 100 200 400 1000 φ00.98 0.98 0.90 0.90 0.90 0.90 Table I: The numerically found critical amplitude φ0for escape dynamics for various numbers Kof lattice sites. Here, Φ0,crit = 1.11 is the corresponding K-independent analytical prediction. 25 Multiplying (4.66) by ¯ φnand summing over n(in the infinite lattice or in the finite lattice subject to Dirichlet or periodic boundary conditions), we derive the equation 1 2 d dt||φ||2 ℓ2+γ 2||φ||2 ℓ2= 0, implying the decay of the ℓ2-norm ||φ(t)||2 ℓ2=e−γt||φ(0)||2 ℓ2,(4.67) for all solutions of the approximating DNLS equation (4.66). Furthermore, (4.67) and the norm relations (2.14) and (2.15) imply for both the cases of the infinite or the finite lattice, the sup-norm decay estimate ||φ(t)||2 ∞≤e−γt||φ(0)||2 ℓ2.(4.68) Hence limt→∞ ||φ(t)||∞= 0, and all solutions of (4.66) decay at an exponential rate, independently of the initial data. However, the spatially uniform decay estimate (4.68) does not exclude a possible transient growth of perturbations, and consequently, a transient modulational instability behavior within the frame of exponential decay. This transient modulational instability is especially expected to be noticeable for small values of damping. The above suggestions have been tested numerically, by considering the evolution of the damped DKG chain (3.5) with perturbed plane wave initial data (4.62). The values of parameters for h,β,ω2 dand Q, are the same as in the numerical study performed for the Hamiltonian system γ= 0. The amplitude is φ0= 0.15, inducing modulation instability in the Hamiltonian case. The two top panels of Fig. 11 show the contour plots of the energy density Hn of the dissipative system for γ= 2 (left panel) and γ= 0.02 (right panel), respectively. Although in both cases the solutions decay to zero as the approximating DNLS estimate (4.68) suggests, in the right panel we observe the formation of a transient breather lattice, which survives the exponential decay up to t≃100. The same breather lattice lives only up to t≃1, as shown in the top left panel, due to the stronger damping γ= 2. The two bottom panels show the contour plot of the difference Hn[Un|γ>0]− Hn[Un|γ=0 exp(−γt)], i.e., of the energy density Hn[Un|γ>0] for the damped system (3.5) from the energy density Hn[Un|γ=0 exp(−γt)], where Un|γ=0 denotes the solution of the Hamiltonian system (2.2). In the bottom right panel the time evolution of the energy density difference is shown for the damping value γ= 0.02. The peaks observed therein, within the time interval [20,150], illustrate a transient modulation instability regime, where localization of energy and formation of a lattice of localized modes occurs. In the case of stronger damping γ= 2, the transient instability regime is hardly observable. Numerical simulations for smaller