Redalyc.Considerations on Undistorted-Progressive X-Waves

Brazilian Journal of Physics ISSN: 0103-9733 [email protected] Sociedade Brasileira de Física Brasil

Mesquita, Marcus V.; Vasconcellos, Áurea R.; Luzzi, Roberto Considerations on undistorted-progressive X-Waves and davydov solitons, Fröhlich-Bose-Einstein condensation, and Cherenkov-like effect in biosystems Brazilian Journal of Physics, vol. 34, núm. 2A, june, 2004, pp. 489-503 Sociedade Brasileira de Física Sâo Paulo, Brasil

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Considerations on Undistorted-Progressive X-Waves and Davydov Solitons, Frohlich-Bose-Einstein¨ Condensation, and Cherenkov-like effect in Biosystems

Marcus V. Mesquita,1 Aurea´ R. Vasconcellos,2 and Roberto Luzzi2 1. Instituto de F´ısica, Universidade Federal do Ceara,´ 60455-760, Fortaleza, CE, Brazil 2. Instituto de F´ısica ‘Gleb Wataghin’, Universidade Estadual de Campinas, Unicamp, 13083-970 Campinas, SP, Brazil

Received on 3 June, 2003

Research in ultrasonography has evidenced the propagation of a peculiar kind of excitation in fluids. Such excitation, dubbed a X-wave, has characteristics resembling that of a solitary-wave type. We reconsider the problem in a medium consisting of a biological material of the like of α-helix proteins. It can be shown that in this case is expected an excitation of the Davydov’s solitary wave type, however strongly damped in normal conditions. The case of acetanilide, an organic polymer which resembles biopolymers, is considered, and the infrared spectrum analyzed. Davydov’s soliton is evidenced as a coherent state of polar vibrations. The case of acoustic (sound) vibrations is also considered, where, also, a damped Davydov-like solitary wave may be excited. However, it is shown that when traveling in conditions sufficiently away from equilibrium, more precisely, when the soliton is embeded in the resulting Frohlich-Bose-Einstein¨ condensate, the lifetime of the solitary wave is largely enhanced. Moreover, a soliton moving in bulk with a velocity larger than that of the group velocity of the normal vibrational waves would produce a Cherenkov-like emission of phonons giving rise to the observed X-wave-like pattern. This paper is a modified and extended version of an earlier publication on the subject.

1 Introduction polymers with very large conductivity for, e. g., use in very light, almost two-dimensional (sheets) batteries [7, 8, 9], and Experiments in ultrasonography have evidenced a particular the case of propagation of light in optical fibers [10]. An- kind of wave propagation, dubbed as X-waves [1, 2]. They other example is that of the so-called Davydov’s solitons have the characteristic of propagating without dispersion, [11, 12], which may have a quite relevant role in bioener- and may result of relevance for ultrasound medical imag- getics. ing. Later on, they were ascribed to some kind of the so- Davydov’s theory has received plenty of attention, and a called undistorted propagating waves [3]. They are waves long list of results published up to the first half of 1992 are traveling in a material media, and the reported characteris- discussed in a comprehensive review due to A. C. Scott [13]. tics point to the possibility of they belonging to the category As pointed out in that review, one question concerning of solitary waves. Soliton is the name coined to describe Davydov’s soliton is that of its stability at normal physi- a pulse-like nonlinear wave (the solitary wave referred to ological conditions, that is, the ability of the excitation to above) which emerges from a collision with a similar pulse transport energy (and so information) at long distances in the having unchanged shape or speed. Its relevance in applied living organism, in spite of the relaxation mechanisms that sciences has been described in a 1973 review paper by A. C. are expected to damp it out at very short (micrometers) dis- Scott et al. [4] Because of the above mentioned technologi- tances. On the assumption that the X-waves in ultrasonog- cal/medical relevance we reanalyze the question in the case raphy may belong to the category of solitary waves in the of propagation in biological materials. material media, we consider this question on the basis of a The original category of solitary waves consists in the model in nonequilibrium thermodynamic conditions, since one observed by the Scottish engineer Scott-Russell in Au- any biological system is an open system out of thermal equi- gust 1834 in an English water channel, and reported in a librium. For that purpose we resort to an informational sta- 1844 meeting of the British Society for the Advancement of tistical thermodynamics (see for example [14-19]), based on Science [5, 6]. During the second half of this century many the Nonequilibrium Statistical Operator Method (NESOM) other types of solitary waves have been associated to a num- [20-23]. The NESOM, which provides microscopic foun- ber of physical situations in condensed matter physics. Sev- dations to phenomenological irreversible thermodynamics eral, seemingly, have a fundamental role in important tech- [19], also allows for the construction of a nonlinear gen- nological areas of large relevance for contemporary society. eralized quantum transport theory — a far-reaching gener- Among them we may highlight the case of doped organic alization of the Chapman-Enskog’s and Mori’s methods — 490 Marcus V. Mesquita, Aurea´ R. Vasconcellos, and Roberto Luzzi which describes the evolution of the system at the macro- acoustic-like vibrations, which is described by the so-called scopic level in arbitrary nonequilibrium situations [20,23- Frohlich-Davydov¨ Hamiltonian given in [11] and [27]: This 25], a formalism to be used in what follows. We notice Hamiltonian is given in the so-called Random Wave Ap- that the method has been applied with particular success to proximation, while the full Hamiltonian — to be used in the study of optical and transport properties in solids [26] what follows — is given by and also to modelled biopolymers [27, 28], of the kind we consider here, and to techno-industrial processes [29]. Con- H = H0 + HI = H0S + H0B + HI , (1) sequently, we do not go over the details of the formalism, just adapting those earlier results for the present discussion, where giving in each case the reference where they are reported. µ ¶ We separate the presentation into two parts: After a general X 1 H = ~ω a†a + , (2a) derivation of the equations of evolution, a first analysis is 0S ~q ~q ~q 2 concentrated on the case of polar vibrations (with frequen- ~q cies in the infrared region of the spectrum), for which are available experimental results, particularly the case of ac- X 1 etanilide, which we analyze, comparing theory and experi- H = ~Ω (b†b + ) , (2b) ment. This is done in subsection 2.1, while in subsection 2.2 0B ~q ~q ~q 2 ~q we deal with the case of longitudinal acoustic vibrations. As stated, we first consider a system where modes of polar vibrations are excited by a continuous supply of energy. X X These polar modes are coupled through a nonlinear kinet- † (1) † HI = Z~qϕ a + V a~q b~q b ics with a bath consisting of a continuous medium modelled ~q ~q ~q1~q2 1 2 ~q1+~q2 by a system of acoustic-like vibrations. The equations of ~q ~q1~q2 evolution for the population of the vibrational modes are de- X X rived resorting to the nonlinear quantum kinetic theory that + V (1) a b† b + V (1) a b b ~q1~q2 ~q1 ~q2 −~q1+~q2 ~q1~q2 ~q1 ~q2 −~q1−~q2 the NESOM provides. This corresponds to the description ~q1~q2 ~q1~q2 of polar vibrations of the CO-stretching type (Amide-I) in, X X for example, α-helix proteins [11-13]. Experiments, of the (1) † † (2) † + V a~q b b + V a~q a~q b class of Raman or neutron scattering or radiation absorp- ~q1~q2 1 ~q2 ~q1−~q2 ~q1~q2 1 2 ~q1+~q2 ~q ~q ~q ~q tion, in active biological systems are particularly difficult. 1 2 1 2 X X For that reason, the comparison of the theory of next sec- (2) (2) † + V a~q a~q b−~q −~q + V a a~q b~q −~q tion with experiment is done in the case of the organic poly- ~q1~q2 1 2 1 2 ~q1~q2 ~q1 2 1 2 mer acetanilide, which constitutes a good mimic of biopoly- ~q1~q2 ~q1~q2 mers [30]. The infrared spectrum in the frequency region X + V (2) a a† b† + H.c. (2c) corresponding to the CO-stretching oscillation is derived, ~q1~q2 ~q1 ~q2 ~q1−~q2 and compare very well with the experimental measurements, ~q1~q2 confirming the presence of Davydov’s soliton. Finally, we comment on the possibly very large increase of the lifetime Figure 1 in [27] describes a particular biological system of Davydov’s soliton when propagating in an open medium and the mechanical analog we are using. The Hamiltonian sufficiently far from equilibrium. consists of the energy of the free subsystems, namely, that In subsection 2.2, we consider a system of longitudinal of the free vibrations, with ω~q being their frequency disper- acoustic vibrations in interaction with the above said ther- sion relation (~q is a wave-vector running over the reciprocal- mal bath. We find a behavior of the acoustic modes quite space Brillouin zone), and that of the thermal bath composed similar to the one evidenced for the optical modes, namely, by oscillations with frequency dispersion relation Ω~q, with existence of the solitary-wave excitation, damped near equi- a Debye cut-off frequency ΩD. The interaction Hamiltonian librium conditions but whose lifetime is greatly enhanced HI contains the interaction of the system of polar vibrations when propagating in a highly excited background. with an external source [which pumps energy on the system Finally, it is demonstrated the possible emergence of a and is the first term on the right side of Eq. (2c)], and, finally, particular phenomenon, which we call Frohlich-Cherenkov¨ the anharmonic interactions between both subsystems. The effect, consisting in that when the soliton is propagating with latter are composed of several contribution, namely, those a velocity larger than the group velocity of the normal modes associated with three quasi-particle (phonons) collisions in- of vibration in the medium, a large number of phonons are volving one of the system and two of the thermal bath (we call V (1) the corresponding matrix element), and two of the emitted at a certain angle with the direction of propagation ~qq~0 of the soliton. system and one of the bath (we call V (2) the corresponding ~qq~0 † † matrix element). Finally, a~q (a~q), b~q (b~q), are, as usual, anni- 2 The Solitary Wave hilation (creation) operators of, respectively, normal-mode vibrations in the system and bath in mode ~q, and the one Let us consider a model biosystem which can sustain longi- corresponding to the quantum excitations in the pumping tudinal vibrations and in interactions with a thermal bath of source, with Z being the coupling strength. Brazilian Journal of Physics, vol. 34, no. 2A, June, 2004 491

