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On the asymptotic solutions of the coupled quasiparticle-oscillator system
Chaos, Solitons and Fractals 15 (2003) 597–610 www.elsevier.com/locate/chaos
On the asymptotic solutions of the coupled quasiparticle-oscillator system L.S. Kuzmenkov b
a,*
, S.G. Maximov b, J.L. Guardado Zavala
b
a Moscow State University, Vorobjovy Gory, 119899 Moscow, Russia Instituto Tecnol ogico de Morelia, P.O. Box 262, Morelia, Mich., Mexico
Accepted 23 May 2002
Abstract A coupled quasiparticle-oscillator system is considered for an arbitrary number of excitons. The exciton dynamics is described in terms of the second quantization (i.e. by means the bosonic operators). As a consequence a radius of a Bloch sphere is obtained different to the previous results. Some integrals of motion are obtained that allowed to reduce the system of equations of motion to a single nonlinear ordinary differential equation of the fourth order. This equation contains the energy of the system as a parameter. The fixed points are found as a functions of the energy of the system, and its stability properties are investigated. It is demonstrated that a bifurcation is presented for the energies H < 1=2p. An asymptotic quasiclassical solution around fixed point for the case H > 1=2p is obtained. The solutions around other stable fixed points can be obtained analogously. The expression for the evolution operator of the quasiparticle-oscillator system is obtained as a functional on the classical solutions. Ó 2002 Published by Elsevier Science Ltd.
1. Introduction The correspondence between classical nonlinear systems on one side and their fully quantized counterparts on the other has been intensively investigated in the last decade (see [1,2,4–7]). One step quantization of a given system is sometimes complicated. As a rule such systems are divided into interacting subsystems. Then a mixed description is applied, in which one of the subsystems is considered classical and the other is treated in a quantum context. This quantization is the basic idea on which the Born–Oppenheimer approximation [3], used in many situations, is based. For nonadiabatic coupled systems this approximation must be complemented into a rigorous quantum description. The nonlinearities generated in this case can give rise to nonintegrability and chaos [4], and the problem of the quantumclassical correspondence arises for the dynamical properties of the mixed and fully quantized schemes. Schanz and Esser [3] stressed that chaos in the mixed description is produced by coupling one-dimensional integrable adiabatic reference states. In this paper the relation between the dynamical properties of the mixed and fully quantized descriptions for the model of a quasiparticle moving between two sites and coupled to an oscillator is considered. As previously reported [4] we also consider a coupled quasiparticle-oscillator system. In our model N excitons moves between two sites of the dimer composed by two molecular monomers. The excitons are coupled to intramolecular vibrations. The interest in the dynamical properties of the quasiparticle-oscillator system is based on the wide range of applications that includes a variety of phenomena in molecular and solid state physics, e.g. excitons in molecular
*
Corresponding author. E-mail addresses: [email protected] (L.S. Kuzmenkov), [email protected] (S.G. Maximov).
0960-0779/03/$ - see front matter Ó 2002 Published by Elsevier Science Ltd. PII: S 0 9 6 0 - 0 7 7 9 ( 0 2 ) 0 0 1 1 9 - 4
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aggregates and coupled to vibrations [8]. The dynamic properties of this model in the mixed description have been previously investigated [4,6,7]. Analysing the dynamical properties of such systems the authors have focused on the coupling parameter ranges. It has been demonstrated the presence of a separatrix structure underlying the phase space for overcritical coupling, and of chaos developing from the hyperbolic point at the center of this structure [7]. The problem of relation between the dynamics in the mixed and fully quantum levels of description of the coupled quasiparticle-oscillator motion were considered by Schanz and Esser [3]. They focused on the adiabatic parameter range to justify best the mixed description. The aim of the present paper is to find some asymptotic quasiclassical solutions around the fixed elliptic points for the case of weak coupling, for the system of N excitons coupled to the molecular dimer. In Section 2 we specify the model in detail considering an arbitrary number of quasiparticles. In Section 3 using integrals of motion the system of equations of motion is reduced to the single ordinary differential equation that contains the energy of the system as a parameter. Further investigation is based completely in this equation. In Section 4 a problem of evolution operator for the oscillator-excitons system is considered and the quasiclassical approximation is justified. In Section 5 fixed points of our system are investigated focusing on the energy as a parameter. In the last section the solving scheme for the case H > 1=2p is justified and the solution for the quasiparticle-oscillator system is given in form of expansion with respect to the small adiabatic parameter r. 2. Model Let us consider a system of an arbitrary number of molecules with the Hamiltonian X1 X X b ¼ H ðp^n2 þ x2n q^2n Þ þ n aþ Vnm aþ n an þ n am ; 2 n n n;m
ð1Þ
þ ¼ Vmn , Vnn ¼ 0, and the nth state energy n is a function of the coordinate qn . aþ where Vmn n and an are excitonic bosonic operators:
½an ; aþ m ¼ dn;m ;
½an ; am ¼ 0:
ð2Þ
It is obvious that the exciton number X b ¼ N aþ n an n
is an integral of motion, i.e. b;N b ¼ 0: ½H Using the first-order expansion in qn of the exciton energy n ðqn Þ: n ðqn Þ ¼ ð0Þ n þ cn qn þ we reduce the Hamiltonian to the following form: b ¼ H
X X X X1 þ ð0Þ cn qn aþ Vnm aþ ðp^n2 þ x2n q^2n Þ þ n an an þ n an þ n am : 2 n n n n;m
ð3Þ
Let us consider the simplest case of a symmetric two site system, e.g. an exciton in a molecular dimer constituted by two ð0Þ ð0Þ identical monomers. We set 1 ¼ 2 , x1 ¼ x2 x, c1 ¼ c2 c and V12 ¼ V21 ¼ V , V > 0. Then by introducing for the vibronic subsystem the coordinates and momenta q ¼
q2 q1 pffiffiffi ; 2
p ¼
p2 p1 pffiffiffi ; 2
ð4Þ
we obtain two independent Hamiltonians: b ¼H bþ þ H b ; H where b þ ¼ 1 ðp^2 þ x2 q^2 Þ þ þ pcffiffiffi q^þ N b; H þ 2 þ 2
ð5Þ
L.S. Kuzmenkov et al. / Chaos, Solitons and Fractals 15 (2003) 597–610 þ þ þ b ¼ 1 ðp^2 þ x2 q^2 Þ þ pcffiffiffi q^ ðaþ H 2 a2 a1 a1 Þ V ða1 a2 þ a2 a1 Þ: 2 2
599
ð6Þ
b þ is not coupled to the exciton, and will be omitted. It is an oscillator Hamiltonian with the The Hamiltonian H displaced fixed point: 0 " #2 1 2 b2 b N c 1 c N A: b þ N bþ h^þ H ¼ @p^þ2 þ x2 q^þ þ pffiffiffi 2 4x2 2x2 b in units of 2 V and replacing q^ and p^ by Passing in Eq. (6) to dimensionless variables by measuring H pffiffiffiffiffiffi b ¼ 2V q ; Pb ¼ p1ffiffiffiffiffiffi p ; Q 2V
ð7Þ
one finally obtains 1=2 b ðaþ a2 aþ a1 Þ: b 2 Þ 1 ðaþ a2 þ aþ a1 Þ þ p b ¼ 1 ð Pb 2 þ r2 Q rQ H 1 2 2 1 2 2 2
ð8Þ
It is obvious that (h 1) b ; Pb ¼ i; ½Q
b ¼ ½an ; Pb ¼ 0: ½an ; Q
ð9Þ
The Heisenberg quantum evolution equations b ; a1 ; ia_ 1 ¼ ½ H
b ; a2 ; ia_ 2 ¼ ½ H
i
d b b ; Pb ; P ¼ ½H dt
i
d b b b;Q Q ¼ ½H dt
and the commutational relationships (2) and (9) result in the following equations: rffiffiffi 1 p b r Q a1 ; ia_ 1 ¼ a2 þ 2 2 rffiffiffi 1 p b r Q a2 ; ia_ 2 ¼ a1 2 2 rffiffiffi d b b prðaþ a2 aþ a1 Þ; P ¼ r2 Q 1 dt 2 2 d b Q ¼ Pb : dt
ð10Þ
ð11Þ ð12Þ ð13Þ ð14Þ
Substituting new operators þ x^ ¼ aþ 2 a1 þ a1 a2 ; þ y^ ¼ iða2 a1 aþ 1 a2 Þ;
ð15Þ
þ ^z ¼ aþ 2 a2 a1 a1
in Eqs. (11)–(14) we obtain the following system of equations: pffiffiffiffiffi d b y^; x^ ¼ 2pr Q dt pffiffiffiffiffi d b x^; y^ ¼ ^z 2pr Q dt d ^z ¼ y^; dt
ð16Þ ð17Þ ð18Þ
d2 b b ¼ r Q þ r2 Q dt2
rffiffiffi p ^z 2
ð19Þ
with new commutational relationships analogous to the algebra of the Pauli matrixes: ½^ x; y^ ¼ 2i^z;
½^ y ; ^z ¼ 2i^ x;
½^z; x^ ¼ 2i^ y:
ð20Þ
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A quantum evolution of any system we can consider in both Heisenberg or Schr€ odinger patterns. Let us work in the frame of first of them, i.e. the wave function jWi we’ll consider constant.