values of γdepict that the transient modulation instability regime is increased, as it is expected when the Hamiltonian limit γ→0 is approached. V. DISCUSSION AND CONCLUSIONS In this work, we have studied the escape problem in the discrete repulsive φ4-model. We have considered both the Hamiltonian (γ= 0) and the dissipative (γ > 0) variants of the model. We have assumed different types of initial conditions, ranging from single-excited units and small segments to plane waves. Our results address several relevant points related to the problem, as is briefly described below. In the Hamiltonian case, we have discussed how the system falls within the class of abstract second order evolution equations of divergent structure. Such systems can be treated, with respect to global non-existence, by the energy methods of Ref. [16], involving the formulation of suitable differential inequalities enabling the identification of conditions for blow-up. For initial data in the form of a single-unit excitation (all other units being initially located in the minimum Umin = 0), it is intuitively expected that escape occurs if the total energy of this unit is (sufficiently) greater than the energy barrier Emax, i.e., the initial position of the unit is located outside the potential well. Indeed, we have derived a suitable analytical value for the initial position of the single unit outside the well, which serves as a sufficient condition for the absence of global existence. However, these single-site initial data are very relevant for two important questions related to escape dynamics (already posed in Refs. [19, 20]): is it possible that the neighbors coupled to the escaping unit also get driven over the barrier and give rise to a concerted escape of the entire chain ? As an alternative scenario, can this escaping unit even be pulled back into the domain of attraction by the restoring coupling forces ? We have definitively shown that both the drive over and the pull back effects can be realized in numerical experiments of such chains. At this point, it should be remarked that the autonomy of the system allows to discuss the derived conditions in terms of the fate of the unit of the critical mode which, at a specific time, escapes from the well. In other words, time-invariance allows for posing as initial condition the position and momenta of the chain at an escaping snapshot, and consider as an initial time, any time referred to this barrier-crossing snapshot. 32 With the above inequality at hand, we estimate the right-hand side of (5.27) from below as 1 2 +∞ X n=−∞ (Un+1 −Un)2+ω2 d 2 +∞ X n=−∞ U2 n≥ω2 d 2 +∞ X n=−∞ U2 n. ≥ω2 d 2 +∞ X n=−∞ U4 n! 1 2 .