† a~q and a~q. Finally, since the thermal bath is taken as re-

maining constantly in a stationary state at a temperature T0, via an efﬁcient homeostatic mechanism, we introduce its

§§ Hamiltonian, HB. The average of these quantities over the

" #%$& ('*),+.-¢/ NESOM

§ ! nonequilibrium ensemble constitute the basic set of

" $&

§12),+435-6/

0 macrovariables, which we designate as

&29

-6/8 3

7 b

\ n o

†

_ ` a

^ ν~q(t), ha~q|ti, ha~q|ti,EB , (3)

]

[M\

WYXZ ¡

¥ that is, they deﬁne Gibbs’s space of nonequilibrium thermo-

U V

dynamic states. This is the thermodynamic state space in

P Informational Statistical Thermodynamics (IST for short).

R S

Q IST is the thermodynamic theory for irreversible processes

:<;>=*?.@ ACBD?*EGF based on the nonequilibrium ensemble formalism NESOM

HJIK?GLM@6N [20, 21, 22]. The basic thermodynamic macrovariables of

Eq. (3) are then given by

¥ ¥

¢¡¤£¦¥ §¡¨¡©¥ ¢¡ ¥ ¢¡¨ ©¥

ν~q(t) = Tr {νˆ~q%(t)} ; (4a)

¢t cd:%egfh:jilk>mon6pqBsr

ha~q|ti = Tr {a~q%(t)} ; (4b) n o † ∗ † Figure 1. Optical vibrations: The IR absorption spectrum in ac- ha~q|ti = ha~q|ti = Tr a~q%(t) ; (4c) etanilide at 80 K. Points are from the experimental data reported in n o [38], and the full line the calculation after Eq. (23). EB = Tr HˆB%B (4d)

%(t) Next step consists in the choice, within the tenets of NE- where is the nonequilibrium statistical operator of the % SOM, of the basic set of dynamical variables relevant for system, and B the canonical distribution of the thermal bath the present problem. Since we are dealing with excitation at temperature T0. The statistical operator %(t) is taken in Zubarev’s approach [20,21], in this case given by of vibrations in modes ~q (with energy ~ω~q), we need to in- † troduce the number of excitations in each mode, νˆ = a a . ˆ ~q ~q ~q %(t) = e−Sε(t) (5) Moreover, once the formation of a coherent state is expected (Davydov’s soliton), we must introduce the ﬁeld amplitudes where c