3. Integrals of motion There are two integrals of motion we will use to solve the system (16)–(19). One of them is offered to us by the Hamiltonian which with the transformations (15) obtains the following form: rffiffiffi 1 b_ 2 1 p b 2 b2 b H ¼ ð Q þ r Q Þ x^ þ r Q^z: ð21Þ 2 2 2 Other integral can be obtained directly from Eqs. (16)–(18). We deduce that x^2 þ y^2 þ ^z2 ¼ const: Using the definition of the operators x^, y^ and ^z and the commutational relationships (2), it is easy to find b þN b 2; x^2 þ y^2 þ ^z2 ¼ 2 N
ð22Þ
b is the operator of number of excitons: where N b ¼ aþ a2 þ aþ a1 : N 2 1 Let us consider the system (16)–(19). From the integral (21) we immediately obtain pffiffiffiffiffi b_ 2 þ r2 Q b 2H b 2 Þ þ 2pr^z Q b; x^ ¼ ð Q and from Eq. (19) sffiffiffi 1 2 b € b Þ: ð Q þ r2 Q ^z ¼ r p Substituting (24) into (17) and taking into account Eq. (18) we arrive at the following equation pffiffiffiffiffi _ 2 pffiffiffiffiffi €^z þ ^z ¼ 2prf Q b Q b 2 2H b g: b þ r2 Q b 3 þ 2pr^z Q bQ
ð23Þ
ð24Þ
ð25Þ
ð26Þ
Eqs. (25) and (26) can be reduced to the following single nonlinear equation of the fourth order: € € b 2 b_ 2 b { b b ¼ r2 pð2 Q b b 3 Þ: b pÞ Q þ r2 ð1 þ 2 H Q Q Q þ r2 Q b þ ð1 þ r2 Þ Q Q
ð27Þ
The stationary properties of the system brings the stationary Schr€ odinger equation: b jCi ¼ EjCi; H where, for example, in the representation jn; mi ¼ jni jmi, where n is the number of excitons in one of the molecules and m is the oscillator state, the Hamiltonian matrix elements have the form: pffiffiffiffiffi pffiffiffiffi pffiffiffi rp b jn0 ; m0 i ¼ r m þ 1 dðm m0 Þdðn n0 Þ þ i hn; mj H ðN 2nÞð m0 dðm m0 þ 1Þ mdðm m0 1ÞÞdðn n0 Þ 2 2 ffi 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi0ffi ð28Þ ðN n0 Þndðn n0 1Þ ðN nÞn dðn n0 þ 1Þ: 2 2
4. Quasiclassical approximation Let us consider a problem of evolution operator for the oscillator-exciton system. The Hamilton in terms of canonical variables has the form: rffiffiffi 1 p b 2 b 2 Þ 1 ð^ b ¼ 1 ð Pb 2 þ r2 Q H p2 p^1 þ q^2 q^1 Þ þ r Q ðp^2 þ q^22 p^12 q^21 Þ: ð29Þ 2 2 2 2
L.S. Kuzmenkov et al. / Chaos, Solitons and Fractals 15 (2003) 597–610
601
To pass from the Hamiltonian (8) to the form (29) we should make the following transformation 1 q1;2 Þ; aþ p1;2 þ i^ 1;2 ¼ pffiffiffi ð^ 2
1 a1;2 ¼ pffiffiffi ðp^1;2 þ i^ q1;2 Þ: 2
ð30Þ
The evolution operator matrix elements are found in the following form: ( ) 3N =2 NY 1 Z N 1 hN dQk i X N i3p N 2p b 4 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dq1k dq2k exp Lðqk ; q_ k ÞDtk hqj U ðt t0 Þjq0 i lim 2 e N !1 t t0 h k¼0 1 þ 2pr2 Q2k k¼0 Z t Z Y i ¼ N1 ðt t0 Þ D3 ½qðtÞ exp Lðq; q_ Þ ds h t0 t
ð31Þ
where we denoted dQðtÞ D3 ½qðtÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dq1 ðtÞ dq2 ðtaÞ; 1 þ 2pr2 Q2 ðtÞ
ð32Þ
and the classical Lagrangian has the form rffiffiffi pffiffiffiffiffi 2prQ 1 2q_ 1 q_ 2 1 1 p 2 2 _ _ q rQðq22 q21 Þ: Lðq; q_ Þ ¼ ðQ_ 2 r2 Q2 Þ þ ð q q Þ q þ 1 2 2 1 þ 2pr2 Q2 1 2 2 2 1 þ 2pr2 Q2 2
ð33Þ
The continual integral (31) has a nontrivial measure (32). Let us set pffiffiffiffiffi 1 Q ¼ pffiffiffiffiffi shð 2prq3 Þ; 2pr
ð34Þ
then dQ dq3 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 1 þ 2pr2 Q2 In terms of new variable the integral (31) obtains the following form Z t Z Y i b ðt t0 Þjq0 i ¼ N1 hqj U dq1 ðtÞ dq2 ðtÞ dq3 ðtÞ exp Lðqi ; q_ i Þ ds ; h t0 t
ð35Þ
where ði ¼ 1; 2; 3Þ Lðqi ; q_ i Þ ¼
pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi 1 2 1 sinhð 2prq3 Þ 2 2q_ 1 q_ 2 1 ðq_ 1 q_ 22 Þ q1 q2 q_ 3 cosh2 ð 2prq3 Þ sinh2 ð 2prq3 Þ þ pffiffiffiffiffi pffiffiffiffiffi 2 2p cosh2 ð 2prq3 Þ 2 cosh2 ð 2prq3 Þ pffiffiffiffiffi 1 þ sinhð 2prq3 Þðq22 q21 Þ: 4
ð36Þ
According to the Feynman’s theory the evolution operator (35) is a functional on the classical solutions. There is a classical action functional in the exponent in the formula (35), that can be expanded around classical solution into the series by the following way: Z t Att0 ¼ Lðq; q_ Þ ds; ð37Þ t0
Att0
¼
Att0 jqcl
1 þ 2
Z
Z
t
ds1 t0
t
ds2 t0
d2 Att0 dqa ðs1 Þdqb ðs2 Þ
! dqa ðs1 Þdqb ðs2 Þ þ ;
ð38Þ
qcl
where the classical solutions are performed by the Lagrange equations: t dAt0 ¼ 0: dqa qcl
ð39Þ
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L.S. Kuzmenkov et al. / Chaos, Solitons and Fractals 15 (2003) 597–610
Substituting the expansion (38) into the integral (31) we obtain 8 Z Z t Z
d2 Att0 a dq ðs1 Þdqb ðs2 Þ
! dqa ðs1 Þdqb ðs2 Þ þ qcl
9 = ;
: ð40Þ
Then we obtain an asymptotic formula for the evolution operator matrix elements in quasiclassical approximation: 0vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! , u u i i d2 A 1 t t @ b þ : ð41Þ A j 2p det hqj U ðt t0 Þjq0 i ¼ N exp h t0 qcl h dqa ðt1 Þdqb ðt2 Þ qcl The classical solutions of the oscillator-quasiparticles system can be substituted into the formula (41). We can also find other terms of the expansion (41) reaching a better accordance with the quantum dynamics of the system.
5. Fixed points and stability problem We consider a coupled oscillator-quasiparticles problem in quasiclassical approximation treating the Eq. (27) as a classical. So we have the following nonlinear equation: v € þ r2 ð1 þ 2HpÞQ ¼ r2 pð2Q2 Q € QQ_ 2 þ r2 Q3 Þ; Q þ ð1 þ r2 ÞQ
ð42Þ
where the adiabatic parameter r is small (r 1). Setting all the derivatives in Eq. (42) to zero we obtain the following stationary equation r2 ð1 þ 2HpÞQ ¼ r4 pQ3 : Fixed points are represented by three solutions of this equation Q ¼ 0; Q2 ¼
1 þ 2Hp : r2 p
ð43Þ ð44Þ
According to the magnitude of energy H the fixed points can be classified into two groups. 5.1. Case 1 þ 2Hp > 0 There exist three different cases to consider. First of them is the situation described by the following inequality 1 þ 2Hp > 0:
ð45Þ
For this case Eq. (44) has not real solutions, therefore there is only one fixed point Q ¼ 0. The stability property of this point can be obtained from a homogeneous equation. The homogeneous equation results in the following characteristical equation (we substitute eixt ): x4 ð1 þ r2 Þx2 þ r2 ð1 þ 2HpÞ ¼ 0:
ð46Þ
All the solutions of Eq. (46) are real x1 ¼ ð1 r2 HpÞ þ Oðr4 Þ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 ¼ r 1 þ 2Hp þ Oðr3 Þ:
ð47Þ
So the fixed point Q ¼ 0 is stable (stable elliptic). 5.2. Case 1 þ 2Hp < 0 Another case is when the energy magnitude satisfies the inequality 1 þ 2Hp < 0:
ð48Þ
L.S. Kuzmenkov et al. / Chaos, Solitons and Fractals 15 (2003) 597–610
603
In this situation Eq. (42) can be rewritten in the following form {
€ r2 j1 þ 2HpjQ ¼ r2 pð2Q2 Q € QQ_ 2 þ r2 Q3 Þ: Q þ ð1 þ r2 ÞQ
ð49Þ
The bifurcation has occurred, and Eq. (49) has three fixed points Q ¼ 0; sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 j1 þ 2Hpj Q ¼
: r p
ð50Þ ð51Þ
The first of them (Q ¼ 0) leads to the following characteristical equation (we substitute eixt ) x4 ð1 þ r2 Þx2 r2 j1 þ 2Hpj ¼ 0
ð52Þ
which has the following solutions x21 ¼ 1 þ r2 ð1 þ j1 þ 2HpjÞ þ Oðr4 Þ; x22 ¼ r2 j1 þ 2Hpj þ Oðr4 Þ: From (53) we obtain r2 x1 1 þ ð1 þ j1 þ 2HpjÞ ; 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 ir j1 þ 2Hpj:
ð53Þ
ð54Þ
Because of the frequency x2 in Eq. (54) obtains imaginary values the fixed point Q ¼ 0 is unstable hyperbolic. To investigate the stability properties of the fixed points (51) we have to pass to displaced variables qðtÞ by the following transformation QðtÞ ¼ Q þ qðtÞ:
ð55Þ
Substituting this form into Eq. (49) and taking into account (51) we arrive at the following two equations: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi { q þ ð1 þ 2j1 þ 2Hpj þ r2 Þ€q þ 3r2 j1 þ 2Hpjq ¼ r pj1 þ 2Hpjf4q€ q qq_ 2 þ r2 q3 g: q q_ 2 þ 3r2 q2 g r2 pf2q2 € ð56Þ It is easy to find from Eq. (56) that the fixed points Q are stable. Equating to zero the nonlinear part of Eq. (56) we obtain a respective characteristical equation which has the following form x4 ð1 þ 2j1 þ 2Hpj þ r2 Þx2 þ 3r2 j1 þ 2Hpjq ¼ 0: All its solutions are real numbers 3 j1 þ 2Hpj x21 1 þ 2j1 þ 2Hpj þ r2 1 ; 2 1 þ 2j1 þ 2Hpj 3 j1 þ 2Hpj x22 ¼ r2 : 2 1 þ 2j1 þ 2Hpj
ð57Þ
ð58Þ
Therefore the fixed points Q of Eq. (49) (q ¼ 0 for Eq. (56) respectively) are stable elliptic. 5.3. Case 1 þ 2Hp ¼ 0 The case 1 þ 2Hp ¼ 0 leads to the following differential equation { € ¼ r2 pð2Q2 Q € QQ_ 2 þ r2 Q3 Þ: Q þ ð1 þ r2 ÞQ
The point Q ¼ 0 for this case is unstable hyperbolic.
ð59Þ
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L.S. Kuzmenkov et al. / Chaos, Solitons and Fractals 15 (2003) 597–610
6. Asymptotic solution for the case H > 1=2p Let us consider Eq. (42) for the energy magnitudes H >
1 : 2p
As we have seen for this case Eq. (42) has only one fixed point Q ¼ 0. We will search the solution of Eq. (42) around this fixed point. We have find that the homogeneous equation has two positive frequencies x1 and x2 (54). Moreover, one of the frequencies is far above the other frequency: x2 x1 :
ð60Þ
Therefore the solution of the homogeneous equation is a superposition of two slowly and quickly oscillating functions with the frequencies x1 and x2 : 1 1 Qh ðtÞ ¼ ðAeix2 t þ A eix2 t Þ þ ðaeix1 t þ a eix1 t Þ: 2 2 It is natural to search the solution of the nonlinear equation (42) in the following quasiperiodical form: X QðtÞ ¼ Qðn1 ; n2 Þeiðn1 x1 þn2 x2 Þt ; Qðn1 ; n2 Þ ¼ Q ðn1 ; n2 Þ;
ð61Þ
ð62Þ
n1 ;n2
where x2 r, and, in general, the frequencies x1 and x2 must have the form (54) of the expansion with respect to the small parameter r. The frequencies of the type n1 x1 þ n2 x2 appear in the solution due to the mixing of frequencies in the right hand side of Eq. (42). The terms of the expansion (54) and the coefficients Qðn1 ; n2 Þ in the formula (62) have to be found by setting to zero all the secularities that appear in the nonlinear part of Eq. (42). The solution (62) can be divided in two rapidly and slowly oscillating parts by the following way: QðtÞ ¼ Qq ðtÞ þ Qs ðtÞ;
ð63Þ
where Qq ðtÞ ¼ Qð1; 0Þeix1 t þ Q ð1; 0Þeix1 t þ
X
Qðn1 ; n2 Þeiðn1 x1 þn2 x2 Þt ;
ð64Þ
n1 ;n2 n1 ;n2 6¼0
Qs ðtÞ ¼
X
Qð0; n2 Þein2 x2 t :
ð65Þ
n2
Therefore a double derivative of Qq ðtÞ is of the same order that the function Qq ðtÞ: X % & € q ¼ x2 Qð1; 0Þeix1 t þ Q ð1; 0Þeix1 t ðn1 x1 þ n2 x2 Þ2 Qðn1 ; n2 Þeiðn1 x1 þn2 x2 Þt ; Q 1 n1 ;n2 n1 ;n2 6¼0
€ q Qq , and for the slow part we obtain i.e. Q € s r2 Qs ¼ Oðr2 Þ: Q Substituting (63) into Eq. (19) one can find rffiffiffi rffiffiffi rffiffiffi € s þ r2 Qs Þ ¼ r pz þ Oðr2 Þ ) Q € q þ r2 Qq ¼ r pz ðQ € q ¼ r pz þ Oðr2 Þ: Q 2 2 2 Because of the integral (23), z is of the order of 1 (or less) that results in that the high frequency part of the solution is of the order of the small parameter r, i.e. Qq r. Thus, the quickly changing part of the solution of Eq. (42) should have the following form: X r aðn1 ; n2 Þeiðn1 x1 þn2 x2 Þt : ð66Þ Qq ðtÞ ¼ ðaeix1 t þ a eix1 t Þ þ r2 2 n1 ;n2 n1 ;n2 6¼0
In particular the amplitude a can be zero. In the second term in Eq. (66) we put small parameter r2 because these terms appear in the solution due to the small nonlinear part of Eq. (42).
