j.a. tuszynski et al- excition self-trapping in the ginzburg-landau framework

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\LOExciton Self-Trapping in the Ginzburg-Landau FrameworkPhysica Scripta. Vo!' 51,423-430, 1995

J. A. Tuszynski and M. L. A. NipDepartment of Physics, University of Alberta, Edmonton, Alberta, Canada T6G 211P. L. Christiansen and M. RoseInstitute for Mathematical Modelling, Technical University of Denmark, DK-2800 Lyngby, DenmarkandOle BangLaboratoire de Physique, Ecole NormaJe Superieure de Lyon, 46 Allee d'Italie, 69364 Lyon Cedex 07, FranceReceived August ]7, 1994; accepted October 31,1994

AbstractIn this paper we reformulate the Oavydov theory for exciton self-trappingin the Ginzburg-Landau language of phase transitions. In the parameterrange of the Davydov system, where both the nonlinearity- and the dipole-dipole coupling parameters are positive, explicit expressions for tbe wave-function and energy per pulse, are derived for all possible extended ellipticwave solutions. The corresponding partition function, free energy and spe-cific heat are calculated numerically. It is shown that for small proteins,and at physiological temperatures, these extended waves may play a rolethermodynamically.

1. IntroductionAs early as in 1933 Landau (1] discussed the possibility ofexciton self-trapping in solids. He showed that, provided theelectron-phonon interaction is sufficiently strong, electrontrapping by the lattice, with an associated lattice distortion,may become energetically favorable over electronic Blochstates.Subsequently a vast body of literature was published

dealing both with the theoretical and experimental aspectsof electron self-trapping in solids (2] as well as liquids anddense gases (3]. Particular attention was given to exciton-phonon interactions leading to self-trapping in lower-dimensional systems. Numerous possible scenarios wereinvestigated, involving electrons as well as holes, and acous-tic as well as optical phonons. A particularly fruitful conceptstudied in this context was the polaron state, which requiredinteractions with long-wave polarized optical phonons [2].A ground-breaking application of the self-trapping phe-nomenon to biophysics was proposed by Davydov in 1973[4]. His idea was to employ solitons to the models of bio-logical energy transfer and storage, especially in chain-likestructures of peptides and proteins. In the nearly twodecades that followed, an intense effort on both experimen-tal and theoretical fronts has been made to answer ques-tions related to biological soliton formation.In 1989 a NATO Advanced Research Workshop was

organized in Denmark in an attempt to summarize the com-bined effort to date. While several types of experimentalresults indicate the presence of features consistent with theexistence of a localized mode, a direct and convincing proofthat a Davydov soliton is indeed present in biomolecular

cal effort concentrated around the question of solitonity under quantum and/or thermal fluctuations. Thecomprehensive study completed thus far in this area apto be that of Forner [5], whose conclusions do not ruthe existence of Davydov solitons dose to roomperature, provided fairly specific structural characteriare present. However, precise verification of the constrimposed on the structure constants seems to be beexperimental reach at the present time. For an exhauoverview of the Davydov soliton problem, the readreferred to the Hanstholm Conference Proceedings [6a review paper by Scott [7J.The present paper has been motivated by a penetr

observation by Rashba [2], that in solid-state sy(especially low-dimensional ones), self-trapped stateshave a tendency to coexist with the nearly free exstates. Indeed, depending on model parameters, thesemay be metastable with a potential barrier separatingfrom the free exciton band as shown in Figs 1 and 2.Note that Fig. 2, which is qualitatively similar to Fig

Ref. [2], indicates different regimes of behavior, fromfree excitons to self-trapped states, depending onstrength of the coupling constant. Incidentally, this tysituation is characteristic of a first order phase trans(8]. It is interesting that an effective energy functionaself-trapping derived by Rashba [2],H(lj!) =~ f 1 V t / l l 2 dD r - c f I ' " 1 4 dDr,takes the fonn of a Ginzburg-Landau Hamiltonian,plays a prominent role in the theory of phase transi

1 ( 1 ) (b)Potential Potential

Freeexcitons

Fig. 1. Schematic illustration of the shape of adiabatic potent ialThe exciton band region is shaded, and the point Q = 0 (Q is dcoordinate of the figure) corresponds to free states: (a) Possible b



-- Metastable--- Stable

CouplingconstantFig. 2. Schematic dependence of the positions of the bottom of the freeexciton and self-trapped exciton band on the coupling constant, followingRd. [2].