Z t ˆ ˆ 0 ε(t0−t) d ˆ 0 0 Sε(t) = S(t, 0) − dt e 0 S(t , t − t), (6a) −∞ dt with X£ ¤ ˆ ∗ † S(t, 0) = φ(t) + F~q(t)ˆν~q + f~q(t)a~q + f~q (t)a~q (6b) ~q being the so-called informational-statistical-entropy operator [31], © 1 ª © 1 ª Sˆ(t0, t0 − t) = exp − (t0 − t)H Sˆ(t, 0) exp (t0 − t)H , (6c) i~ i~

d and ε is a positive infinitesimal that goes to zero after the −1 calculation of averages has been performed [20, 21]. More- where β0 = (kBT0) , since the thermal bath remains in over, φ(t) — playing the role of a nonequilibrium partition a stationary state at fixed temperature T0, and the total sta- function — ensures the normalization of the statistical oper- tistical operator is the direct product of %(t) and %B; kB is ator. Boltzmann universal constant. In Eq. (6b) are present the Lagrange multipliers that the In continuation we proceed to derive, in the correspond- method introduces, namely those we have called ing NESOM nonlinear quantum kinetic theory [20-25] the equations of motion for the basic variables of Eq. (2). Since © ∗ ª F~q(t), f~q(t), f~q (t), β0 , (7) β0 is assumed to be constant in time, and so is EB, we 492 Marcus V. Mesquita, Aurea´ R. Vasconcellos, and Roberto Luzzi are simply left to calculate the equations of evolution for semble [27]. In compact form, the one for the population is the population of the vibrational modes, ν~q(t), and of the 5 amplitude ha |ti and its complex conjugate. As noticed, d X ~q ν (t) = I + J (t) + ζ (t), (8) these equations are derived resorting to the nonlinear quan- dt ~q ~q ~q(j) ~q j=1 tum generalized transport theory that the NESOM provides. We introduce an approximated treatment, however appropriate for the present case since the anharmonic interactions where the first term on the right hand side is the one as- are weak, consisting of the so-called second order approx- sociated to the pumping source, where I~q is then the rate imation in relaxation theory, SOART for short [24]. It is of population increase that it generates, and the remaining five collisions operators J (t) are those arising out of the usually referred to as the quasi-linear theory of relaxation ~q(i) [32], which is a Markovian approximation involving only anharmonic interactions. The first two correspond to col- the second order in the interaction strengths [24], in the lisional events involving a single vibration and two of the (1) bath, and gives rise to a pure dissipative term which takes present case involving contributions proportional to |V |2 ~q~q0 the form (2) 2 and |V~q~q0 | . h i The calculation shows that, because of the symmetry 1 (0) J~q(1) (t) + J~q(2) (t) = − ν~q(t) − ν~q , (9) properties of the system and the selected choice of basic τ~q variables, several contributions in NESOM-SOART vanish in (0) this case: The surviving one corresponds to the Golden Rule where ν~q is the population in equilibrium at temperature of quantum mechanics averaged over the nonequilibrium en- T0, and τ plays the role of a relaxation time given by

X 4π 1 (1) £ ¤ −1 2 B B β~Ω˜q0 τ = |V | ν 0 ν 0 δ(Ω 0 + Ω 0 − ω ) + 2e δ(Ω 0 − Ω 0 + ω ) , (10) ~q ~2 (0) ~q~q0 ~q ~q−~q ~q ~q−~q ~q ˜q ˜q−˜q ˜q ν~q ~q0

B where ν~q is the population of the phonons in the bath, namely, the Planck distribution

B −1 ν~q = [exp(β0~Ω~q) − 1] , (11) and the delta functions account for conservation of energy in the scattering events. The other terms, J~q(i) (t) with i = 3, 4, 5, are X 8π (2) 2 £ B ¤ J (t) = |V | ν 0 (ν 0 − ν ) − ν (1 + ν 0 ) δ(Ω 0 + ω 0 − ω ) , (12a) ~q(3) ~2 ~q~q0 ~q−~q ~q ~q ~q ~q ~q−~q ~q ~q ~q0 X 8π (2) 2 £ B ¤ J (t) = |V | ν 0 (ν 0 − ν ) + ν 0 (1 + ν ) δ(Ω 0 − ω 0 + ω ) , (12b) ~q(4) ~2 ~q~q0 ~q−~q ~q ~q ~q ~q ~q−~q ~q ~q ~q0 X 8π (2) 2 £ B B ¤ J (t) = |V | ν 0 (1 + ν 0 ) − (ν 0 − ν 0 )ν δ(Ω 0 − ω 0 − ω ) , (12c) ~q(5) ~2 ~q~q0 ~q+~q ~q ~q ~q+~q ~q ~q+~q ~q ~q ~q0 and, ﬁnally, the term ζ~q(t) is the one which couples the populations with the amplitudes, namely X 4π (1) 2 2© B B ζ (t) = |V | |ha |ti| (1 + ν 0 + ν 0 )δ(Ω 0 + Ω 0 − ω 0 ) ~q ~2 ~q~q0 q ~q ~q−~q ~q−~q ~q ~q ~q0

B B ª +2(ν~q0 − ν~q−~q0 )δ(Ω~q−~q0 − Ω~q0 − ω~q0 )

X 8π (2) 2© 2 B 2 B ª + |V | |ha |ti| (1 + ν 0 + ν 0 ) − |ha 0 |ti| (ν − ν 0 ) δ(Ω 0 + ω 0 − ω ) ~2 ~q~q0 q ~q ~q−~q q ~q ~q−~q ~q−~q ~q ~q ~q0 X 8π (2) 2© 2 B 2 B ª − |V | |ha |ti| (ν 0 − ν 0 ) − |ha 0 |ti| (1 + ν + ν 0 ) δ(Ω 0 − ω 0 + ω ) ~2 ~q~q0 q ~q ~q−~q q ~q ~q−~q ~q−~q ~q ~q ~q0 X 8π (2) 2© 2 B 2 B ª + |V | |ha |ti| (ν 0 − ν 0 ) − |ha 0 |ti| (ν − ν 0 ) δ(Ω 0 − ω 0 − ω ) . (12d) ~2 ~q~q0 q ~q ~q+~q q ~q ~q+~q ~q+~q ~q ~q ~q0 Brazilian Journal of Physics, vol. 34, no. 2A, June, 2004 493

The collision operator J~q(5) is also a relaxation term through the nonlinear terms, of the transfer of energy to the (containing contributions nonlinear in the mode popula- polar modes lowest in frequency. In fact, they contain non- tions); and the ﬁrst two terms are those responsible for the linear contributions proportional to so-called Frohlich¨ effect, as a result that they account for,

c X (2) 2 £ ¤ |V~q~q0 | ν~q(t)ν~q0 (t) δ(Ω~q−~q0 − ω~q0 + ω~q) − δ(Ω~q−~q0 + ω~q0 − ω~q) , (13) ~q0

0 and we may notice that for modes ~q such that ω~q0 > ω~q Eq. (12d) the term ζ~q(t) acts as a source coupling the pop- the energy conservation as required by the first delta func- ulations of the vibrational modes with the amplitudes of the tion is satisfied, while this is not possible for the second: expected coherent excitation (Davydov’s soliton as shown a hence this nonlinear contribution tends to increase the pop- posteriori). ulation in mode ~q at the expenses of the other modes higher in frequency. Reciprocally, for ω~q0 < ω~q, the mode ~q trans- On the other hand, the equations of evolution for the field fers energy to the modes lower in frequency. Moreover, in amplitudes are