L.S. Kuzmenkov et al. / Chaos, Solitons and Fractals 15 (2003) 597–610
605
There is another question we have to clarify before solving Eq. (42). One can observe from Eq. (42) that the third term in the left hand side of that equation has the same order (with respect to the small parameter r) that the nonlinearity in the right hand side. Therefore we have to investigate what contribution brings the right hand side of Eq. (42) in the low-frequency part of solution. To do this let us substitute (64) into Eq. (42) and average the equation over the small period T1 ¼ 2p=x1 . The low-frequency part of solution will not change dramatically and we obtain: {
€ qi þ Q € s þ r2 ð1 þ 2HpÞQs ¼ r2 pð2hQ2 Q € i hQQ_ 2 iÞ þ Oðr4 Þ; hQ q i þ hQ
ð67Þ
where €q þ Q € s Þi ¼ OðrÞ € i ¼ hðQ2 þ 2Qq Qs þ Q2 ÞðQ hQ2 Q q s € s r2 , Qq r and Q € q r, and analogously because Q hQQ_ 2 i ¼ hðQq þ Qs ÞðQ_ 2q þ 2Q_ q Q_ s þ Q_ 2s Þi ¼ OðrÞ: Taking into account the expansion (47), from Eq. (66) we find X { € q i ¼ x2 r2 2Hp r ðaeix1 t þ a eix1 t Þ þ r2 x2 hQ q þ Q 2pn2 ðn1 x1 þ n2 x2 Þ½ðn1 x1 þ n2 x2 Þ2 1aðn1 ; n2 Þeiðn1 x1 þn2 x2 Þt 1 2 x1 n1 ;n2 n1 6¼0
þ Oðr4 Þ ¼ Oðr3 Þ: Then the equation for the slow part takes the form € s þ r2 ð1 þ 2HpÞQs ¼ Oðr3 Þ; Q and therefore the frequency x2 has the following form pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 ¼ r 1 þ 2Hp þ oðrÞ: Thus, we search the solution of Eq. (42) in the form (62). This form is the expansion of the solution of Eq. (42) over nonorthogonal vectors eiðn1 x1 þn2 x2 Þt . Equating the coefficients near the exponents with the same frequencies ðn1 x1 þ n2 x2 Þ one can find: ½ðn1 x1 þ n2 x2 Þ4 ð1 þ r2 Þðn1 x1 þ n2 x2 Þ2 þ r2 ð1 þ 2HpÞQðn1 ; n2 Þ X X dðn1 m1 l1 k1 Þdðn2 m2 l2 k2 Þ½2ðk1 x1 þ k2 x2 Þ2 ðl1 x1 þ l2 x2 Þðk1 x1 þ k2 x2 Þ r2 ¼ r2 p m1 ;l1 ;k1 m2 ;l2 ;k2
Qðm1 ; m2 ÞQðl1 ; l2 ÞQðk1 ; k2 Þ;
ð68Þ
where the coefficients Qðn1 ; n2 Þ and the frequencies x1;2 we will search in the form of the expansion with respect to the small parameter r: Qðn1 ; n2 Þ ¼
1 X
Qðf Þ ðn1 ; n2 Þrf ;
f ¼0
x1;2 ¼
1 X
ðf Þ
x1;2 rf ;
ð69Þ
f ¼0 ð0Þ x2
¼ 0;
ð0Þ
x1 6¼ 0:
Substituting (69) into (68) and equating all the terms with the same powers of the small parameter r, after some simple but labor consuming transformations one can find some infinite system of algebraic equations at the point ðn1 ; n2 Þ. ð0Þ Because x2 ¼ 0, the zero order perturbation equation takes the form: ð0Þ
ð0Þ
ðn1 x1 Þ2 ½ðn1 x1 Þ2 1Qð0Þ ðn1 ; n2 Þ ¼ 0;
ð70Þ
wherefrom we immediately obtain: 1 Qð0Þ ðn1 ; n2 Þ ¼ Að0Þ ðn2 Þdðn1 Þ þ ðað0Þ ðn2 Þdðn1 1Þ þ að0Þ ðn2 Þdðn1 þ 1ÞÞ: 2
ð71Þ
606
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However according to (64) the solution with n1 ¼ 1 appears only in the third order with respect to the parameter r. Therefore Qð0Þ ðn1 ; n2 Þ ¼ Að0Þ ðn2 Þdðn1 Þ;
aðn2 Þ 0:
ð72Þ
From the first order perturbations one can deduce: ð0Þ
ð0Þ
ðn1 x1 Þ2 ½ðn1 x1 Þ2 1Qð1Þ ðn1 ; n2 Þ ¼ 0:
ð73Þ
The solution of this equation has the form 1 Qð1Þ ðn1 ; n2 Þ ¼ ðað1Þ ðn2 Þdðn1 1Þ þ að1Þ ðn2 Þdðn1 þ 1ÞÞ; 2
ð74Þ
ð0Þ
x1 ¼ 1;
ð75Þ
where we took into account that the solution with n2 ¼ 1 has to appear in the second iteration in Eq. (42). Substituting the solutions (72) and (74) into the second order iteration in Eq. (68) we obtain: ð1Þ
ð1Þ
ð1Þ
ð1Þ
n21 ðn21 1ÞQð2Þ ðn1 ; n2 Þ þ að1Þ ðn2 Þðx1 þ n2 x2 Þdðn1 1Þ þ að1Þ ðn2 Þðx1 þ n2 x2 Þdðn1 þ 1Þ þ ½1 þ 2Hp
ð1Þ ðn2 x2 Þ2 Að0Þ ðn2 Þdðn1 Þ
¼ 0:
ð76Þ
For to avoid the resonance at n1 ¼ 0 and n1 ¼ 1 we have to equate to zero the following two secular terms: ð1Þ
½1 þ 2Hp ðn2 x2 Þ2 Að0Þ ðn2 Þ ¼ 0; ð1Þ
ð1Þ
að1Þ ðn2 Þðx1 þ n2 x2 Þ ¼ 0: We will not consider trivial cases. Therefore the solution of Eq. (77) has the form: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1Þ x2 ¼ 1 þ 2Hp; 1 Að0Þ ðn2 Þ ¼ ðAdðn2 1Þ þ A dðn2 þ 1ÞÞ: 2
ð77Þ ð78Þ
ð79Þ ð80Þ
Eq. (78) brings several solutions: ð1Þ
x1 ¼ 0 if að1Þ ðn2 Þ ¼ adðn2 Þ;
ð81Þ
ð1Þ x1
ð82Þ
¼
ð1Þ x2
if að1Þ ðn2 Þ ¼ adðn2 1Þ:
Both frequencies are contained in the solution of Eq. (42) in the following linear form ð1Þ
ð1Þ
n1 x1 þ n2 x2 ;
ð83Þ
where we have to put n1 ¼ 1 because the results (81) and (82) have been obtained just for the case n1 ¼ 1. Substituting ð1Þ ð1Þ ð1Þ from (81) x1 ¼ 0 and n2 ¼ 0 into the linear form (83) we obtain n1 x1 þ n2 x2 ¼ 0. The same result we obtain ð1Þ ð1Þ substituting into (83) x1 ¼ x2 and n2 ¼ 1, i.e. the results from (82). Therefore both cases (81) and (82) are identical. Let us use the result (81). So we can rewrite the solutions (72) and (74) in the following form: 1 Qð0Þ ðn1 ; n2 Þ ¼ dðn1 ÞðAdðn1 1Þ þ A dðn1 þ 1ÞÞ; 2
ð84Þ
1 Qð1Þ ðn1 ; n2 Þ ¼ dðn2 Þðadðn1 1Þ þ a dðn1 þ 1ÞÞ: 2
ð85Þ
After setting to zero all the secular members in Eq. (76) we obtain: n21 ðn21 1ÞQð2Þ ðn1 ; n2 Þ ¼ 0:
ð86Þ
A general solution of Eq. (86) is performed by the following equation 1 Qð2Þ ðn1 ; n2 Þ ¼ Að2Þ ðn2 Þdðn1 Þ þ ðað2Þ ðn2 Þdðn1 1Þ þ að2Þ ðn2 Þdðn1 þ 1ÞÞ: 2
ð87Þ
The third iteration in Eq. (68), in terms of the results (75), (79), (81), (84), (85) and (87) is represented by the following equation:
L.S. Kuzmenkov et al. / Chaos, Solitons and Fractals 15 (2003) 597–610
607
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2Þ n21 ðn21 1ÞQð3Þ ðn1 ; n2 Þ þ n2 1 þ 2Hpðað2Þ ðn2 Þdðn1 1Þ að2Þ ðn2 Þdðn1 þ 1ÞÞ þ dðn2 Þðx1 þ HpÞðadðn1 1Þ p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ð2Þ þ a dðn1 þ 1ÞÞ x2 1 þ 2Hpdðn1 ÞðAdðn2 1Þ þ A dðn2 þ 1ÞÞ ¼ fA2 adðn1 1Þdðn2 2Þ 4 þ ðA2 aÞ dðn1 þ 1Þdðn2 þ 2Þ þ A2 a dðn1 þ 1Þdðn2 2Þ þ A2 adðn1 1Þdðn2 þ 2Þg:
ð88Þ
Equating to zero the secular term with dðn1 Þ in Eq. (88) we immediately find ð2Þ
x2 ¼ 0:
ð89Þ
Other secularities in Eq. (88) are presented by the terms with dðn1 1Þ. Setting to zero all these terms we obtain the equation pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ð2Þ ð90Þ n2 1 þ 2Hpað2Þ ðn2 Þ ¼ aðx1 þ HpÞdðn2 Þ þ aðA2 dðn2 2Þ þ A2 dðn2 þ 2ÞÞ: 4 where we have to set ð2Þ
x1 ¼ Hp;
ð91Þ
for to avoid another secularity which appears in Eq. (90) at n2 ¼ 0. Taking into considering the result (91) we find the general solution of Eq. (90) pa ð92Þ að2Þ ðn2 Þ ¼ lð2Þ dðn2 Þ þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi fA2 dðn2 2Þ A2 dðn2 þ 2Þg: 8 1 þ 2Hp In the solution of Eq. (42) the term with lð2Þ will be included together with the amplitude a by the following way: 1 fðra þ r2 lð2Þ Þeix1 t þ ðra þ r2 lð2Þ Þ eix1 t g: 2 It leads only to redefining of the amplitude a from a to a þ rlð2Þ . Therefore without sacrifice of generality we can set lð2Þ ¼ 0, and substituting (92) into (87) we obtain: p Qð2Þ ðn1 ; n2 Þ ¼ Að2Þ ðn2 Þdðn1 Þ þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi faA2 dðn1 1Þdðn2 2Þ þ ðaA2 Þ dðn1 þ 1Þdðn2 þ 2Þ 16 1 þ 2Hp aA2 dðn1 1Þdðn2 þ 2Þ a A2 dðn1 þ 1Þdðn2 2Þg: ð93Þ Thus, Eq. (88) after setting to zero all the secularities takes the following simple form: n21 ðn21 1ÞQð3Þ ðn1 ; n2 Þ ¼ 0 with general solution of the type 1 Qð3Þ ðn1 ; n2 Þ ¼ Að3Þ ðn2 Þdðn1 Þ þ fað3Þ ðn2 Þdðn1 1Þ þ að3Þ ðn2 Þdðn1 þ 1Þg: 2