and critical phenomena [8]. In eq. (1.1) J jJ is the electronicwave function in the adiabatic approximation, m is themolecular mass, c is the velocity of propagation of the self-trapped states, and J dDr is short for the integral over Dspatial dimensions. The Euler-Lagrange equation, describ-ing the functional extrema of H ( t J i ) , is

which can be classified as a stationary cubic nonlinearSchrodinger (NLS) equation [9]. In the case of one-dimensional physical space with a coordinate x, a particularsolution discussed by Rashba [2] is the self-trapped"soliton",

r-1R

Fig. 3. Schematic illustration of the molecular chain discussed.

where B! and B~denote creation and annihilation operatoof exciton quanta at site n, respectively, and Do anddenote the ground state energy and molecular site energrespectively. The dipole-dipole interaction energy, J, is potive in the Davydov models of protein and peptide chai[7]. The phonon energy operator is- 1 " [ 1 . 2 , . .,1" ) 2 ]Hplt = 2 ' ; ; - ; ; P n + "'\Un+1 - Un , (2.

(1.2) where w is the elasticity coefficient and U n is the displacment operator, with the conjugate momentum operator, pFinally, the exciton-phonon interaction energy is givenfirst order as(2

where the coupling constant is defined as X(1.3) -(aD/au,,),u.~o. The best currently available estimatesthe parameter values are [7J

A solution of this type in fact provides the basis for theDavydov theory.Rather curiously, to the best of our knowledge, no othertypes of solutions of eq. (1.2) have been given prominentexposure, either in the general theory of exciton-phononinteraction, or in the more specific applications to bio-molecular solitons. This is paradoxial in viewof the fact thatin one-dimensional chains, a complete set of solutions to eq.(1.2) can be obtained and rigorously analyzed. Possibly anexception to this rule is Ref. [1OJ in which in addition tosech-solitons, elliptic wave solutions (sn, cn and dn Jacobifunctions) to the exciton-phonon problem have been listed.However, the role the latter solutions may play thermody-namically has not been investigated in detail. In this paperwe will look for travelling wave solutions to the NLS equa-tion corresponding to the stationary version of eq. (1.2).Wewill show that travelling solutions of extended-wave typemay be of importance for the understanding of the system'sbehavior.2. From Davydov to Ginzburg-LandauIn the simple form, where bending and twisting areneglected, the quantum Hamiltonian, that Davydov pro-posed to describe the dynamics of peptide chains (see Fig. 3),is composed of three distinct contributions [4],Ii = iI



I)

Here the energy parameter, A, is given byA = = Do - D + W - 21, (2.12)where W is the total phonon energy of the chain,

(2.13)

The probability amplitude for finding an exciton at site nat time t is I an(t ) 1 2 , and from eq. (2.10) we have that the totalprobability, N,

(2.14)is a conserved quantity. N is also known as the excitonnumber. We thus take the continuum limit by writinga~(t) ......jR x a(x, t), pft(t) -'> fJ(x, t) , (2 .15)where x = nR and R is the distance between adjacent sites.Consequently eqs (2.10) and (2.11) become

[ . a D Jh O l - A + JR 2 o x 2 + XRp(x , t) a(x, l) = 0, (2.16)(2.17)

where u, = R.jWi i i i is the sound velocity, and p = = - ( } f J l o x .Equations (2.16) and (2.17) are known as the Zakharov

system because they first arose under that name in thecontext of plasma physics [11]. The full solution to thelinear inhomogeneous wave equation, eq. (2.17), is com-posed of two parts, the homogeneous solution and the par-ticular solution. The homogeneous solution is unboundedand does not lead to self-trapping. It describes linear per-turbative effects of the phonons on the excitons, due to thescattering term, (XRp} a, in eq. (2.16). The effect of this solu-tion has been investigated in the past (see e.g. Ref. [5]), andin the limit where Ia 1 2 varies slowly in time compared withp , it may be removed by integrating over a period of oscil-lations, 2n/va. Thus it will not be considered in this context.Instead our interest will be focussed on the particular solu-tion, and thus the nonlinear effects of the self-trapping,where the two wave functions, a and p , are dynamicallycoupled.A solution to eq. (2.17) is easily found

(2.18)

where s = = vlv~, and v is the soliton velocity. Inserting thissolution in eq. (2.16) we arrive at the nonlinear Schrodinger(NLS) equationiha , - Aa + JR 2axx + GR l a l2a = 0,where the nonlinearity parameter, G, is given by

