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Discrete gap breathers in chains with strong hydrogen bonding
Physica B 296 (2001) 251}258
Discrete gap breathers in chains with strong hydrogen bonding A.V. Zolotaryuk *, P. Maniadis, G.P. Tsironis Bogolyubov Institute for Theoretical Physics, 03143 Kyiv, Ukraine Department of Physics, University of Crete and Foundation for Research and Technology } Hellas (Forth), P.O. Box 2208, 71003 Heraklion, Crete, Greece
Abstract We consider a diatomic chain of heavy ions coupled by hydrogen bonds which are su$ciently strong compared with other interactions in the system. In this case, each proton in the hydrogen bond is subject to a single-minimum potential resulting from its interaction with nearest-neighbor heavy ions through the Morse potential that contains soft anharmonicity. This diatomic chain of nonlinearly coupled masses admits discrete breather solutions in the gap of the phonon spectrum. Simple analytical arguments accompanying explicit solutions that demonstrate the existence of the gap breather with only one type of symmetry, namely the odd-parity pattern centered at a hydrogen-bonded proton, are present. These arguments are supported by the numerically exact procedure using the anticontinuous limit. Some other multi-breather solutions in the gap are also obtained exactly from the anticontinuous limit. 2001 Elsevier Science B.V. All rights reserved. PACS: 63.20.Ry; 63.20.Pw; 63.70.#h; 03.20.#i Keywords: Hydrogen-bonded chains; Diatomic lattices; Discrete breathers; Anticontinuous limit
1. Introduction Intrinsic localized modes or discrete breathers (for a review see, e.g., [1]) are nonlinear collective excitations that seem to play a very important role in condensed matter physics. Interest in these modes has been intensi"ed recently due to experimental generation and observation in some chemical compounds [2] and antiferromagnets [3], coupled arrays of Josephson junctions [4,5], and
* Correspondence address: Department of Physics, University of Crete, P.O. Box 2208, 71003, Heraklion, Crete, Greece. Fax: #30-81-394201. E-mail address: [email protected], [email protected] (A.V. Zolotaryuk).
even possibly in myoglobin [6]. Another type of condensed matter systems in which discrete breathers could possibly exist and could be detected experimentally are hydrogen-bonded (HB) systems such as ice, quasi-one-dimensional HB crystals, and one-dimensional (1D) HB chains (for a review see, e.g., [7]). The basic idea in the nonlinear model for proton dynamics in a HB chain stems from the fact that the proton in each H-bond of the chain can be constructed as the sum of two-body ion}proton potentials, e.g., the Morse potentials [8]. If this (asymmetric) two-body potential is su$ciently strong compared to other interactions (e.g., between heavy ions or with an external on-site coupling), the resulting potential for the HB proton has only one minimum and the H-bond in this case is referred to as the strong H-bond [9]. Otherwise,
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this potential has two degenerate minima separated by a barrier [8,10,11] and the breather solutions have been studied before in the long-wavelength approximation [12]. Schematically, a chain with strong hydrogen bonding can be represented by the diatomic sequence 2 }X}H}X}H}X}H}X} 2, where X denotes a heavy ion and H a proton. In general, a 1D anharmonic diatomic lattice that has a gap in its phonon band, admits standing (anharmonic gap modes) and moving (gap solitons) solutions which were studied using di!erent approximate techniques [13}20]. In particular, the diatomic chain with realistic (asymmetric soft) two-body nearest-neighbor potentials was studied recently both analytically and numerically [18,21,22]. However, on one side, the gap breather solutions obtained in these studies have not been treated in a rigorous way and therefore they cannot be referred to as really exact and stable solutions, and on the other hand, di!erent analytical procedures (like asymptotic expansion and others) are too sophisticated to demonstrate which symmetry of the intrinsic gap modes (gap breathers) does exist being stable. Therefore some, possibly oversimpli"ed arguments or procedures should be developed to gain a better understanding of the origin of the existence and stability of the gap breathers (say, on the level of linear and/or quadratic algebraic equations, like the equation explaining the structure of the phonon band gap), before going to the numerically exact procedure of "nding the gap breathers from the anticontinuous limit, discovered recently in a series of papers [23}27]. Therefore the present paper aims to "nd in chains with strong hydrogen bonding the exact solutions for discrete gap breathers and to prove their stability, using the anticontinuous approach [26]. This numerical work is accompanied by analytical arguments based on the similarity [28] between the massimpurity mode [29,30] and intrinsic localized modes.
2. The diatomic model and localization length of gap breathers Let the heavy ions and the protons (light masses) in a HB chain, label according to the sequence
2, Q ,q ,Q , q , Q ,q , where L\ L\ L L L> L> 2 Q is the displacement of the nth ion and q , the L L displacement of the nth proton from their equilibria. We denote by and , the phonon frequencies of the heavy and the light masses at these equilibria, respectively. These two lattice displacement "elds can be replaced by one "eld 2, u , u , u ,2 with local characteristic H\ H H> frequency according to the relations u "Q and H H L " if j"2n, and u "q and " if H H L H j"2n#1. Then the equations of motion of this system can be written in the form of one discrete equation as uK "[=(u !u )!=(u !u )], H H H> H H H\ j"0,$1,2,
(1)
where =(r) is an asymmetric two-body potential of the standard type and normalized by =(0)"1; the overdot and the prime denote the di!erentiation with respect to time t and the relative distance r, respectively. In the case of the Morse potential we have =(r)"(\/2)[1!exp(!r)], '0.
