ap-3-10.dvi Acta Polytechnica Vol. 50 No. 3/2010 Maxwell-Chern-Simons Models: Their Symmetries, Exact Solutions and Non-relativistic Limits J. Niederle, A. G. Nikitin, O. Kuriksha Abstract Two Maxwell-Chern-Simons (MCS) models in the (1+3)-dimensional space-space are discussed and families of their exact solutions are found. In contrast to the Carroll-Field-Jackiw (CFE) model [2] these systems are relativistically invariant and include the CFJ model as a particular sector. Using the Inönü-Wigner contraction aGalilei-invariant non-relativistic limit of the systems is found,whichmakespossible to find a Galilean formulation of the CFJ model. 1 Introduction There are three motivations of the present paper. First we search for four-dimensional formulations of Maxwell-Chern-Simon models [1]. Secondly, we look for relativistic- and Galilei-invariant versions of the Carroll-Field-Jackiwelectrodynamics [2]. At the third place we construct a relativistic counterpart of the Galilei-invariant equations for vector fields proposed in our paper [3]. There are two symmetries of Maxwell’s electrody- namics that have dominated all fundamental physical theories, namely, gauge and Lorentz invariance. They provide physical principles that guide the invention of models describing fundamental phenomena. At the first place, the properties of the electromagnetic ra- diation (in a natural setting and in HE accelerators) are described by Lorentz-invariant dynamics. On the other hand, the gauge invariance is possible for mass- less field only, and so it is validated by the stringent limits on the photon mass mγ. Possible breaking of the Lorentz and gauge invari- ance has been tested within a theoretical framework with symmetry breaking parameters. The violation of gauge invariance was tested in frames of two modifi- cations of the Maxwell theory. In the first of them the Lagrangian of the free e.m. field LEM = −Fμν F μν is modified to LEM → LEM + m2γ 2 Aν A ν where Fμν = ∂μAν − ∂ν Aμ is the tensor of e.m. field and Aν is the four-vector of the photon field. This field Aν is massive and so the gauge invariance is lost. The breaking of gauge invariance caused by pres- ence ofmassive term m2γ has been testedwith geomag- netic and galacticmagnetic data. As a result, the lim- its for parameter m2γ have been obtained in the form mγ ≤ 3 ·10−24 and mγ ≤ 3 ·10−36, correspondingly. The othermodification of the Maxwell Lagrangian was proposed by Carroll, Field and Jackiw: LEM → L = LEM +LCS (1) where LCS is a four-dimensional version of the Chern- Simons term: LCS = 1 4 εμνρσF μAν F ρσ. (2) Here pμ is a constant vector which causes violation of the Lorentz-invariance. The CFJ model presents a rather elegant and convenient way for testing possible violation of the Lorentz-invariance,which causes its large impact. But this model has a principle disadvantage, namely, the breaking of theLorentz-invariance“byhands” and the additional constants pμ have no physical meaning. In addition, this model is invariant with respect to nei- ther the Lorentz nor Poincaré group, i.e. it does not satisfy any relativity principle accepted in physics. In the following sections we discuss two dynamical versions of the CFJ model which contain the ordinary CFJ model as a particular sector. One of them is very similar to axion electrodynamics in which, however, the axion rest mass is zero. The other includes axion electrodynamics as a limiting case corresponding to a small coupling constant. 