Theoretical and Mathematical Physics, 152(2): 1173–1182 (2007)
BREATHER RESONANT PHASE LOCKING BY AN EXTERNAL PERTURBATION E. M. Maslov,∗ L. A. Kalyakin,† and A. G. Shagalov‡
We study the effect of the resonant phase locking in the problem of the sine-Gordon equation breather under the action of a small oscillating external force with slowly varying frequency. We obtain equations determining the time evolution of the parameters of the perturbed breather. We describe the regular asymptotic procedure of averaging such equations and show that the averaged equations in the leading order already well describe the phenomenon of resonant phase locking in which the breather oscillations are strongly excited. We obtain necessary and sufficient conditions for the phase locking relating the rate of the perturbation frequency variation and its amplitude to the initial data of the breather.
Keywords: averaging
sine-Gordon equation, breather, phase locking, perturbation theory, asymptotic behavior,
1. Introduction The majority of nonlinear wave equations have a broad set of exact time-periodic solutions describing both localized perturbations and the propagation of finite-amplitude waves in various physical systems. It is essential in practice that one or another periodic solution can be selectively controlled by small perturbations contained in the equations, which permits purposefully varying the amplitude–phase characteristics of such solutions. One of the promising methods in this direction is based on the phenomenon of resonant phase locking (or, in another terminology, on autoresonance). This method was successfully used to generate both simple knoidal waves [1] and multiphase solutions [2] in the periodic problem for the Korteweg–de Vries equation, multiphase waves in the nonlinear Schr¨ odinger equation [3], and plasma oscillations in the sineGordon (SG) equation [4]. The idea of the method goes back to Veksler’s and MacMillan’s studies of using the self-phasing effect to accelerate relativistic particles [5]. Chirikov proposed the adiabatic theory of phase locking for a one-dimensional nonlinear oscillator excited by an external force with a slowly varying frequency [6]. Modern investigations of the asymptotic theory of this phenomenon are described in [7]. The resonant phase locking in the nonlinear oscillator occurs if the excitation amplitude exceeds some critical value depending on the rate of the frequency variation. In this case, the oscillation frequency starts to tend automatically toward the pumping frequency. Because the oscillation frequency in nonlinear systems depends on the amplitude, any variations in the external pumping frequency also cause variations in the amplitude of the phase locked oscillations, i.e., there is an efficient energy exchange between the system and the pump. In particular, in the case of a nonlinear pendulum, a slow decrease in the frequency can result in exciting oscillations with amplitudes of the order of unity even for an exciting force with a very small amplitude. Moreover, such a process can be used to control the already existing large-amplitude oscillations ∗
Pushkov Institute of Terrestrial Magnetism, Ionosphere, and Radiowave Propagation, RAS, Troitsk, Moscow Oblast, Russia, e-mail:
[email protected]. † ‡
Institute of Mathematics, Ufa Scientific Center, RAS, Ufa, Russia, e-mail:
[email protected]. Institute of Metal Physics, Ural Branch, RAS, Ekaterinburg, Russia, e-mail:
[email protected].
Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 152, No. 2, pp. 356–367, August, 2007. c 2007 Springer Science+Business Media, Inc. 0040-5779/07/1522-1173
1173
because the phase locking by an external action ensures total control over both the amplitude and phase of the oscillations. The external pumping frequency is a control parameter in this case. In this paper, we study the phase locking effect in the finite-amplitude solution of the SG equation with a small spatially homogeneous quasiperiodic pumping: utt − uxx + sin u = ε cos ψ(t),
0 < ε 1.
(1)
We assume that the pumping frequency Ω = ψ˙ varies slowly in time: Ω˙ = O(α), |α| 1. If there is no perturbation (for ε = 0), then Eq. (1) admits a localized solution in the breather form κ0 cos(ω0 t + θ0 ) , u = φb (x, t) = −4 arctan ω0 cosh κ0 (x − x0 )
ω02 = 1 − κ02 ,
(2)
which is characterized by the two constants κ0 and θ0 . The coordinate of the breather center x0 is inessential in our problem because Eq. (1) is invariant under translations in x. Turning on the perturbation in the system results in exciting small spatially homogeneous background oscillations, which interact with the breather. As a result of this interaction, in a certain range of the perturbation parameters ε and α and under certain conditions on the initial data κ0 and θ0 , the breather oscillation frequency starts to follow the varying pumping frequency Ω. In this case, the breather energy increases or decreases depending on the sign ˙ In this paper, we find these values and these conditions and obtain a system of equations describing of Ω. the dynamics of the phase locked oscillations up to large times (of the order of ε−1 ). For this, we use the perturbation theory based on the inverse scattering method [8]–[10] in Sec. 2, the generalized Krylov– Bogoliubov–Mitropolskii averaging method [11] in Sec. 3, and the nonlinear pendulum approximation [6] in Sec. 4.