(2)
The phonon band of the system is obtained directly from the linearized equation of motion (1): it has the gap (2 ))(2 splitting the band into the lower (acoustic) and the upper (optical) branches [see below Eq. (17) at "1]. Here we will deal with strongly localized vibrational modes the amplitude of which has an exponential asymptotic behavior. Imposing for the amplitudes of oscillations to have at nP$R the exponential factor exp(! n / ),p L , 0(p(1, with being the localization length of the mode and substituting this ansatz into the linearized equation (1), one obtains the relation / "1# $((1! p)(1! /p), "m/M" / , (3) that couples the breather frequency with the localization parameter . This relation has two (lower and upper) branches that merge at the critical point "!1/ln as shown in Fig. 1. The upper (lower) branch bifurcates from the out-ofphase oscillations of light (heavy) masses and
A.V. Zolotaryuk et al. / Physica B 296 (2001) 251}258
Fig. 1. Two (upper and lower) branches of frequency as functions of localization length within phonon gap that merge at critical point . Region 1 corresponds to acoustic band, region 2 to phonon gap and region 3 to optical band.
standing heavy (light) masses. Note that for stronger localization than the exponential one, like the breathers with compact support in the continuum limit [31], the region * (see Fig. 1), in which the gap breather solutions are possible, would be extended to the left. However, for our qualitative arguments given below, Eq. (3) can be adopted as a `working curvea with the inequality ( e\ H"p to be valid.
(heavy or light) particles oscillate symmetrically and out-of-phase with large amplitudes. The rest particles of the chain are assumed to perform small-amplitude oscillations. We denote the breather mode with a "xed ion or proton by light even-parity (LE) or heavy even-parity (HE), respectively. Using the rotating wave approximation in this way, we get three equations with respect to the three amplitudes: one for the central particles and the other two for the decaying light and heavy tails of the chain. The solution to these equations, including Eq. (3), appears to be very simple, so that its behavior with varying the breather frequency within the phonon band gap can be treated in very simple terms. For the LO mode centered at the site n"0, the central proton is assumed to oscillate with large amplitudes: q "a cos( t), where the amplitude a is to be determined. The rest of the heavy and light particles are assumed to oscillate in a symmetric way with an exponential decay: Q "(!1)L>ApL\ cos( t), n"1,2,2, L Q "(!1)LAp\L cos( t), n"0,!1,!2,2, L q "(!1)Lap L cos( t), n"$1,$2,2, L
3. Analytical arguments obtained from similarity between impurity and intrinsic gap modes To simplify analytical calculations as much as possible, we restrict ourselves, in Eq. (1), to the symmetric quartic potential =(r)"r/2# r/4 with the anharmonicity parameter being either positive (hard anharmonicity) or negative (soft anharmonicity). Similarly to the mass-impurity mode, we assume that only one mass (a proton or a heavy ion), supposed to be a center of the breather with odd-parity symmetry, performs large-amplitude oscillations, whereas the rest masses oscillate with small amplitudes, so that they approximately obey the linearized equations of motion for which Eq. (3) is valid. We denote each of these localized modes by light odd-parity (LO) and heavy oddparity (HO), respectively, for a proton-centered and an ion-centered breather. For the breathers with even-parity symmetry, we suppose the central (light or heavy) particle to be "xed, whereas the lateral
253
(4)
so that the LO pattern schematically can be represented as 2; pa,!pA;!pa, A; a ; A,!pa;!pA, pa;2. (5) The two equations of motion for the central and one of the lateral (e.g., the right one) particles with n"0 can be written approximately in the form qK K2 [Q !q # q (3Q !q )], Q$ K [q !2Q #q # q (q !3Q )],
(6)
where only the coordinate q is assumed to have large-amplitude values. The rest of the linearized equations of motion (1) yields relation (3) and the amplitude ratio a / !2 p\!1 "! " . A 1!p / !2
(7)
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Using Eqs. (3) and (7), from the equations of motion (6) we get the solution for the central proton 2(p\!p) a " 3 [1#p! / !3(1!p) /2 ]
(8)
as a function of the breather frequency and the amplitude ratio A / !2!3 a /2 . "! a 2#9 a /2
(9)
Similarly, for the HO pattern we assume the ansatz which schematically can be represented as 2; pa, pA;!pa,!pA; a; A , a; !pA,!pa; pA, pa;2.
(10)
Therefore the solution for the HO pattern can be obtained from Eqs. (7) to (9) by the substitution a PA , aA, and . Consider now the LE mode centered at a "xed heavy ion and assume that the lateral protons perform large-amplitude out-of-phase oscillations, i.e., Q ,0 and q "!q "a cos( t), where the \ amplitude a is to be determined. The LE ansatz can be represented schematically as 2; pa,!pA;!pa, pA;!a , 0, a ; !pA, pa; pA,!pa;2.