ð94Þ
The fourth order iteration in Eq. (68), in terms of the results obtained above, takes the form: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n21 ðn21 1ÞQð4Þ þ 1 þ 2Hpn2 fað3Þ ðn2 Þdðn1 1Þ að3Þ ðn2 Þdðn1 þ 1Þg þ ð1 þ 2HpÞð1 n22 ÞAð2Þ ðn2 Þdðn1 Þ 5p pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 2HpfðaA2 dðn1 1Þdðn2 2Þ þ ðaA2 Þ dðn1 þ 1Þdðn2 þ 2ÞÞ ðaA2 dðn1 1Þdðn2 þ 2Þ þ 4 ð3Þ þ a A2 dðn1 þ 1Þdðn2 2ÞÞg þ x1 dðn2 Þðadðn1 1Þ þ a dðn1 þ 1ÞÞ " # ð3Þ x2 þ ð1 þ 2HpÞdðn1 Þ Hp pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðAdðn2 1Þ þ A dðn2 þ 1ÞÞ 1 þ 2Hp ¼
p2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi fðaA4 dðn1 1Þdðn2 4Þ þ ðaA4 Þ dðn1 þ 1Þdðn2 þ 4ÞÞ þ 2jAj2 ðaA2 dðn1 1Þdðn2 2Þ 32 1 þ 2Hp þ ðaA2 Þ dðn1 þ 1Þdðn2 þ 2ÞÞ ðaA4 dðn1 1Þdðn2 þ 4Þ þ a A4 dðn1 þ 1Þdðn2 4ÞÞ 2jAj2 ðaA2 dðn1 1Þdðn2 þ 2Þ þ a A2 dðn1 þ 1Þdðn2 2ÞÞg p pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 2HpfðaA2 dðn1 1Þdðn2 2Þ þ ðaA2 Þ dðn1 þ 1Þdðn2 þ 2ÞÞ ðaA2 dðn1 1Þdðn2 þ 2Þ þ 4 p þ a A2 dðn1 þ 1Þdðn2 2ÞÞg þ ð1 þ 2HpÞdðn1 ÞfðA3 dðn2 3Þ þ A3 dðn2 þ 3ÞÞ 8 þ 7jAj2 ðAdðn2 1Þ þ A dðn2 þ 1ÞÞg:
ð95Þ
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In this equation we must equate to zero the secular term which appears at n1 ¼ 0: " # ð3Þ x2 2 ð2Þ ð1 n2 ÞA ðn2 Þ þ Hp pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðAdðn2 1Þ þ A dðn2 þ 1ÞÞ 1 þ 2Hp p ð96Þ ¼ fðA3 dðn2 3Þ þ A3 dðn2 þ 3ÞÞ þ 7jAj2 ðAdðn2 1Þ þ A dðn2 þ 1ÞÞg: 8 Another secular members appear in this equation at n2 ¼ 1. So we have to equate the terms with dðn2 1Þ (and dðn2 þ 1Þ respectively). As a results we obtain the following equation: " # ð3Þ x2 7 Hp pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A ¼ pjAj2 A: 8 1 þ 2Hp Because A 6¼ 0 we immediately find that pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 ð3Þ x2 ¼ 1 þ 2Hp Hp jAj2 : 8
ð97Þ
Taking into considering (97) we obtain from Eq. (96): p Að2Þ ðn2 Þ ¼ ðA3 dðn2 3Þ þ A3 dðn2 þ 3ÞÞ: 64
ð98Þ
There are another secular terms in Eq. (95) at n1 ¼ 1 that result in the equation pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð3Þ 5p pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ð3Þ 1 þ 2HpðaA dðn2 2Þ aA2 dðn2 þ 2ÞÞ þ x1 adðn1 1Þ 1 þ 2Hpn2 a þ 4 p2 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi faA4 dðn2 4Þ þ 2jAj2 aA2 dðn2 2Þ aA4 dðn2 þ 4Þ 2jAj2 aA2 dðn2 þ 2Þg 32 1 þ 2Hp p pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 1 þ 2HpfaA2 dðn2 2Þ aA2 dðn2 þ 2Þg: 4
ð99Þ
From Eq. (99) we find ð3Þ
x1 ¼ 0;
ð100Þ "
að3Þ ðn2 Þ ¼
#
pa pjAj2 p2 a pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðA4 dðn2 4Þ þ A4 dðn2 þ 4ÞÞ: 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðA2 dðn2 2Þ þ A2 dðn2 þ 2ÞÞ þ 2 16 1 þ 2Hp 128 1 þ 2Hp ð101Þ
Summarising all the results we can write p p Qð2Þ ðn1 ; n2 Þ ¼ dðn1 ÞðA3 dðn2 3Þ þ A3 dðn2 þ 3ÞÞ þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi fðaA2 dðn1 1Þdðn2 2Þ 64 16 1 þ 2Hp þ ðaA2 Þ dðn1 þ 1Þdðn2 þ 2ÞÞ ðaA2 dðn1 1Þdðn2 þ 2Þ þ a A2 dðn1 þ 1Þdðn2 2ÞÞg; ( " # pa pjAj2 ð3Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ðA2 dðn2 2Þ þ A2 dðn2 þ 2ÞÞ Q ðn1 ; n2 Þ ¼ ðadðn1 1Þ þ a dðn1 þ 1ÞÞ 4 16 1 þ 2Hp ) p2 a 4 4 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðA dðn2 4Þ þ A dðn2 þ 4ÞÞ : þ 256 1 þ 2Hp
ð102Þ
ð103Þ
Thus, the time dependent solution of Eq. (42) has the following form: 1 1 p QðtÞ ¼ ðAeix2 t þ A eix2 t Þ þ r ðaeix1 t þ a eix1 t Þ r2 ðA3 e3ix2 t þ A3 e3ix2 t Þ 2" 2 64 # ! p p jAj2 2 3p pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ r pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r ðaA2 eiðx1 þ2x2 Þt þ ðaA2 Þ eiðx1 þ2x2 Þt Þ þ r 4 1 þ 2Hp 4 16 1 þ 2Hp # ! " p p jAj2 2 3p pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 r pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ r ðaA2 eiðx1 2x2 Þt þ a A2 eiðx1 2x2 Þt Þ r 4 1 þ 2Hp 4 16 1 þ 2Hp þ r3
p2 ðaA4 eiðx1 þ4x2 Þt þ ðaA4 Þ eiðx1 þ4x2 Þt þ aA4 eiðx1 4x2 Þt þ a A4 eiðx1 4x2 Þt Þ þ Oðr4 Þ; 256ð1 þ 2HpÞ
ð104Þ
L.S. Kuzmenkov et al. / Chaos, Solitons and Fractals 15 (2003) 597–610
609
where for the frequencies x1 and x2 we have obtained the following expansions x1 ¼ 1 r2 Hp þ Oðr4 Þ;
ð105Þ
x2 ¼ r
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 1 þ 2Hp þ r3 1 þ 2Hp Hp pjAj2 8
þ Oðr5 Þ:
ð106Þ
From Eqs. (25) and (104) we can find zðtÞ in the form of expansion with respect to the coupling parameter. Naturally, in the expression for z we have to take into account only the terms of the order up to r2 . Finally we obtain: pffiffiffiffiffi 1 zðtÞ ¼ 2pHrðAeix2 t þ A eix2 t Þ þ pffiffiffiffiffi ½1 r2 ð1 þ 2HpÞðaeix1 t þ a eix1 t Þ 2p ! pffiffiffiffiffi 2p p jAj2 þ r pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ r pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðaA2 eiðx1 þ2x2 Þt þ ðaA2 Þ eiðx1 þ2x2 Þt Þ 4 1 þ 2Hp 16 1 þ 2Hp ! rffiffiffi# pffiffiffiffiffi 2p p jAj2 p 2 ðaA2 eiðx1 2x2 Þt þ a A2 eiðx1 2x2 Þt Þ r pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 r pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ r 4 1 þ 2Hp 2 16 1 þ 2Hp "
þ r2
pffiffiffiffiffi p 2p ðaA4 eiðx1 þ4x2 Þt þ ðaA4 Þ eiðx1 þ4x2 Þt þ aA4 eiðx1 4x2 Þt þ a A4 eiðx1 4x2 Þt Þ þ Oðr3 Þ: 256ð1 þ 2HpÞ
ð107Þ
It seems clear that for the dynamics of QðtÞ the frequency x2 is a basic and the contribution of the frequency x1 appears only as a correction with a correction factor r. On the contrary, for zðtÞ the frequency x1 is a basic and the frequency x2 appears in the correction of the order r. The expression (104)–(106) is an asymptotic solution of Eq. (42) for the case 1 þ 2Hp > 0. In the case 1 þ 2Hp < 0 we have to solve Eq. (56) that have the same form that Eq. (42). Therefore the method used to solve Eq. (42) for the case 1 þ 2Hp > 0 can be analogously applied to the case 1 þ 2HP < 0. The behaviour of the quantities xðtÞ and yðtÞ we can obtain from Eqs. (18) and (24). Substituting the expressions for xðtÞ, yðtÞ and the formula (107) to Eq. (22) we have to obtain the connection of the parameters A, a, H and p with the integral of motion x2 þ y 2 þ z2 . The solution (104) has been found with the assumption that QðtÞ has a classical nature. The method we have applied is a Krylov-Bogolubov method extended for the case of two frequencies x1 and x2 , where one of the frequencies is far above the other frequency. The solution we have obtained can be substituted into the functional expansion (38) of the action, taking into account the transformation (34). The expansion obtained this way can be used to evaluate the evolution operator such it is performed in Eqs. (40) and (41). The behaviour of the quantities QðtÞ and zðtÞ demonstrates good accordance with the numerical results performed by Schanz and Esser [4].
Acknowledgements The authors wish to thank Dr. B. Esser for his valuable comments of this paper. S.G.M. wish to acknowledge the financial support provided by the National Council of Science and Technology of Mexico (CONACYT).
