Exotic localized structures based on the symmetrical lucas function of the (2+1)-dimensional generalized Nizhnik-Novikov-Veselov system

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1 Turk J Phs 35 (11), c TÜBİTAK doi:1.396/fiz-19-1 Eotic localized structures based on the smmetrical lucas function of the (+1)-dimensional generalized Nizhnik-Novikov-Veselov sstem Emad A-B. ABDEL-SALAM 1, andzeidi.a.al-muhiameed 1 Assiut Universit, Department of Mathematics, New Valle Facult of Education El-Kharga, New Valle-EGYPT emad abdelsalam@ahoo.com Qassim Universit, Department of Mathematics, Facult of Science Buraida-SAUDI ARABIA ksapr6@ahoo.com Received: Abstract Based on the Lucas Riccati method and a linear variable separation method, new variable separation solutions of the (+1)-dimensional generalized Nizhnik-Novikov-Veselov sstem are derived. Then, we give a positive answer for the following question: Are there an localized ecitations derived b the use of another functions? For this purpose, some attention will be paid to dromion, peakon, multi dromion-solitoff ecitations, regular fractal dromions, stochastic fractal dromion structure and combined structures including bell-like compactons, peakon-like compactons and compacton-like semi-foldons based on the golden main. With the help of the modified Weierstrass function, we discuss the stochastic fractal dromion structure both analticall and graphicall. Finall, we conclude the paper and give some features and comments. Ke Words: Lucas functions, localized ecitations, variable separation solutions, generalized Nizhnik- Novikov-Veselov sstem PACS:.3.Jr, 5.45.Yv,.3.Ik 1. Introduction Nonlinear wave phenomena pla an important role in the phsical world and have been challenging topic of research for both mathematicians and phsicists. Man phenomena in phsics and other fields are described b nonlinear partial differential equations (NLPDEs). When we want to understand the phsical mechanism of phenomena in nature, described b NLPDEs, eact solutions for NLPDEs have to be eplored. There eists an etensive literature dealing with NLPDEs, in which eact solitar wave, kink wave and periodic wave solutions are discussed. Man powerful methods have been proposed b mathematicians and phsicists to obtain 41

2 special solutions of NLPDEs, including the inverse scattering method, Bäcklund and Darbou transformations [1 4], Hirota bi-linear method [5], homogeneous balance method [6], Jacobi elliptic function method [7, 8], tanh-function method [9, 1], etended tanh-function method [11 14], modified etended tanh-function method [15 17], F-epansion method, and so on [18 9]. There is well-known fact that two mathematical constants of Nature, the values π and e, plaagreat role in mathematics and phsics. Their importance lie in that the generate the main classes of so-called elementar functions : sine, cosine (the number π), eponential, logarithmic and hperbolic functions (the number e). However, there is the one more mathematical constant plaing a great role in modeling of processes in living nature termed the Golden Section, Golden Proportion, Golden Ratio, Golden Mean [3 36]. However, we should certif that a role of this mathematical constant is sometimes undeservedl humiliated in modern mathematics and mathematical education. There is the well-known fact that the basic smbols of esoteric (pentagram, pentagonal star, platonic solids, etc.) are connected to the Golden Section closel. Moreover, the materialistic science together with it s materialistic education had decided to throw out the Golden Section. However, in modern science, an attitude towards the Golden Section and connected to its Fibonacci and Lucas numbers is changing ver quickl. The outstanding discoveries of modern science based on the Golden Section have a revolutionar importance for development of modern science. These are enough convincing confirmation of the fact that human science approaches that of uncovering one of the most complicated scientific notions, namel, the notion of Harmon. Harmon is opposed to Chaos, hence meaning the organization of the Universe. In Euclid s The Elements we find a geometric problem called the problem of division of a line segment in the etreme and middle ratio. This problem is often called the golden section problem [3 36]. Solution of the golden section problem reduces to the algebraic equation = + 1, to which there are two roots. The positive root is α = 1+ 5, known as the golden proportion, golden mean, or golden ratio. El Naschie s works [3 36] develop the Golden Mean applications into modern phsics. In [36], devoted to the role of the Golden Mean in quantum phsics El Naschie concludes: In our opinion it is ver worthwhile enterprise to follow the idea of Cantorian space-time with all its mathematical and phsical ramifications. The final version ma well be a snthesis between the results of quantum topolog, quantum geometr and ma be also Rossler s endorphsics which like Nottale s latest work makes etensive use of the ideas of Nelson s stochastic mechanism. Thus, in the Shechtman s, Butusov s, Mauldin and Williams, El Naschie s, Vladimirov s works, the Golden Section occupied a firm place in modern phsics and it is impossible to imagine the future progress in phsical researches without the Golden Section. In our present paper, we review smmetrical Lucas functions [5 7] and we find new solutions of the Riccati equation b using these functions. Also, we devise an algorithm called Lucas Riccati method to obtain new eact solutions of NLPDEs. For a given NLPDE with independent variables =( = t, 1,, 3,..., s ) and dependent variable u, P ( u, u t, u i, u i j,...)=, (1) where P is in general a polnomial function of its argument and the subscripts denote the partial derivatives, in order to derive some new solutions with certain arbitrar functions, we assume that its solutions in the form u() = n a i () F i (ϕ()), () i= 4