2 The MCS model in the (1+3)-dimensional Minkovski space Let us start with the following Lagrangian L=− 1 4 Fμν F μν − 1 2 FμF μ − κ 4 εμνρσ F μAν F ρσ + ejμA μ + qj4A 4. (3) Here the Greek indices run over the values 0,1,2,3, Aν and F μν = ∂μAν − ∂ν Aμ are the four-vector po- tential and tensor of the electromagnetic field, respec- tively. In addition, Lagrangian (3) includes a scalar 96 Acta Polytechnica Vol. 50 No. 3/2010 potential A4 and its derivatives F μ = ∂A4 ∂xμ as well as a scalar current j4. The Lagrangian (3) has the following nice proper- ties. • It is transparently invariant w.r.t. Lorentz trans- formations and shifts of independent variables. Moreover, since j4 is a scalar, it is possible to introduce an additional charge q which is not nec- essarily equal to the electric charge e but in gen- eral to a coupling constant corresponding to some interaction which is not necessary purely electro- magnetic. • The corresponding energy-momenta tensor does not depend on parameter κ and so is not affected by the term − κ 4 εμνρσF μAν F ρσ. More precisely, this tensor has the following form T00 = 1 2 (E2 +B2 + F20 +F 2), T0a = 1 2 (εabcEbBc + F 0F a), (4) where E, B and F are three-vectors whose com- ponents are: Ea = F0a, Ba = 1 2 εabcFbc and Fa. • Lagrangian(3) includesboth thefieldcomponents F μν , F ν andpotentials Aμ. But in spite of the ex- plicit dependence on potentials, the Lagrangian admits gauge transformations Aμ → Aμ + ∂μϕ since via them it is changed only by a mere sur- face term: L → L + ∂μ(ϕεμνρσ F νρF σ). In addition, this Lagrangian is not affected by the change A4 → A4 + C, where C is a constant. • In non-relativistic approximation, Lagrangian (3) is reduced to the Galilei-invariant Lagrangian for the irreducible Galilean field discussed in [3]. 3 Field equations Let us consider the Euler-Lagrange equations which correspond to Lagrangian (3): ∂ν F μν + κFν F̃ μν = ejμ, (5) ∂ν F ν + κ 2 Fλν F̃ λν = qj4. (6) Here, F̃ μν = 1 2 εμνρσFρσ is the dual tensor of the elec- tromagnetic field F μν. Field variables F̃ μν and Fμ satisfy the following conditions: ∂ν F̃ μν =0, ∂ν Fμ − ∂μFν =0 (7) in accordance with their definitions as derivatives of the potential. Equations (5)–(7) involve ten field variables, i.e., six-component tensor F μν and four-vector F μ. All these variables are dynamical and of equal value. If q =0 then equations (5)–(7) reduce to the field equa- tions of an axion electrodynamics [9] with zero axion mass. First, let us note that equations (5)–(7) cover the system of equations proposed by Carroll, Field and Jackiw [2]. Indeed, the system (5)–(7) is compatible with the following additional condition ∂ν Fμ =0. (8) If the condition (8) is imposed then Fμ = pμ where pμ are constants. Substituting this solution into (5) we obtain the CFJ equations. Concerning our addi- tional equation (6), for constant pμ it is reduced to a definition of j4. Note that equations (5) with variable (i.e., non- constant) pμ were discussed in [7] in frames of the pre- metric approach [8]. However, F μ is treated there as an external axion field, while in our model it is a dy- namical variable satisfying evolution equation (6) and constraints (7). 4 MCS model with nonliner Bianchy indentity The system (5), (6) generates the tensor conserved current whose components are given in equation (4). Moreover, the energy-momentumtensor (4) has a very simple formand does not depend on the coupling con- stant κ. On the other hand, for κ =0 this tensor can be segregated to two parts, each of which is conserved separately: T μν = T μν1 + T μν 2 , (9) where T001 = 1 2 (E2 +B2), T0a1 = 1 2 εabcEbBc, (10) T002 = F 2 0 +F 2, T0a2 = 1 2 F0F a. (11) However, for κ nonzero either tensor (10) or (11) is not conserved, but only their sum (9) is a conserved quantity. In this section we propose an MCS model which causes conservation of the standard energy-momenta tensor ofMaxwell field given by expressions (10). The related field equations for antisymmetric tensor F μν are: ∂ν F μν + κFν F̃ μν = ejμ, (12) ∂ν F̃ μν − κFν F μν = 0 (13) where F μ = ∂μA 4 and A4 is a scalar potential satisfy- ing the following nonlinear