2. Adiabatic approximation As already noted, homogeneous background oscillations χ(t) ∼ ε arise in the system under the action of an external perturbation. Because these perturbations are small, we can assume that they are described by the linear equation χ ¨ + χ = ε cos ψ(t). (3) Therefore, setting u(x, t) = φ(x, t) + χ(t)
(4)
with φ(x, t) → 0, |x| → ∞, in (1), we obtain a perturbed SG equation in the form [9] φtt − φxx + sin φ = χ(t)(1 − cos φ),
(5)
where terms of the order of ε2 are neglected. Precisely this equation describes the dynamics of the localized breather under the action of homogeneous background oscillations. In what follows, we assume that there are no natural linear oscillations in the system and the external excitation frequency Ω is not too close to unity. For simplicity, we can then set ε cos ψ χ(t) = (6) 1 − Ω2 in the right-hand side of Eq. (5). It is easy to verify that this expression satisfies Eq. (3) up to O(αε). We now apply the perturbation theory based on the inverse scattering method to Eq. (5). We here follow [10], where the equations determining the evolution of the breather parameters under the action of 1174
an arbitrary perturbation were obtained. According to this approach, the solution describing the perturbed breather can be represented as φ(x, t) = φb (x, t) + O(ε), (7) φb (x, t) = −4 arctan
where
κ(t) cos θ(t) , ω(t) cosh[κ(t)(x − x0 )]
ω2 = 1 − κ2 .
(8)
Here, θ is the breather phase, and ω and κ −1 are its characteristic frequency and dimension (0 < ω < 1, 0 < κ < 1). The parameterization π (9) ω = cos γ, κ = sin γ, 0 < γ < , 2 determines the breather characteristic amplitude γ = arctan(κ/ω). The equations for these parameters can be obtained in an approximation of any arbitrary order in ε by successive approximations from the equations for the scattering data corresponding to the discrete spectrum in the associated scattering problem. The correction terms O(ε) in formula (7) correspond to the continuous part of the spectrum. They describe the breather shape distortion on the scale κ −1 and the radiation that starts to propagate from the breather after the perturbation is turned on. In this paper, we consider only the first term in Eq. (7). Such an approximation is usually said to be adiabatic. In what follows, we show that this approximation is quite sufficient to describe phase locking with a high accuracy. We regard the right-hand side of Eq. (5) as a perturbation, calculate the corresponding integrals (see [10]), and in the first-order approximation in ε obtain the system of equations of the adiabatic approximation γ˙ = ε
F (γ, θ) cos ψ, 1 − Ω2
G(γ, θ) θ˙ = ω(γ) + ε cos ψ, 1 − Ω2
(10) (11)
where 4ω 2 + κ 2 cos2 θ π 2 κ sin θ cos2 θ , 4 (1 − κ 2 sin2 θ)5/2 ω 2 κ cos θ κ πω κ 2 cos2 θ(4ω 2 + κ 2 cos2 θ) cos θ , G(γ, θ) = 1 + − arcsinh 4κ 2 (1 − κ 2 sin2 θ)3/2 ω ω 4 (1 − κ 2 sin2 θ)
F (γ, θ) =
(12)
(13)
and ω and κ are determined by the amplitude γ according to (9). These equations contain two fast variables, the breather phase θ and the given phase of the external perturbation ψ, and two slow variables, ˙ The functions F (γ, θ) and G(γ, θ) the breather amplitude γ and the given perturbation frequency Ω = ψ. are periodic in θ with the period 2π. In this case, F (γ, θ) is bounded, and G(γ, θ) has singularities at γ = 0, π/2. We assume that the amplitude γ is not too close to these values, and G(γ, θ) is therefore also assumed to be bounded. We note that Eqs. (10)–(13) coincide with the equations obtained in [9] up to notation. By definition, phase locking means that the difference δ = θ − ψ is a slowly varying bounded function of time. In this case, the dynamical system γ˙ = ε