3 with F = A + BF, F = d F, (3) dφ where = ( = t, 1,, 3,..., n )and A, Bare constants and the prime denotes differentiation with respect to ϕ ϕ(). To determine u eplicitl, one ma take the following steps: First, similar to the usual mapping approach, determine n b balancing the highest non-linear term(s) and the highest-order partial term(s) in the given NLPDE. Second, substituting equation () and equation (3) into the given NLPDE and collecting coefficients of polnomials of F, then eliminating each coefficient to derive a set of partial differential equations of a i (i =, 1,,..., n)andϕ. Third, solving the sstem of partial differential equations to obtain a i and ϕ. Substituting these results into equation (1), then a general formula of solutions of equation (1) can be obtained. Choose properl A and B in ODE equation (3) such that the corresponding solution F (ϕ) is one of the smmetrical Lucas function given bellow. Some definitions and properties of the smmetrical Lucas function are given in appendi A. Case 1: IfA =lnαand B = ln α, then equation (3) possesses solutions tls(φ), cotls(φ). Case : IfA = ln α and B = ln α, then equation (3) possesses the solution tls(ϕ) 1 ± secls(ϕ). Case 3: If A =lnα and B = 4 lnα, then equation (3) possesses the solution tls(ϕ) 1 ± tls (ϕ). Case 4: If A = then equation (3) possesses the solution 1 BF(ϕ). In section, we appl the Lucas Riccati method to obtain new localized ecitations. Also in section 3, we pa attention to dromion, peakon, multi dromion-solitoff ecitations, regular fractal dromions, stochastic fractal dromion structure and combined structures including bell-like compactons, peakon-like compactons and compacton-like semi-foldons based on the golden mean and the smmetrical hperbolic and triangular Lucas functions. Finall, we give conclusions and comments.. New variable separation solutions of the (+1)-dimensional generalized Nizhnik-Novikov-Vrselov sstem We consider here the (+1)-dimensional generalized Nizhnik-Novikov-Veselov (GNNV) sstem u t + ρu + βu + γu + δu 3α(uv) 3β(uw) =, v = u, w = u, (4) where ρ, β, γ, α and δ are arbitrar constants. For γ = δ =, the GNNV sstem degenerates to the usual (+1)-dimensional NNV sstem, which is an isotropic La etension of the classical (1+1)-dimensional shallow 43