equation ∂ν ∂ ν A4=κ(Fμν F μν sin(κA4)− Fμν F̃ μν cos(κA4)). (14) 97 Acta Polytechnica Vol. 50 No. 3/2010 It is easy to verify that tensor (10) satisfies the continuity equation ∂μT μν 1 =0provided F μν solve the field equations (12)–(13) with zero current jμ =0. Like (5)–(7), equations (12)–(14) admit La- grangian formulation. To construct the related Lagrangian we express F μν via potential A = (A0, A1, A2, A3, A4) in a non-linear fashion: F μν =(∂μAν − ∂ν Aμ)cos(κA4)− 1 2 εμν λσ(∂ λAσ − ∂σAλ)sin(κA4). (15) TheAnsatz (15) converts equation (13) to identity, which plays the role of Bianchi identity in our model. Note that this identity appears to be essentially non- linear. Using definition (15), we can write a Lagrangian for the system (12)–(14): L= 1 4 (Fμν F μν cos(κA4)+ Fμν F̃ μν sin(κA4))+ 1 2 FμF μ. (16) Variating (16)w.r.t. Aμ and A4 one obtains equations (12) and (14) correspondingly. For small values of parameter κ and bounded A4 it is possible to expand Lagrangian (16) and the re- lated equations (12)–(14) in power series of κ. Then neglecting terms whose order in κ is higher than one we obtain Lagrangian (3). In other words, the model considered in Sections 4 and 5 can be treated as a first approximationof themodel based onLagrangian (16). 5 Continuous and discrete symmetries Equations (5)–(7) (or (24)) are transparently invariant w.r.t. the Poincaré group. Nevertheless we examined them using tools of Lie analysis and found their max- imal invariance group. We will not present here de- tails of this routine procedure,whose algorithmcanbe found in [10], but will formulate its result: equations (5)–(7) are invariant w.r.t. 11-parametrical extended Poincaré group P̃(1,3), whose infinitesimal generators are Pμ = ∂μ, Jμν = xμ∂ν − xν ∂μ + Sμν , D = xμ∂ μ − I − jμ∂jμ . (17) Here I = Fμ∂Fμ + Fμν ∂Fμν is an operator which acts on the field variables as the unit one, Sμν are generators of the Lorentz group acting on the field variables and currents: Sab =Kab − Kba, Kab =Fa∂Fb + Ea∂Eb + Ha∂Hb + ja∂jb , a, b �=0, S0a =F0∂Fa + Fa∂F0 + εabc(Ea∂Hb − Ha∂Eb Hb∂Ea). (18) Thus, in contrast to the CFJ model, the considered CSM model in (1 + 3)-dimensional Minkovski space is invariant w.r.t. the extended Poincaré group. Note that the additional condition (8) also is invariantw.r.t. this group; violation of Lorentz invariance takes place only after fixingparticular constant solutions Fμ = pμ. Equations (24) are also invariant w.r.t. discrete transformations of space reflection P and time inver- sion T . Moreover, F μ is a pseudovector and so the potential A4 is a pseudoscalar. In an analogous way we have found the maximal Lie groupanddiscrete symmetries admittedby system (12)–(14). It happens that the symmetry of this sys- tem is completely analogous to the symmetry of equa- tions (5)–(7). Namely, system (12)–(14) is invariant w.r.t. extended Poincaré group P̃(1,3), whose gener- ators are given in equations (17), and admits discrete symmetry transformations P, C and T. 6 Non-relativistic limit The correct definition of the non-relativistic limit is by no means a simple problem in general and in the case of theories ofmassless fields in particular, see, for example, [12]. A necessary (but not sufficient) con- dition for obtaining consistent non-relativistic limit of a relativistic theory is to take care that the limiting theory be in agreement with the principle of Galilean relativity [11]. To find a non-relativistic limit of equations (24) we use the Inönü-Wigner contraction [13], which guarantees Galilean symmetry of the limiting theory. Namely, we shall start with the representation of the Poincaré algebra, which is realized on the set of solu- tions of equations (24), andcontract it to the represen- tation of the Galilei algebra. Then, using the contrac- tion matrix