(11)
In the way similar to the LO pattern, one obtains the solution for this case. It is given by the quadratic equation
3 1 / !2 8 # # a # a 3! 9 2 1!p\ 16 ! p\"0 27
(12)
and the relations A / !2!3 a /2 , " a p(1#9 a /4)
(13)
a / !2 1!p " " . A 1!p\ / !2
(14)
Finally, the ansatz for the HE mode is given by q ,0, Q "!Q "A cos( t) with the ampli tude A to be determined, and the sequence 2; pA,!pa;!pA, pa;!A , 0, A ; !pa, pA; pa,!pA;2. (15) The solution for this mode is obtained from Eqs. (12) and (13) by the same substitution as in the case of the odd-parity modes. In order to "nd which of the four solutions obtained above is right one in the case of soft anharmonicity, we accept the rule that such a solution bifurcating from the upper or lower gap edge, must continuously pertain the signs of all its amplitudes A, a, and A or a along the curve (3) and pass the region of strong localization, i.e., the critical point . Note that Eqs. (7) and (14) keep the sign of A/a within the whole curve, but this is not a case for the other equations that determine the breather solutions. First consider the solution for the LO mode. Since the expression in the square brackets of Eq. (8) is negative for all the gap frequencies , the LO pattern can exist only if (0 (soft anharmonicity). This mode bifurcates from the upper gap edge, where it has the asymptotics given by a P2 (1!p)/3 (1! ) and A/a P! (1!p)/ 2(1! )(0. Next, it follows from Eq. (7) that A/a(0 and therefore A(0 and a'0 if a '0. These signs are kept up to some point on the lower branch, overcoming the central region of strong localization. In particular, at the gap middle where " # , the exact solution a "4/ 3(2#3 ), A"! a /2, and a"a /2, demon strates strong localization of the LO mode. The similar analysis of the equations which are obtained from Eqs. (7) to (9) as described above shows that the HO solution bifurcating from the lower edge fails before reaching the middle point along curve (3). Moreover, it exists only for hard anharmonicity and its pattern coincides with that found by Chubykalo and Kivshar [16,17], but it seems to be unstable because it cannot reach the middle point. In the case of the LE solution that bifurcates from the upper edge, we "nd from the quadratic equation (12) the asymptotics a P! (1!p)/
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3(1! )(0 as pP1 (therefore (0) and from Eq. (13) A/a P (1!p)/2(1! )'0. Since a/A(0 [see Eq. (14)], a implies the inequalities A'0 and a(0. However, these signs change on the way along the upper branch before reaching the critical point . Therefore, the LE mode is not acceptable. The similar analysis of the HE solution bifurcating from the lower edge and obtained from Eqs. (12) to (14) by the substitution described above, yields the asymptotics A P(1!p)/3(1! )'0 and a/A P!(1!p)/2(1! )(0 as pP1, and the inequality a/A'0. Consequently, this pattern exists if the anharmonicity is hard and A (0 implies A'0 and a'0; this also agrees with the pattern found by Chubykalo and Kivshar [16,17]. In the gap middle, the solution is A "4/3 '0 and a/A "!(1# )/ (3# )(0. This solution is well de"ned and its signs are persistent along the upper branch; therefore it is expected to be stable if the anharmonicity is hard.
255
the LO pattern survives. Here we support this statement using the Newton method in the numerical calculation of the gap breathers which is based on the concept of the anticontinuous (AC) limit [23}27]. In order to introduce the AC in our system, we rewrite the equation of motion (1) in terms of the relative displacement "eld r "u !u H H> H [26]: rK "!(# )=(r ) H H H> H #[=(r )# =(r )] (16) H H\ H> H> with the new parameter , 0))1. Then Eq. (16) can be thought as a Klein}Gordon system with nonlinear dispersion and the nearest-neighbor coupling, . In the limit P0, we obtain the system of uncoupled nonlinear oscillators and in the other
4. Exact breather solutions obtained from the anticontinuous limit Thus, from analysis given above, we expect that in the case of the realistic Morse potential (2), only
Fig. 2. Phonon dispersion relation as a function of parameter . Acoustic band (k; ) is bounded by solid lines and optical \ band (k; ) by the dashed lines. At "0 both bands merge at > frequency "( # ( "0.1 and "0.7).
Fig. 3. Exact breather solution (a) and its Floquet analysis (b) for the chain with "0.1 and "0.7. This solution corres ponds to asymmetric even-parity breather centered at a proton (LO mode). The Floquet analysis illustrates stability of this breather solution.
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limit P1, we have the original system we want to study. Using the standard Newton method, "rst we can "nd the breather solution for small values of and then continue it up to "1. Unfortunately, due to resonances with the phonons, this procedure fails and the reason of this can be illustrated on the dispersion relation of the system (16) given as a function of " # $( #2 cos(k) # , ! 0)k) (17) and depicted in Fig. 2 for "0.1 and "0.7 . As can be seen from this "gure, in the AC limit (P0) all the normal modes of the system have the same frequency, namely "( # . Indeed, as the coupling increases, the frequencies change continuously to form the two (acoustic and optical) bands and when "1, we retrieve the well-known dispersion law for the diatomic lattice. However, at "0 there is no a gap in the phonon spectrum in which we could initially "nd a breather solution.