(5.30) The left-hand side of (5.30) is the sum of the coupling and quadratic Un-dependent terms of the Hamiltonian. Since the kinetic energy term is always 1 2P+∞ n=−∞ ˙ U2 n≥0, for all t∈[0, T ], we have due to (5.30) that H(t) = 1 2 +∞ X n=−∞ ˙ U2 n+1 2 +∞ X n=−∞ (Un+1 −Un)2+ω2 d 2 +∞ X n=−∞ U2 n−βω2 d 4 +∞ X n=−∞ U4 n ≥+1 2 +∞ X n=−∞ (Un+1 −Un)2+ω2 d 2 +∞ X n=−∞ U2 n−βω2 d 4 +∞ X n=−∞ U4 n ≥ω2 d 2 +∞ X n=−∞ U4 n! 1 2 −βω2 d 4 +∞ X n=−∞ U4 n. The claim (5.26) is proved. Step 3 (Derivation of the conditions (3.3) and (3.4) and their consequences): The right-hand side of (5.26) defines a functional for the scalar continuous function x(t) = ||U(t)||ℓ4=P+∞ n=−∞ U4 n 1 4. This functional is of the repulsive φ4-type. For instance, (5.26) can be written in the short-hand notation H(t)≥ω2 d 2||U(t)||2 ℓ4−βω2 d 4||U(t)||4 ℓ4,for all t∈[0, T],(5.31) and motivated form the right-hand side of (5.31), we consider the function J(x) := ω2 d 2x2−βω2 d 4x4, x ≥0.(5.32) Employing J, we will work out the conditions for x(t) = ||U(t)||ℓ4; we are interested on its unique positive maximum J(x∗) = ω2 d 4βat x∗=1 √β.(5.33) Now, let us assume that the initial data U(0) and ˙ U(0) are chosen to satisfy H(0) < J(x∗) = ω2 d 4βand ||U(0)||ℓ4> x∗=1 √β,(5.34) i.e., we assume that conditions (3.3) and (3.4) are satisfied. Let us remark that the second of (5.34) and the continuity of x(t) = ||U(t)||ℓ4, implies that there exists 0 < t1< T, such that |||U(t)||ℓ4> x∗=1 √β,for all t∈[0, t1].(5.35) Due to the energy identity (5.25), we have the bound H(t)≤ H(0),for all t∈[0, T].(5.36) Then, from the first of (5.34) and (5.36), we derive that the Hamiltonian should be bounded by the maximum of J, H(t)< J(x∗) = ω2 d 4β,for all t∈[0, T].(5.37) 33 Then (5.36), together with (5.31), imply that J(x(t)) ≤ H(t)< J(x∗) = ω2 d 4β,for all t∈[0, T].(5.38) The consequences (5.37) and (5.38) can be used as in [35, pg. 212], in order to prove that in (5.35) t1=T, i.e., (5.35) holds for all t∈[0, T ]. To see this, we assume (for the contradiction) that t1< T . Then, there exists some t∗> t1for which x(t∗) = |||U(t∗)||ℓ4=x∗=1 √β. Inserting this t∗∈[0, T ] in (5.38), the upper and lower bounds for H(t), J(x∗) = J(x(t∗)) ≤ H(t)< J(x∗) = ω2 d 4β,for all t∈[0, T] follow, meaning that J(x∗) = H(t) for all t∈[0, T]. This is a contradiction with (5.37). Thus: |||U(t)||ℓ4> x∗=1 √β,for all t∈[0, T].(5.39) It should be noted that the strict inequality of the assumptions (3.3) and (3.4) is crucial for the derivation of the validity of (5.37) and (5.39). The consequence (5.38) has the geometric implementation in view of “Lyapunov functions”, that under the assumptions (5.34), both Jand Hcalculated along the orbits {U, ˙ U} ∈ ℓ2×ℓ2are always less that the maximum J(x∗) of the function J. Step 4 (Definition of a “blow-up” functional F(t)and the properties of its derivatives): We shall derive the differential inequality (5.18) for the function, F(t) : = ||U(t)||2 ℓ2+γZt 