~q1~q2 ∂ ha†|ti = the c.c. of the r.h.s. of Eq. (14a), (14b) ∂t ~q where ω˜~q = ω~q + W~q, with W~q being a term of renormalization of frequency which will not be of interest in the following analysis, and the lengthy expression for R is given elsewhere [28]; its detailed expression is unnecessary for the analysis ~q1~q2 here. Finally, Γ~q(t), which has a quite relevant role in what follows, is given by

X 1 −1 4π (2) 2 £ B ¤ Γ (t) = τ (t) + |V | 1 + ν 0 + ν 0 δ(Ω 0 + ω 0 − ω ) + ~q 2 ~q ~2 ~q~q0 ~q ~q−~q ~q−~q ~q ~q ~q0 X 4π (2) 2 £ B ¤ − |V | ν 0 − ν 0 δ(Ω 0 − ω 0 + ω ) + ~2 ~q~q0 ~q ~q−~q ~q−~q ~q ~q ~q0 X 4π (2) 2 £ B ¤ |V | ν 0 − ν 0 δ(Ω 0 − ω 0 − ω ) . (15) ~2 ~q~q0 ~q ~q+~q ~q+~q ~q ~q ~q0

The coupled equations (14) contain linear and tri-linear operator X terms. Ignoring the latter, the resulting linearized equation ψ(x, t) = ha |tieiqx. (16) has as solutions the normal damped wave motion, proceed- ~q −1 ~q ing with a renormalized frequency and lifetime Γ~q . The complete equations, i.e. including the nonlinear terms, are for linear propagation along the, say, x direction on bulk or of the Davydov’s soliton type, but with damping, or more along the one-dimensional polymer. precisely, are nonlinear damped Schrodinger-like¨ equations [4, 33]. We introduce a representation in direct space, defin- ing the averaged (over the nonequilibrium ensemble) field 2.1 Polar (Optical) Vibrational Modes At this point we introduce a first type of analysis, speci- fying the vibrational modes as being of the class of lon- 494 Marcus V. Mesquita, Aurea´ R. Vasconcellos, and Roberto Luzzi gitudinal polar-modes (optical modes with frequencies in curvature at this centre, respectively. Next, using Eqs. (14), the infrared), as in Davydov’s and Frohlich’s¨ works [11, after neglecting the coupling terms with the conjugated am- 12, 34, 35]. Their frequency dispersion relation is approxi- plitude (what can be shown to be the case when we introduce mated (to a good degree of accuracy) by the parabolic-law from the outset a truncated Hamiltonian in the so-called Ro- 2 ω~q = ω0 − αq , where ω0 and α are constants, standing for tating Wave Approximation [36]), it follows that the average the frequency at the zone centre (the maximum one) and the field amplitude satisfies the equation

2 Z L 0 ∂ ∂ dx 0 0 i~ ψ(x, t) = (~ω0 + ~α 2 )ψ(x, t) − i~ Γ(x − x )ψ(x , t)+ ∂t ∂x 0 L Z Z L dx0 L dx00 + R(x − x0, x − x00)ψ(x0, t)ψ(x00, t)ψ∗(x, t), (17) 0 L 0 L

d where Γ and R are the back-transforms to direct space of Equation (17) is a nonlinear Schrodinger-type¨ equation

Γ~q in Eq. (15) and R~q1~q2 in Eq. (14a), and L is the length with damping [33]. Introducing a local approximation, that of the sample. Moreover, we have taken a time-independent is, neglecting space correlations, after using the expressions population ν~q, that is, according to Eq. (8) it is either the equilibrium distribution at temperature T0 when no external 0 00 0 00 pumping source is present (i. e. I~q(ω~q) = 0), or when in R(x − x , x − x ) = ~Gδ(x − x )δ(x − x ), (18a) the presence of a constant pumping source leading, after a short transient has elapsed, to a steady state and, moreover, 0 0 the term ζ~q is weakly dependent on time (a condition to be Γ(x − x ) = γsδ(x − x ), (18b) characterized a posteriori). we obtain that Eq. (17) becomes

∂ψ(x, t) ∂2 i − (ω − iγ )ψ(x, t) − α ψ(x, t) − G |ψ(x, t)|2 ψ(x, t) = 0, (19) ∂t 0 s ∂x2

Equations (14) and (19) are of the form of the equa- lifetimes in the order of a few picoseconds. For the case of tions derived by Davydov in an alternative way, but, with a Gaussian signal impinged at the beginning of the polymer the present thermodynamic treatment clearly showing the chain, which can be approximated to a good degree of accu- damping effects. In equilibrium conditions at temperature racy by a hyperbolic secant shape, and next using the Inverse 300 K, the damping constants have values corresponding to Scattering Method [37] we obtain the solution c ½ · ¸¾ " µ ¶ # υ θ |G| 1/2 ψ(x, t) = A exp i x − (ω − iγ ) t − sech A (x − υt) , (20) 2α s s 2 2α

d where γs is, evidently, the reciprocal lifetime of the excita- Hence, it is proved the possible presence of Davydov’s tion (taking γs = 0 and ω0 = 0, Eq. (20) is the expression solitons in polymers, like the α-helix protein in biological for Davydov’s soliton in its original version [11, 12]), and matter, but, we stress, of a damped character. The mech- we used G = |G|eiθ. Moreover, anism for the formation of the soliton is in this case interpreted as follows [13]: Vibrational energy of the CO- υ2 |G| A2 ωs = ω0 − + , (21) stretching (Amide-I) oscillators that is localized on the 4α 2 quasi-periodic helix acts — through a phonon coupling ef- is the reciprocal period of the solitary wave and A and υ are fect — to distort the structure of the helix. The helical dis- an amplitude and a velocity of propagation ﬁxed by the ini- tortion reacts — again through phonon coupling — to trap tial condition of excitation imposed by the external source. Brazilian Journal of Physics, vol. 34, no. 2A, June, 2004 495 the Amide-I oscillation and prevents its dispersion in a self- trapping. Let us consider the experimental observation of this ex-

citation. As already noticed, experimental observation is

¥ §©

difﬁcult in active biological materials. A way around this

#"$

difﬁculty consists into the experimental study of polymers

whose vibronic characteristics resemble those of biopoly- %

¨§ ¤

mers, and a quite favorable one is acetanilide. Experiments (

879 -

of infrared absorption in acetanilide showed an “anomalous” ,+

G H I F

band in the IR spectrum, which was ascribed to a Davydov’s E ¨§©£

soliton [38], and later on reproduced in other experiments, BDC

56 A

and also observed in Raman scattering experiments [39-43]. ? @ >

We analyze the experiment of Careri et. al [38], re- =

¨§ ¢

sorting to a response function theory consistently derived 34 ; in the framework of NESOM [20, 44, 45]. Without going :

into details (see [46]), the absorbance in the region of the 102

¨§ ¡ /. CO-stretching mode has the expression

α(ω) = αn(ω) + αs(ω), (22)