F (γ, ψ + δ) cos ψ, 1 − Ω2
(14)
G(γ, ψ + δ) δ˙ = ∆ω(γ, Ω) + ε cos ψ, 1 − Ω2
(15)
ψ˙ = Ω,
(16) 1175
where ∆ω = ω(γ) − Ω,
Ω˙ = O(α),
(17)
contains only one fast variable ψ. Such solutions can exist in system (14)–(16) only for some initial conditions and with certain restrictions imposed on α and ε. This already follows because the detuning ∆ω must be small in the phase locking regime and this smallness must be preserved in the entire time interval of
variations in Ω, i.e., until t ∼ |α|−1 . This implies that, first, ω γ(0) ≈ Ω(0) and, second, the rate of variation in Ω must be of the order of the average (over the phase ψ) of the rate of variation in γ, i.e., α = O(ε). We therefore set α = βε, |β| = O(1), (18) where β is a fixed constant. The smallness of ∆ω therefore depends only on the smallness of ε and on the smallness of ∆ω for t = 0. Because the latter quantity is in no way related to ε, we assume that ∆ω is the second independent small parameter, just as Garifullin [12] assumed in the nonlinear oscillator problem.
3. Averaged equations Taking everything said above into account, we now pass to the averaged description of the phase locking process. We use the generalized Krylov–Bogoliubov–Mitropol’skii method (see, e.g., [11]). According to this approach, we consider an almost identical transformation of the variables γ and δ to the new “averaged” variables γ¯ and δ¯ and write this transformation as the asymptotic series γ δ
M−1 N −1 ¯ Ω, ψ)
γ¯ γ , δ,
umn (¯ m n = + + O εM , ε( ∆ω )N . ε ( ∆ω ) ¯ Ω, ψ) δ¯ vmn (¯ γ , δ,
(19)
m=1 n=0
Instead of ∆ω for the second independent small parameter, we here use the quantity ∆ω = ∆ω + O(ε) defined as ∆ω = ω(¯ γ ) − Ω. (20) We now require that the rate of variation in the slow variables γ¯ and δ¯ be independent of the fast variable ψ, i.e., we set d dt
M−1 N −1 ¯ Ω)
Amn (¯ γ¯ 0 γ , δ,
m n ε ( ∆ω ) = + + O εM , ε( ∆ω )N . ¯ ¯ δ Bmn (¯ ∆ω γ , δ, Ω)
(21)
m=1 n=0
The procedure for averaging system (14)–(16) thus reduces to constructing asymptotic expansions (19) and (21), i.e., to finding the quantities umn , vmn , Amn , and Bmn . We note that these expansions do not contain terms with m = 0, because the system becomes unperturbed for ε = 0. We now substitute (19) and (21) in (14)–(16) and equate the coefficients of like εm ( ∆ω )n . To be ˙ = α, take expressions (18) and (20) definite, we here choose the linear law of variation in Ω(t) by setting Ω into account, and use the expansion
d∆ω = ε A10 ω (¯ γ ) − β + ε∆ωA11 ω (¯ γ ) + ε2 A20 ω (¯ γ) + . . . . dt
(22)
As a result, we obtain a system of recursive first-order differential equations (in the fast variable ψ) for the functions umn and vmn . We find the expressions for Amn and Bmn from the conditions that umn and vmn are bounded in ψ. Then the solutions umn and vmn are 2π-periodic functions of the fast variable ψ, which ¯ and Ω. The choice of these functions are determined up to arbitrary functions of the slow variables γ¯, δ, 1176