4 water-wave KdV model. For ρ =1, β = γ = δ = in sstem (4), we get the smmetric NNV equation, which ma be considered as a model for an incompressible fluid. Some tpes of the soliton solutions of the GNNV equation have been studied b man authors. For instance, Boiti et al. [37] solved the GNNV equation via the inverse scattering transformation. Zhang obtained man eact solution of this sstem based on an etended homogeneous balance approach [38]. In [39], discussed new tpes of interactions between the multi-valued and the single-valued solitons of sstem (4). It is worth mentioning that this sstem has been widel applied in man branches of phsics, such as plasma phsics, fluid dnamics, nonlinear optics, etc. So, a good understanding of more solutions of the (+1)-dimensional GNNV sstem (4) is ver helpful. Now we appl the Lucas Riccati method to sstem (4). Balancing the highest order derivative term with the nonlinear term in sstem (4), we have the following ansätz: u(,, t) =a (,, t) +a 1 (,, t)f (ϕ(,, t)) + a (,, t)f (ϕ(,, t)), v(,, t) =b (,, t) +b 1 (,, t)f (ϕ(,, t)) + b (,, t)f (ϕ(,, t)), w(,, t) =c (,, t) +c 1 (,, t)f (ϕ(,, t)) + c (,, t)f (ϕ(,, t)), (5) where a (,, t) a,a 1 (,, t) a 1,a (,, t) a, b (,, t) b, b 1 (,, t) b 1, b (,, t) b,c (,, t) c, c 1 (,, t) c 1,c (,, t) c and ϕ(,, t) ϕ are functions to be determined. Substituting equation (5) with equation (3) into equation (4), and equating each of the coefficients of F (ϕ) to zero, we obtain a sstem of PDEs. Solving this sstem with the help of Maple, and selecting the variable separation ansätz ϕ(,,t) = f(,t) + g(,t), we obtain the solution: a = AB f g, a 1 =, a = B f g, b = ρf+ρabf 3 +ft+γf 3ρf, b 1 = Bf, b = B f, (6) c = βg+βabg3 +gt+δg 3βg, c 1 = Bg, c = B g, where f and g are two arbitrar functions of {, t} and {, t}, respectivel. Now, based on the solutions of equation (3), one can obtain new tpes of localized ecitations of the (+1)-dimensional GNNV sstem. We obtain the general formulae of the solutions of the (+1)-dimensional GNNV sstem u = AB f g + B f g F (f + g), (7) v = ρf +ρabf 3 + f t + γf 3ρf + Bf F (f + g)+b f F (f + g), (8) w = βg +βabg 3 + g t + δg 3βg + Bg F (f + g)+b g F (f + g), (9) B selecting the special values of the A, B and the corresponding function F, we have the following solutions of (+1)-dimensional GNNV sstem: u 1 = f g [ln α] ( 1+tLs (f + g)), (1) v 1 = ρf ρf 3 [ln α] + f t + γf 3ρf f ln αtls(f + g) + [ln α] f tls (f + g), (11) 44

5 w 1 = βg β[ln α] g 3 + g t + δg 3βg g ln αtls(f + g) + [ln α] g tls (f + g), (1) u = f g [ln α] ( 1+cotLs (f + g)), (13) v = ρf ρf 3 [ln α] + f t + γf 3ρf f ln αcotls(f + g) + [ln α] f cotls (f + g), (14) w = βg βg 3 [ln α] + g t + δg 3βg g ln αcotls(f + g) + [ln α] g cotls (f + g), (15) v 3 = ρf 1 ρf 3 [ln α] + f t + γf 3ρf w 3 = βg 1 βg3 [ln α] + g t + δg 3βg v 4 = ρf 8ρf 3 [ln α] + f t + γf 3ρf w 4 = βg 8βg 3 [ln α] + g t + δg 3βg u 3 = 1 [ ] ) 4 f g [ln α] ( 1+ tls(f + g), (16) 1 ± secls(f + g) 1 [ f ln α 1 [ g ln α [ u 4 =4f g [ln α] ( 1+4 [ 4f ln α [ 4g ln α u 5 = ] tls(f + g) + 14 [ 1 ± secls(f + g) f [ln α] tls(f + g) ]+ 14 [ 1 ± secls(f + g) g[ln α] ] tls(f + g), 1 ± secls(f + g) (17) ] tls(ξ), (18) 1 ± secls(ξ) ] ) tls(f + g) 1 ± tls, (19) (f + g) ] tls(f + g) 1 ± tls +16f [ln [ α] (f + g) ] [ tls(f + g) 1 ± tls +16g [ln α] (f + g) ] tls(f + g) 1 ± tls, (f + g) () ] tls(f + g) 1 ± tls, (f + g) (1) f g (f + g), () v 5 = ρf + f t + γf 3ρf f (f + g) + f (f + g), (3) w 5 = βg + g t + δg 3βg g (f + g) + g (f + g), (4) where f f(, t) andg g(, t) are two arbitrar variable separation functions. The potential U = u 5 has the form U = f g (f + g). (5) 45