X 2G=~ . : . . : : . . . . _ ~w(1 - S2)

(2.19)

Exciton Self- Trapping in the Ginzburq-Landau FrameworkThe adiabatic limit, s -'> 0, has e.g. been discussed by R[2].Rather than proceed with a conventional discussio

the Davydov soliton, which can be found elsewhere (sRefs [4~7J), we now prefer to adopt a more gapproach including both the band of localized solutioeq. (2.19), and several classes of extended nonlinearsolutions in the description. However, before we discdetail the large variety of solutions to eq. (2.19), we maimportant conceptual link with the Ginzburg-LanHamiltonian.It can be readily verified that eq. (2.19) can be d

from Hamilton's canonical equations for the Hamiltodensity,

Clearly .K(x) is in the Ginzburg-Landau form! Howthere are particular aspects relevant to the Davydovthat need to be emphasized here:1. The Davydov soliton is subsonic, and hence G >2. The angle between neighbouring dipoles is small

Oavydov model of protein, and hence J > O .Thus we have a situation which is opposite to tha

mally studied in critical phenomena, i.e. the quartic near term, I a 14, is multiplied by a negative coefficient -This case leads to the possibility of creating ellipticexcitations, in addition to the existence of the better klocalized seen-soliton solutions. We will discuss theand the relative importance of these two types of solutthe sections that follow.

3. Solving the equation of motionIntroducing dimensionless quantities according toa .... .aR - 1/2 exp ( - i~) ' x .....xR,the NLS equation, eq. (2.19), becomesia t + a X . < + V 1 a 1 2 a = 0 ,

ht .....t-J'

where the nonlinearity parameter, V = G/J, is positivshall look for solutions to eq. (3.2) in the travelling-form

where the envelope function, , is real. The coordinateand ~1 are given by

Dc and Vc being the envelope and carrier velocity, rtively.Inserting eqs (3.3) and (3.4) in eq. (3.2), and separating

and imaginary parts, yieldsVC Y ~ 2 + q,~l~l - q , Y Z 2 + V3 = 0,

(2.20) -V O . Thus we are in the regime, wherethe NLS equation has bright soliton solutions [9J, the

where subscripts ~1 and ~2 denote differentiationrespect to the respective coordinate. We first investiga(3.6). Multiplying with 2 q , on both sides and integwith respect to x, the equation reduces to

, , J



(e) u>O , Co=OP ( < I

(d) 0>0, -o'j2V 0 is chosen, corresponding to the Davydov system.

where f(t}, t c c ( I- (z), is an integration constant. Due tothe form of this equation it can be demonstrated that onlytwo acceptable solutions exists.The evident kind of solution has the form

where Cl> Cz , C3 and C t are real constants, However, this isan unphysical solution, since c f J diverges at 00, dependingon the sign of C t, unless of course CI= 0 or C( = 0 is chosen,reducing < P to a constant.

sech{x)I r - - - - - - - - - - A-+__~----~~------~ __----~~ x

(b)

; -10 o 10

dn(x,k) (c ),~/\_-j"\_~~./. X

-10 0 10f-- 2X(k)____'

Fig. 5. Graphical illustration of the relevant wave-like solutions for V >O.(a) en-waves. (b) sech-solitons. (e) dn-waves.Physic Scripta 51

( 3 . 9 )and th e envelope function, 4 > , satisfies the equation~1~1= (0 - - V 2).Here the parameter, G, is given by

(3.10)

(3.11)Multiplying with O . The analytical characteristics of these solutions arelisted below.

(i ) For Co > 0 two types of elliptic en-waves [12] exist[see Fig. 4{a) and (b)]; which we denote cn.- and cnj-waves,corresponding to 0 - >0 and 0 - < 0, respectively. They aredescribed byr P { e1)=l cn(J!V[i + ~Hl' k), (3 .1 3)where c f J 1 and i c f J 2 are the two symmetric real and ima-ginary roots of the polynomial P ( < p ) , respectively. TheJacobi modulus, k, is given in terms of c f J I and zk2 = i .4,,2A,,2 = 2Co .4,,2 _ .4,,2 = 20- (3.14)q, i+4>r o/\o/z V' 0/1 0/2 VThe wavelength of this class of solutions isT . : I I = 4K(k)/J! V[i + n. (3.1S)where K(k) is an elliptic integral of the first kind [12]. Thetwo types of en-waves differ in the range of k: For cnz-waves 0 - l ' O ; k ~ (1/J2), while for COl-waves (1/fi) ~ k ~ 1.