does not affect the accuracy and is dictated only by convenience (see [11]). We choose them such that the solutions umn and vmn averaged over ψ are zero. The quantities γ¯ and δ¯ in this case gain the natural meaning of the average of γ and δ. Moreover, the expressions for Amn and Bmn become significantly simpler under this choice. For example, A10 and B10 can be simply obtained by calculating the averages over ψ in the right-hand sides of Eqs. (14) and (15), and A11 and B11 become zero. The problem of finding solutions of system of equations (10) and (11) in the regime of the resonant phase locking thus reduces to integrating system (21), which is usually said to be averaged. The procedure described above permits obtaining averaged systems with any finite accuracy in the two small parameters ε and ∆ω. It was shown for systems with a fast rotating phase and only one small parameter that the limit error arising in such averaging procedures is exponentially small in the inverse small parameter [13]. In this connection, we note that in the context of our original problem, preserving the terms whose order of magnitude is greater than O(ε) in the averaged equations gives an excess of the prescribed accuracy because in solution (7), we analyze only the first adiabatic term, whose parameters satisfy Eqs. (10) and (11), where terms of the order of ε2 are omitted. Moreover, Eqs. (5) and (6) are also equations approximated with the same accuracy. For the same reasons, we do not analyze the corrections umn and vmn here; for us, it suffices to know that they exist and are bounded functions of time. This approach, which is usually called adiabatic, is typical of such problems (see, e.g., [10]). Taking all this into account, we preserve only the leading terms determined by the coefficients A10 and B10 in averaged equations (21). We calculate them using the specific expressions (12) and (13) and obtain the averaged system of the first order in ε and ∆ω,
Here, g(¯ γ) =
g(¯ γ) d¯ γ ¯ =ε sin δ, dt 1 − Ω2 (t)
(23)
dδ¯ γ) g (¯ ¯ = ∆ω + ε cos δ. dt 1 − Ω2 (t)
(24)
1 K(κ) ¯ − E(κ) ¯ ≥ 0, 2
g (¯ γ) =
1 E(κ) ¯ tan γ¯ ≥ 0, 2
(25)
K(κ) ¯ and E(κ) ¯ are complete elliptic integrals of the respective first and second kinds, κ ¯ = sin γ¯, Ω(t) = Ω0 + αt, and ∆ω is defined by expression (20). The phase locking regime is associated with solutions ¯ containing the bounded functions δ(t). We note that the obtained system has the Hamiltonian γ) ¯ Ω) = γ¯ Ω − sin γ¯ − ε g(¯ ¯ H(¯ γ , δ; cos δ. 1 − Ω2
(26)
In Fig. 1, we present the time dependences of the average frequency and the average energy of the ¯ ¯ breather respectively determined as dθ/dt = Ω(t) + dδ/dt and E b = 16 sin γ¯, which were obtained by ¯ integrating system (23), (24) numerically with the initial conditions γ¯ (0) = 0.5, δ(0) = 0, and ∆ω(0) = 0. For each fixed α, the passage to the phase locking regime was observed for ε exceeding some critical value. In particular, for the initial conditions chosen above and for α = −0.0003, we obtained εcr = 0.00178. To verify the result, we integrated the original SG equation (1) numerically with the initial conditions u(x, 0) = φb (x, 0) + χ(0) and ut (x, 0) = 0. In this case, we took the same initial conditions for the breather parameters and perturbations as in system (23), (24). The breather frequency is determined by the instants at which the field u(x, t) passes through zero, and the energy is obtained by integrating the energy density over the breather width. Comparing the results thus obtained, we see that the corresponding curves practically coincide such that they can hardly be distinguished in Fig. 1. We only note that the true value of εcr obtained from the solution of Eq. (1) is equal to 0.00179. 1177
a
b
Fig. 1. Time dependence of (a) the average frequency and (b) the average energy of the breather: curves 1 are “no” phase locking with ε = 0.0016 and curves 2 are the phase locking regime with ε = 0.0019. The dotted line is the frequency Ω(t) = Ω0 + αt, where Ω0 = cos γ¯ (0) 0.8776 and α = −0.0003.