we find the Galilean limits of Lagrangian (3) and system (24). The tensor field F μν and vector field F μ transform in accordance with the representation [D(0,1)⊕ D(1,0)]⊕ D(1/2,1/2) (19) of the Lorentz group. Contractions of this repre- sentation (and also of all irreducible representations involved into the direct sum (19)) to indecompos- able representations of the homogeneousGalilei group hg(1,3) were discussed in [14] and [15]. The contraction of (19) to the indecomposable rep- resentation of hg(1,3) is reduced to the following pro- cedure. First, let us represent the field variables as a ten component vector Ψ=column(F01, F02, F03, F23, F31, F12, F1, F2, F3, F0) (20) then Lorentz generators (18) act onΨas the following matrices 98 Acta Polytechnica Vol. 50 No. 3/2010 Sab =εabc ⎛ ⎜⎜⎜⎜⎝ Sc 0 0 0 0 Sc 0 0 0 0 Sc 0 0 0 0 0 ⎞ ⎟⎟⎟⎟⎠ ; S0a = ⎛ ⎜⎜⎜⎜⎝ 0 −Sa 0 0 Sa 0 0 0 0 0 0 K†a 0 0 −Ka 0 ⎞ ⎟⎟⎟⎟⎠ . (21) Here Sa are spin-one matrices whose elements are (Sa)bc = iεabc, and Ka are 1×3 matrices of the form K1 =(i,0,0) , K2 =(0, i,0) , K3 =(0,0, i) , (22) and 0 denote the zero matrices of an appropriate di- mension. The Inönü-Wigner contraction consists of transfor- mation to a new basis Sab → Sab, S0a → εS0a fol- lowed by a similarity transformation of basis elements Sμν → S′μν = V Sμν V −1 with a matrix V depend- ing on contracting parameter ε. Moreover, V should depend on ε in a tricky way, such that all the trans- formed generators S′ab and εS ′ 0a are kept non-trivial and non-singular when ε → 0 [13]. In accordancewith [14, 15], representation (19) can be contracted either to the indecomposable represen- tation of hg(1,3) or to a direct sum of such represen- tations. To obtain an indecomposable representation, contractionmatrix V has to be chosen in the following form: V = ⎛ ⎜⎜⎜⎜⎝ ε 2 0 ε 2 0 0 I 0 0 −ε−1 0 ε−1 0 0 0 0 1 ⎞ ⎟⎟⎟⎟⎠ , V −1= ⎛ ⎜⎜⎜⎜⎜⎝ ε−1 0 − ε 2 0 0 I 0 0 ε−1 0 ε 2 0 0 0 0 1 ⎞ ⎟⎟⎟⎟⎟⎠ . (23) To apply the contraction procedure to the field equations (5)–(7) we first write them in vector nota- tions ∂0E−∇×B = κ(F0B−F×E)+ ej, (24) ∇·E = κF ·B+ ej0, (25) ∂0B+∇×E = 0, (26) ∇·B = 0, (27) ∂0F0 −∇·F = −κE ·B+ qj4, (28) ∂0F−∇F0 = 0, (29) ∇×F = 0. (30) Taking half sums and half divergences of pairs of equation (25) and (28), (24) and (29), (26) and (30) we come to a system equivalent to (24)–(30): ∂0F0 −∇· (F−E)= κ(F−E) ·B+ e ( j0 + q e j4 ) , ∂0F0 −∇· (F+E)= κ(F+E) ·B+ e ( j0 − q e j4 ) , ∂0(E+F)−∇×B−∇F0 = (31) κ(F0B− 1 2 (F−E)× (F+E))+ ej, ∂0(E−F)−∇×B+∇F0 = κ(F0B− 1 2 (F−E)× (F+E))+ ej, ∂0B+∇× (E+F)= 0, ∂0B+∇× (E−F)= 0, ∇·B=0. Defining Ψ′ = column(F′,B′,E′, F ′0) = V Ψ we obtain from (20), (23): E+F=2ε−1F′, B′ =B, F−E= εF′, F0 = F ′0 2ε−1j′4 = (q e j4 − j0 ) , εj′0 = q e j4 + j0, j′ = j. (32) Substituting (32) into (31), equating terms with low- est powers of ε and taking into account that relativis- tic variable x0 is related to non-relativistic time t as x0 = ct = ε −1t we obtain the following system: ∂tF ′ 0 −∇·E ′ + κB′ ·E′ = ej′0, ∂tF ′ +∇×B′ + κ(F ′0B ′ +F′ ×E′)= ej′, ∇·F′ + κF′ ·B′ = ej′4, ∂tB ′ +∇×E′ =0, ∇·B′ =0, ∇×F′ =0, ∂tF′ = ∇F ′0. (33) System of equations (33) coincides with theGalilei invariant system for the indecomposable ten compo- nent field deduced in [3]. Like the corresponding rel- ativistic equations (5), (6), system (33) admits a La- grangian formulation. The related Lagrangian can be obtained from (3) using the contractionprocedure and has the following form L = 1 2 (F ′0 2 −B′2)−E′ ·F′ + κ(A0B′ ·F′ −A′ · (B′F ′0 +F ′ ×E′))− (34) e(A′0j′4 + A′4j′0 − j′ ·A′). Just the system(33) andLagrangian(34) represent the Galilean limit of the model discussed in Sections 4 and 5. Exactly this lagrangianwas found in [3] start- ing with the Galilei invariance condition. 