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On the existence of moving breathers in one-dimensional anharmonic lattices
Physica D 181 (2003) 215–221
On the existence of moving breathers in one-dimensional anharmonic lattices Jacob Szeftel a,∗ , Guoxiang Huang b , Vladimir Konotop c a
Laboratoire de Physique Théorique de la Matière Condensée, Case 7020, 2 Place Jussieu, 75251 Paris Cedex 05, France b Department of Physics, East China Normal University, Shanghai 200062, PR China c Universidade de Lisboa, Complexo Interdisciplinar, Avenida Prof. Gama Pinto, 2, Lisbon P-1649-003, Portugal Received 9 April 2002; received in revised form 23 November 2002; accepted 25 November 2002 Communicated by R.E. Goldstein
Abstract The existence of moving breathers is discussed in several anharmonic, atomic lattices, including the Fermi–Pasta–Ulam and Frenkel–Kontorova models. Previous work on this topic is also reviewed. It is concluded that, apart from the integrable case, moving breathers are unlikely to arise over significant time-durations in infinite, one-dimensional crystals. © 2003 Elsevier Science B.V. All rights reserved. PACS: 63.20 Ry Keywords: Moving breathers; Anharmonic lattices
1. Introduction Over the past decade many efforts have been undertaken to unravel the dynamics of anharmonic lattices because of its significance on a score of issues in solid matter, such as heat transport, energy relaxation and structural or magnetic instabilities [1–5]. The dynamics of perfectly periodic, harmonic crystals is completely accounted for by phonons which are time-periodic plane waves, characterized by a dispersion relation between frequency and wave-vector. Nevertheless the anharmonicity of the interatomic potential gives rise to additional, lattice dynamical excitations, namely stationary and moving breathers [6,7] which correspond to localized and traveling vibrational fields, respectively, and solitons [8–10]. As ∗
Corresponding author. E-mail address: [email protected] (J. Szeftel).
it is generally not possible to compute the associated displacement patterns by solving directly the equation of motion of the anharmonic crystal, the issue of the very existence of those excitations can be secured by numerical simulation. It consists of integrating the equation of motion in as large as possible a crystal with initial conditions taken from the calculated displacement pattern. The latter is regarded a stable solution provided the outcome of the simulation does keep the same shape without undergoing any deformation over arbitrarily long time-duration. Such a test has already been carried out for stationary breathers, solitons and anharmonic phonons [11–15], leading to the conclusion that those excitations are stable in the Fermi–Pasta–Ulam (FPU) and Frenkel–Kontorova (FK) models [16,17]. The FPU and FK potentials have been chosen for illustrative purposes because of their strong bearing on such nonlinear phenomena as energy relaxation [18] in solids, solitons [9,19]
0167-2789/03/$ – see front matter © 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0167-2789(03)00116-7
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and breathers [20]. The present work is aimed at completing this series by addressing the case of the moving breather. It turns out that this issue has been tackled by resorting to a number of schemes [21–29] in mono- and diatomic lattices as well, but different definitions have been taken for the moving breather in each case. It is intended here to investigate whether a new kind of moving breather is sustained by the infinite FPU and FK lattices and to make a critical review of previous results. Here is the outline: our own approach is presented in Section 2 while Section 3 deals with a comparative discussion of all available results.
2. Analysis of the moving breather 2.1. The FPU model
Fig. 1. Dispersion relations ω(τ), T(τ) of the FPU moving breather.