References [1] Heller EJ. In: Giannoni MJ, Voros A, Zinn-Justin J, editors. Chaos and quantum physics. Les Houches summer school session I, vol. II. Amsterdam: North-Holland; 1991. [2] Bohigas O, Tomsovic S, Ullmo D. Manifestations of classical phase space structures in quantum mechanics. Phys Rep 1993;223:43. [3] Born M, Oppenheimer R. Zur quantentheorie der molekeln. Ann Phys (Leipzig) 1927;84:457. [4] Schanz H, Esser B. Mixed quantum–classical versus full quantum dynamics: Coupled quasiparticle-oscillator system. Phys Rev A 1997;55(5):3375–87. [5] Koch Ch, Esser B. Spectrum, lifetime distributions and relaxation in a dimer with strong excitonic–vibronic coupling. J Lumin 1999;81:171–81. [6] Koch Ch, Esser B. Spin-boson Hamiltonian and optical absorption of molecular dimer. Phys Rev A 1999;61:0225XX.
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[7] Hennig D, Esser B. Transfer dynamics of a quasiparticle in a nonlinear dimer coupled to an intersite vibration: Chaos on Bloch sphere. Phys Rev A 1992;46(8):4569–76. [8] Sonnek M, Eiermann H, Wagner M. Squeezed excited states in exciton–phonon systems. Phys Rev B 1995;51:905.
On the asymptotic solutions of the coupled quasiparticle-oscillator system L.S. Kuzmenkov b
a,*
, S.G. Maximov b, J.L. Guardado Zavala
b
a Moscow State University, Vorobjovy Gory, 119899 Moscow, Russia Instituto Tecnol ogico de Morelia, P.O. Box 262, Morelia, Mich., Mexico
Accepted 23 May 2002
Abstract A coupled quasiparticle-oscillator system is considered for an arbitrary number of excitons. The exciton dynamics is described in terms of the second quantization (i.e. by means the bosonic operators). As a consequence a radius of a Bloch sphere is obtained different to the previous results. Some integrals of motion are obtained that allowed to reduce the system of equations of motion to a single nonlinear ordinary differential equation of the fourth order. This equation contains the energy of the system as a parameter. The fixed points are found as a functions of the energy of the system, and its stability properties are investigated. It is demonstrated that a bifurcation is presented for the energies H < 1=2p. An asymptotic quasiclassical solution around fixed point for the case H > 1=2p is obtained. The solutions around other stable fixed points can be obtained analogously. The expression for the evolution operator of the quasiparticle-oscillator system is obtained as a functional on the classical solutions. Ó 2002 Published by Elsevier Science Ltd.
1. Introduction The correspondence between classical nonlinear systems on one side and their fully quantized counterparts on the other has been intensively investigated in the last decade (see [1,2,4–7]). One step quantization of a given system is sometimes complicated. As a rule such systems are divided into interacting subsystems. Then a mixed description is applied, in which one of the subsystems is considered classical and the other is treated in a quantum context. This quantization is the basic idea on which the Born–Oppenheimer approximation [3], used in many situations, is based. For nonadiabatic coupled systems this approximation must be complemented into a rigorous quantum description. The nonlinearities generated in this case can give rise to nonintegrability and chaos [4], and the problem of the quantumclassical correspondence arises for the dynamical properties of the mixed and fully quantized schemes. Schanz and Esser [3] stressed that chaos in the mixed description is produced by coupling one-dimensional integrable adiabatic reference states. In this paper the relation between the dynamical properties of the mixed and fully quantized descriptions for the model of a quasiparticle moving between two sites and coupled to an oscillator is considered. As previously reported [4] we also consider a coupled quasiparticle-oscillator system. In our model N excitons moves between two sites of the dimer composed by two molecular monomers. The excitons are coupled to intramolecular vibrations. The interest in the dynamical properties of the quasiparticle-oscillator system is based on the wide range of applications that includes a variety of phenomena in molecular and solid state physics, e.g. excitons in molecular
*
Corresponding author. E-mail addresses: [email protected] (L.S. Kuzmenkov), [email protected] (S.G. Maximov).
0960-0779/03/$ - see front matter Ó 2002 Published by Elsevier Science Ltd. PII: S 0 9 6 0 - 0 7 7 9 ( 0 2 ) 0 0 1 1 9 - 4
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aggregates and coupled to vibrations [8]. The dynamic properties of this model in the mixed description have been previously investigated [4,6,7]. Analysing the dynamical properties of such systems the authors have focused on the coupling parameter ranges. It has been demonstrated the presence of a separatrix structure underlying the phase space for overcritical coupling, and of chaos developing from the hyperbolic point at the center of this structure [7]. The problem of relation between the dynamics in the mixed and fully quantum levels of description of the coupled quasiparticle-oscillator motion were considered by Schanz and Esser [3]. They focused on the adiabatic parameter range to justify best the mixed description. The aim of the present paper is to find some asymptotic quasiclassical solutions around the fixed elliptic points for the case of weak coupling, for the system of N excitons coupled to the molecular dimer. In Section 2 we specify the model in detail considering an arbitrary number of quasiparticles. In Section 3 using integrals of motion the system of equations of motion is reduced to the single ordinary differential equation that contains the energy of the system as a parameter. Further investigation is based completely in this equation. In Section 4 a problem of evolution operator for the oscillator-excitons system is considered and the quasiclassical approximation is justified. In Section 5 fixed points of our system are investigated focusing on the energy as a parameter. In the last section the solving scheme for the case H > 1=2p is justified and the solution for the quasiparticle-oscillator system is given in form of expansion with respect to the small adiabatic parameter r. 2. Model Let us consider a system of an arbitrary number of molecules with the Hamiltonian X1 X X b ¼ H ðp^n2 þ x2n q^2n Þ þ n aþ Vnm aþ n an þ n am ; 2 n n n;m
ð1Þ
þ ¼ Vmn , Vnn ¼ 0, and the nth state energy n is a function of the coordinate qn . aþ where Vmn n and an are excitonic bosonic operators:
½an ; aþ m ¼ dn;m ;
½an ; am ¼ 0:
ð2Þ
It is obvious that the exciton number X b ¼ N aþ n an n
is an integral of motion, i.e. b;N b ¼ 0: ½H Using the first-order expansion in qn of the exciton energy n ðqn Þ: n ðqn Þ ¼ ð0Þ n þ cn qn þ we reduce the Hamiltonian to the following form: b ¼ H
X X X X1 þ ð0Þ cn qn aþ Vnm aþ ðp^n2 þ x2n q^2n Þ þ n an an þ n an þ n am : 2 n n n n;m
ð3Þ
Let us consider the simplest case of a symmetric two site system, e.g. an exciton in a molecular dimer constituted by two ð0Þ ð0Þ identical monomers. We set 1 ¼ 2 , x1 ¼ x2 x, c1 ¼ c2 c and V12 ¼ V21 ¼ V , V > 0. Then by introducing for the vibronic subsystem the coordinates and momenta q ¼
q2 q1 pffiffiffi ; 2
p ¼
p2 p1 pffiffiffi ; 2
ð4Þ
we obtain two independent Hamiltonians: b ¼H bþ þ H b ; H where b þ ¼ 1 ðp^2 þ x2 q^2 Þ þ þ pcffiffiffi q^þ N b; H þ 2 þ 2
ð5Þ
L.S. Kuzmenkov et al. / Chaos, Solitons and Fractals 15 (2003) 597–610 þ þ þ b ¼ 1 ðp^2 þ x2 q^2 Þ þ pcffiffiffi q^ ðaþ H 2 a2 a1 a1 Þ V ða1 a2 þ a2 a1 Þ: 2 2
599
ð6Þ
b þ is not coupled to the exciton, and will be omitted. It is an oscillator Hamiltonian with the The Hamiltonian H displaced fixed point: 0 " #2 1 2 b2 b N c 1 c N A: b þ N bþ h^þ H ¼ @p^þ2 þ x2 q^þ þ pffiffiffi 2 4x2 2x2 b in units of 2 V and replacing q^ and p^ by Passing in Eq. (6) to dimensionless variables by measuring H pffiffiffiffiffiffi b ¼ 2V q ; Pb ¼ p1ffiffiffiffiffiffi p ; Q 2V
ð7Þ
one finally obtains 1=2 b ðaþ a2 aþ a1 Þ: b 2 Þ 1 ðaþ a2 þ aþ a1 Þ þ p b ¼ 1 ð Pb 2 þ r2 Q rQ H 1 2 2 1 2 2 2
ð8Þ
It is obvious that (h 1) b ; Pb ¼ i; ½Q
b ¼ ½an ; Pb ¼ 0: ½an ; Q
ð9Þ
The Heisenberg quantum evolution equations b ; a1 ; ia_ 1 ¼ ½ H
b ; a2 ; ia_ 2 ¼ ½ H
i
d b b ; Pb ; P ¼ ½H dt
i
d b b b;Q Q ¼ ½H dt
and the commutational relationships (2) and (9) result in the following equations: rffiffiffi 1 p b r Q a1 ; ia_ 1 ¼ a2 þ 2 2 rffiffiffi 1 p b r Q a2 ; ia_ 2 ¼ a1 2 2 rffiffiffi d b b prðaþ a2 aþ a1 Þ; P ¼ r2 Q 1 dt 2 2 d b Q ¼ Pb : dt
ð10Þ
ð11Þ ð12Þ ð13Þ ð14Þ
Substituting new operators þ x^ ¼ aþ 2 a1 þ a1 a2 ; þ y^ ¼ iða2 a1 aþ 1 a2 Þ;
ð15Þ
þ ^z ¼ aþ 2 a2 a1 a1
in Eqs. (11)–(14) we obtain the following system of equations: pffiffiffiffiffi d b y^; x^ ¼ 2pr Q dt pffiffiffiffiffi d b x^; y^ ¼ ^z 2pr Q dt d ^z ¼ y^; dt
ð16Þ ð17Þ ð18Þ
d2 b b ¼ r Q þ r2 Q dt2
rffiffiffi p ^z 2
ð19Þ
with new commutational relationships analogous to the algebra of the Pauli matrixes: ½^ x; y^ ¼ 2i^z;
½^ y ; ^z ¼ 2i^ x;
½^z; x^ ¼ 2i^ y:
ð20Þ
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L.S. Kuzmenkov et al. / Chaos, Solitons and Fractals 15 (2003) 597–610
A quantum evolution of any system we can consider in both Heisenberg or Schr€ odinger patterns. Let us work in the frame of first of them, i.e. the wave function jWi we’ll consider constant.