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¡ ¢ £ ¤ ¥¦ 'YZ which describes two bands: one centered around ω~q, the fre- KMLONQP4RTSVUXW quency of the normal mode, (~q being equal to the IR-photon wave-vector) and with intensity proportional to the population in equilibrium ν~q at temperature T0, since no external pump is present, I = 0 in Eq. (8); and the other is centered Figure 2. Optical vibrations: Profile of the soliton energy (propor- ~q tional to |ψ(x, t)|2) along a few picoseconds after application of around ωs, the frequency of the soliton. The band widths are −1 −1 1/2 −1 the initial Gaussian-like perturbation (in the experimental condi- γn = τ0 and γs = τ0 + A(|G|/2α) , where τ0 is the tions of Fig. 1 for T = 80 K). one of Eq. (10), for ~q near the zone center. Let us consider the experiments of reference [38], and take, for example, the the modes at intermediate and high frequencies remain case of T0 = 80 K; on the basis of the red shift of the band nearly constant on further increase of the intensity of the due to the soliton in relation to the normal CO-stretching −1 pump. As expressed above, this is similar to a Bose-Einstein band, that is, ω0 − ωs ≈ 16 cm , and that γs − γn ≈ condensation, but, it must be stressed, not in equilibrium but 3.6 cm−1, we find that A(|G|/2α)1/2 ≈ 2.3 × 106 cm−1 4 −1 in nonequilibrium conditions, and then the phenomenon — and υ ≈ 3×10 cm s . The calculated spectra is shown as Frohlich¨ effect — may be considered a kind of emergence of a full line in Fig. 1, while the dots are experimental points, a dissipative structure in Prigogine’s sense [47, 48, 49]. This evidencing a satisfactory agreement. is illustrated in Fig. 3, where the steady state populations In Fig. 2 it is shown the propagation of the energy ac- in terms of the intensity of the pumping source are shown. 2 companying the soliton (proportional to |ψ(x, t)| ) along a We have used numerical parameters characteristic of a poly- few picoseconds after the application of the initial Gaussian- mer of the α-helix protein type [27]. The figure evidences like excitation. It is clearly evidenced the conservation of the large increase (after an intensity threshold has been at- the shape characteristic of the soliton, but accompanied, as tained) of a mode lowest in frequency (the one labelled 1), at already described, with a decay in amplitude in the picosec- the expenses of other modes higher in frequency. In Fig. 4 it ond range. Hence, a pulse signal impinged on the system is evidenced the Frohlich-Bose-Einstein-like¨ condensate at would be carried a few micrometers, since the velocity of the lowest frequencies in the vibrational spectrum. 4 −1 propagation is ≈ 3 × 10 cm s . The relevant point to be stressed is that Frohlich¨ effect However, the situation may be substantially modified if and Davydov soliton are phenomena arising out of the same the excitation propagates in a nonequilibrium background, nonlinear kinetic effects that are present in Eq. (9) for the namely the one provided by the presence of a constant populations ν~q(t), and in Eq. (15) for the reciprocal life- pumping source [I~q 6= 0 in Eq. (8)] leading the system to times Γ~q. As a consequence of the fact that, because of a steady state, with populations, say, ν¯~q, constant in time but Frohlich¨ effect, the population of the modes lowest in fre- much larger than the population in equilibrium. At a suf- quency largely increase, concomitantly their lifetime also ficient distance from equilibrium, as proposed by Frohlich¨ largely increases (i.e. the reciprocal lifetime Γ~q in Eq. (15) near thirty years ago [34, 35], and later on verified by sev- largely decreases), while for the modes at intermediate to eral authors (in the framework of IST in [27]), is expected to high frequencies their lifetime largely decreases (the recip- arise a particular complex behavior in the system, namely a rocal of Γ~q largely increases). This is illustrated in Fig. 5 phenomenon akin to a Bose-Einstein condensation, consist- [28]. Hence, in the expression for the average field ampli- ing in that [because of the presence of the terms of Eq. (9)], tude of Eq. (16), after a fraction of picosecond following and as already noticed, the energy stored in the polar vibra- the application of the exciting pulse has elapsed, there sur- tional modes is preferentially channeled to the modes lowest vive for a long time the contributions from the modes lowest in frequency. The latter greatly increase in population, while in frequency, a survival time that keeps increasing as the 496 Marcus V. Mesquita, Aurea´ R. Vasconcellos, and Roberto Luzzi intensity I increases [28]. This implies that it may be ex-

pected that an excitation composed by a coherent interplay

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2.2 Acoustical Vibrational Modes Figure 3. Optical vibrations: The steady-state populations vs. the intensity of the pumping source, for a selected set of modes: After So far we have considered propagation of vibronic waves in an intensity S = 103 is reached, there follows a large increase in biological media, via Eq. (16), but restricted to the case of the population ν6 of the mode lowest in frequency in the said set polar modes. We brieﬂy consider next the case of longitudi- (After Ref. [27]). nal acoustic modes. For that purpose we return to Eqs. (14), where now we take into account that the dispersion relation

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that the excitation is expected to be a closed-packet solitary ¦ ¦ ¦¥¦ wave, we arrive at the equivalent of Eq. (19), in this case

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(longitudinal acoustic) vibrations. However, a remarkable ¦

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¢¡ coefficient in front of the second derivative in space is deter- ¨ !"$#&% mined, through α, by the bandwidth of the LO (longitudinal optical) vibrations dispersion relation, in this case, as shown in Appendix A, it depends through the pseudo-mass Ms on the characteristics of the experiment, that is, depends on the Figure 4. Optical vibrations: The steady-state populations for width of the solitary wave packet which is determined by the S = 104, showing a “two fluid” separation consisting in the initial condition. The solution for a given hyperbolic secant- “condensate” at low frequencies and the “normal” contributions at profile signal impinged on the system, say, the same as in higher frequencies (After Ref. [27]). the previous subsection, is given by Brazilian Journal of Physics, vol. 34, no. 2A, June, 2004 497

c n o n o £M v θ ¤ £|G |M ¤1/2 ψ(x, t) = A exp i s x − (ω − iγ )t − sech A s s (x − vt) , (24) ~ s s 2 ~