An infinite, monoatomic chain is considered. There is a single coordinate per atom uj designating the displacement at site j with respect to equilibrium. The atoms of mass unity are coupled through the FPU pair potential W1 (uj − uj+1 ) written as
The ω(τ), T(τ) dependences inferred from Eq. (4) have been plotted in Fig. 1. There is no solution outside the range τ ∈ [τm = 0.55, τM = 7.75]. In between T −1 , ω grow roughly linearly with τ −1 . The data in Fig. 1 rule out any possibility that the moving breather might turn continuously into a stationary breather or a soliton, which corresponds, respectively, to τ −1 = 0 and ω = 0 because those latter limiting cases do conspicuously not pertain to the pictured curves ω(τ), T(τ). Inversely the anharmonic phonon limit could be reached for T → ∞, which corresponds to τ → τM . As a matter of fact ω(τM ) turns out in Fig. 1 to be slightly bigger than the lower bound [14] for anharmonic phonon frequencies equal to 2 sin (ω(τM )τM /2) as given by the harmonic phonon dispersion associated with the potential W1 . Taking account of Eq. (3), Eq. (2) is recast into
W1 (x) = 21 x2 + 41 x4 ,
(1)
where x = uj − uj+1 . The equations of motion read dW1 dW1 (uj−1 − uj ) − (uj − uj+1 ). (2) dx dx The displacement field associated with a moving breather is chosen to be defined here as a solution {uj∈Z (t)} of Eq. (2) characterized by u¨ j∈Z (t) =
uj+1 (t) = uj (t + τ),
uj (t → ±∞) ∝ eλ|t| ,
(3)
where τ −1 denotes the propagation velocity of the moving breather (the lattice parameter is set equal to 1), and λ = −1/T + iω with T > 0. Replacing then W1 by its harmonic limit in Eq. (2), which is vindicated by uj (t → ±∞) → 0 in Eq. (3), while taking advantage of Eq. (3) gives rise to the following dispersion relations: τ 1 2 − ω = 2 cos (ωτ) cosh − 1 , T T2 τ ω = sin (ωτ) sinh . (4) T T
u(t) ¨ =
dW1 (u(t − τ) − u(t)) dx dW1 − (u(t) − u(t + τ)). dx
(5)
Eq. (5) is recognized to be a second-order differential equation involving retarded [u(t − τ) − u(t)] and advanced [u(t) − u(t + τ)] terms. Furthermore odd (even) solutions u(t) such that u(t) = −u(−t)
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(u(t) = u(−t)) are consistent with Eq. (5) because of W1 (x) = W1 (−x). Each solution depends on a single parameter τ. In order to work out a solution of Eq. (5) consistent with Eq. (3), time t is first discretized according to the finite difference procedure, so that Eq. (5) turns into a system of finite difference equations for the unknown vector U having n + 1 components {Uj=1,...,n+1 } Uj−1 + Uj+1 − 2Uj θ2 dW1 dW1 = (Uj−m − Uj ) − (Uj − Uj+m ), dx dx
(6)
where the time-duration θ 1 and the integer m are related by mθ = τ, Uj=1,...,n = u((j − 1)θ) so that U1 = u(0) and Un = u(tM ), which entails 0 ≤ t ≤ tM = (n − 1)θ. Eq. (6) has been solved by Newton’s method, enforcing the boundary conditions in the n + 1 × n + 1 Jacobian matrix at t = 0 and t = tM by setting u(t < 0) = −u(−t) (u(t < 0) = u(−t)) for the odd (even) case and u(t > tM ) = Un e−(t−tM )/T(τ) cos (ω(τ)(t − tM ) + Un+1 )/ cos (Un+1 ) where T(τ), ω(τ) are given by Eq. (4) and Un+1 is an unknown phase-shift. Accordingly tM has been taken large enough to ensure that the asymptotic regime in Eq. (3) has set in for t > tM . The starting assignment of U results from the choice u(t) = e−t/3T(τ) sin (ω(τ)t) [u(t) = e−t/3T(τ) cos (ω(τ)t)] in the odd (even) case. Multiplying T by 3 warrants a fast convergence of Newton’s method, considered to be achieved if Eq. (6) gets satisfied within 10−11 with n = 2000, θ = 0.008. The computed u(t)’s have been plotted in Figs. 2 and 3 for the odd and even cases, respectively. Increasing tM causes the maximum value of |u(t ∈ [0, tM ])| to decrease steadily, which casts doubt on whether the tM -dependent patterns u(t), depicted in Figs. 2 and 3 and obtained by solving Eq. (6) are indeed solutions of Eq. (5). To settle this issue we perform a simulation by integrating iteratively over [tM , 0] v¨ j (t) =
dW (vj−1 (t − τ) − vj (t)) dx dW − (vj (t) − vj−1 (t + τ)), dx
(7)
Fig. 2. Plot of the displacement pattern u(t) for an odd FPU moving breather.
for the sequence of unknown functions {vj (t)} taking as initial conditions vj (tM ) = vj−1 (tM ), v˙ j (tM ) = v˙ j−1 (tM ). The iteration is started for v1 (t), setting v0 (t) = u(t). Note that Eq. (7) is an ordinary differential equation with respect to the unknown vj (t) since the previously determined vj−1 (t) plays the role of a
Fig. 3. Plot of the displacement pattern u(t) for an even FPU moving breather.