4. Nonlinear pendulum approximation Equations (23) and (24) describe the dynamics of the average γ and δ on the large time interval 0 ≤ t ε−1 . But the phase locking itself, as follows from Fig. 1, occurs at the initial instants, almost immediately after the perturbation is turned on. Following [6], we use this fact to obtain the phase locking ¯ Differentiating conditions. For this, we pass from system (23), (24) to the second-order equation for δ. Eq. (24) and using (23), we obtain
d2 δ¯ γ ) − g (¯ γ )∆ω g(¯ γ )ω (¯ sin δ¯ − εβ + O ε2 , ε( ∆ω )2 . = ε dt2 1 − Ω2 (t)
(27)
We here take into account that because A11 = B11 = 0, the error in the right-hand sides of Eqs. (23)
and (24) is of the order O ε2 , ε( ∆ω )2 . In Eq. (27), we therefore preserve the term of the order O(ε∆ω ) obtained by differentiating the second-order term in the right-hand side of (24). We now consider the initial time interval 0 ≤ t ε−1 on which the phase locking indeed occurs. In the right-hand side of Eq. (27), we can then neglect the time dependence of the coefficient of sin δ¯ by taking its value, for example, at t = 0. On the time scale under study, we thus obtain the nonlinear pendulum equation d2 δ¯ = λ sin δ¯ − β, 0 ≤ τ ε−1/2 , (28) dτ 2 where τ = ε1/2 t is the slow time and the constant λ is calculated with the required accuracy from the initial values of γ = γ¯ + O(ε) and Ω:
g γ(0) ω γ(0) − g γ(0) ∆ω(0) λ . (29) 1 − Ω20 In particular, if the initial detuning is zero or sufficiently small, ∆ω(0) O(ε), then we have
g γ(0) . λ− sin γ(0) 1178
(30)
a Fig. 2.
b Phase portrait of Eq. (28): (a) |β/λ| > 1, (b) |β/λ| < 1.
We note that λ is always negative. The general properties of all possible solutions of Eq. (28) can be obtained by analyzing different phase trajectories determined by the first integral (the energy) ¯ δ¯τ ) = E(δ,
1 ¯2 ¯ δ + V (δ), 2 τ
¯ = λ cos δ¯ + β δ. ¯ V (δ)
(31)
It is clear that the phase portrait depends essentially on the relation between the parameters β and λ. If |β/λ| > 1, then there are no fixed points. The phase trajectories are periodically deformed parabolas ¯ ) = −βτ 2 /2 + O(τ ), (Fig. 2(a)). For any initial conditions, all the solutions increase rapidly with time: δ(τ τ → ±∞. The phase locking is therefore impossible if the pumping frequency varies too fast compared with the value of its amplitude. But if the pumping frequency varies sufficiently slowly such that β < 1, λ
(32)
then two infinite sequences of fixed points, stable and unstable, appear on the phase portrait (Fig. 2(b)). A pair of separatrices going to infinity and a separatrix loop surrounding the neighboring stable point issue from each unstable point. The closed trajectories inside the loop correspond to periodic solutions, i.e., to the phase locking regime. The breather amplitude for such solutions increases on the average for β < 0 or decreases for β > 0. This can be verified by substituting sin δ¯ from Eq. (28) in Eq. (23) and averaging (23) ¯ over the oscillations of δ(t). Obviously, inequality (32) is a necessary, but not sufficient, condition for the phase locking. It is also
(1) (2) ¯ necessary that the initial point δ(0), δ¯τ (0) be inside one of the separatrix loops. Let δ¯n and δ¯n be the (0) minimum and maximum values of δ¯ on the nth separatrix loop, and let δ¯n be the value at the stable point (1) (0) (2) ¯ ), it is then necessary and sufficient that the inside this loop: δ¯n < δ¯n < δ¯n . For the periodicity of δ(τ inequalities ¯ < δ¯(2) , δ¯n(1) < δ(0) n
¯ δ¯τ (0) < En(s) En(0) ≤ E δ(0),
(33) (34) 1179
(s)
(0)
be satisfied for some n, where En and En are the values of the first integral on the nth separatrix loop (2)
(1)
(0)
(s) (0) and at the corresponding stable point: En = V δ¯n = V δ¯n and En = V δ¯n . Taking λ < 0 into (i) account, we determine the quantities δ¯n , i = 0, 1, 2, n = 0, ±1, ±2, . . . , as follows: 1. for β < 0, β δ¯n(0) = 2nπ + arcsin , λ
β δ¯n(2) = (2n + 1)π − arcsin , λ
(35)
2 β β β 1− + (2n + 1)π − arcsin λ λ λ
(36)
(1) and δ¯n is the root of the equation
β cos δ + δ = − λ (2) (1) (0) in the interval δ¯n−1 < δ¯n < δ¯n ; and
2. for β > 0, β δ¯n(0) = (2n + 2)π + arcsin , λ
β δ¯n(1) = (2n + 1)π − arcsin , λ
(37)
(2) (0) (2) (1) and δ¯n is the root of Eq. (36) in the interval δ¯n < δ¯n < δ¯n+1 .