d where for this case of acoustical vibrations: In fact, pumping of |G |A2 M v2 ω = s − s , (25) the modes in a restricted ultrasonic band (in the present case s 2 4~ in the interval 4.5 × 105 Hz ≤ ω ≤ 2.8 × 107 Hz), leads at As in the case of LO vibrations, the “acoustic” solitary wave sufficiently high intensity of excitation to the transmission is damped, and one may wonder if, as in the case of the “op- of the pumped energy in these modes to those with lower tical” solitary wave, this lifetime may be largely extended by frequencies (ω < ω2), while those with larger frequencies the action of nonlinear kinetic terms enhanced by the pump- (ω > 2.8 × 107 Hz) remain in near equilibrium, as shown ing of energy on the system. We reconsider Eqs. (11), now in Fig. 7. It may be noticed that for the given value of 23 specialized for the LA vibrations, and look for the station- τ¯, for S = 10 , the flux power provided by the external ary states when a constant exciting source is continuously source, in the given interval of ultrasound frequencies being applied. excited, is of the order of milliwatts. Modes in the interval 4 5 To perform numerical calculations we choose 5.6 × 10 ≤ ω~q ≤ 4.5 × 10 , those lowest in frequency, a set of parameters in a typical order of mag- have large populations in comparison with those higher in nitude approximation. We take for the Bril- frequency. As already noticed, because of these charac- 7 −1 louin zone-end wavenumber qB = 3.14 × 10 cm , teristics of Frohlich’s¨ effect, it is sometimes referred-to as 5 −1 ω~q = sq with s = 1.8 × 10 cm s , Ω~q = sBq with a Bose-Einstein-like condensation. However, it must be 5 −1 sB = 1.4 × 10 cm s . Moreover, the matrix elements stressed that not in equilibrium, but in nonequilibrium con- V (1) and V (2) are proportional to the square roots of the ditions, then being a kind of nonequilibrium phase transition (1,2) (1,2) 1 or better to say, a kind of emergence of a dissipative struc- wavenumbers [50], say V 0 = K [|~q1||~q2||~q1 − ~q2|] 2 , ~q~q ture in Prigogine’s sense [47-49]. Moreover, we may say, K(1) and is determined from a typical value of 10 ps for the in a descriptive way, that it is present a kind of a “two fluid lifetime of Eq. (10) (for any system it can be determined system”, the normal one and the Frohlich¨ condensate. from the linewidth in Raman scattering experiments). An open parameter λ = |K(2)/K(1)|2 is introduced, and we Another relevant result is that, also as in the case of take λ = 1 to draw Fig. 6. Finally, L, the length of the the optical vibrations, the modes in the condensate largely sample in the direction of propagation is taken as 10 cm. increase their lifetimes; this is shown in Fig. 8. There- Therefore, the permitted wavenumbers for propagation of fore, the soliton, composed by the coherent interplay of low- vibrations are contained in the interval π/L ≤ q ≤ qB. For frequency acoustical modes, travels nearly undamped in the these characteristic values it follows that, because of en- Frohlich¨ condensate. ergy and momentum conservation in the scattering events, We proceed now to consider another possible novel phe- the set of equations of evolution, Eqs. (8), which in prin- nomenon in this kind of systems. ciple couple all modes among themselves, can be sepa- rated into independent sets each one having nine modes. For example, taking the mode with the lowest wavenumber π/L, the set to which it belongs contains the modes n−1 κ π/L, where κ = (s + sB)/(s − sB) = 8 in this case, and n = 2, 3,..., 9. Let us call ν1, . . . , ν9 the correspond- 2.3 The accompanying Cherenkov-like emis- 4 ing populations, having frequencies ω1 = 5.6 × 10 Hz, 5 6 7 sion ω2 = 4.5×10 Hz. ω3 = 3.6×10 Hz, ω4 = 2.9×10 Hz, 8 9 10 ω5 = 2.3×10 Hz, ω6 = 1.8×10 Hz, ω7 = 1.5×10 Hz, 11 11 ω8 = 1.2 × 10 Hz, ω9 = 9.5 × 10 Hz. Moreover, for Considering either an “optical” or an “acoustical” soliton illustration, the open parameter λ is taken equal to 1, and we of the Davydov type respectively described in the previous consider that only the modes 2 and 3 (in the ultrasonic re- subsections, we recall that the amplitude and the velocity of gion) are pumped with the same constant intensity S = Iτ¯, propagation are determined by the initial condition of exci- where I2 = I3 = I, and I1 and In with n = 4,..., 9 are tation (that is, the energy and the momentum transferred in null, and τ¯ is a characteristic time used for scaling purposes the process of interaction with the external source). For ex- (as in [27]) here equal to 0.17 s. The results are shown in ample, in the case of the acetanilide we have considered in Fig. 6, where it is evident the large enhancement of the pop- subsection 2.1, and in the conditions of the experiment of 19 ulation in the mode lowest in frequency (ν1), for S0 ' 10 , Careri et al. [38], the velocity of propagation is larger than at the expenses of the two pumped modes ν2 and ν3, while the group velocity of the phonons in the optical branch cor- the modes ν4 to ν9 (higher in frequency) are practically unal- responding to the CO-stretching vibrations, which is small tered. The emergence of Frohlich¨ effect is clearly evidenced because the dispersion relation is flat.

498 Marcus V. Mesquita, Aurea´ R. Vasconcellos, and Roberto Luzzi

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Figure 6. Acoustic vibrations: The steady state populations of the Figure 8. Acoustic vibrations: The lifetime of the same modes as three relevant modes in the set — as described in the main text in Fig. 7, in terms of the intensity of the pumping source (After —, with increasing values of the intensity of the external source Ref. [46]). pumping modes labeled 2 and 3 in the ultrasonic region. (After Ref. [46]). When the soliton velocity of propagation, say v, is larger than the group velocity of the normal vibrations, (the velocity of sound s when the acoustic modes are involved), it may follow a Cherenkov-like effect. We recall that originally it was observed in electromagnetic radiation (e.g. [51]) by Cherenkov in 1934. It is a result that in a material media 16 1016 10 with an index of refraction n, the velocity of propagation 23 S 10 15 1015 = 10 of light is c/n, smaller than the velocity c in vacuum (since T0 300 K = n > 1), and if an electron with velocity u > c/n (but with 14 1 14 10 = 10 the relativistic limitation of u < c) travels in this medium _ 13 () 1013 10 then, along a cone deﬁned by the angle cos θ = c/n u is emitted the so-called Cherenkov radiation: that is, along 12 12 10 10 such direction photons are strongly emitted. This is the so-

11 1011 10 called superluminal radiation [52, 53]. Something similar is present in the case of phonons in 10 10 10 10 the photoinjected plasma in semiconductors in the presence 9 109 10 of an electric field: when the drift velocity of the carriers ex- FRÖHLICH CONDENSATE ceeds the group velocity of the ~q-mode optical phonon, then 8 8 10 10 along a cone whose axis is along the electric field, and with 7 107 10 an aperture with angle θ defined by STEADY STATE MODE POPULATIONS ~q