Therefore, in order to avoid the resonance with the phonons at "0, we consider at each lattice site [in Eq. (16)] a local mass m "1#(!1)L , where L
is some small quantity. This modi"cation of Eq. (16) creates a gap in the phonon spectrum even in the limit "0. Now it is easy to "nd the breather solution in this gap and then to continue it for non-zero values of . After some iterations, we can slowly set "0 and continue the breather solution up to "1, getting the breather solution for the original system. After we have found the breather solution for some frequency, we vary the frequency with small steps and using the Newton method, we "nd the breather solution for any frequency within the phonon gap. The breather solution and the Floquet stability analysis are presented in Fig. 3, and in Fig. 4, we can see the time evolution of the breather solution. Using this method, one can "nd not only single breathers, but also multi-breather solutions of the system. Thus, Fig. 5 depicts the `101a multi-breather (according to the notation given by Aubry [24}26]), and Fig. 6 illustrates its time evolution.
Fig. 4. Time evolution of the breather solution shown in Fig. 3.
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5. Summary and outlook
Fig. 5. Exact `101a multi-breather solution (a) and its Floquet analysis (a) for the with "0.1 and "0.7. As illustrated by the Floquet analysis, two pairs of eigenvalues collide and escape from the unit circle, so that the solution becomes unstable.
The diatomic chain with an asymmetric interatomic coupling of the standard type (Morse, Lennard}Jones, etc.) that contains soft anharmonicity seems to be an appropriate theoretical model for one-dimensional hydrogen-bonded systems in which the H-bond is su$ciently strong compared to other interactions in the chain (like the coupling between the nearest heavy ions or an interaction with a substrate), so that the total on-site potential for the HB proton appears to have a single equilibrium position. This diatomic chain of nonlinearly coupled masses is known to admit discrete breather solutions in the gap of the linear phonon band [21,22,32,33]. However, yet these solutions have not obtained in the framework of the AC approach [23}26], the procedure which presently is considered as a rigorous method to "nd discrete breathers and to investigate their stability properties. We have also developed the simplest analytical procedure that allows us to "nd which symmetry of possible gap breather ansatzen can be accepted.
Fig. 6. Time evolution of the multi-breather shown in Fig. 5.
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This approach is based on the similarity between the linear mass-impurity mode and the possible intrinsic gap modes. We impose an exponential decay of the gap modes, "nd the relation (3) between the gap frequency and the localization length using it as a `working curvea, and next allow only one or two particles at the breather center to perform large-amplitude oscillations. As a result, we get the set of three simple algebraic equations which can be solved explicitly. Finally, we accept the rule the breather solution is appropriate if while moving along curve (3) starting at the upper or lower gap edge, it reaches and passes the region of strong localization near the critical point at the middle of the gap, not changing all the amplitude signs. Thus, in our system, the odd-parity pattern centered at a HB proton (we call it the LO mode) is shown to be an appropriate solution using such an approach. This result has been supported by the numerically exact procedure using the AC limit. Some other multi-breather solutions in the gap have also been obtained exactly from the AC limit.
Acknowledgements The work was supported from INTAS grant No. 96-0156 of the European Community. The authors are grateful to S. Aubry for helpful discussions and valuable suggestions. We also thank Prof. E.N. Economou for numerous discussions and fruitful collaboration over the course of many years on topics related to the subjects of nonlinearity and localization and wish to dedicate this work to him on the occasion of his 60th birthday.
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[3] U.T. Schwarz, L.Q. English, A.J. Sievers, Phys. Rev. Lett. 83 (1999) 223. [4] E. Trias, J.J. Mazo, T.P. Orlando, Phys. Rev. Lett. 84 (2000) 741. [5] P. Binder, D. Abraimov, A.V. Ustinov, S. Flach, Y. Zolotaryuk, Phys. Rev. Lett. 84 (2000) 745. [6] A. Xie, L. van der Meer, W. Ho!, R.H. Austin, Phys. Rev. Lett. 84 (2000) 5435. [7] J.F. Nagle, S. Tristram-Nagle, J. Membr. Biol. 74 (1983) 1. [8] X. Duan, S. Scheiner, J. Mol. Struct. 270 (1992) 173. [9] N.D. Sokolov, M.V. Vener, V.A. Savel'ev, J. Mol. Struct. 222 (1990) 365. [10] A.V. Zolotaryuk, St. Pnevmatikos, A.V. Savin, Physica D 51 (1991) 407. [11] G. Kalosakas, A.V. Zolotaryuk, G.P. Tsironis, E.N. Economou, Phys. Rev. E 56 (1997) 1088. [12] St. Pnevmatikos, Y.S. Kivshar, M.J. Velgakis, A.V. Zolotaryuk, Phys. Lett. A 173 (1993) 43. [13] St. Pnevmatikos, N. Flytzanis, M. Remoissenet, Phys. Rev. B 33 (1986) 2308. [14] Y.S. Kivshar, N. Flytzanis, Phys. Rev. A 46 (1992) 7972. [15] O.A. Chubykalo, A.S. Kovalev, O.V. Usatenko, Phys. Rev. B 47 (1993) 3153. [16] O.A. Chubykalo, Y.S. Kivshar, Phys. Rev. E 48 (1993) 4128. [17] O.A. Chubykalo, Y.S. Kivshar, Phys. Rev. E 49 (1994) 5906. [18] G. Huang, Phys. Rev. B 51 (1995) 12 347. [19] R. Livi, M. Spicci, R.S. MacKay, Nonlinearity 10 (1997) 1421. [20] T. Cretegny, R. Livi, M. Spicci, Physica D 119 (1998) 88. [21] S.A. Kiselev, S.R. Bickham, A.J. Sievers, Phys. Rev. B 48 (1993) 13 508. [22] S.A. Kiselev, S.R. Bickham, A.J. Sievers, Phys. Rev. B 50 (1994) 9135. [23] R.S. MacKay, S. Aubry, Nonlinearity 7 (1994) 1623. [24] S. Aubry, Physica D 71 (1994) 196. [25] S. Aubry, Physica D 103 (1996) 201. [26] J.L. Marin, S. Aubry, Nonlinearity 9 (1994) 1501. [27] D. Chen, S. Aubry, G.P. Tsironis, Phys. Rev. Lett. 77 (1996) 4776. [28] P. Maniadis, Phys. Lett. A (2000), submitted for publication. [29] Q. Li, St. Pnevmatikos, E.N. Economou, C.M. Soukoulis, Phys. Rev. B 37 (1988) 3534. [30] T. Fraggis, St. Pnevmatikos, E.N. Economou, Phys. Lett. A 142 (1989) 361. [31] M. Eleftheriou, B. Dey, G.P. Tsironis, Phys. Rev. E 2000, to appear. [32] G. Huang, B. Hu, Phys. Rev. B 57 (1998) 5746. [33] B. Hu, G. Huang, M.G. Velarde, Phys. Rev. E 62 (2000) 2827.