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[0, tM ], as actually observed for stationary breathers, solitons and anharmonic phonons [13–15], this would point out toward long-term stability of the FPU moving breather. However quite the opposite is observed, i.e. |vj (t) − u(t)| is actually seen to diverge from the very first iteration j = 1, which speaks against the existence of moving breathers in the FPU model. 2.2. The FK model The pair potential has been chosen here to read W2 (ui , ui+1 ) = 21 ((ui − ui+1 )2 − cos (ui ) − cos (ui+1 )) + 41 λ(ui − ui+1 )4 ,
Fig. 4. Simulation carried out for an odd FPU moving breather.
known parameter field. Each iteration labeled by its j value mimics the moving breather propagating over one unit-cell. The results of this simulation procedure have been represented in Figs. 4 and 5 for the odd and even moving breathers, respectively. If the resulting discrepancy |vj (t) − u(t)| were found to vanish within numerical accuracy for j = 1, . . . , jf 1 and t ∈
where λ ∈ R is taken as a disposable parameter. Actually all previous authors have assumed λ = 0 but this case has been confirmed to sustain no soliton nor stationary breather or anharmonic phonon [13,15]. The equations of motion read for a moving breather u(t) ¨ = u(t − τ) + u(t + τ) − 2u(t) − sin (u(t)) + λ((u(t − τ) − u(t))3 − (u(t) − u(t + τ))3 ), (8) where τ −1 stands again for the velocity of the moving breather. Odd [u(t) = −u(−t)] and even [u(t) = u(−t)] solutions are both consistent with Eq. (8) because of W2 (ui , ui+1 ) = W2 (−ui , −ui+1 ). As done in the FPU case, bringing linearized Eq.(8) in keeping with Eq. (3) owing to u(t → ±∞) → 0 yields the dispersion relations of the moving breather τ 1 − ω2 = 2 cos (ωτ) cosh − 3, 2 T T τ ω = sin (ωτ) sinh . T T
Fig. 5. Simulation carried out for an even FPU moving breather.
(9)
As the dispersion is completely determined by the harmonic limit of W2 , it is independent of λ. It has been plotted in Fig. 6. The dispersion data can be seen to exhibit the same qualitative features as in Fig. 1 whence they are inferred not to allow for the moving breather unfolding continuously into a stationary one or a soliton either. All our attempts to solve Eq. (8) with λ ∈ [0, 10] under the constraints in Eqs. (3) and (9) by combining
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The traveling field {ψj } in Eq. (11) and the moving breather {uj } studied in the foregoing section differ from each other in two respects: • {ψj } depends on two parameters k, µ whereas {uj } is assessed by a single one τ; • {ψj } is at odds with Eq. (3), in particular because it involves two distinct velocities µv, ω/k instead of a single one τ −1 for {uj }. Likewise looking for a solution {ψj∈Z } of Eq. (10) fulfilling Eq. (3) leads to the dispersion relations τ 1 = −2 sin (ωτ) sinh , T τ T ω = 2 cos (ωτ) cosh . T
3. Comparing with previous work
These relations do allow for no T > 0, which confirms that the model defined in Eq. (10) has no solution obeying Eq. (3) although it has indeed the solution written in Eq. (11). Similarly there is another integrable discrete anharmonic potential [31]. Nevertheless it is known to sustain no excitation but cnoidal waves and solitons, which thence rules out any possibility of having moving breathers. An extension of Eq. (11) to cope with nonintegrable lattice dynamical models has been reported [21]. It consists of seeking the unknown moving breather field {uj∈Z (t)} as having the form
3.1. The integrable case
uj =
Fig. 6. Dispersion relations ω(τ), T(τ) of the FK moving breather.
the finite-difference and Newton methods as done here above in the FPU model did not result in any solution except u(t) = 0, ∀t. This seems to indicate that the FK model cannot sustain moving breathers. Conversely stable solitons, stationary breathers and anharmonic phonons have been shown [13,15] to arise in the FK model provided λ ≥ 0.05.
uj,k ,
uj,k (t → ±∞) ∝ ei((kω+qk v)t−jqk ) ,
k∈Z
The discrete nonlinear Schrödinger model [7] and its variations [30] have been shown to be integrable with respect to moving breathers. It is defined as ˙ j∈Z = i(ψj−1 + ψj+1 )(1 + |ψj |2 ). ψ
(10)
The field {ψj } associated with the integrable solution reads ψj =
sinh (µ) ei(ωt−jk) , cosh(µ(j − vt))
(11)
where the frequency ω = 2cosh(µ) cos (k) and the velocity v = 2 sinh (µ) sin (k)/µ are functions of two independent parameters k, µ.
where qk ∈ C. Linear analysis carried out for the FPU potential in Eq. (1) yields Re(qk ), Im(qk ) as functions of ω, v by solving (kω + qk v)2 = 2(1 − cosh(iqk )). This allows also for a continuous path linking moving (v = 0) to stationary (v = 0) breathers. A fixed point procedure [21] has been applied to solve the equation of motion for the moving breather but has failed to bring about any clear-cut conclusion on the existence of the moving breather because solutions appeared to be teeming around the fixed point.