6 106 10 cos θ = ω~q/vq, (26) 5 105 10 there follows a large emission of ~q-mode optical phonons 4 104 10 5 6 7 8 [54]. 10 10 10 10 FREQUENCY (Hz) This is also the case when the soliton, either optical or acoustical, travels in bulk with a velocity v larger than the Figure 7. Acoustic vibrations: The population in the steady state 23 group velocity of the normal vibronic waves. This is de- for a pumping intensity S = 10 , of the modes along the spectrum scribed elsewhere [46], and next we brieﬂy outline the re- of frequencies of the acoustic modes. Dots indicate the modes in the ﬁrst set (the remaining part of the spectrum up to the highest sults. Inspection of Eq. (8) tells us that the presence of the 11 Brillouin frequency ωB = 9.5×10 Hz has been omitted). (After direct coupling to the external source via I~q, and indirectly Ref. [46]). through ζ depending on the squared amplitude of the soli- Brazilian Journal of Physics, vol. 34, no. 2A, June, 2004 499 ton, tends to increase the population of phonons. But, as al- Eq. (4a) leads to the result that ready noticed, because of Frohlich¨ effect, such pumped en- ¯f (t)¯2 ergy tends to concentrate in the modes lowest in frequency, F˜q −1 ˜q ν~q(t) = [e (t) − 1] + ¯ ¯ · (27) those at the Brillouin zone boundary in the case of optical F˜q vibrations and around the zone center in the case of acoustic vibrations. Evidently, in the absence of the perturbation, that is, I~q = 0 and ha~qi = 0 and then f~q = 0, it follows that F~q(t) = ~sq/kBT0, and we recover the usual Planck dis- Take the case of acoustic phonons, when there should tribution in equilibrium. In the presence of the perturbation be a large increase in the population of the modes with we need to obtain both F~q(t) and f~q(t). On the one hand, a very small wavenumber. A straightforward calculation of direct calculation tell us that

c Z L dx 2 2 2 2 2 2 −γst |f~q(t)/F~q(t)| = |ha~q|ti| ≈ |ψ(x, t)| ≈ (A w /L )e , (28) 0 L

where µ~q plays the role of a quasi-chemical potential (this kind of choice was done by Frohlich¨ [35] and Landsberg

1018 1019 1020 1021 1022 1023 [55]). In Fig. 9 is shown the dependence of the quasi- 1.2 1.2 chemical potential, corresponding to the modes in Fig. 6, T0 300 K = with the pumping intensity. Another is /ω 1 1 1 ) ∗ q = 1.0 1.0 F~q = ~sq/kBT~q (30) /ω q introducing a quasitemperature T ∗, as it is done in semicon- ( 0.8 0.8 2/ω2 ductor physics [26]. In Fig. 10 is shown the dependence of the dependence of the quasitemperature, corresponding to the modes in Fig. 6, with the pumping intensity. 0.6 0.6

0.4 0.4 1018 1019 1020 1021 1022 1023 7 107 10 3/ω3

QUASI-CHEMICAL POTENTIAL T0 300 K 0.2 0.2 6 = 6 10 1 10 =

5 5 ) 0.0 0.0 0 10 T ÍT 10

/T 1 0 ? (T 4 4 18 19 20 21 22 23 10 10 10 10 10 10 10 10 SOURCE INTENSITY (S)

3 103 10

2 102 Figure 9. The quasi-chemical potential of the modes labeled 1 to 10 T3ÍT0

3 in Fig 6, with mode 1 corresponding to the one with the lowest QUASITEMPERATURE

1 frequency in the given set: it is evident the emergence of a “Bose- 101 T ÍT 10 Einstein-like condensation” for S approaching a critical value of 2 0 19 the order of 10 . 0 100 10

-1 where we have used Eqs. (16) and (20), and, we recall, under 10-1 10 2 1018 1019 1020 1021 1022 1023 a sufﬁciently intense excitation γs is small and then |haq|ti| becomes near time independent; we have called w the width SOURCE INTENSITY (S) of the solitary wave packet. On the other hand, F~q in steady state conditions after ap- Figure 10. The quasi-temperature, deﬁned in Eq. (30) for the plication of the constant external excitation, depends on the modes in Fig. 6. intensity of the pumping source. This Lagrange multiplier Let us take the choice of Eq. (30), then the quasitemper- may be rewritten in either of two alternative forms, which ature T ∗ is given by resemble well known results in equilibrium theory. One is · ¸ ∗ 1 kBT~q = ~sq ln 1 + 2 (31) F~q(t) = [~sq − µ~q]/kBT0, (29) ν~q − |ha~qi| 500 Marcus V. Mesquita, Aurea´ R. Vasconcellos, and Roberto Luzzi and we recall that Eq. (8)). Using Eqs. (29) to (32) we have that [46]

2 2 2 2 |ha~qi| ≈ A w /L , (32) µ~q = ~vq cos η~q (33)

(for γs → 0, with ν~q determined in each case solving where

s £ kBT0 2 −1 ¤ s T0 cos η~q = 1 − ln[1 + (ν~q − |ha~qi| ) ] = [1 − ∗ ] . (34) v ~sq v T~q

These results imply in this case in a phenomenon of a pe- Vacuum Crystal Vacuum culiar character which we call Frhlich-Cherenkov-effect. In fact, we note, ﬁrst, that there follows a large enhancement of phonons in mode ~q for µ~q approaching ~sq, and second, the φ k linear motion of the soliton deﬁnes a particular direction, the (1) vgr one given by its velocity of propagation ~v. Therefore, there v θ O d (2) z is a preferential direction of production of vibrational waves vgr given by φ