Discrete gap breathers in chains with strong hydrogen bonding A.V. Zolotaryuk *, P. Maniadis, G.P. Tsironis Bogolyubov Institute for Theoretical Physics, 03143 Kyiv, Ukraine Department of Physics, University of Crete and Foundation for Research and Technology } Hellas (Forth), P.O. Box 2208, 71003 Heraklion, Crete, Greece
Abstract We consider a diatomic chain of heavy ions coupled by hydrogen bonds which are su$ciently strong compared with other interactions in the system. In this case, each proton in the hydrogen bond is subject to a single-minimum potential resulting from its interaction with nearest-neighbor heavy ions through the Morse potential that contains soft anharmonicity. This diatomic chain of nonlinearly coupled masses admits discrete breather solutions in the gap of the phonon spectrum. Simple analytical arguments accompanying explicit solutions that demonstrate the existence of the gap breather with only one type of symmetry, namely the odd-parity pattern centered at a hydrogen-bonded proton, are present. These arguments are supported by the numerically exact procedure using the anticontinuous limit. Some other multi-breather solutions in the gap are also obtained exactly from the anticontinuous limit. 2001 Elsevier Science B.V. All rights reserved. PACS: 63.20.Ry; 63.20.Pw; 63.70.#h; 03.20.#i Keywords: Hydrogen-bonded chains; Diatomic lattices; Discrete breathers; Anticontinuous limit
1. Introduction Intrinsic localized modes or discrete breathers (for a review see, e.g., [1]) are nonlinear collective excitations that seem to play a very important role in condensed matter physics. Interest in these modes has been intensi"ed recently due to experimental generation and observation in some chemical compounds [2] and antiferromagnets [3], coupled arrays of Josephson junctions [4,5], and
* Correspondence address: Department of Physics, University of Crete, P.O. Box 2208, 71003, Heraklion, Crete, Greece. Fax: #30-81-394201. E-mail address: [email protected], [email protected] (A.V. Zolotaryuk).
even possibly in myoglobin [6]. Another type of condensed matter systems in which discrete breathers could possibly exist and could be detected experimentally are hydrogen-bonded (HB) systems such as ice, quasi-one-dimensional HB crystals, and one-dimensional (1D) HB chains (for a review see, e.g., [7]). The basic idea in the nonlinear model for proton dynamics in a HB chain stems from the fact that the proton in each H-bond of the chain can be constructed as the sum of two-body ion}proton potentials, e.g., the Morse potentials [8]. If this (asymmetric) two-body potential is su$ciently strong compared to other interactions (e.g., between heavy ions or with an external on-site coupling), the resulting potential for the HB proton has only one minimum and the H-bond in this case is referred to as the strong H-bond [9]. Otherwise,
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this potential has two degenerate minima separated by a barrier [8,10,11] and the breather solutions have been studied before in the long-wavelength approximation [12]. Schematically, a chain with strong hydrogen bonding can be represented by the diatomic sequence 2 }X}H}X}H}X}H}X} 2, where X denotes a heavy ion and H a proton. In general, a 1D anharmonic diatomic lattice that has a gap in its phonon band, admits standing (anharmonic gap modes) and moving (gap solitons) solutions which were studied using di!erent approximate techniques [13}20]. In particular, the diatomic chain with realistic (asymmetric soft) two-body nearest-neighbor potentials was studied recently both analytically and numerically [18,21,22]. However, on one side, the gap breather solutions obtained in these studies have not been treated in a rigorous way and therefore they cannot be referred to as really exact and stable solutions, and on the other hand, di!erent analytical procedures (like asymptotic expansion and others) are too sophisticated to demonstrate which symmetry of the intrinsic gap modes (gap breathers) does exist being stable. Therefore some, possibly oversimpli"ed arguments or procedures should be developed to gain a better understanding of the origin of the existence and stability of the gap breathers (say, on the level of linear and/or quadratic algebraic equations, like the equation explaining the structure of the phonon band gap), before going to the numerically exact procedure of "nding the gap breathers from the anticontinuous limit, discovered recently in a series of papers [23}27]. Therefore the present paper aims to "nd in chains with strong hydrogen bonding the exact solutions for discrete gap breathers and to prove their stability, using the anticontinuous approach [26]. This numerical work is accompanied by analytical arguments based on the similarity [28] between the massimpurity mode [29,30] and intrinsic localized modes.