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3.2. The perturbational schemes Moving breathers have been also studied within the quasidiscreteness approach (QDA) [20,22–25], which the rotating wave approximation (RWA) [26–28] is a variation of. In QDA and RWA the vibrational field of the breather is worked out from a truncated perturbation expansion. The results are believed to be reliable in case of weak anharmonicity. At the lowest order of QDA, the vibrational pattern reads for a moving breather uj∈Z (t) = ξj + φj cos (ω(k, κ0 )t − kj), where the field {uj (t)} depends on two independent real parameters k, κ0 whereby it differs from the moving breather addressed in the preceding section. Furthermore it does not fulfil Eq. (3) either. Within the limit t → ±∞, we obtain for ξj , φj ξj (t → ±∞), φj (t → ±∞) ∝ e−|κ0 (j−vg (k)t)| , where k, vg stand for the wave-vector and group velocity of harmonic phonons, respectively. Inserting these formulae into the linearized equation of motion due to uj (t → ±∞) → 0 results, for the FPU potential in Eq. (1), in three constraints on k, κ0 (κ0 vg )2 = 2(cosh(κ0 ) − 1), (κ0 vg + iω)2 = 2(cosh(κ0 + ik) − 1). Such a system of three equations relating two unknowns k, κ0 has in general no solution k, κ0 . Due to this inconsistency, simulations based on the QDA and RWA results are bound to decay in the long run by radiating spurious oscillations referred to as nanopterons [6] whence it is concluded that QDA and RWA fail to yield stable moving breathers.
the coupling within a finite coding sequence of initially uncoupled oscillators. The continuation process is likely to break down for strong coupling. As the interatomic potential involves necessarily an on-site contribution, it is suited to the FK model but not the FPU one. This scheme has been extended to moving breathers by setting stationary ones to overcome the Peierls–Nabarro barrier [3,29]. By construction there is a continuous path between stationary and moving breathers. The boundary conditions used for a finite crystal are very different from the case of the infinite one because no matching with an asymptotic regime is undertaken at the crystal edges. Therefore finite-size effects including nanopterons show up so that UOL based moving breathers turn out to lack long-term stability too.
4. Conclusion The existence of moving breathers has been discussed in anharmonic, one-dimensional lattices. There is no unique definition of this dynamical excitation. Nevertheless requiring its displacement field to vanish at both ends of a crystal of infinite size, consistent with the equation of motion, has led to dispersion relations in terms of one or two independent parameters, depending on the chosen definition of the moving breather. These constraints cannot be satisfied in the QDA treatment, which seemingly points toward an inconsistency. But even though they may be fulfilled in other approaches, the resulting vibrational patterns have displayed instability features in simulations so that apart from the integrable case [7] the existence of moving breathers remains doubtful in anharmonic crystals.
3.3. Finite systems
Acknowledgements
All approaches discussed so far have attempted to cope with moving breathers in infinite crystals. However the uncoupled oscillator limit (UOL) [32] has been devised primarily to investigate breathers [33] in finite lattices. It consists of switching on continuously
We thank François-Xavier Girod and Olivier Huet for providing invaluable help at an early stage of this work. One of us (G.H.) is indebted to the French Ministry of Research and Université Paris 7 for two visiting grants.
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References [1] P. Bruesch, Phonons: Theory and Experiments, vol. 3, Springer, Berlin, 1986. [2] T. Dauxois, et al., Phys. Rev. Lett. 70 (1993) 3935. [3] D. Chen, et al., Phys. Rev. Lett. 77 (1996) 4776. [4] A.D. Bruce, R.A. Cowley, Structural Phase Transitions, Taylor & Francis, London, 1981. [5] N. Theodorakopoulos, M. Peyrard, Phys. Rev. Lett. 83 (1999) 2293. [6] S. Flach, C.R. Willis, Phys. Rep. 295 (1998) 181. [7] M.J. Ablowitz, J.F. Ladik, J. Math. Phys. 17 (1976) 1011. [8] P.G. Drazin, R.S. Johnson, Solitons: an Introduction, Cambridge University Press, Cambridge, 1989. [9] M. Remoissenet, Waves Called Solitons, Springer, New York, 1996. [10] A.R. Bishop, T. Schneider, Solitons and Condensed Matter Physics, Springer, New York, 1981. [11] J. Szeftel, P. Laurent, Phys. Rev. E 57 (1998) 1134. [12] J. Szeftel, P. Laurent, E. Ilisca, Phys. Rev. Lett. 83 (1999) 3982. [13] J. Szeftel, et al., Physica A 288 (2000) 225.