(ii) Sech-solitary waves occur for 0- > 0 and Co = 0 [seeFig. 4(c)]. Thus the sech-pulse is the limit between theperiodic cn.- and dn-waves (see below), for which the periodbecomes infinity. The analytical form of this type of solutionis

(3,16)where 0 and

-(0-2/2V) < Co < 0 [see Fig. 4(d)]. In this case P { < p ) , hasfour real roots, cfJl and 2, where 0 < cfJz < I c f J I I ,These solutions are described by



Table L Pulse period (units of R) and energy per pulse forelliptic cn- and dn-waces, and sech-solitons.found as solutionsto the nonlinear Schrodinger equation, eq. (2.19),Jor V > 0Solution Period Energy per pulseenl wavesCo> 0 ..1.=T " " (en = iCo.u + 1 \ [ . p f + < 1 J D ~ - .p~)A.{fsech solitonsCo = 0 A . = 00 g J arI f"cb =WA.adn -waves

q2Sdo = tCo)J + iM P ~ ~ Aofr- II!20L ' f: ~ I10 IIo I i I-2 0 2 4 6

Co-2 0 4 6

Con

i) 14r--.~-------- 1.0 ,..-.....,...-=--~--~-----,0.8d 1 1 :

.s 0.6I 0.4

5l) 2.._--,-_~_~_

-2 0o 0 ' -'-~ __ ~ __ ~---'-2 0 26 4

3)Co Co

Fig. 6. The dependence or the energy per pulse, 8, the total energy. E. thepulse period, A, and the Jacobi modulus, k, on the integration constant, Co.J =0.97 meV, X = 50 pN, w = 49 Nj'rn and G = 0.29 meV, which are typicalvalues for the Davydov system in tbe adiabatic limit. A = 0, v.= 2 andv< = O . The length or the protein is L = lOOK. where R is the distanceL _ . _ _ _ _ _ . . . _ _ ~ . I '" ~-

Exciton Self- Trapping in the Ginzburq-Landau Framework4. Energy calculationsIn order to gain better understanding of the travelling-solutions obtained in Section 3, we will calculaterespective energies, for stationary solutions first. Thetonian energy density, Jf'(x), is given by eq. (2.21). Saccording to eq. (3.1), and inserting the solution, giveqs (3 .3) , (3 .4 ) and (3.9), the energy density becomesJf'(1) = JR - 1 ( - V 4 > 4 + [ 7 + a + tV ; J c P 2 + c o ) .where we have used eq. (3.12) to reduce the expressiothe solutions of interest to us allow for exact calculatithe energy per pulse [13], defined as

f +( 1/2 ) .1 .tS ;;: Jf'(' I)R d l'-(1/2)lwhere A . is the dimensionless pulse period. For en-;,= t T . : n ' for sech-solitons A . = CIJ, and for dn-waves1 d n . The results are listed in Table J , where the efenergy parameter, A.rr, is defined as

Exact analytical expressions of the total energysystem cannot be found, except in the case where theL, is an integer number of pulse periods. In order tothe extent of the consequent numerical calculations,the following approximation of the total energy, in tethe energy per pulse

_ n 1 1 1 _ {L/O.R) for L > ARH ~ ptS , p\Co) - 1 for L :!( AR'which is a good approximation for L~R.In Fig. 6 we have shown the Co-dependence of Gand the Jacobi modulus, k , for J = 0.97 meV, X =

OJ = 49N/m, and G = 0.29 meV, which are typical valuthe Davydov system in the adiabatic limit [see eqs (2(2.20)]. Choosing A = 0, v. = 2 and Vo = 0, we get Va = 1 and Acf r = J.We see that the sech-soliton solution has the lowesenergy, corresponding to a sharp dip in H at Co = Othat J(Co) is continuous with the value Sb = 7.84mCo =0, as we know it should be.The qualitative behavior seen in Fig. 6 does not d

on the specific value of the parameters, as long as V >a> O. Thus we conclude that the sech-soliton is thegetically stable excitation in the Davydov system. Thground state excitation of the Davydov system is locfor L ~ R has recently been shown more strictCruzeiro-Hansson and Kenkre [14]. However, notH(Co ) is continuous, and thus en- and dn-waves willrole thermodynamically, e.g, contributing to the sheat. This will be considered in more detail in the follo