0||U(s)||2 ℓ2ds +γ(T0−t)||U(0)||2 ℓ2+δ(t+t0)2 = (U(t), U(t))ℓ2+γZt 0 (U(s), U(s))ℓ2ds +γ(T0−t)||U(0)||2 ℓ2+δ(t+t0)2,(5.40) by applying the “quadratic form argument” of Pucci and Serrin [35, cf. Theorem 1, pg. 206]. This argument will be explained in detail in Step 5. For its implementation it is necessary to discuss the derivatives ˙ F(t), ¨ F(t). In (5.40), the constants t0, T0, δ are positive and will be determined later. The functional Fis well defined for all t∈[0, T ]. The first derivative ˙ F(t) is found to be ˙ F(t) = 2(U, ˙ U)ℓ2+d dt Zt 0||U(s)||2 ℓ2ds−γ||U(0)||2 ℓ2+ 2δ(t+t0) = 2(U, ˙ U)ℓ2+γ||U(t)||2 ℓ2−γ||U(0)||2 ℓ2+ 2δ(t+t0) = 2(U, ˙ U)ℓ2+γ{(U(t), U(t))ℓ2−(U(0), U(0))ℓ2}+ 2δ(t+t0) = 2(U, ˙ U)ℓ2+Zt 0 d ds(U(s), U(s))ℓ2ds + 2δ(t+t0) = 2(U, ˙ U)ℓ2+γZt 0n(U(s),˙ U(s))ℓ2+ (U(s),˙ U(s))ℓ2ods + 2δ(t+t0). Therefore, the derivative ˙ F(t) is ˙ F(t) = 2(U, ˙ U)ℓ2+ 2γZt 0 (U(s),˙ U(s))ℓ2ds + 2δ(t+t0).(5.41) Differentiating (5.41), we find a first expression for the second derivative ¨ F(t), ¨ F(t) = 2( ˙ U, ˙ U)ℓ2+ 2(U, ¨ U)ℓ2+ 2γ(U, ˙ U)ℓ2+ 2δ = 2|| ˙ U||2 ℓ2+ 2(U, ¨ U)ℓ2+γd dt||U||2 ℓ2+ 2δ. (5.42) 34 Writing (3.1) in the shorthand notation ¨ U= ∆2U−γ˙ U−ω2 dU+βω2 dU3, and substituting the derivative ¨ Uin the second term of (5.42), we see that the expression for ¨ F(t) takes the form: ¨ F(t) = 2|| ˙ U||2 ℓ2+ 2(U, ∆2U)ℓ2−2γ(U, ˙ U)ℓ2−2ω2 d(U, U)ℓ2+ 2βω2 d(U, U3)ℓ2+γd dt||U||2 ℓ2+ 2δ = 2|| ˙ U||2 ℓ2−2 +∞ X n=−∞ (Un+1 −Un)2−γd dt||U||2 ℓ2−2ω2 d +∞ X n=−∞ U2 n+ 2βω2 d +∞ X n=−∞ U4 n+γd dt||U||2 ℓ2+ 2δ = 2|| ˙ U||2 ℓ2−2 +∞ X n=−∞ (Un+1 −Un)2−2ω2 d +∞ X n=−∞ U2 n+ 2βω2 d +∞ X n=−∞ U4 n+ 2δ. Hence: 1 2¨ F(t) = || ˙ U||2 ℓ2− +∞ X n=−∞ (Un+1 −Un)2−ω2 d +∞ X n=−∞ U2 n+βω2 d +∞ X n=−∞ U4 n+δ. (5.43) On the other hand, from the Hamiltonian (1.5), we see that the quartic term equals to βω2 d +∞ X n=−∞ U4 n= 2 +∞ X n=−∞ ˙ U2 n+ 2 +∞ X n=−∞ (Un+1 −Un)2+ 2ω2 d +∞ X n=−∞ U2 n−4H(t).(5.44) Then, replacing the quartic term of (5.43) by the right-hand side of (5.44), we write 1 2¨ F(t) as 1 2¨ F(t) = 3|| ˙ U||2 ℓ2+ +∞ X n=−∞ (Un+1 −Un)2+ω2 d +∞ X n=−∞ U2 n−4H(t) + δ. (5.45) Recall that by the energy identity (5.25), H(t) = H(0) −γZt 0|| ˙ U(s)||2 ℓ2ds. (5.46) By replacing the H(t)-term of (5.45) with the right-hand side of (5.46), 1 2¨ F(t) becomes: 1 2¨ F(t) = 3|| ˙ U||2 ℓ2+ +∞ X n=−∞ (Un+1 −Un)2+ω2 d +∞ X n=−∞ U2 n−4H(0) −γZt 0|| ˙ U(s)||2 ℓ2ds+δ. (5.47) Furthermore, multiplying (5.30) by 2, namely, +∞ X n=−∞ (Un+1 −Un)2+ω2 d +∞ X n=−∞ U2 n≥ω2 d +∞ X n=−∞ U4 n! 1 2 ,for all t∈[0, T],(5.48) and from (5.39), it follows that |||U||2 ℓ4satisfies ω2 d|||U||2 ℓ4=ω2 d +∞ X n=−∞ U4 n! 1 2 > ω2 dx∗2=ωd2 β,for all t∈[0, T].(5.49) Therefore, by estimating the right-hand side of (5.48) as it is suggested by (5.49), we have +∞ X n=−∞ (Un+1 −Un)2+ω2 d +∞ X n=−∞ U2 n≥ω2 dx∗2,for all t∈[0, T].