6 3. Novel localized structures of the (+1)-dimensional GNNV sstem All rich localized coherent structures, such as non-propagating solitons, dromions, peakons, compactons, foldons, instantons, ghostons, ring solitons, and the interactions between these solitons [4 5], can be derived b the quantit U epressed b equation (5) with the help of the hperbolic and triangular functions. It is known that, for the (+1)-dimensional integrable models, there are man more abundant localized structures than in (1+1)-dimensional case because some tpes of arbitrar functions can be included in the eplicit solution epression [4 5]. Moreover, the periodic waves also have been studied b some authors. In this paper, we tr to give an answer for the following question: Are there an localized ecitations derived b the use of another functions? Fortunatel, the answer is still positive due to some arbitrariness of the functions f(, t) andg(, t)in the potential U given b the equation (5). In order to answer this question, some attention will be paid to dromion, peakon, dromion lattice, multi dromion-solitoff ecitations, regular fractal dromions, stochastic fractal dromion structure, combined structures, including bell-like compactons, peakon-like compactons and compacton-like semi-foldons based on the golden main and the smmetrical hperbolic and triangular Lucas functions for the potential field U in (+1) dimensions Dromion, peakon and dromion lattice ecitations AccordingtosolutionU, we first discuss its dromion ecitation which is one of the significant localized ecitations localized eponentiall in all directions are driven b multiple straight-line ghost solitons with some suitable dispersion relation. Also, multiple dromion solutions are driven b curved line and straight line solitons. When the simple selections of the functions f(, t) andg(, t) = g() aregiventobe M N f(, t) =1+ a i tls[k i ( + c i t)+ i ], g() =1+ b j tls(k j + j ), (6) i=1 where a i,k i,c i,b j,k j, i, j are arbitrar constants, and M and N are integers. We can obtain a twodromion ecitation for the phsical quantit U, as shown in Figure 1, and the parameter selections as j=1 M =, N = k 1 = k = c = c 1 = K 1 =1, 1 = 1 = =, a 1 = a =b 1 =. (7).1 a b.8 c U.1 U.6 U Figure 1. Time evolutional plots of an interaction between two travilling dromions for the potential U with functional forms (6) and (7); when (a) t = 5, (b) t =,and(c) t =5. 46