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[14] J. Szeftel, et al., Phys. Lett. A 297 (2002) 218. [15] J. Szeftel, G. Huang, Eur. Phys. J. B 27 (2002) 329. [16] E. Fermi, J. Pasta, S. Ulam, The many-body problem, in: D.C. Mattis (Ed.), World Scientific, Singapore, 1993. [17] B. Hu, B. Li, Physica A 288 (2000) 81. [18] T. Prosen, D.K. Campbell, Phys. Rev. Lett. 84 (2000) 2857. [19] M. Peyrard, M.D. Kruskal, Physica D 14 (1984) 88. [20] G. Huang, et al., Phys. Rev. B 58 (1998) 9194. [21] S. Flach, K. Kladko, Physica D 127 (1999) 61. [22] G. Huang, et al., Phys. Rev. B 47 (1993) 14561. [23] G. Huang, B. Hu, Phys. Rev. B 57 (1998) 5746. [24] V.V. Konotop, Phys. Rev. E 53 (1996) 2843. [25] S. Jiménez, V.V. Konotop, Phys. Rev. B 60 (1999) 6465. [26] S.R. Bickham, et al., Phys. Rev. B 47 (1993) 14206. [27] K.W. Sandusky, J.B. Page, Phys. Rev. B 50 (1994) 866. [28] V.V. Konotop, S. Takeno, Phys. Rev. E 54 (1996) 2010. [29] S. Aubry, T. Crétegny, Physica D 119 (1998) 34. [30] V.V. Konotop, et al., Phys. Rev. E 48 (1993) 563. [31] M. Toda, Theory of Non-Linear Lattices, Springer, New York, 1988. [32] R. Mac Kay, S. Aubry, Nonlinearity 7 (1994) 1623. [33] J.L. Marin, S. Aubry, Nonlinearity 9 (1996) 1501.
On the existence of moving breathers in one-dimensional anharmonic lattices Jacob Szeftel a,∗ , Guoxiang Huang b , Vladimir Konotop c a
Laboratoire de Physique Théorique de la Matière Condensée, Case 7020, 2 Place Jussieu, 75251 Paris Cedex 05, France b Department of Physics, East China Normal University, Shanghai 200062, PR China c Universidade de Lisboa, Complexo Interdisciplinar, Avenida Prof. Gama Pinto, 2, Lisbon P-1649-003, Portugal Received 9 April 2002; received in revised form 23 November 2002; accepted 25 November 2002 Communicated by R.E. Goldstein
Abstract The existence of moving breathers is discussed in several anharmonic, atomic lattices, including the Fermi–Pasta–Ulam and Frenkel–Kontorova models. Previous work on this topic is also reviewed. It is concluded that, apart from the integrable case, moving breathers are unlikely to arise over significant time-durations in infinite, one-dimensional crystals. © 2003 Elsevier Science B.V. All rights reserved. PACS: 63.20 Ry Keywords: Moving breathers; Anharmonic lattices
1. Introduction Over the past decade many efforts have been undertaken to unravel the dynamics of anharmonic lattices because of its significance on a score of issues in solid matter, such as heat transport, energy relaxation and structural or magnetic instabilities [1–5]. The dynamics of perfectly periodic, harmonic crystals is completely accounted for by phonons which are time-periodic plane waves, characterized by a dispersion relation between frequency and wave-vector. Nevertheless the anharmonicity of the interatomic potential gives rise to additional, lattice dynamical excitations, namely stationary and moving breathers [6,7] which correspond to localized and traveling vibrational fields, respectively, and solitons [8–10]. As ∗
Corresponding author. E-mail address: [email protected] (J. Szeftel).
it is generally not possible to compute the associated displacement patterns by solving directly the equation of motion of the anharmonic crystal, the issue of the very existence of those excitations can be secured by numerical simulation. It consists of integrating the equation of motion in as large as possible a crystal with initial conditions taken from the calculated displacement pattern. The latter is regarded a stable solution provided the outcome of the simulation does keep the same shape without undergoing any deformation over arbitrarily long time-duration. Such a test has already been carried out for stationary breathers, solitons and anharmonic phonons [11–15], leading to the conclusion that those excitations are stable in the Fermi–Pasta–Ulam (FPU) and Frenkel–Kontorova (FK) models [16,17]. The FPU and FK potentials have been chosen for illustrative purposes because of their strong bearing on such nonlinear phenomena as energy relaxation [18] in solids, solitons [9,19]
0167-2789/03/$ – see front matter © 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0167-2789(03)00116-7
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and breathers [20]. The present work is aimed at completing this series by addressing the case of the moving breather. It turns out that this issue has been tackled by resorting to a number of schemes [21–29] in mono- and diatomic lattices as well, but different definitions have been taken for the moving breather in each case. It is intended here to investigate whether a new kind of moving breather is sustained by the infinite FPU and FK lattices and to make a critical review of previous results. Here is the outline: our own approach is presented in Section 2 while Section 3 deals with a comparative discussion of all available results.