(5.50) Estimating further (from below) the sum P+∞ n=−∞(Un+1 −Un)2+ω2 dP+∞ n=−∞ U2 nwhich appears in (5.47) by the right hand side of (5.50), we eventually derive the following inequality for 1 2¨ F(t), 1 2¨ F(t)≥3|| ˙ U||2 ℓ2+ 4γZt 0|| ˙ U(s)||2 ℓ2ds + [ω2 dx2 ∗−4H(0)] + δ. (5.51) 35 We require that the term ω2 dx2 ∗−4H(0) is positive. Hence, if ω2 dx2 ∗−4H(0) >0, in terms of the initial Hamiltonian, the above requirement reads: H(0) <ω2 dx2 ∗ 4=ω2 d 4β=J(x∗), which is exactly the assumption (3.3). Since this is satisfied as an assumption on the initial data, we may select 2δ= [Cωdx2 ∗−4H(0)].(5.52) With the choice (5.52) for δ, (5.51) can be rewritten as 1 2¨ F(t)≥3|| ˙ U||2 ℓ2+ 4γZt 0|| ˙ U(s)||2 ℓ2ds + 3δ, or equivalently ¨ F(t)≥6|| ˙ U||2 ℓ2+δ+ 8γZt 0|| ˙ U(s)||2 ℓ2ds. (5.53) Step 5 (Blow-up by non-continuation to [0,∞)):We argue by contradiction and assume that the solution U(t) of (3.1) exists for all t∈[0,∞), i.e., Tmax =∞. Then, at first, by choosing an arbitrary T0∈[0,∞) and since δ, t0>0 (recall that a specific range for t0will be fixed later), we have that F(t)>0,for all t∈[0, T0].(5.54) By the expression for the derivative ˙ F(t) given in (5.41), we see that at t= 0: ˙ F(0) = 2(U(0),˙ U(0))ℓ2+ 2δt0.(5.55) The choice of t0will be determined by the necessity to assume ˙ F(0) >0. This will be guaranteed by selecting t0>−1 δ(U(0),˙ U(0))ℓ2.(5.56) Let us note that (5.56) is valid for any t0>0 if (U(0),˙ U(0))ℓ2≥0, while the choice (5.56) for t0is needed in the case where (U(0),˙ U(0))ℓ2<0. With the above choice for t0,˙ F(0) >0 and by (5.53), which implies that ¨ F(t)>0, we get that ˙ F(t) is strictly increasing in [0, T0]. Together with the continuity of ˙ F(t), we derive that ˙ F(t)>0 on [0, T0]. Summarizing, we have derived that Fand its derivatives are positive, i.e., F(t),˙ F(t),¨ F(t)>0,for all t∈[0, T0] for some T0∈[0,∞).(5.57) With (5.53) at hand, we may apply the “quadratic form argument” of [35]: recall that for any (χ, y)∈R2, the quadratic form Aχ2+ 2Bχy +Cy2,A,B,C∈R,A,C>0, is positive definite (≥0), if and only if AC −B2≥0.(5.58) The idea is to derive the differential inequality (5.18) for F(t) from (5.58), by proving that a quadratic form for suitable choices of real coefficients A,B,Cinvolving F(t),˙ F(t),¨ F(t) is positive definite. 36 To apply this argument, we consider the quadratic form with the coefficients A:= (U(t), U(t))ℓ2+γZt 0 (U(s), U(s))ℓ2ds +δ(t+t0)2(5.59) =||U||2 ℓ2+γZt 0||U(s)||2 ℓ2ds +δ(t+t0)2, B:= 1 2˙ F(t) = (U(t),˙ U(t))ℓ2+γZt 0 (U(s),˙ U(s))ℓ2ds +δ(t+t0),(5.60) C:= ( ˙ U(t),˙ U(t))ℓ2+γZt 0 (˙ U(s),˙ U(s))ℓ2ds +δ(5.61) =|| ˙ U||2 ℓ2+γZt 0|| ˙ U(s)||2 ℓ2ds +δ. Note that A,B>0 in [0, T0], the first requirement for a positive definite quadratic form. Also, B>0 on [0, T0]. With these coefficients, we have Aχ2= (χU(t), χU(t))ℓ2+γZt 0 (χU(s), χU(s))ℓ2ds +χ2δ(t+t0)2,(5.62) 2Bχy =˙ F(t)χy = 2(χU(t), y ˙ U(t))ℓ2+ 2γZt 0 (χU(s), y ˙ U(s))ℓ2ds + 2χyδ(t+t0),(5.63) Cy2= (y˙ U(t), y ˙ U(t))ℓ2+γZt 0 (y˙ U(s), y ˙ U(s))ℓ2ds +y2δ. (5.64) Before summing (5.62)-(5.64) to recover the quadratic form Aχ2+ 2Bχy +Cy2, we remark that 0≤ ||χU +y˙ U||2 ℓ2= (χU +y˙ U, χU +y˙ U)ℓ2 = (χU, χU)ℓ2+ (χU, y ˙ U)ℓ2+ (y˙ U, χU)ℓ2+ (y˙ U, y ˙ U)ℓ2 = (χU, χU)ℓ2+ 2(χU, y ˙ U)ℓ2+ (y˙ U, y ˙ U)ℓ2.(5.65) Summing now (5.62)-(5.64), the right hand side of (5.65) appears as the sum of the inner product terms (outside and inside the integral). The sum of the remaining terms satisfies χ2δ(t+t0)2+ +2χyδ(t+t0) + y2δ=δ(χ(t+t0) + y)2. Therefore, Aχ2+ 2Bχy +Cy2=||χU +y˙ U||2 ℓ2+γZt 0||χU(s) + y˙ U(s)||2 ℓ2ds +δ(χ(t+t0) + y)2≥0, and therefore (5.58) should hold, i.e., AC ≥B2.(5.66) To derive the differential inequality (5.18) for F(t) with the help of (5.66), we note the following: by comparing the definition of F(t) in (5.40) and the definition (5.62) of Awe have: F(t)>A,for all t∈[0, T0].(5.67) Next, to compare ¨ F(t) and C, we use (5.53) and the definition of C(5.64), showing that ¨ F(t)≥6|| ˙ U||2 ℓ2+ 6δ+ 8γZt 0|| ˙ U(s)||2 ℓ2ds ≥6|| ˙ U||2 ℓ2+ 6δ+ 6γZt 0|| ˙ U(s)||2 ℓ2ds = 6C Thus, we have the differential inequality 1 6¨ F(t)≥C,for all t∈[0, T0].(5.68) 37 All the quantities in (5.67) and (5.68) are positive in [0, T0], hence 1 6F(t)¨ F(t)>AC,for all t∈[0, T0].(5.69) By the definition of Bin (5.60), B2=1 4[˙ F(t)]2,for all t∈[0, T0].(5.70) Since (5.66) and (5.69), (5.70) hold, we have that 1 6F(t)¨ F(t)>1 4[˙ F(t)]2, i.e., the differential inequality F(t)¨ F(t)>3 2[˙ F(t)]2,for all t∈[0, T0].(5.71) The functions F(t),˙ F(t),¨ F(t)>0 in [0, T0] due to (5.57), and (5.71) can be rewritten in the form of (5.18), ¨ F(t) ˙ F(t)>3 2 ˙ F(t) F(t),for all t∈[0, T0].(5.72) Manipulating (5.72) exactly as in Step 5 of the proof of Theorem II.1, we derive that the function F(t) cannot be continued further in time than [0, T ∗), where T∗is given by T∗= 2F(0) ˙ F(0). For instance, by choosing T0=T∗, the blow-up of the function F(t) in [0, T0) contradicts the fact that T0can be arbitrarily chosen in [0,∞) and that the solution Uof (3.1)-(3.2) can be continued further than [0, T0). Appendix C: Proof of Theorem IV.1 Step 1 (Definition of the energy functional for the study of global existence): For convenience, we will denote the quartic term of the Hamiltonian (1.5) by R(U(t)) := βω2 d 4 +∞ X n=−∞ U4 n.(5.73) Obviously, from the conservation of energy, we have 2H(t) = 2H(0), and this can be rewritten by using (4.49) and (5.73) as: E(t)−2R(U(t)) = E(0) −2R(U(0)),for all t∈[0, Tmax].(5.74) Step 2 (An inequality for R(U(t)) and E(t)): The energy functionals E(t)and R(U(t)) satisfy the inequality 2R(U(t)) ≤β 2ω2 d E2(t),for all t∈[0, Tmax].