7 It is well-known that the interactions of solitons in (1+1)-dimensional nonlinear models are usuall considered elastic. That means there is no echange of an phsical quantities like the energ and the momentum among interacting solitons. That is, the amplitude, velocit and wave shape of a soliton do not undergo an change after the nonlinear interaction. But in higher dimensional non-linear models the interactions between solitar waves ma be completel elastic or non-completel elastic. From Figure 1, we show that the interaction between a dromion-dromion is non-completel elastic since their shapes and amplitudes are not completel preserved after interaction. Similarl, based on the field U we can obtain some important weak localized ecitations such as peakons (weak continuous solution) which is discontinuous at its crest [u(, t) = k + ce ( ct ), k ], was first found in the celebrated (1+1)-dimensional Camassa-Holm equation [53] u t +ku u t +3uu =u u + uu. We find man (+1)-dimensional nonlinear models also possess these soliton ecitations [15 33]. We select f(, t) andg() to be piecewise smooth functions f(, t) =1+ { M i=1 f i ( + c i t) + c i t, f i ( c i t) + c i t>, g() =1+ N { gi (), g i ( ) >, i=1 (8) where the functions f i ( + c i t)and g i () are differentiable functions of the indicated arguments and possess boundar condition f i (± ) =C ±i,i=1,,...,m and g i (± ) =D ±i,i=1,,...,n with C ±i and D ±i being constants, even infinit. For instance, when choosing f 1 =.1α +t, f =.5α t g() =.1α, M =N =, (9) we can derive a propagating two-peakon ecitation for the potential field U. The corresponding two-peakon ecitation profile is depicted in Figure. Also, a simple eample of peaked solitar waves with periodic behavior is depicted in Figure 3, in which the selection function is f 1 =.1α +t, f =.5α t g() =α st Ls(), (3) where st Ls() = 1 i (αi α i ), (i = 1 )is the smmetrical triangular Lucas sine function [33]..6.5 a b.4 c Figure. Time evolutional plots of the two peaked solitar waves for the potential U with functional forms (8) and (9); when (a) t = 5, (b) t =,and(c)t =5. 47

8 From figure, we show that the interaction between a peakon-peakon is non-completel elastic since their shapes and amplitudes are not completel preserved after interaction. Also with the help of figure 3, one can easil sa that the multi-peakon ecitation possess periodic behavior..8 a b c Figure 3. Time evolutional plots of the two peaked solitar waves with periodic behavior for the potential U with the functional forms (8) and (3); when (a) t = 5, (b) t =,and(c)t = Multi dromion-solitoff ecitations According to the potential U in the form of its multi dromion-solitoff ecitations. That can be epressed b means of Lucas functions in the form N M f(, t) =. secls( +5n + t), g() =. secls( +5m). (31) n= N m= M If m = n =, we can obtain a multi dromion-solitoff ecitation for the phsical quantit U depicted in Figure 4witht =..1.5 U Figure 4. A plot of a special tpe of multi dromion-solitoff structure for the phsical quantitu with the choice of relation (31) and t =. 48

9 For instance, when f(, t)and g() are f(, t) = g() = N n= N M m= M [. secls( +5n + t)+. secls( +5n t)],. secls( +5m), (3).1.5 U a U b U c Figure 5. Time-evolution plots of an interaction between two multi dromion-solitoffs for the potential U with functional form (3); when (a) t = -5; (b) t =,and(c)t = 5. We can obtain the interactions between two multi dromion-solitoffs. Figure 5 shows an evolutional profile corresponding to the phsical quantit U. From Figure 5 and through detailed analsis, we find that the shapes, amplitudes, and velocities of the two multi dromion-solitoffs are completel conserved after their interactions. Consequentl, the interactions between two multi dromion-solitoffs are completel elastic. Now we focus our attention on the intriguing evolution of a dromion in a background wave for the potential field U. For instance, if we choose f(, t) andg() as f(, t) =3+.1 tls(.5 t)+. sn(.4,.3), g() =3+.1 tls(.5)+. sn(.4,.3), (33) where, sn denotes the Jacobi sine function, we can obtain an evolutional profile of single-dromion in the background wave for the phsical quantit U presented in Figure 6 at different times when (a) t = 1, (b) t =, and(c) t = 1. From Figure 6 and through detailed analsis, this figure shows the corresponding profile of the comple wave ecitation presenting the propagation of a dromion moving on the determined double periodic wave background. However, its wave shape and wave velocit do not suffer an change, which is ver close to man actual phsical processes in the natural world. 49