2. The diatomic model and localization length of gap breathers Let the heavy ions and the protons (light masses) in a HB chain, label according to the sequence
2, Q ,q ,Q , q , Q ,q , where L\ L\ L L L> L> 2 Q is the displacement of the nth ion and q , the L L displacement of the nth proton from their equilibria. We denote by and , the phonon frequencies of the heavy and the light masses at these equilibria, respectively. These two lattice displacement "elds can be replaced by one "eld 2, u , u , u ,2 with local characteristic H\ H H> frequency according to the relations u "Q and H H L " if j"2n, and u "q and " if H H L H j"2n#1. Then the equations of motion of this system can be written in the form of one discrete equation as uK "[=(u !u )!=(u !u )], H H H> H H H\ j"0,$1,2,
(1)
where =(r) is an asymmetric two-body potential of the standard type and normalized by =(0)"1; the overdot and the prime denote the di!erentiation with respect to time t and the relative distance r, respectively. In the case of the Morse potential we have =(r)"(\/2)[1!exp(!r)], '0.
(2)
The phonon band of the system is obtained directly from the linearized equation of motion (1): it has the gap (2 ))(2 splitting the band into the lower (acoustic) and the upper (optical) branches [see below Eq. (17) at "1]. Here we will deal with strongly localized vibrational modes the amplitude of which has an exponential asymptotic behavior. Imposing for the amplitudes of oscillations to have at nP$R the exponential factor exp(! n / ),p L , 0(p(1, with being the localization length of the mode and substituting this ansatz into the linearized equation (1), one obtains the relation / "1# $((1! p)(1! /p), "m/M" / , (3) that couples the breather frequency with the localization parameter . This relation has two (lower and upper) branches that merge at the critical point "!1/ln as shown in Fig. 1. The upper (lower) branch bifurcates from the out-ofphase oscillations of light (heavy) masses and
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Fig. 1. Two (upper and lower) branches of frequency as functions of localization length within phonon gap that merge at critical point . Region 1 corresponds to acoustic band, region 2 to phonon gap and region 3 to optical band.
standing heavy (light) masses. Note that for stronger localization than the exponential one, like the breathers with compact support in the continuum limit [31], the region * (see Fig. 1), in which the gap breather solutions are possible, would be extended to the left. However, for our qualitative arguments given below, Eq. (3) can be adopted as a `working curvea with the inequality ( e\ H"p to be valid.
(heavy or light) particles oscillate symmetrically and out-of-phase with large amplitudes. The rest particles of the chain are assumed to perform small-amplitude oscillations. We denote the breather mode with a "xed ion or proton by light even-parity (LE) or heavy even-parity (HE), respectively. Using the rotating wave approximation in this way, we get three equations with respect to the three amplitudes: one for the central particles and the other two for the decaying light and heavy tails of the chain. The solution to these equations, including Eq. (3), appears to be very simple, so that its behavior with varying the breather frequency within the phonon band gap can be treated in very simple terms. For the LO mode centered at the site n"0, the central proton is assumed to oscillate with large amplitudes: q "a cos( t), where the amplitude a is to be determined. The rest of the heavy and light particles are assumed to oscillate in a symmetric way with an exponential decay: Q "(!1)L>ApL\ cos( t), n"1,2,2, L Q "(!1)LAp\L cos( t), n"0,!1,!2,2, L q "(!1)Lap L cos( t), n"$1,$2,2, L
3. Analytical arguments obtained from similarity between impurity and intrinsic gap modes To simplify analytical calculations as much as possible, we restrict ourselves, in Eq. (1), to the symmetric quartic potential =(r)"r/2# r/4 with the anharmonicity parameter being either positive (hard anharmonicity) or negative (soft anharmonicity). Similarly to the mass-impurity mode, we assume that only one mass (a proton or a heavy ion), supposed to be a center of the breather with odd-parity symmetry, performs large-amplitude oscillations, whereas the rest masses oscillate with small amplitudes, so that they approximately obey the linearized equations of motion for which Eq. (3) is valid. We denote each of these localized modes by light odd-parity (LO) and heavy oddparity (HO), respectively, for a proton-centered and an ion-centered breather. For the breathers with even-parity symmetry, we suppose the central (light or heavy) particle to be "xed, whereas the lateral
253
(4)
so that the LO pattern schematically can be represented as 2; pa,!pA;!pa, A; a ; A,!pa;!pA, pa;2. (5) The two equations of motion for the central and one of the lateral (e.g., the right one) particles with n"0 can be written approximately in the form qK K2 [Q !q # q (3Q !q )], Q$ K [q !2Q #q # q (q !3Q )],
(6)
where only the coordinate q is assumed to have large-amplitude values. The rest of the linearized equations of motion (1) yields relation (3) and the amplitude ratio a / !2 p\!1 "! " . A 1!p / !2
(7)
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Using Eqs. (3) and (7), from the equations of motion (6) we get the solution for the central proton 2(p\!p) a " 3 [1#p! / !3(1!p) /2 ]
(8)
as a function of the breather frequency and the amplitude ratio A / !2!3 a /2 . "! a 2#9 a /2
(9)
Similarly, for the HO pattern we assume the ansatz which schematically can be represented as 2; pa, pA;!pa,!pA; a; A , a; !pA,!pa; pA, pa;2.