~sq ≈ ~vq cos η~q = µ~q (35) or Figure 11. The direction of propagation of the waves of Cherenkov radiation, when spatial dispersion is taken into account, for the or- cos η~q ≈ s/v. (36) dinary wave (subscript 1) and the anomalous wave (subscript 2) (Adapted from Ref. [53]). Equation (35) defines the direction of propagation ~q of the longitudinal vibration and its modulus. Since η~q depends only on the modulus of ~q, there follows two Cherenkov-like privileged directions of emission of ~q-mode phonons, one forward and one backwards, like the normal and anomalous Cherenkov cones in radiation theory as illustrated in Fig. 9 adapted from [53]. In the present case both directions are symmetrical on both sides of the centre defined at each time by the position occupied by the soliton. This may account for the observed so-called X-waves [1, 56, 3]. In Fig. 12 is illustrated the cases of propagation of the normal sound wave (upper figure) and of the, presumably, solitary wavepacket selectively excited by the transducer (lower figure) with velocity larger than the sound velocity in the medium. The figure has appeared in [3]. Given the angle η~q (called the Figure 12. Normal sound propagation (upper figure), and the exci- axicon angle in [1]), then v is larger than s in the percentage tation interpreted as a supersonic soliton (lower figure); from ref- [(v/ cos η ) − s]/v. Same arguments are valid for the case erence [3] (We thank W. A. Rodrigues and J. E. Maiorino for pro- ~q viding us with a postscript file of this picture). of the optical soliton, when ∇~qω~q (the group velocity of the normal mode) enters in place of s. In the case of Fig. 12, ◦ a rough estimate gives η ∼ 13 and vs/s ∼ 1.02, that is, anharmonic interactions between the system and the sur- the velocity of propagation of the soliton, v, is roughly 2% roundings) it is expected that complex behavior shall arise. larger than the velocity of sound in the medium. Resorting to an appropriate thermo-mechanical statistical approach, we have shown that such complex behavior consists of four particular phenomena. One is that the nor- 3 Concluding remarks mal vibrational modes are accompanied by another type of excitation, consisting in the propagation of solitary waves of We have considered the propagation of vibronic excitations the Schrodinger-Davydov¨ type. They are undeformed waves in nonlinear condensed matter media, like biological mate- composed by a coherent state of normal modes. Although rial. Because of the nonlinearities in the kinetic equations the wavepacket is spatially undeformed, it presents, as it that describe the evolution of the macroscopic collective should, decay in time with a given lifetime resulting from the modes (nonlinearities having their origin in the microscopic dissipative effects that develop in the excited sample. The Brazilian Journal of Physics, vol. 34, no. 2A, June, 2004 501 amplitude, velocity, and frequency of the solitary wave are We conclude with the remark that solitary waves appear determined by the initial and boundary conditions. Another to be ubiquitous, and having large relevance in a number phenomenon, arising out of the same nonlinearities that al- of important situations. Some at the technological level, lows for the creation of the soliton, consists in that, under like propagation in optical fibers (e.g., in a projected trans- conditions of excitation which lead the system sufficiently Atlantic cable), in conducting polymers (for electric-car bat- away from equilibrium, there follows a large increase of the teries; microcircuits; etc.), and the case of biological sys- population of the modes lowest in frequency. This effect has tems (long range propagation of nervous signals; the here a reminiscence of a Bose-Einstein condensation but here in mentioned case of medical imaging; etc.). nonequilibrium conditions, and we have termed it Frohlich¨ effect. Acknowledgments A third phenomenon consists in that, and again because of the nonlinearities which are responsible for both, Frohlich¨ We gratefully acknowledge financial support to our effect and formation of a Schrodinger-Davydov¨ soliton, the Group, provided in different opportunities by the Sao˜ Paulo latter acquires a very long lifetime, that is, the soliton be- State Research Foundation (FAPESP), the National Research comes nearly undamped, when travelling in the Frohlich-¨ Council (CNPq), the Ministry of Planning (Finep), Unicamp Bose-Einstein-like condensate. Foundation (FAEP), IBM Brasil, the John Simon Guggen- Finally, the fourth phenomenon refers to a situation heim Memorial Foundation (New York, USA). when, because of appropriate initial and boundary conditions, the soliton travels with a speed larger than that of the normal vibronic modes. In this case, as shown, there follows Appendix A The Acoustic Solitary what can be termed as Frohlich-Cherenkov¨ effect: along two Wave symmetrical privileged directions centered on the position of the soliton, is produced a large number of long wavelengths In direct space, after the terms that couple the amplitude phonons. This could be the origin of the so-called X-waves ha~qi with its conjugate are neglected, what, as noticed in observed in experiments of ultrasonography, as noticed in the main text is accomplished using the rotating wave ap- previous sections. proximation, Eq. (14) takes the form:

Z 0 Z 0 ∂ X dx 0 X dx 0 i~ ψ(x, t) = −i ~ω eiq(x−x )ψ(x0, t) − i~ Γ eiq(x−x )ψ(x0, t) ∂t ~q L ~q L q ~q

Z 0 Z 00 X dx dx 0 00 + R eiq1(x−x )eiq2(x−x )ψ(x0, t)ψ(x00, t)ψ∗(x, t), (A.1) ~q1~q2 L L q1q2

d where, we recall, ω~q = sq. Considering that it is expected ∂ the formation of a highly localized packet (the soliton), cen- ψ(x0, t) ≈ ψ(x, t) − ξ ψ(x, t), (A.2) tered in point x and with a Gaussian-like profile with a ∂x width, say, w (fixed by the initial condition of excitation) where ξ = x − x0 is roughly restricted to be smaller or extending along a certain large number of lattice parameter at most of the order of w. The first term on the right of a (i.e. w À a), in Eq. (A.1) we make the expansion Eq. (A.1) is

Z 0 X dx 0 i s|q| eiq(x−x )ψ(x0, t) L q

Z π Z L 0 h i Ls ∂ a dx 0 0 = − dq eiq(x−x ) − eiq(x−x ) ψ(x0, t) 2π ∂x 0 0 L

Z π Z x+w/2 · ¸ is ∂ a ∂ ≈ − dq dξ sin(qξ) ψ(x, t) − ξ ψ(x, t) π ∂x 0 x−w/2 ∂x

Z x+ 1 w is ∂ £ 2 π dξ ¤ = (1 − cos ξ) ψ(x, t) π ∂x 1 a ξ x− 2 w 502 Marcus V. Mesquita, Aurea´ R. Vasconcellos, and Roberto Luzzi

Z x+ 1 w is£ 2 π dξ ¤ ∂ + (1 − cos ξ) ψ(x, t) π 1 a ξ ∂x x− 2 w " # Z x+ 1 w is 2 π ∂ − (1 − cos ξ) dξ ψ(x, t) π 1 a ∂x x− 2 w " # Z x+ 1 w is 2 π ∂2 − (1 − cos ξ) dξ 2 ψ(x, t) . (A.3) π 1 a ∂x x− 2 w

But, of the four terms after the last equal sign in this But, we notice that the width of the packet is w À a, and Eq. (A.3), the second and third are null, because of the the cosine in Eq. (A.4) has a period 2a, and then it oscillates Ansatz that a soliton would follow, since the derivative at very many times in w, and with amplitude (2a/πw) ¿ 1, the center of the packet is null. Consider now the last term, and can be neglected. Similarly, the ﬁrst term becomes pro- which after the integrations are performed becomes portional to

isw £ 2a π w π ¤ ∂2 − 1 − sin cos x ψ(x, t) . (A.4) π πw a 2 a ∂x2 c

is ψ(x) n h ³πx´ ³πw ´i ³πx´ ³πw ´o w 1 − cos cos − 2x sin sin , (A.5) π x2 − (w/2)2 a 2a a 2a

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