6 5. Finite-temperature effects5.1. General commentsFinite-temperature effects are related to the stabilitytons. Although our approach to this question will beon thermodynamics, we will give a brief overview of s



system and perform dynamical calculations for soliton solu-tions. Forner performed very accurate calculations at finitetemperatures, in order to obtain a threshold for soliton for-mation and pinning [5]. In terms of the coupling constants,X and 0), he found parameter ranges at T = 0 K andT = = 300 K, within which solitons can propagate.It should be mentioned in this context, that disorder may

set in through a number of effects [15], such as (a) massdisorder, (b) spring constant disorder, (c) dipole-dipole dis-order, (d) exciton-phonon coupling disorder, and (e) diago-nal disorder. Variations in the respective values of the modelparameters, exceeding rather narrowly specified ranges,result in the destruction of solitons. Disorder in the dipole-dipole interaction energies, and in the site energies, appearto be of greatest concern [15].Cruzeiro et al. [16] examined temperature effects on the

Davydov soliton assuming an anisotropic model with differ-ent values of the left- and right-sided coupling constants, X+and X-, respectively. The results obtained indicate the exis-tence of a critical temperature, r ; " above which solitonscease to exist as stable excitations. Moreover, the depen-dence of the threshold value of the coupling constant, X,canbe approximated by X t : : ; : : X O + X l ( T - T , J + X 2 ( T - T . : f+ ....Davydov discussed temperature effects in terms of effec-tive model parameters [17]. In particular, he found that,

due to thermal population of the phonon states, the effectivenonlinearity coefficient becomes

X 2 ( T )(T) =- 1 - - ,0) To

whereTo =!0_ f ! .2k B -i:is estimated to be between lOOK and 120K.In this framework we must certainly treat the dipole-

dipole coupling constant, J, as temperature dependent. Wethen note that the role of thermal fluctuations is two-fold:(a) to elongate the chain, and (b) to cause orientational dis-order among the dipoles. Clearly then, the value of J mustdecrease with temperature, eventually becoming negligiblysmall. In the simplest case we could use the approximationJ(T) ::;::JO (T l - T ). The positive sign of J below the charac-teristic temperature, T1, results in the exciton energy havinga tendency to be distributed inhomogeneously. Thisexplains the predominance of the localized sech-solitons,and the extended elliptic waves found in the earlier sections.Above Ti,on the other hand, J becomes negative, resultingin a diffusive behavior of the exciton energy, and its even-tual homogenization (thermalization),In what follows we do not presuppose any particular tem-

perature dependence of the effective Hamiltonian param-eters, but calculate statistical properties based on the energycalculations in Section 4.5.2. Partition junction and specific heatsHaving obtained analytical expressions and plots for theenergies of the various classes of solitions, we turn to sta-tistical properties that might give us clues as to how thesePhysica Scripta 51

want to point out that elliptic waves may playa role in thethermodynamics of the Davydov system, we will only con-sider the effect of en-waves, and sech-soliton solutions.Regarding the integration constant, Co . as a continuous

parameter, the partition functions becomeZ m h = exp ( _ : : e ~ }

r o o (H.~[CoJ)z., = J o exp - k8 T dCo,.where the total energy, HeD is defined in eq. (4.4). From eq(4.4) it is also found that H.ech = Io'.ech. The specific heat ithen calculated as

(5.2

(5.3The Boltzmann probability factor for the en-waves i

defined as

(5.l)

( Hcn[CoJ)Pcn(Co) = exp - k8 T 'and thus p.ech = Pcn(O). In Fig. 7 we show P(Co ) aT = 300 K for different values of the length of the system, LWe see that the thermodynamic influence of en-wavesdepends critically on L. For systems longer than a fewhundred sites only the sech-soliton will contribute to thepartition function. while for smaller systems the en-wavesmay begin to playa role.The temperature dependence of the partition function,

and the specific heat has been plotted in Fig. 8. The lengthof the system is chosen to L = 20R, in order to clearly seethe influence of en-waves. Since C,.ch = 0 we see that therelative influence of en-waves increases with temperature(Con increases with 0.3meV/K when T is increased to300 K). For larger systems the increase in Cen will be lessand the role of en-waves will be negligible.