(10)
Therefore the solution for the HO pattern can be obtained from Eqs. (7) to (9) by the substitution a PA , aA, and . Consider now the LE mode centered at a "xed heavy ion and assume that the lateral protons perform large-amplitude out-of-phase oscillations, i.e., Q ,0 and q "!q "a cos( t), where the \ amplitude a is to be determined. The LE ansatz can be represented schematically as 2; pa,!pA;!pa, pA;!a , 0, a ; !pA, pa; pA,!pa;2.
(11)
In the way similar to the LO pattern, one obtains the solution for this case. It is given by the quadratic equation
3 1 / !2 8 # # a # a 3! 9 2 1!p\ 16 ! p\"0 27
(12)
and the relations A / !2!3 a /2 , " a p(1#9 a /4)
(13)
a / !2 1!p " " . A 1!p\ / !2
(14)
Finally, the ansatz for the HE mode is given by q ,0, Q "!Q "A cos( t) with the ampli tude A to be determined, and the sequence 2; pA,!pa;!pA, pa;!A , 0, A ; !pa, pA; pa,!pA;2. (15) The solution for this mode is obtained from Eqs. (12) and (13) by the same substitution as in the case of the odd-parity modes. In order to "nd which of the four solutions obtained above is right one in the case of soft anharmonicity, we accept the rule that such a solution bifurcating from the upper or lower gap edge, must continuously pertain the signs of all its amplitudes A, a, and A or a along the curve (3) and pass the region of strong localization, i.e., the critical point . Note that Eqs. (7) and (14) keep the sign of A/a within the whole curve, but this is not a case for the other equations that determine the breather solutions. First consider the solution for the LO mode. Since the expression in the square brackets of Eq. (8) is negative for all the gap frequencies , the LO pattern can exist only if (0 (soft anharmonicity). This mode bifurcates from the upper gap edge, where it has the asymptotics given by a P2 (1!p)/3 (1! ) and A/a P! (1!p)/ 2(1! )(0. Next, it follows from Eq. (7) that A/a(0 and therefore A(0 and a'0 if a '0. These signs are kept up to some point on the lower branch, overcoming the central region of strong localization. In particular, at the gap middle where " # , the exact solution a "4/ 3(2#3 ), A"! a /2, and a"a /2, demon strates strong localization of the LO mode. The similar analysis of the equations which are obtained from Eqs. (7) to (9) as described above shows that the HO solution bifurcating from the lower edge fails before reaching the middle point along curve (3). Moreover, it exists only for hard anharmonicity and its pattern coincides with that found by Chubykalo and Kivshar [16,17], but it seems to be unstable because it cannot reach the middle point. In the case of the LE solution that bifurcates from the upper edge, we "nd from the quadratic equation (12) the asymptotics a P! (1!p)/
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3(1! )(0 as pP1 (therefore (0) and from Eq. (13) A/a P (1!p)/2(1! )'0. Since a/A(0 [see Eq. (14)], a implies the inequalities A'0 and a(0. However, these signs change on the way along the upper branch before reaching the critical point . Therefore, the LE mode is not acceptable. The similar analysis of the HE solution bifurcating from the lower edge and obtained from Eqs. (12) to (14) by the substitution described above, yields the asymptotics A P(1!p)/3(1! )'0 and a/A P!(1!p)/2(1! )(0 as pP1, and the inequality a/A'0. Consequently, this pattern exists if the anharmonicity is hard and A (0 implies A'0 and a'0; this also agrees with the pattern found by Chubykalo and Kivshar [16,17]. In the gap middle, the solution is A "4/3 '0 and a/A "!(1# )/ (3# )(0. This solution is well de"ned and its signs are persistent along the upper branch; therefore it is expected to be stable if the anharmonicity is hard.