Hence, inequalities (32)–(34) are necessary and sufficient conditions for the breather phase locking by an external perturbation. These inequalities mean that the phase portrait of Eq. (28) has separatrix loops
¯ and that the initial point δ(0), δ¯τ (0) is inside one of these loops. As |β/λ| → 0, the separatrix loops expand, and their dimensions along the axis δ¯ tends to 2π. As |β/λ| → 1, the loops contract to the points sgn(β/λ)π/2 + 2nπ. We assume that the initial point is inside the nth separatrix loop. We now slowly increase the rate of variation in the pumping frequency. The critical value of β at which the phase locking breaks down is reached when the contracting loop passes through the initial point. Hence, βcr is found as a root of the equation
¯ E δ(0), δ¯τ (0); β, λ = E (s) (β, λ). (38) n
If the initial detuning is sufficiently small, ∆ω(0) = ∆ω(0) + O(ε) = o(ε1/2 ), then by (24), we can neglect the kinetic term in the left-hand side of Eq. (38). In this case, (38) takes the form of Eq. (36), where ¯ + O(ε). This equation now determines βcr /λ implicitly as a function of the initial phase δ = δ(0) = δ(0) difference δ(0). As an example, we consider the case where the pumping frequency decreases, i.e., β < 0, the initial detuning ∆ω(0) = 0, and the initial difference between the breather and pumping phases is distributed in the interval from 0 to 2π. In this case, we must take n = 0 (the zeroth loop) for 0 ≤ δ(0) < π in Eq. (36) and n = 1 (the first loop) for π < δ(0) ≤ 2π. Obviously, we have βcr /λ = 0 for δ(0) = π, i.e., the phase locking never occurs in the antiphase. For δ(0) = π/2, the phase locking occurs for all β up to the fastest variation in the frequency when β = λ. The graph of βcr /λ depending on δ(0) is shown in Fig. 3. The phase locking regime is associated with the values of β/λ below the solid curve. In the same figure, we preset the values of βcr /λ obtained by numerical integration of SG equation (1). We note that in numerical experiments in a narrow region near the value δ(0) = π, we observed that the breather intensively radiates waves and the phase locking does not occur even for very low rates of variation in the pumping frequency. It is clear that the adiabatic approximation cannot be used in this region. Nevertheless, despite this defect and a certain roughness of the nonlinear pendulum approximation, Fig. 3 shows that the obtained results agree sufficiently well with the solution of the SG equation obtained by numerical integration. 1180
Fig. 3. The dependence of βcr /λ on the initial phase difference: the solid curve results from solving Eq. (36), denotes the values obtained by integrating the SG equation numerically for α = −0.0003, and denotes the same for α = −0.00003.
5. Conclusion In this paper, we used an example of the breather under the action of a small oscillating external force with a slowly varying frequency to study the effect of the resonant phase locking in the SG equation. We used the perturbation theory based on the inverse scattering method to obtain system of equations (14)–(16) for the breather parameters. This system describes the phase locking effect in the adiabatic approximation. A particular feature of this system belonging to the class of dynamical systems with a fast rotating phase is that it has two independent small parameters: the external force amplitude ε and the frequency detuning ∆ω. We described a regular procedure for averaging such systems based on expanding it asymptotically in both small parameters and obtained the explicit averaged system in the first-order approximation (Eqs. (23) and (24)), which is a Hamiltonian system. Comparing the solutions obtained by integrating this system numerically and the corresponding numerical solutions of the original SG equation showed that the results coincide with a high degree of accuracy. We used nonlinear pendulum approximation (28) to obtain the phase locking conditions and the critical perturbation parameters. As a result, we have necessary and sufficient conditions for the phase locking in the form of inequalities (32)–(34) and Eq. (38) for the critical values of β, which takes the form of Eq. (36) in the case of a small initial detuning. The solution of this equation agrees well with the results obtained by integrating the SG equation numerically. Acknowledgments. The authors thank the organizers of the workshop “Nonlinear Physics: Theory and Experiment IV” for the possibility to present the results of this work as a report. The authors also express their thanks to the participants of the conference, especially to L. Friedland and O. Kiselev, for the useful discussions. This work is supported in part by INTAS (Grant No. 03-51-4286) and the Russian Foundation for Basic Research (Grant Nos. 06-01-00124 and 06-01-92052).
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