3. Integrals of motion There are two integrals of motion we will use to solve the system (16)–(19). One of them is offered to us by the Hamiltonian which with the transformations (15) obtains the following form: rffiffiffi 1 b_ 2 1 p b 2 b2 b H ¼ ð Q þ r Q Þ x^ þ r Q^z: ð21Þ 2 2 2 Other integral can be obtained directly from Eqs. (16)–(18). We deduce that x^2 þ y^2 þ ^z2 ¼ const: Using the definition of the operators x^, y^ and ^z and the commutational relationships (2), it is easy to find b þN b 2; x^2 þ y^2 þ ^z2 ¼ 2 N
ð22Þ
b is the operator of number of excitons: where N b ¼ aþ a2 þ aþ a1 : N 2 1 Let us consider the system (16)–(19). From the integral (21) we immediately obtain pffiffiffiffiffi b_ 2 þ r2 Q b 2H b 2 Þ þ 2pr^z Q b; x^ ¼ ð Q and from Eq. (19) sffiffiffi 1 2 b € b Þ: ð Q þ r2 Q ^z ¼ r p Substituting (24) into (17) and taking into account Eq. (18) we arrive at the following equation pffiffiffiffiffi _ 2 pffiffiffiffiffi €^z þ ^z ¼ 2prf Q b Q b 2 2H b g: b þ r2 Q b 3 þ 2pr^z Q bQ
ð23Þ
ð24Þ
ð25Þ
ð26Þ
Eqs. (25) and (26) can be reduced to the following single nonlinear equation of the fourth order: € € b 2 b_ 2 b { b b ¼ r2 pð2 Q b b 3 Þ: b pÞ Q þ r2 ð1 þ 2 H Q Q Q þ r2 Q b þ ð1 þ r2 Þ Q Q
ð27Þ
The stationary properties of the system brings the stationary Schr€ odinger equation: b jCi ¼ EjCi; H where, for example, in the representation jn; mi ¼ jni jmi, where n is the number of excitons in one of the molecules and m is the oscillator state, the Hamiltonian matrix elements have the form: pffiffiffiffiffi pffiffiffiffi pffiffiffi rp b jn0 ; m0 i ¼ r m þ 1 dðm m0 Þdðn n0 Þ þ i hn; mj H ðN 2nÞð m0 dðm m0 þ 1Þ mdðm m0 1ÞÞdðn n0 Þ 2 2 ffi 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi0ffi ð28Þ ðN n0 Þndðn n0 1Þ ðN nÞn dðn n0 þ 1Þ: 2 2
4. Quasiclassical approximation Let us consider a problem of evolution operator for the oscillator-exciton system. The Hamilton in terms of canonical variables has the form: rffiffiffi 1 p b 2 b 2 Þ 1 ð^ b ¼ 1 ð Pb 2 þ r2 Q H p2 p^1 þ q^2 q^1 Þ þ r Q ðp^2 þ q^22 p^12 q^21 Þ: ð29Þ 2 2 2 2
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601
To pass from the Hamiltonian (8) to the form (29) we should make the following transformation 1 q1;2 Þ; aþ p1;2 þ i^ 1;2 ¼ pffiffiffi ð^ 2
1 a1;2 ¼ pffiffiffi ðp^1;2 þ i^ q1;2 Þ: 2
ð30Þ
The evolution operator matrix elements are found in the following form: ( ) 3N =2 NY 1 Z N 1 hN dQk i X N i3p N 2p b 4 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dq1k dq2k exp Lðqk ; q_ k ÞDtk hqj U ðt t0 Þjq0 i lim 2 e N !1 t t0 h k¼0 1 þ 2pr2 Q2k k¼0 Z t Z Y i ¼ N1 ðt t0 Þ D3 ½qðtÞ exp Lðq; q_ Þ ds h t0 t
ð31Þ
where we denoted dQðtÞ D3 ½qðtÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dq1 ðtÞ dq2 ðtaÞ; 1 þ 2pr2 Q2 ðtÞ
ð32Þ
and the classical Lagrangian has the form rffiffiffi pffiffiffiffiffi 2prQ 1 2q_ 1 q_ 2 1 1 p 2 2 _ _ q rQðq22 q21 Þ: Lðq; q_ Þ ¼ ðQ_ 2 r2 Q2 Þ þ ð q q Þ q þ 1 2 2 1 þ 2pr2 Q2 1 2 2 2 1 þ 2pr2 Q2 2
ð33Þ
The continual integral (31) has a nontrivial measure (32). Let us set pffiffiffiffiffi 1 Q ¼ pffiffiffiffiffi shð 2prq3 Þ; 2pr
ð34Þ
then dQ dq3 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 1 þ 2pr2 Q2 In terms of new variable the integral (31) obtains the following form Z t Z Y i b ðt t0 Þjq0 i ¼ N1 hqj U dq1 ðtÞ dq2 ðtÞ dq3 ðtÞ exp Lðqi ; q_ i Þ ds ; h t0 t
ð35Þ
where ði ¼ 1; 2; 3Þ Lðqi ; q_ i Þ ¼
pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi 1 2 1 sinhð 2prq3 Þ 2 2q_ 1 q_ 2 1 ðq_ 1 q_ 22 Þ q1 q2 q_ 3 cosh2 ð 2prq3 Þ sinh2 ð 2prq3 Þ þ pffiffiffiffiffi pffiffiffiffiffi 2 2p cosh2 ð 2prq3 Þ 2 cosh2 ð 2prq3 Þ pffiffiffiffiffi 1 þ sinhð 2prq3 Þðq22 q21 Þ: 4
ð36Þ
According to the Feynman’s theory the evolution operator (35) is a functional on the classical solutions. There is a classical action functional in the exponent in the formula (35), that can be expanded around classical solution into the series by the following way: Z t Att0 ¼ Lðq; q_ Þ ds; ð37Þ t0
Att0
¼
Att0 jqcl
1 þ 2
Z
Z
t
ds1 t0
t
ds2 t0
d2 Att0 dqa ðs1 Þdqb ðs2 Þ
! dqa ðs1 Þdqb ðs2 Þ þ ;
ð38Þ
qcl
where the classical solutions are performed by the Lagrange equations: t dAt0 ¼ 0: dqa qcl
ð39Þ
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L.S. Kuzmenkov et al. / Chaos, Solitons and Fractals 15 (2003) 597–610
Substituting the expansion (38) into the integral (31) we obtain 8 Z Z t Z
d2 Att0 a dq ðs1 Þdqb ðs2 Þ
! dqa ðs1 Þdqb ðs2 Þ þ qcl
9 = ;
: ð40Þ
Then we obtain an asymptotic formula for the evolution operator matrix elements in quasiclassical approximation: 0vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! , u u i i d2 A 1 t t @ b þ : ð41Þ A j 2p det hqj U ðt t0 Þjq0 i ¼ N exp h t0 qcl h dqa ðt1 Þdqb ðt2 Þ qcl The classical solutions of the oscillator-quasiparticles system can be substituted into the formula (41). We can also find other terms of the expansion (41) reaching a better accordance with the quantum dynamics of the system.
5. Fixed points and stability problem We consider a coupled oscillator-quasiparticles problem in quasiclassical approximation treating the Eq. (27) as a classical. So we have the following nonlinear equation: v € þ r2 ð1 þ 2HpÞQ ¼ r2 pð2Q2 Q € QQ_ 2 þ r2 Q3 Þ; Q þ ð1 þ r2 ÞQ
ð42Þ
where the adiabatic parameter r is small (r 1). Setting all the derivatives in Eq. (42) to zero we obtain the following stationary equation r2 ð1 þ 2HpÞQ ¼ r4 pQ3 : Fixed points are represented by three solutions of this equation Q ¼ 0; Q2 ¼
1 þ 2Hp : r2 p
ð43Þ ð44Þ
According to the magnitude of energy H the fixed points can be classified into two groups. 5.1. Case 1 þ 2Hp > 0 There exist three different cases to consider. First of them is the situation described by the following inequality 1 þ 2Hp > 0:
ð45Þ
For this case Eq. (44) has not real solutions, therefore there is only one fixed point Q ¼ 0. The stability property of this point can be obtained from a homogeneous equation. The homogeneous equation results in the following characteristical equation (we substitute eixt ): x4 ð1 þ r2 Þx2 þ r2 ð1 þ 2HpÞ ¼ 0:
ð46Þ
All the solutions of Eq. (46) are real x1 ¼ ð1 r2 HpÞ þ Oðr4 Þ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 ¼ r 1 þ 2Hp þ Oðr3 Þ:
ð47Þ
So the fixed point Q ¼ 0 is stable (stable elliptic). 5.2. Case 1 þ 2Hp < 0 Another case is when the energy magnitude satisfies the inequality 1 þ 2Hp < 0:
ð48Þ
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603
In this situation Eq. (42) can be rewritten in the following form {
€ r2 j1 þ 2HpjQ ¼ r2 pð2Q2 Q € QQ_ 2 þ r2 Q3 Þ: Q þ ð1 þ r2 ÞQ
ð49Þ
The bifurcation has occurred, and Eq. (49) has three fixed points Q ¼ 0; sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 j1 þ 2Hpj Q ¼
: r p
ð50Þ ð51Þ
The first of them (Q ¼ 0) leads to the following characteristical equation (we substitute eixt ) x4 ð1 þ r2 Þx2 r2 j1 þ 2Hpj ¼ 0
ð52Þ
which has the following solutions x21 ¼ 1 þ r2 ð1 þ j1 þ 2HpjÞ þ Oðr4 Þ; x22 ¼ r2 j1 þ 2Hpj þ Oðr4 Þ: From (53) we obtain r2 x1 1 þ ð1 þ j1 þ 2HpjÞ ; 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 ir j1 þ 2Hpj:
ð53Þ
ð54Þ
Because of the frequency x2 in Eq. (54) obtains imaginary values the fixed point Q ¼ 0 is unstable hyperbolic. To investigate the stability properties of the fixed points (51) we have to pass to displaced variables qðtÞ by the following transformation QðtÞ ¼ Q þ qðtÞ:
ð55Þ
Substituting this form into Eq. (49) and taking into account (51) we arrive at the following two equations: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi { q þ ð1 þ 2j1 þ 2Hpj þ r2 Þ€q þ 3r2 j1 þ 2Hpjq ¼ r pj1 þ 2Hpjf4q€ q qq_ 2 þ r2 q3 g: q q_ 2 þ 3r2 q2 g r2 pf2q2 € ð56Þ It is easy to find from Eq. (56) that the fixed points Q are stable. Equating to zero the nonlinear part of Eq. (56) we obtain a respective characteristical equation which has the following form x4 ð1 þ 2j1 þ 2Hpj þ r2 Þx2 þ 3r2 j1 þ 2Hpjq ¼ 0: All its solutions are real numbers 3 j1 þ 2Hpj x21 1 þ 2j1 þ 2Hpj þ r2 1 ; 2 1 þ 2j1 þ 2Hpj 3 j1 þ 2Hpj x22 ¼ r2 : 2 1 þ 2j1 þ 2Hpj
ð57Þ
ð58Þ
Therefore the fixed points Q of Eq. (49) (q ¼ 0 for Eq. (56) respectively) are stable elliptic. 5.3. Case 1 þ 2Hp ¼ 0 The case 1 þ 2Hp ¼ 0 leads to the following differential equation { € ¼ r2 pð2Q2 Q € QQ_ 2 þ r2 Q3 Þ: Q þ ð1 þ r2 ÞQ
The point Q ¼ 0 for this case is unstable hyperbolic.