2. Analysis of the moving breather 2.1. The FPU model
Fig. 1. Dispersion relations ω(τ), T(τ) of the FPU moving breather.
An infinite, monoatomic chain is considered. There is a single coordinate per atom uj designating the displacement at site j with respect to equilibrium. The atoms of mass unity are coupled through the FPU pair potential W1 (uj − uj+1 ) written as
The ω(τ), T(τ) dependences inferred from Eq. (4) have been plotted in Fig. 1. There is no solution outside the range τ ∈ [τm = 0.55, τM = 7.75]. In between T −1 , ω grow roughly linearly with τ −1 . The data in Fig. 1 rule out any possibility that the moving breather might turn continuously into a stationary breather or a soliton, which corresponds, respectively, to τ −1 = 0 and ω = 0 because those latter limiting cases do conspicuously not pertain to the pictured curves ω(τ), T(τ). Inversely the anharmonic phonon limit could be reached for T → ∞, which corresponds to τ → τM . As a matter of fact ω(τM ) turns out in Fig. 1 to be slightly bigger than the lower bound [14] for anharmonic phonon frequencies equal to 2 sin (ω(τM )τM /2) as given by the harmonic phonon dispersion associated with the potential W1 . Taking account of Eq. (3), Eq. (2) is recast into
W1 (x) = 21 x2 + 41 x4 ,
(1)
where x = uj − uj+1 . The equations of motion read dW1 dW1 (uj−1 − uj ) − (uj − uj+1 ). (2) dx dx The displacement field associated with a moving breather is chosen to be defined here as a solution {uj∈Z (t)} of Eq. (2) characterized by u¨ j∈Z (t) =
uj+1 (t) = uj (t + τ),
uj (t → ±∞) ∝ eλ|t| ,
(3)
where τ −1 denotes the propagation velocity of the moving breather (the lattice parameter is set equal to 1), and λ = −1/T + iω with T > 0. Replacing then W1 by its harmonic limit in Eq. (2), which is vindicated by uj (t → ±∞) → 0 in Eq. (3), while taking advantage of Eq. (3) gives rise to the following dispersion relations: τ 1 2 − ω = 2 cos (ωτ) cosh − 1 , T T2 τ ω = sin (ωτ) sinh . (4) T T
u(t) ¨ =
dW1 (u(t − τ) − u(t)) dx dW1 − (u(t) − u(t + τ)). dx
(5)
Eq. (5) is recognized to be a second-order differential equation involving retarded [u(t − τ) − u(t)] and advanced [u(t) − u(t + τ)] terms. Furthermore odd (even) solutions u(t) such that u(t) = −u(−t)
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(u(t) = u(−t)) are consistent with Eq. (5) because of W1 (x) = W1 (−x). Each solution depends on a single parameter τ. In order to work out a solution of Eq. (5) consistent with Eq. (3), time t is first discretized according to the finite difference procedure, so that Eq. (5) turns into a system of finite difference equations for the unknown vector U having n + 1 components {Uj=1,...,n+1 } Uj−1 + Uj+1 − 2Uj θ2 dW1 dW1 = (Uj−m − Uj ) − (Uj − Uj+m ), dx dx
(6)
where the time-duration θ 1 and the integer m are related by mθ = τ, Uj=1,...,n = u((j − 1)θ) so that U1 = u(0) and Un = u(tM ), which entails 0 ≤ t ≤ tM = (n − 1)θ. Eq. (6) has been solved by Newton’s method, enforcing the boundary conditions in the n + 1 × n + 1 Jacobian matrix at t = 0 and t = tM by setting u(t < 0) = −u(−t) (u(t < 0) = u(−t)) for the odd (even) case and u(t > tM ) = Un e−(t−tM )/T(τ) cos (ω(τ)(t − tM ) + Un+1 )/ cos (Un+1 ) where T(τ), ω(τ) are given by Eq. (4) and Un+1 is an unknown phase-shift. Accordingly tM has been taken large enough to ensure that the asymptotic regime in Eq. (3) has set in for t > tM . The starting assignment of U results from the choice u(t) = e−t/3T(τ) sin (ω(τ)t) [u(t) = e−t/3T(τ) cos (ω(τ)t)] in the odd (even) case. Multiplying T by 3 warrants a fast convergence of Newton’s method, considered to be achieved if Eq. (6) gets satisfied within 10−11 with n = 2000, θ = 0.008. The computed u(t)’s have been plotted in Figs. 2 and 3 for the odd and even cases, respectively. Increasing tM causes the maximum value of |u(t ∈ [0, tM ])| to decrease steadily, which casts doubt on whether the tM -dependent patterns u(t), depicted in Figs. 2 and 3 and obtained by solving Eq. (6) are indeed solutions of Eq. (5). To settle this issue we perform a simulation by integrating iteratively over [tM , 0] v¨ j (t) =
dW (vj−1 (t − τ) − vj (t)) dx dW − (vj (t) − vj−1 (t + τ)), dx
(7)
Fig. 2. Plot of the displacement pattern u(t) for an odd FPU moving breather.