10 Ge-5 4e-5 U e-5 a Ge-5 4e-5 U e-5 b Ge-5 5e-5 4e-5 3e-5 U e-5 1e-5-1e-5 c Figure 6. Time evolutional plots of the single dromion in the background wave for the potential U with selection (33), when (a) t = -1, (b) t =,and(c) t = Regular fractal dromions and stochastic fractal dromion structure Recentl, it has been found that man lower-dimensional piecewise smooth functions with fractal structure can be used to construct eact localized solutions of the higher-dimensional soliton sstem which also possesses fractal structures. If we appropriatel select the arbitrar functions f(, t) andg(), we find that some special tpes of fractal dromions for the potential U can be revealed. For eample, if we take f(, t) =1+α t [ct Ls(ln( t) )+st Ls(ln( t) )], g() =1+α [ct Ls(ln() )+st Ls(ln() )], (34) where ct Ls() =α i + α i, (i = 1 ) is the smmetrical triangular Lucas cosine function [33], then we can obtain a simple fractal dromion.. b 1 U -1 a Figure 7. (a) Fractal dromion structure for the potential U with the choice if functional form (34), at t =. (b) Densit of the fractal structure of the dromion in the region {= [-.,.], = [-.,.]}. From Figure 7 and through detailed analsis, Figure 7(a) shows a plot of this special tpe of fractal dromion structure for the potential U given b equation (5) with the selection functions (34) for t =. Figure 5

11 7(b) shows the densit of the fractal structure of the dromion in the region {= [-.,.], = [-.,.]}. To observe the self-similar structure of the fractal dromion clearl, one ma enlarge a small region near the centre of Figure 7(b). For instance, if we reduce the region of figure 7(b) to {= [-.,.], = [-.,.]}, { = [-.1,.1], = [-.1,.1]}, and so on, we find a totall similar structure to that presented in Figure 7(b). In addition to the regular fractal dromion, the lower-dimensional stochastic fractal functions ma also be used to construct high-dimensional irregular stochastic or chaotic fractal localized ecitations. B the help of the smmetrical triangular Lucas sine function [33], we define the modified Weierstrass function as (ξ) = M λ (q )j st Ls(λ j ξ), M, (35) j= where λ>1, q<and the independent variable ξ ma be a suitable function of {, t} or { }, saξ = + vt and ξ = in the functions f(, t) andg(), respectivel, for selections f(, t) = ( + vt)+ ( + vt) + 1, g() = () + + 1, (36) and and f(, t) =. ( + vt) tt Ls(3[ + vt] ) + 1, g() =.4 tt Ls()+.1 tt Ls( +8)+.8 tt Ls( 16) +1, f(, t) =. ( + vt) tt Ls(3[ + vt] ) + 1, g() =. () tt Ls(3 ) + 1. (37) (38) If the modified Weierstrass function is included in the field U, then we can derive some stochastic fractal patterns. Figure 8 shows three plots of tpical stochastic fractal patterns for the potential function U given b equation (5) under the function selections equations (35) (38) with the parameters λ = 1.5, q = 1att =. From Figure 8, one can find that the amplitude of the localized or non-localized patterns is randoml changed. a 1e+1 5e+11-5e+11-1e e b 8e+14 6e+14 4e+14 e+14 -e+14-4e+14-6e+14 1e c Figure 8. Stochastic fractal dromion structure for the potential U : (a) with selections (35) and (36), (b) with selections (35) and (37), (c) with selections (35) and (38) with λ =1.5, q =1 at t = 51