ð59Þ
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L.S. Kuzmenkov et al. / Chaos, Solitons and Fractals 15 (2003) 597–610
6. Asymptotic solution for the case H > 1=2p Let us consider Eq. (42) for the energy magnitudes H >
1 : 2p
As we have seen for this case Eq. (42) has only one fixed point Q ¼ 0. We will search the solution of Eq. (42) around this fixed point. We have find that the homogeneous equation has two positive frequencies x1 and x2 (54). Moreover, one of the frequencies is far above the other frequency: x2 x1 :
ð60Þ
Therefore the solution of the homogeneous equation is a superposition of two slowly and quickly oscillating functions with the frequencies x1 and x2 : 1 1 Qh ðtÞ ¼ ðAeix2 t þ A eix2 t Þ þ ðaeix1 t þ a eix1 t Þ: 2 2 It is natural to search the solution of the nonlinear equation (42) in the following quasiperiodical form: X QðtÞ ¼ Qðn1 ; n2 Þeiðn1 x1 þn2 x2 Þt ; Qðn1 ; n2 Þ ¼ Q ðn1 ; n2 Þ;
ð61Þ
ð62Þ
n1 ;n2
where x2 r, and, in general, the frequencies x1 and x2 must have the form (54) of the expansion with respect to the small parameter r. The frequencies of the type n1 x1 þ n2 x2 appear in the solution due to the mixing of frequencies in the right hand side of Eq. (42). The terms of the expansion (54) and the coefficients Qðn1 ; n2 Þ in the formula (62) have to be found by setting to zero all the secularities that appear in the nonlinear part of Eq. (42). The solution (62) can be divided in two rapidly and slowly oscillating parts by the following way: QðtÞ ¼ Qq ðtÞ þ Qs ðtÞ;
ð63Þ
where Qq ðtÞ ¼ Qð1; 0Þeix1 t þ Q ð1; 0Þeix1 t þ
X
Qðn1 ; n2 Þeiðn1 x1 þn2 x2 Þt ;
ð64Þ
n1 ;n2 n1 ;n2 6¼0
Qs ðtÞ ¼
X
Qð0; n2 Þein2 x2 t :
ð65Þ
n2
Therefore a double derivative of Qq ðtÞ is of the same order that the function Qq ðtÞ: X % & € q ¼ x2 Qð1; 0Þeix1 t þ Q ð1; 0Þeix1 t ðn1 x1 þ n2 x2 Þ2 Qðn1 ; n2 Þeiðn1 x1 þn2 x2 Þt ; Q 1 n1 ;n2 n1 ;n2 6¼0
€ q Qq , and for the slow part we obtain i.e. Q € s r2 Qs ¼ Oðr2 Þ: Q Substituting (63) into Eq. (19) one can find rffiffiffi rffiffiffi rffiffiffi € s þ r2 Qs Þ ¼ r pz þ Oðr2 Þ ) Q € q þ r2 Qq ¼ r pz ðQ € q ¼ r pz þ Oðr2 Þ: Q 2 2 2 Because of the integral (23), z is of the order of 1 (or less) that results in that the high frequency part of the solution is of the order of the small parameter r, i.e. Qq r. Thus, the quickly changing part of the solution of Eq. (42) should have the following form: X r aðn1 ; n2 Þeiðn1 x1 þn2 x2 Þt : ð66Þ Qq ðtÞ ¼ ðaeix1 t þ a eix1 t Þ þ r2 2 n1 ;n2 n1 ;n2 6¼0
In particular the amplitude a can be zero. In the second term in Eq. (66) we put small parameter r2 because these terms appear in the solution due to the small nonlinear part of Eq. (42).
L.S. Kuzmenkov et al. / Chaos, Solitons and Fractals 15 (2003) 597–610
605
There is another question we have to clarify before solving Eq. (42). One can observe from Eq. (42) that the third term in the left hand side of that equation has the same order (with respect to the small parameter r) that the nonlinearity in the right hand side. Therefore we have to investigate what contribution brings the right hand side of Eq. (42) in the low-frequency part of solution. To do this let us substitute (64) into Eq. (42) and average the equation over the small period T1 ¼ 2p=x1 . The low-frequency part of solution will not change dramatically and we obtain: {
€ qi þ Q € s þ r2 ð1 þ 2HpÞQs ¼ r2 pð2hQ2 Q € i hQQ_ 2 iÞ þ Oðr4 Þ; hQ q i þ hQ
ð67Þ
where €q þ Q € s Þi ¼ OðrÞ € i ¼ hðQ2 þ 2Qq Qs þ Q2 ÞðQ hQ2 Q q s € s r2 , Qq r and Q € q r, and analogously because Q hQQ_ 2 i ¼ hðQq þ Qs ÞðQ_ 2q þ 2Q_ q Q_ s þ Q_ 2s Þi ¼ OðrÞ: Taking into account the expansion (47), from Eq. (66) we find X { € q i ¼ x2 r2 2Hp r ðaeix1 t þ a eix1 t Þ þ r2 x2 hQ q þ Q 2pn2 ðn1 x1 þ n2 x2 Þ½ðn1 x1 þ n2 x2 Þ2 1aðn1 ; n2 Þeiðn1 x1 þn2 x2 Þt 1 2 x1 n1 ;n2 n1 6¼0
þ Oðr4 Þ ¼ Oðr3 Þ: Then the equation for the slow part takes the form € s þ r2 ð1 þ 2HpÞQs ¼ Oðr3 Þ; Q and therefore the frequency x2 has the following form pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 ¼ r 1 þ 2Hp þ oðrÞ: Thus, we search the solution of Eq. (42) in the form (62). This form is the expansion of the solution of Eq. (42) over nonorthogonal vectors eiðn1 x1 þn2 x2 Þt . Equating the coefficients near the exponents with the same frequencies ðn1 x1 þ n2 x2 Þ one can find: ½ðn1 x1 þ n2 x2 Þ4 ð1 þ r2 Þðn1 x1 þ n2 x2 Þ2 þ r2 ð1 þ 2HpÞQðn1 ; n2 Þ X X dðn1 m1 l1 k1 Þdðn2 m2 l2 k2 Þ½2ðk1 x1 þ k2 x2 Þ2 ðl1 x1 þ l2 x2 Þðk1 x1 þ k2 x2 Þ r2 ¼ r2 p m1 ;l1 ;k1 m2 ;l2 ;k2
Qðm1 ; m2 ÞQðl1 ; l2 ÞQðk1 ; k2 Þ;
ð68Þ
where the coefficients Qðn1 ; n2 Þ and the frequencies x1;2 we will search in the form of the expansion with respect to the small parameter r: Qðn1 ; n2 Þ ¼
1 X
Qðf Þ ðn1 ; n2 Þrf ;
f ¼0
x1;2 ¼
1 X
ðf Þ
x1;2 rf ;
ð69Þ
f ¼0 ð0Þ x2
¼ 0;
ð0Þ
x1 6¼ 0:
Substituting (69) into (68) and equating all the terms with the same powers of the small parameter r, after some simple but labor consuming transformations one can find some infinite system of algebraic equations at the point ðn1 ; n2 Þ. ð0Þ Because x2 ¼ 0, the zero order perturbation equation takes the form: ð0Þ
ð0Þ
ðn1 x1 Þ2 ½ðn1 x1 Þ2 1Qð0Þ ðn1 ; n2 Þ ¼ 0;
ð70Þ
wherefrom we immediately obtain: 1 Qð0Þ ðn1 ; n2 Þ ¼ Að0Þ ðn2 Þdðn1 Þ þ ðað0Þ ðn2 Þdðn1 1Þ þ að0Þ ðn2 Þdðn1 þ 1ÞÞ: 2
ð71Þ
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However according to (64) the solution with n1 ¼ 1 appears only in the third order with respect to the parameter r. Therefore Qð0Þ ðn1 ; n2 Þ ¼ Að0Þ ðn2 Þdðn1 Þ;
aðn2 Þ 0:
ð72Þ
From the first order perturbations one can deduce: ð0Þ
ð0Þ
ðn1 x1 Þ2 ½ðn1 x1 Þ2 1Qð1Þ ðn1 ; n2 Þ ¼ 0:
ð73Þ
The solution of this equation has the form 1 Qð1Þ ðn1 ; n2 Þ ¼ ðað1Þ ðn2 Þdðn1 1Þ þ að1Þ ðn2 Þdðn1 þ 1ÞÞ; 2
ð74Þ
ð0Þ
x1 ¼ 1;
ð75Þ
where we took into account that the solution with n2 ¼ 1 has to appear in the second iteration in Eq. (42). Substituting the solutions (72) and (74) into the second order iteration in Eq. (68) we obtain: ð1Þ
ð1Þ
ð1Þ
ð1Þ
n21 ðn21 1ÞQð2Þ ðn1 ; n2 Þ þ að1Þ ðn2 Þðx1 þ n2 x2 Þdðn1 1Þ þ að1Þ ðn2 Þðx1 þ n2 x2 Þdðn1 þ 1Þ þ ½1 þ 2Hp
ð1Þ ðn2 x2 Þ2 Að0Þ ðn2 Þdðn1 Þ
¼ 0:
ð76Þ
For to avoid the resonance at n1 ¼ 0 and n1 ¼ 1 we have to equate to zero the following two secular terms: ð1Þ
½1 þ 2Hp ðn2 x2 Þ2 Að0Þ ðn2 Þ ¼ 0; ð1Þ
ð1Þ
að1Þ ðn2 Þðx1 þ n2 x2 Þ ¼ 0: We will not consider trivial cases. Therefore the solution of Eq. (77) has the form: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1Þ x2 ¼ 1 þ 2Hp; 1 Að0Þ ðn2 Þ ¼ ðAdðn2 1Þ þ A dðn2 þ 1ÞÞ: 2
ð77Þ ð78Þ
ð79Þ ð80Þ
Eq. (78) brings several solutions: ð1Þ
x1 ¼ 0 if að1Þ ðn2 Þ ¼ adðn2 Þ;
ð81Þ
ð1Þ x1
ð82Þ
¼
ð1Þ x2
if að1Þ ðn2 Þ ¼ adðn2 1Þ:
Both frequencies are contained in the solution of Eq. (42) in the following linear form ð1Þ
ð1Þ
n1 x1 þ n2 x2 ;
ð83Þ
where we have to put n1 ¼ 1 because the results (81) and (82) have been obtained just for the case n1 ¼ 1. Substituting ð1Þ ð1Þ ð1Þ from (81) x1 ¼ 0 and n2 ¼ 0 into the linear form (83) we obtain n1 x1 þ n2 x2 ¼ 0. The same result we obtain ð1Þ ð1Þ substituting into (83) x1 ¼ x2 and n2 ¼ 1, i.e. the results from (82). Therefore both cases (81) and (82) are identical. Let us use the result (81). So we can rewrite the solutions (72) and (74) in the following form: 1 Qð0Þ ðn1 ; n2 Þ ¼ dðn1 ÞðAdðn1 1Þ þ A dðn1 þ 1ÞÞ; 2
ð84Þ
1 Qð1Þ ðn1 ; n2 Þ ¼ dðn2 Þðadðn1 1Þ þ a dðn1 þ 1ÞÞ: 2
ð85Þ
After setting to zero all the secular members in Eq. (76) we obtain: n21 ðn21 1ÞQð2Þ ðn1 ; n2 Þ ¼ 0:
ð86Þ
A general solution of Eq. (86) is performed by the following equation 1 Qð2Þ ðn1 ; n2 Þ ¼ Að2Þ ðn2 Þdðn1 Þ þ ðað2Þ ðn2 Þdðn1 1Þ þ að2Þ ðn2 Þdðn1 þ 1ÞÞ: 2
ð87Þ
The third iteration in Eq. (68), in terms of the results (75), (79), (81), (84), (85) and (87) is represented by the following equation:
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pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2Þ n21 ðn21 1ÞQð3Þ ðn1 ; n2 Þ þ n2 1 þ 2Hpðað2Þ ðn2 Þdðn1 1Þ að2Þ ðn2 Þdðn1 þ 1ÞÞ þ dðn2 Þðx1 þ HpÞðadðn1 1Þ p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ð2Þ þ a dðn1 þ 1ÞÞ x2 1 þ 2Hpdðn1 ÞðAdðn2 1Þ þ A dðn2 þ 1ÞÞ ¼ fA2 adðn1 1Þdðn2 2Þ 4 þ ðA2 aÞ dðn1 þ 1Þdðn2 þ 2Þ þ A2 a dðn1 þ 1Þdðn2 2Þ þ A2 adðn1 1Þdðn2 þ 2Þg:
ð88Þ
Equating to zero the secular term with dðn1 Þ in Eq. (88) we immediately find ð2Þ
x2 ¼ 0:
ð89Þ