for the sequence of unknown functions {vj (t)} taking as initial conditions vj (tM ) = vj−1 (tM ), v˙ j (tM ) = v˙ j−1 (tM ). The iteration is started for v1 (t), setting v0 (t) = u(t). Note that Eq. (7) is an ordinary differential equation with respect to the unknown vj (t) since the previously determined vj−1 (t) plays the role of a
Fig. 3. Plot of the displacement pattern u(t) for an even FPU moving breather.
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[0, tM ], as actually observed for stationary breathers, solitons and anharmonic phonons [13–15], this would point out toward long-term stability of the FPU moving breather. However quite the opposite is observed, i.e. |vj (t) − u(t)| is actually seen to diverge from the very first iteration j = 1, which speaks against the existence of moving breathers in the FPU model. 2.2. The FK model The pair potential has been chosen here to read W2 (ui , ui+1 ) = 21 ((ui − ui+1 )2 − cos (ui ) − cos (ui+1 )) + 41 λ(ui − ui+1 )4 ,
Fig. 4. Simulation carried out for an odd FPU moving breather.
known parameter field. Each iteration labeled by its j value mimics the moving breather propagating over one unit-cell. The results of this simulation procedure have been represented in Figs. 4 and 5 for the odd and even moving breathers, respectively. If the resulting discrepancy |vj (t) − u(t)| were found to vanish within numerical accuracy for j = 1, . . . , jf 1 and t ∈
where λ ∈ R is taken as a disposable parameter. Actually all previous authors have assumed λ = 0 but this case has been confirmed to sustain no soliton nor stationary breather or anharmonic phonon [13,15]. The equations of motion read for a moving breather u(t) ¨ = u(t − τ) + u(t + τ) − 2u(t) − sin (u(t)) + λ((u(t − τ) − u(t))3 − (u(t) − u(t + τ))3 ), (8) where τ −1 stands again for the velocity of the moving breather. Odd [u(t) = −u(−t)] and even [u(t) = u(−t)] solutions are both consistent with Eq. (8) because of W2 (ui , ui+1 ) = W2 (−ui , −ui+1 ). As done in the FPU case, bringing linearized Eq.(8) in keeping with Eq. (3) owing to u(t → ±∞) → 0 yields the dispersion relations of the moving breather τ 1 − ω2 = 2 cos (ωτ) cosh − 3, 2 T T τ ω = sin (ωτ) sinh . T T
Fig. 5. Simulation carried out for an even FPU moving breather.
(9)
As the dispersion is completely determined by the harmonic limit of W2 , it is independent of λ. It has been plotted in Fig. 6. The dispersion data can be seen to exhibit the same qualitative features as in Fig. 1 whence they are inferred not to allow for the moving breather unfolding continuously into a stationary one or a soliton either. All our attempts to solve Eq. (8) with λ ∈ [0, 10] under the constraints in Eqs. (3) and (9) by combining
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The traveling field {ψj } in Eq. (11) and the moving breather {uj } studied in the foregoing section differ from each other in two respects: • {ψj } depends on two parameters k, µ whereas {uj } is assessed by a single one τ; • {ψj } is at odds with Eq. (3), in particular because it involves two distinct velocities µv, ω/k instead of a single one τ −1 for {uj }. Likewise looking for a solution {ψj∈Z } of Eq. (10) fulfilling Eq. (3) leads to the dispersion relations τ 1 = −2 sin (ωτ) sinh , T τ T ω = 2 cos (ωτ) cosh . T
3. Comparing with previous work
These relations do allow for no T > 0, which confirms that the model defined in Eq. (10) has no solution obeying Eq. (3) although it has indeed the solution written in Eq. (11). Similarly there is another integrable discrete anharmonic potential [31]. Nevertheless it is known to sustain no excitation but cnoidal waves and solitons, which thence rules out any possibility of having moving breathers. An extension of Eq. (11) to cope with nonintegrable lattice dynamical models has been reported [21]. It consists of seeking the unknown moving breather field {uj∈Z (t)} as having the form
3.1. The integrable case
uj =
Fig. 6. Dispersion relations ω(τ), T(τ) of the FK moving breather.