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the LO pattern survives. Here we support this statement using the Newton method in the numerical calculation of the gap breathers which is based on the concept of the anticontinuous (AC) limit [23}27]. In order to introduce the AC in our system, we rewrite the equation of motion (1) in terms of the relative displacement "eld r "u !u H H> H [26]: rK "!(# )=(r ) H H H> H #[=(r )# =(r )] (16) H H\ H> H> with the new parameter , 0))1. Then Eq. (16) can be thought as a Klein}Gordon system with nonlinear dispersion and the nearest-neighbor coupling, . In the limit P0, we obtain the system of uncoupled nonlinear oscillators and in the other
4. Exact breather solutions obtained from the anticontinuous limit Thus, from analysis given above, we expect that in the case of the realistic Morse potential (2), only
Fig. 2. Phonon dispersion relation as a function of parameter . Acoustic band (k; ) is bounded by solid lines and optical \ band (k; ) by the dashed lines. At "0 both bands merge at > frequency "( # ( "0.1 and "0.7).
Fig. 3. Exact breather solution (a) and its Floquet analysis (b) for the chain with "0.1 and "0.7. This solution corres ponds to asymmetric even-parity breather centered at a proton (LO mode). The Floquet analysis illustrates stability of this breather solution.
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limit P1, we have the original system we want to study. Using the standard Newton method, "rst we can "nd the breather solution for small values of and then continue it up to "1. Unfortunately, due to resonances with the phonons, this procedure fails and the reason of this can be illustrated on the dispersion relation of the system (16) given as a function of " # $( #2 cos(k) # , ! 0)k) (17) and depicted in Fig. 2 for "0.1 and "0.7 . As can be seen from this "gure, in the AC limit (P0) all the normal modes of the system have the same frequency, namely "( # . Indeed, as the coupling increases, the frequencies change continuously to form the two (acoustic and optical) bands and when "1, we retrieve the well-known dispersion law for the diatomic lattice. However, at "0 there is no a gap in the phonon spectrum in which we could initially "nd a breather solution.
Therefore, in order to avoid the resonance with the phonons at "0, we consider at each lattice site [in Eq. (16)] a local mass m "1#(!1)L , where L
is some small quantity. This modi"cation of Eq. (16) creates a gap in the phonon spectrum even in the limit "0. Now it is easy to "nd the breather solution in this gap and then to continue it for non-zero values of . After some iterations, we can slowly set "0 and continue the breather solution up to "1, getting the breather solution for the original system. After we have found the breather solution for some frequency, we vary the frequency with small steps and using the Newton method, we "nd the breather solution for any frequency within the phonon gap. The breather solution and the Floquet stability analysis are presented in Fig. 3, and in Fig. 4, we can see the time evolution of the breather solution. Using this method, one can "nd not only single breathers, but also multi-breather solutions of the system. Thus, Fig. 5 depicts the `101a multi-breather (according to the notation given by Aubry [24}26]), and Fig. 6 illustrates its time evolution.
Fig. 4. Time evolution of the breather solution shown in Fig. 3.
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5. Summary and outlook
Fig. 5. Exact `101a multi-breather solution (a) and its Floquet analysis (a) for the with "0.1 and "0.7. As illustrated by the Floquet analysis, two pairs of eigenvalues collide and escape from the unit circle, so that the solution becomes unstable.
The diatomic chain with an asymmetric interatomic coupling of the standard type (Morse, Lennard}Jones, etc.) that contains soft anharmonicity seems to be an appropriate theoretical model for one-dimensional hydrogen-bonded systems in which the H-bond is su$ciently strong compared to other interactions in the chain (like the coupling between the nearest heavy ions or an interaction with a substrate), so that the total on-site potential for the HB proton appears to have a single equilibrium position. This diatomic chain of nonlinearly coupled masses is known to admit discrete breather solutions in the gap of the linear phonon band [21,22,32,33]. However, yet these solutions have not obtained in the framework of the AC approach [23}26], the procedure which presently is considered as a rigorous method to "nd discrete breathers and to investigate their stability properties. We have also developed the simplest analytical procedure that allows us to "nd which symmetry of possible gap breather ansatzen can be accepted.
Fig. 6. Time evolution of the multi-breather shown in Fig. 5.
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This approach is based on the similarity between the linear mass-impurity mode and the possible intrinsic gap modes. We impose an exponential decay of the gap modes, "nd the relation (3) between the gap frequency and the localization length using it as a `working curvea, and next allow only one or two particles at the breather center to perform large-amplitude oscillations. As a result, we get the set of three simple algebraic equations which can be solved explicitly. Finally, we accept the rule the breather solution is appropriate if while moving along curve (3) starting at the upper or lower gap edge, it reaches and passes the region of strong localization near the critical point at the middle of the gap, not changing all the amplitude signs. Thus, in our system, the odd-parity pattern centered at a HB proton (we call it the LO mode) is shown to be an appropriate solution using such an approach. This result has been supported by the numerically exact procedure using the AC limit. Some other multi-breather solutions in the gap have also been obtained exactly from the AC limit.
Acknowledgements The work was supported from INTAS grant No. 96-0156 of the European Community. The authors are grateful to S. Aubry for helpful discussions and valuable suggestions. We also thank Prof. E.N. Economou for numerous discussions and fruitful collaboration over the course of many years on topics related to the subjects of nonlinearity and localization and wish to dedicate this work to him on the occasion of his 60th birthday.
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