99 Acta Polytechnica Vol. 50 No. 3/2010 With the additional Galilei-invariant constraints F′ =0, j′4 =0, F ′0 = p0 = const system (33) is reduced to the form ∂B′ ∂t +∇×E′ =0, ∇·B′ =0, ∇·E′ + κB′ ·E′ = ej′0, ∇×B′ + κp0B′ = ej′. (35) Equations (35) representaGalilei-invariantversion of the CFJ model with time-like pμ. 7 Exact solutions One of important applications of Lie symmetries to partial differential equations (PDE) is connected with constructing their exact solutions. Lie algorithm for constructing exact solutions of differential equations has been known for more than 120 years, see, e.g., [10]. Various applications of this algorithm to relativistic systems can be found in [16]. In this section we present some group solutions of the relativistic system (5)–(7). Since the maximal Lie symmetry of this systemhasbeen foundandpresented in Section 6, it is possible to find its exact solutions using the following algorithm: 1. To find all non-equivalent three-dimensional subalgebras of the Lie algebra of group P̃(1,3) whose generators are given by formulae (17). 2. To find invariants of the related three- parametric Lie groups. 3. To choosenewvariables in suchaway that eight of themcoincideswith these invariants. As a resultwe obtain a system of ordinary differential equations. 4. To solve (if possible) the obtained systems of ordinary differential equations and reconstruct the re- lated solutions of the incoming system. The first step of the algorithm is reduced to using the classification results for the subalgebras of algebra P(1,3), which can be found in [17]. All the remaining steps are rather cumbersomebut algorithmic, and it is possible to find all exact solutions for systems (5)–(7), (12)–(14)which canbe obtainedviaLie reductionpro- cedure. Here we shall present only two examples of such solutions, while the complete list of them can be found in [18]. Let us startwith the subalgebra of p̃(1,3), spanned on the basis elements < J12, P1, P2 > which are given explicitlyby equations (17) and (18). There is the only invariant of the related group depending on x0, x1, x2 and x3, namely, ω = x 2 1 + x 2 2. In addition, there are seven invariants depending on both space-time and field variables, which we denote as ϕ1, ϕ2, · · · , ϕ7. They are supposed to be functions of ω such that B1 = ϕ1cosω − ϕ3 sinω, B2 = ϕ1 sinω + ϕ2 cosω, E1 = ϕ3 cosω − ϕ4 sinω, E2 = ϕ3 sinω + ϕ4cosω, B3 = ϕ5, E3 = ϕ6, A 4 = ϕ7. (36) Substituting (36) into (5)–(7)we reduce it to a sys- temof ordinarydifferential equationswhichappears to be integrable. Moreover, its general solution depends on six arbitrary parameters. A particular solution has the following form: B1 = −c1x2/ω, B2 = c1x1/ω, B3 = c3A4, (37) E1 = c2x1/ω, E2 = c2x2/ω, E3 = c3, (38) F0 = F3 =0, Fα = ∇αA4, α =1,2 (39) where A4 = c4J0(c3 √ kω)+ c5Y0(c3 √ kω), J0 and Y0 are the Bessel functions of the first and second kind, respectively, c1, · · · c5 are arbitrary parameters. It is interesting to note that functions (37) and (38) also solve the standard (linear)Maxwell equations with bounded currents. Namely, the electric field (38) coincides with the field of an infinite straight charged line coinciding with the third coordinate axis supple- mented by the constant electric field E3 = c3. The magnetic field (37) is a superposition of the field of a straight line current directed along the third coordi- nate axis and the field E3 = c3A4 generated by the current whose components are j1 = − x2√ ω A′4, j2 = x1√ ω A′4, j3 =0, where A ′ 4 = ∂A4 ∂ω . Let us present one more exact solution of the sys- tem (5)–(7): B1 = c x2 ω3/2 , B2 = c −x1 ω3/2 , B3 =0, E1 = c x1 ω3/2 , E2 = c x2 ω3/2 , E3 =0, F1 = x2 ω , F2 = − x1 ω , F3 = F0 =0, jμ =0, μ =0, · · · ,4. (40) In contrast with (37)–(39) vectors B and E defined in (40) do not solve linear Maxwell equations with bounded currents. However, they solve the system of nonlinear equations (5)–(7) for ω > 0. A complete set of reductions and exact solutions for equations (5)–(7) can be found in [20]. 