(5.75) To prove (5.75), we first apply on (5.73) the inequality (5.28) for p= 4 and q= 2 to see that 2R(U) = βω2 d 2 +∞ X n=−∞ U4 n=βω2 d 2||U||4 ℓ4 ≤βω2 d 2||U||4 ℓ2.(5.76) On the other hand, the energy E(t) can be estimated from below as E(t)≥ω2 d +∞ X n=−∞ U2 n.(5.77) 38 Recalling that ||U||4 ℓ2=P+∞ n=−∞ U2 n 1 24 =P+∞ n=−∞ U2 n2, it follows from(5.77) that E(t)2satisfies E2(t)≥ω4 d +∞ X n=−∞ U2 n!2 =ω4 d||U||4 ℓ2, which can be written equivalently as ||U||4 ℓ2≤E2(t) ω4 d .(5.78) Then, estimation of the ||U|||4 ℓ2-term in the right-hand side of (5.76) by (5.78), results in 2R(U)≤βω2 d 2·E2(t) ω4 d =β 2ω2 d E2(t),for all t∈[0, Tmax], which is (5.75). Step 3 (Local in time estimates for E(t)−2R(U(t))): Some useful local in time estimates for the energy E(t)−2R(U(t)) can be derived with the help of (5.75). By writing the conservation of energy (5.74) as E(t) = E(0) −2R(U(0)) + 2R(U(t)),for all t∈[0, Tmax], and estimating the 2R(U(t))-term by using (5.75), we derive a first estimate, namely E(t)≤E(0) −2R(U(0)) + β 2ω2 d E(t)2,for all t∈[0, Tmax].(5.79) Note that (5.75) is valid for t= 0. Hence 2R(U(0)) ≤β 2ω2 d E2(0).(5.80) Then, using again the conservation (5.74) and the positivity of R(U), we obtain E(t)−2R(U(t)) = E(0) −2R(U(0)) ≤E(0) −2R(U(0)) + 4R(U(0)) = E(0) + 2R(U(0)). Estimating the 2R(U(0))-term in the right-hand side by (5.80), leads to the second –local in time– upper bound, in terms of E(0): E(t)−2R(U(t)) ≤E(0) + β 2ω2 d E2(0),for all t∈[0, Tmax].(5.81) Consequently, since (5.81) is valid at t= 0, the initial data satisfy E(0) −2R(U(0)) ≤E(0) + β 2ω2 d E2(0).(5.82) Step 4 (Derivation of the condition on E(0) for global existence). We first assume that the initial data are small in the sense E(0) <1.(5.83) Under condition (5.83), we have that E2(0) < E(0), as well as that E2(0) <1. Therefore, from the inequality (5.82) it follows that E(0) −2R(U(0)) ≤E(0) + β 2ω2 d E2(0) ≤E(0) + β 2ω2 d E(0), 39 or in other words, E(0) −2R(U(0)) ≤E(0) 1 + β 2ω2 d.(5.84) On the other hand, the estimate (5.79) can be written as 0≤E(0) −2R(U(0)) + β 2ω2 d E2(t)−E(t),for all t∈[0, Tmax].(5.85) Estimating the E(0)−2R(U(0))-term in the right-hand side of (5.85) from above, by using (5.84), yields the quadratic inequality for the derivation of conditions for small-data global existence Θ(E(t)) := β 2ω2 d E2(t)−E(t) + E(0) 1 + β 2ω2 d≥0,for all t∈[0, Tmax].(5.86) The quadratic equation Θ(E) = 0 defined from (5.86) has the two positive roots (4.50), if E(0) satisfies the condition (4.51). Then, if (4.51) holds, inequality (5.86) implies that E(t)∈[0, E−]∪[E+,+∞).(5.87) However, the function E(t) is continuous on [0,∞), therefore E(t)∈[0, E−],(5.88) implying that solutions exist globally, and are uniformly bounded in time. [1] M. J. Ablowitz, B. Prinari, and A. D. 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