the finite-difference and Newton methods as done here above in the FPU model did not result in any solution except u(t) = 0, ∀t. This seems to indicate that the FK model cannot sustain moving breathers. Conversely stable solitons, stationary breathers and anharmonic phonons have been shown [13,15] to arise in the FK model provided λ ≥ 0.05.
uj,k ,
uj,k (t → ±∞) ∝ ei((kω+qk v)t−jqk ) ,
k∈Z
The discrete nonlinear Schrödinger model [7] and its variations [30] have been shown to be integrable with respect to moving breathers. It is defined as ˙ j∈Z = i(ψj−1 + ψj+1 )(1 + |ψj |2 ). ψ
(10)
The field {ψj } associated with the integrable solution reads ψj =
sinh (µ) ei(ωt−jk) , cosh(µ(j − vt))
(11)
where the frequency ω = 2cosh(µ) cos (k) and the velocity v = 2 sinh (µ) sin (k)/µ are functions of two independent parameters k, µ.
where qk ∈ C. Linear analysis carried out for the FPU potential in Eq. (1) yields Re(qk ), Im(qk ) as functions of ω, v by solving (kω + qk v)2 = 2(1 − cosh(iqk )). This allows also for a continuous path linking moving (v = 0) to stationary (v = 0) breathers. A fixed point procedure [21] has been applied to solve the equation of motion for the moving breather but has failed to bring about any clear-cut conclusion on the existence of the moving breather because solutions appeared to be teeming around the fixed point.
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3.2. The perturbational schemes Moving breathers have been also studied within the quasidiscreteness approach (QDA) [20,22–25], which the rotating wave approximation (RWA) [26–28] is a variation of. In QDA and RWA the vibrational field of the breather is worked out from a truncated perturbation expansion. The results are believed to be reliable in case of weak anharmonicity. At the lowest order of QDA, the vibrational pattern reads for a moving breather uj∈Z (t) = ξj + φj cos (ω(k, κ0 )t − kj), where the field {uj (t)} depends on two independent real parameters k, κ0 whereby it differs from the moving breather addressed in the preceding section. Furthermore it does not fulfil Eq. (3) either. Within the limit t → ±∞, we obtain for ξj , φj ξj (t → ±∞), φj (t → ±∞) ∝ e−|κ0 (j−vg (k)t)| , where k, vg stand for the wave-vector and group velocity of harmonic phonons, respectively. Inserting these formulae into the linearized equation of motion due to uj (t → ±∞) → 0 results, for the FPU potential in Eq. (1), in three constraints on k, κ0 (κ0 vg )2 = 2(cosh(κ0 ) − 1), (κ0 vg + iω)2 = 2(cosh(κ0 + ik) − 1). Such a system of three equations relating two unknowns k, κ0 has in general no solution k, κ0 . Due to this inconsistency, simulations based on the QDA and RWA results are bound to decay in the long run by radiating spurious oscillations referred to as nanopterons [6] whence it is concluded that QDA and RWA fail to yield stable moving breathers.
the coupling within a finite coding sequence of initially uncoupled oscillators. The continuation process is likely to break down for strong coupling. As the interatomic potential involves necessarily an on-site contribution, it is suited to the FK model but not the FPU one. This scheme has been extended to moving breathers by setting stationary ones to overcome the Peierls–Nabarro barrier [3,29]. By construction there is a continuous path between stationary and moving breathers. The boundary conditions used for a finite crystal are very different from the case of the infinite one because no matching with an asymptotic regime is undertaken at the crystal edges. Therefore finite-size effects including nanopterons show up so that UOL based moving breathers turn out to lack long-term stability too.
4. Conclusion The existence of moving breathers has been discussed in anharmonic, one-dimensional lattices. There is no unique definition of this dynamical excitation. Nevertheless requiring its displacement field to vanish at both ends of a crystal of infinite size, consistent with the equation of motion, has led to dispersion relations in terms of one or two independent parameters, depending on the chosen definition of the moving breather. These constraints cannot be satisfied in the QDA treatment, which seemingly points toward an inconsistency. But even though they may be fulfilled in other approaches, the resulting vibrational patterns have displayed instability features in simulations so that apart from the integrable case [7] the existence of moving breathers remains doubtful in anharmonic crystals.
3.3. Finite systems
Acknowledgements
All approaches discussed so far have attempted to cope with moving breathers in infinite crystals. However the uncoupled oscillator limit (UOL) [32] has been devised primarily to investigate breathers [33] in finite lattices. It consists of switching on continuously
We thank François-Xavier Girod and Olivier Huet for providing invaluable help at an early stage of this work. One of us (G.H.) is indebted to the French Ministry of Research and Université Paris 7 for two visiting grants.
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