Other secularities in Eq. (88) are presented by the terms with dðn1 1Þ. Setting to zero all these terms we obtain the equation pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ð2Þ ð90Þ n2 1 þ 2Hpað2Þ ðn2 Þ ¼ aðx1 þ HpÞdðn2 Þ þ aðA2 dðn2 2Þ þ A2 dðn2 þ 2ÞÞ: 4 where we have to set ð2Þ
x1 ¼ Hp;
ð91Þ
for to avoid another secularity which appears in Eq. (90) at n2 ¼ 0. Taking into considering the result (91) we find the general solution of Eq. (90) pa ð92Þ að2Þ ðn2 Þ ¼ lð2Þ dðn2 Þ þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi fA2 dðn2 2Þ A2 dðn2 þ 2Þg: 8 1 þ 2Hp In the solution of Eq. (42) the term with lð2Þ will be included together with the amplitude a by the following way: 1 fðra þ r2 lð2Þ Þeix1 t þ ðra þ r2 lð2Þ Þ eix1 t g: 2 It leads only to redefining of the amplitude a from a to a þ rlð2Þ . Therefore without sacrifice of generality we can set lð2Þ ¼ 0, and substituting (92) into (87) we obtain: p Qð2Þ ðn1 ; n2 Þ ¼ Að2Þ ðn2 Þdðn1 Þ þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi faA2 dðn1 1Þdðn2 2Þ þ ðaA2 Þ dðn1 þ 1Þdðn2 þ 2Þ 16 1 þ 2Hp aA2 dðn1 1Þdðn2 þ 2Þ a A2 dðn1 þ 1Þdðn2 2Þg: ð93Þ Thus, Eq. (88) after setting to zero all the secularities takes the following simple form: n21 ðn21 1ÞQð3Þ ðn1 ; n2 Þ ¼ 0 with general solution of the type 1 Qð3Þ ðn1 ; n2 Þ ¼ Að3Þ ðn2 Þdðn1 Þ þ fað3Þ ðn2 Þdðn1 1Þ þ að3Þ ðn2 Þdðn1 þ 1Þg: 2
ð94Þ
The fourth order iteration in Eq. (68), in terms of the results obtained above, takes the form: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n21 ðn21 1ÞQð4Þ þ 1 þ 2Hpn2 fað3Þ ðn2 Þdðn1 1Þ að3Þ ðn2 Þdðn1 þ 1Þg þ ð1 þ 2HpÞð1 n22 ÞAð2Þ ðn2 Þdðn1 Þ 5p pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 2HpfðaA2 dðn1 1Þdðn2 2Þ þ ðaA2 Þ dðn1 þ 1Þdðn2 þ 2ÞÞ ðaA2 dðn1 1Þdðn2 þ 2Þ þ 4 ð3Þ þ a A2 dðn1 þ 1Þdðn2 2ÞÞg þ x1 dðn2 Þðadðn1 1Þ þ a dðn1 þ 1ÞÞ " # ð3Þ x2 þ ð1 þ 2HpÞdðn1 Þ Hp pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðAdðn2 1Þ þ A dðn2 þ 1ÞÞ 1 þ 2Hp ¼
p2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi fðaA4 dðn1 1Þdðn2 4Þ þ ðaA4 Þ dðn1 þ 1Þdðn2 þ 4ÞÞ þ 2jAj2 ðaA2 dðn1 1Þdðn2 2Þ 32 1 þ 2Hp þ ðaA2 Þ dðn1 þ 1Þdðn2 þ 2ÞÞ ðaA4 dðn1 1Þdðn2 þ 4Þ þ a A4 dðn1 þ 1Þdðn2 4ÞÞ 2jAj2 ðaA2 dðn1 1Þdðn2 þ 2Þ þ a A2 dðn1 þ 1Þdðn2 2ÞÞg p pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 2HpfðaA2 dðn1 1Þdðn2 2Þ þ ðaA2 Þ dðn1 þ 1Þdðn2 þ 2ÞÞ ðaA2 dðn1 1Þdðn2 þ 2Þ þ 4 p þ a A2 dðn1 þ 1Þdðn2 2ÞÞg þ ð1 þ 2HpÞdðn1 ÞfðA3 dðn2 3Þ þ A3 dðn2 þ 3ÞÞ 8 þ 7jAj2 ðAdðn2 1Þ þ A dðn2 þ 1ÞÞg:
ð95Þ
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In this equation we must equate to zero the secular term which appears at n1 ¼ 0: " # ð3Þ x2 2 ð2Þ ð1 n2 ÞA ðn2 Þ þ Hp pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðAdðn2 1Þ þ A dðn2 þ 1ÞÞ 1 þ 2Hp p ð96Þ ¼ fðA3 dðn2 3Þ þ A3 dðn2 þ 3ÞÞ þ 7jAj2 ðAdðn2 1Þ þ A dðn2 þ 1ÞÞg: 8 Another secular members appear in this equation at n2 ¼ 1. So we have to equate the terms with dðn2 1Þ (and dðn2 þ 1Þ respectively). As a results we obtain the following equation: " # ð3Þ x2 7 Hp pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A ¼ pjAj2 A: 8 1 þ 2Hp Because A 6¼ 0 we immediately find that pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 ð3Þ x2 ¼ 1 þ 2Hp Hp jAj2 : 8
ð97Þ
Taking into considering (97) we obtain from Eq. (96): p Að2Þ ðn2 Þ ¼ ðA3 dðn2 3Þ þ A3 dðn2 þ 3ÞÞ: 64
ð98Þ
There are another secular terms in Eq. (95) at n1 ¼ 1 that result in the equation pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð3Þ 5p pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ð3Þ 1 þ 2HpðaA dðn2 2Þ aA2 dðn2 þ 2ÞÞ þ x1 adðn1 1Þ 1 þ 2Hpn2 a þ 4 p2 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi faA4 dðn2 4Þ þ 2jAj2 aA2 dðn2 2Þ aA4 dðn2 þ 4Þ 2jAj2 aA2 dðn2 þ 2Þg 32 1 þ 2Hp p pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 1 þ 2HpfaA2 dðn2 2Þ aA2 dðn2 þ 2Þg: 4
ð99Þ
From Eq. (99) we find ð3Þ
x1 ¼ 0;
ð100Þ "
að3Þ ðn2 Þ ¼
#
pa pjAj2 p2 a pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðA4 dðn2 4Þ þ A4 dðn2 þ 4ÞÞ: 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðA2 dðn2 2Þ þ A2 dðn2 þ 2ÞÞ þ 2 16 1 þ 2Hp 128 1 þ 2Hp ð101Þ
Summarising all the results we can write p p Qð2Þ ðn1 ; n2 Þ ¼ dðn1 ÞðA3 dðn2 3Þ þ A3 dðn2 þ 3ÞÞ þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi fðaA2 dðn1 1Þdðn2 2Þ 64 16 1 þ 2Hp þ ðaA2 Þ dðn1 þ 1Þdðn2 þ 2ÞÞ ðaA2 dðn1 1Þdðn2 þ 2Þ þ a A2 dðn1 þ 1Þdðn2 2ÞÞg; ( " # pa pjAj2 ð3Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ðA2 dðn2 2Þ þ A2 dðn2 þ 2ÞÞ Q ðn1 ; n2 Þ ¼ ðadðn1 1Þ þ a dðn1 þ 1ÞÞ 4 16 1 þ 2Hp ) p2 a 4 4 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðA dðn2 4Þ þ A dðn2 þ 4ÞÞ : þ 256 1 þ 2Hp
ð102Þ
ð103Þ
Thus, the time dependent solution of Eq. (42) has the following form: 1 1 p QðtÞ ¼ ðAeix2 t þ A eix2 t Þ þ r ðaeix1 t þ a eix1 t Þ r2 ðA3 e3ix2 t þ A3 e3ix2 t Þ 2" 2 64 # ! p p jAj2 2 3p pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ r pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r ðaA2 eiðx1 þ2x2 Þt þ ðaA2 Þ eiðx1 þ2x2 Þt Þ þ r 4 1 þ 2Hp 4 16 1 þ 2Hp # ! " p p jAj2 2 3p pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 r pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ r ðaA2 eiðx1 2x2 Þt þ a A2 eiðx1 2x2 Þt Þ r 4 1 þ 2Hp 4 16 1 þ 2Hp þ r3
p2 ðaA4 eiðx1 þ4x2 Þt þ ðaA4 Þ eiðx1 þ4x2 Þt þ aA4 eiðx1 4x2 Þt þ a A4 eiðx1 4x2 Þt Þ þ Oðr4 Þ; 256ð1 þ 2HpÞ
ð104Þ
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where for the frequencies x1 and x2 we have obtained the following expansions x1 ¼ 1 r2 Hp þ Oðr4 Þ;
ð105Þ
x2 ¼ r
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 1 þ 2Hp þ r3 1 þ 2Hp Hp pjAj2 8
þ Oðr5 Þ:
ð106Þ
From Eqs. (25) and (104) we can find zðtÞ in the form of expansion with respect to the coupling parameter. Naturally, in the expression for z we have to take into account only the terms of the order up to r2 . Finally we obtain: pffiffiffiffiffi 1 zðtÞ ¼ 2pHrðAeix2 t þ A eix2 t Þ þ pffiffiffiffiffi ½1 r2 ð1 þ 2HpÞðaeix1 t þ a eix1 t Þ 2p ! pffiffiffiffiffi 2p p jAj2 þ r pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ r pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðaA2 eiðx1 þ2x2 Þt þ ðaA2 Þ eiðx1 þ2x2 Þt Þ 4 1 þ 2Hp 16 1 þ 2Hp ! rffiffiffi# pffiffiffiffiffi 2p p jAj2 p 2 ðaA2 eiðx1 2x2 Þt þ a A2 eiðx1 2x2 Þt Þ r pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 r pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ r 4 1 þ 2Hp 2 16 1 þ 2Hp "
þ r2
pffiffiffiffiffi p 2p ðaA4 eiðx1 þ4x2 Þt þ ðaA4 Þ eiðx1 þ4x2 Þt þ aA4 eiðx1 4x2 Þt þ a A4 eiðx1 4x2 Þt Þ þ Oðr3 Þ: 256ð1 þ 2HpÞ
ð107Þ
It seems clear that for the dynamics of QðtÞ the frequency x2 is a basic and the contribution of the frequency x1 appears only as a correction with a correction factor r. On the contrary, for zðtÞ the frequency x1 is a basic and the frequency x2 appears in the correction of the order r. The expression (104)–(106) is an asymptotic solution of Eq. (42) for the case 1 þ 2Hp > 0. In the case 1 þ 2Hp < 0 we have to solve Eq. (56) that have the same form that Eq. (42). Therefore the method used to solve Eq. (42) for the case 1 þ 2Hp > 0 can be analogously applied to the case 1 þ 2HP < 0. The behaviour of the quantities xðtÞ and yðtÞ we can obtain from Eqs. (18) and (24). Substituting the expressions for xðtÞ, yðtÞ and the formula (107) to Eq. (22) we have to obtain the connection of the parameters A, a, H and p with the integral of motion x2 þ y 2 þ z2 . The solution (104) has been found with the assumption that QðtÞ has a classical nature. The method we have applied is a Krylov-Bogolubov method extended for the case of two frequencies x1 and x2 , where one of the frequencies is far above the other frequency. The solution we have obtained can be substituted into the functional expansion (38) of the action, taking into account the transformation (34). The expansion obtained this way can be used to evaluate the evolution operator such it is performed in Eqs. (40) and (41). The behaviour of the quantities QðtÞ and zðtÞ demonstrates good accordance with the numerical results performed by Schanz and Esser [4].
Acknowledgements The authors wish to thank Dr. B. Esser for his valuable comments of this paper. S.G.M. wish to acknowledge the financial support provided by the National Council of Science and Technology of Mexico (CONACYT).
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