8 Conclusion We have discused two MCS models in the (1 + 3)- dimensional space-time. One of them is presented in Sections 2 and 3. It generalizes the axion electrody- namics toa theorywithafive-componentcurrent. The other model includes axion electrodynamics as a lim- iting case corresponding to a small coupling constant. Thismodel includes anon-linearversionof theBianchi 100 Acta Polytechnica Vol. 50 No. 3/2010 identity. The other specific feature of thismodel is the existence of two conservedparts of the energy density, one of which corresponds to the tensor of the electro- magnetic field while the other one is formed by the additional four-vector field F μ. In contrast to the CFJ model, ourmodels are rela- tivistically invariant and include this model as a par- ticular sector corresponding to constant solutions for vector field F μ. Both the models have good non- relativistic limit, coinciding with the Galilei invariant system discussed in [3]. To obtain this limit we apply the Inönü-Wigner contraction,whichmakes it possible to find a Galilean formulation of the CFJ model. Using the classical Lie approach, we find continu- ous symmetries of our models and construct multipa- rameter families of their exact solutions. Two of these solutionsarepresented inSection7,while thecomplete list of group solutions can be found in preprint [18]. Note that solutions (37)–(39) and (40) give rise to new exactly solvable models for Dirac fermions. One of such models is presented in [19]. References [1] Snonfeld, J.: Nucl. Phys., B175 157, 1991; Deser, S., Jackiw, R., Templeton, S.: Ann. Phys. 140 372, 1982. [2] Carrol, S. M., Field, J. B., Jackiw, R.: Phys. Rev. D 41 1231, 1990. [3] Niederle, J., Nikitin, A. G.: J. Phys. A: Mat. Theor. 42 105207, 2009. [4] Chern, S. S., Simons, J.: Annals Math. 99 48, 1974. [5] Horvaty, P. A.: Lections on (Abelian) Chern- Simons vortices. ArXiv 0704.3220 [6] Hariton,A. J., Legnert,R.: Phys. Lett.A367 11, 2007. [7] Itin, Ya.: Phys. Rev. D 70 025012, 2004; Itin, Ya.: Wave propagation in axion electrody- namics, ArXiv 0706.2991. [8] Obukhov, Y. N., Hehl, F. W.: Phys. Lett. B 458, 466, 1999; Obukhov,Y.N., Rubilar,G.F.: Phys. Rev.D66, 2002. [9] Wilczek, F.: Phys. Rev. Lett. 58, 1799, 1987. [10] Olver, P.: Application of Lie groups to Differen- tial equations. Springer-Verlag, N.Y., 1986. [11] Le Bellac, M., Lévy-Leblond, J.-M.: Nuovo Ci- mento B 14, 217, 1973. [12] Holland,P., Brown,H.R.: Studies inHistory and Philosophy of Science 34, 161, 2003. [13] Inönü, E., Wigner, E. P.: Proc. Nat. Acad. Sci. U.S. 39, 510, 1953. [14] de Montigny, M., Niederle, J., Nikitin, A. G.: J. Phys. A: Mat. Theor. 39, 1, 2006. [15] Niederle, J., Nikitin, A. G.: Czech. J. Phys. 56, 1243, 2006. [16] Fushchich, W. I., Nikitin, A. G.: Symmetries of Equations of Quantum Mechanics. New York, Allerton Press, 1994. [17] Fushchich, W. I., Barannyk, A. F., Baran- nyk, L. F.: Subgroup structure of Lie groups. Naukova Dumka, Kiev, 1993. [18] Kuriksha, O., Nikitin, A. G.: Arxiv preprint arXiv:1002.0064, 2010. [19] Ferraro, E., Messina, N., Nikitin, A. G.: Phys. Rev. A 81, 042108, 2010, ArXiv 0909.5543. [20] Kuriksha,O.: Group analysis of (1+3)-dimensio- nal Maxwell-Chern-Simons models and their ex- act solutions. ArXiv 0911.3220. J. Niederle E-mail: niederle@fzu.cz Institute of Physics of the Academy of Sciences of the Czech Republic Na Slovance 2, 182 21 Prague, Czech Republic A. G. Nikitin E-mail: nikitin@imath.kiev.ua Institute of Mathematics National Academy of Sciences of Ukraine 3 Tereshchenkivs’ka Street, Kyiv-4, Ukraine, 01601 O. Kuriksha Institute of Mathematics National Academy of Sciences of Ukraine 3 Tereshchenkivs’ka Street, Kyiv-4, Ukraine, 01601 101