12 3.4. Special localized structures Based on the phsical quantit (5), multi-valued localized structures can be discussed, when the function g() is a single-valued function and f(, t) is selected via the relations M M f = F j (ξ + ω j t), = ξ + X j (ξ + ω j t), (39) j=1 j=1 where ω j are all arbitrar constants and F j,x j are all localized ecitations with properties F j (± ) =, X j (± )=. From the second equation of equation (39), one knows that ξ ma be a multi-valued function in some suitable regions of b selecting the functions X j appropriatel. Therefore, the function f, which is obviousl an interaction solution of M localized ecitations since the propert ξ ma be a multi-valued function of in these area though it is a single valued functions of ξ. Actuall, most of the known multi-loop solutions are the special situations of equation (39). Now, some attention will be paid to combined structures, including bell-like compactons, peakon-like compactons and compacton-like semi-foldons based on the golden main and the smmetrical hperbolic Lucas functions: f =.5secLs (ξ), = ξ A tls(ξ), (4) ln α g() =8+, π, sin +1, π << π, (41) > π, where A is a characteristic parameter, determining the localized structure. Figure 9 describe three localized structures, i.e. bell-like compacton, peakon-like compacton and compacton-like semi-foldon with A =.5,.95, 1.5, respectivel. The localize as a compacton in the -direction and a bell-like soliton, peakon and loop soliton in the -direction, respectivel a.5 b. c Figure 9. (a) Bell-like compacton structure for the potential U with selections (4) and (41) when A =.5,(b) Peakon-like compacton structure for the potential U with selections (4) and (41) when A =.95,(c) Compacton-like semi-foldon structure for the potential U with selections (4) and (41) when A =

13 4. Summar and discussion In conclusion, the Lucas Riccati method is applied to obtain variable separation solutions of the (+1)- dimensional GNNV sstem. With the help of quantit (5), some localized ecitations, such as dromion, peakon, multi dromion-solitoff ecitations, regular fractal dromions, combined structures, including bell-like compactons, peakon-like compactons and compacton-like semi-foldons based on the golden main and the smmetrical hperbolic and triangular Lucas functions. With the help of the modified Weierstrass function, we discuss the stochastic fractal dromion structure both analticall and graphicall. We think that all localized solutions of the (+1)-dimensional GNNV sstem can not be constructed in general for all localized solutions of the sstem. We hope that in future eperimental studies these localized ecitations obtained here can be realized in some fields. Actuall, our present short paper is merel a beginning work; more application to other nonlinear phsical sstems should be conducted and deserve further investigation. In our future work, on the one hand, we devote to generalizing this method to other (+1)-dimensional nonlinear sstems such as the ANNV sstem, BKK sstem, KdV sstem, Boiti-Leon-Pempinelle sstem etc. On the other hand, we will look for more interesting localized ecitations. 5. Appende Stakhov and Rozin in [3] introduced a new class of hperbolic functions that unite the characteristics of the classical hperbolic functions and the recurring Fibonacci and Lucas series. The hperbolic Fibonacci and Lucas functions, which are an etension of Binet s formulas for the Fibonacci and Lucas numbers in continuous domain, transform the Fibonacci numbers theor into continuous theor because ever identit for the hperbolic Fibonacci and Lucas functions has its discrete analog in the framework of the Fibonacci and Lucas numbers. Taking into consideration a great role plaed b the hperbolic functions in geometr and phsics, ( Lobatchevski s hperbolic geometr, Four-dimensional Minkowski s world, etc.), it is possible to epect that the new theor of the hperbolic functions will bring to new results and interpretations on mathematics, biolog, phsics, and cosmolog. In particular, the result is vital for understanding the relation between transfinitness i.e. fractal geometr and the hperbolic smmetrical character of the disintegration of the neural vacuum, as pointed out b El Naschie. The definition and properties of the smmetrical Lucas functions The smmetrical Lucas sine function (sls), the smmetrical Lucas cosine function (cls) and the smmetrical Lucas tangent function (tls) are defined [3, 31] as sls() =α α, cls() =α + α, tls() = α α α + α. (I) The are introduced to consider so-called smmetrical representation of the hperbolic Lucas functions and the ma present a certain interest for modern theoretical phsics taking into consideration a great role plaed b the Golden Section, Golden Proportion, Golden ratio, Golden Mean in modern phsical researches [3, 31]. The smmetrical Lucas cotangent function (cotls) is cotls() = 1 tls(), the smmetrical Lucas secant function (secls) is secls() = 1 1, the smmetrical Lucas cosecant function (cscls) is cscls() = cls() sls(). These functions satisf the following relations [3 31] 53

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