(5.4

1p0.1

0.01 1\0.001 \

\ \0.0001

0.00001

0.000001 +------,,--'----r---..,---.....:_,,.---.----,o 15 C 18oFig. 7. The Bol tzmann probabil ity factor, P, vs. the integration constant,C", at T =300 K and ro r (full curve) L = 20R, (dashed curve) L =50 R and(chain-dashed curve) I,= lOOR.

3 6 9 12



0.5N0.40.3020.10.0

0

( a )

50 100 150 200 250 300T [ K ]

0.4;Z~ 0.3 (b ))gU 02

O J

0.0 a 50 100 150 200 250 300T [ K ]

Fig. 8. (a) The partition function, ZeD' and (b) the specific heat., C en' ofelliptic en-waves versus temperature. The length of the system is L = 20R.

In the final subsection we shift our attention to anothertype of thermal excitation, namely equilibrium fluctuations.Thermal fluctuations differ from all the patterns discussedso far in that they do not minimize the energy functional,and thus are not solutions of the equation of motion.However, they may be infinitesimally close to the energybottom, and hence their role in thermodynamics may besubstantial.5.3. 'Thermal fiuctuationsThermal fluctuations do not solve the differential equationof state, such as eq. (2.19) discussed here. Therefore, they donot represent functional extrema of any kind, but ratherlow-lying accessible states, that the system may reside in ata low enough energy cost. It is important to note that as thesystem approaches a bifurcation or an instability, thermalfluctuations (commonly referred to as critical fluctuations)dominate the picture completely. In order to properlyaccount for the strong anharmonities present in this regime,the relevant statistical calculations must be performed withgreat care. To this end we adopt a new method of evalu-ating non-Gaussian averages for near-critical systems [18].The starting point is the effective Hamiltonian density of

eq. (2.21). Next, we represent the fluctuations, a(x), inFourier space as(I(X) = ~ L U k exp (ikx),yL IkloO

t,d where A = 2 7 t 1 R is the cut-off wavelength, and L is again the

Exciton Self- Trapping in the Ginzburq-Landau FrameworkHamiltonian takes the form

IH =- - L (A - JR 2k2)IQkI22L Ikl"').GR+-2 L Qkak,Qk"Q-k-k'-k'"41.; Jkl. In W'I < ~

The only approximation that we make in this derivato retain only paired-up modes in the second term(5.6). This means that only pairs of k and - k are pwhile the remaining contributions are expected tocancel out. Thus the appropriate Hamiltonian thatused in our statistical calculations isH ';;( , L [- -2 1 (A - JR 2k2) 1 ak l 2 + 3G ~ I Qt4]Ikl



increases. A cross-over temperature is expected above whichen-waves dominate the thermodynamic nature of the chain.This should in principle be experimentally verifiable.

AcknowledgementsWe are indebted to Professor A. C Scott for numerous insightful commentsand valuable suggestions. This research has been supported by the NSERC(Canada) and the Danish Research Council for Scientific and IndustrialResearch. J. A. Tuszynski acknowledges the award of a Guest Pro-fessorship at the Technical University of Denmark, which was tenablewhile this research was being carried out. O. Bang, acknowledges TheDanish Technical Research Council for financial support under grant nr.16--5009-1 PG, and CEC for financial support with the contract No. SCI-CT91-0705.

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(North-Hol land 1982), p. 543.3. Hernandez, J. P., Rev. Mod. Phys. 63, 67 5 (1991).4. Davydov, A. S., J. Theor. BioI. 38.559 (1973).5. Forner. W . P hys. Rev. A44, 2694 (1991); Schweitzer. J. W. and Cot-

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Physico Scripta 51

Revisited" (Plenum Press 1990).7. Scott, A. C, Phys. Rep. 117, 1 ( 1992).8. Landau, L. D. and Liffshitz, E. M., "Statistical Physics" (Pergamo

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12. Byrd. P. F. and Friedman. M. D., "Handbook of Elliptic IntegralsEngineers and Scientists" (Springer-Verlag 1971).13 . Winternitz, P., Grundland, A. M. and Tuszynski, 1. A., 1. Phys. C4 9 3 1 ( 19 8 8) .

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16. Cruzeiro, J. , Halding, L., Christiansen, P. L., Skovgaard, O. and ScoA . C ., Phys, Rev. A37, 880 (1988).

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j.a. tuszynski et al- excition self-trapping in the ginzburg-landau framework

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