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Commun. Theor. Phys. (Beijing, China) 52 (2009) pp. 1004–1012c Chinese Physical Society and IOP Publishing Ltd Vol. 52, No. 6, December 15, 2009

Quasi-Periodic Structures Based on Symmetrical Lucas Function of (2+1)-Dimensional

Modified Dispersive Water-Wave System

Emad A-B. ABDEL-SALAM∗

Assiut University, Department of Mathematics, New Valley Faculty of Education, El-Kharga, New Valley, Egypt

(Received December 24, 2008)Abstract By introducing the Lucas–Riccati method and a linear variable separation method, new variable separationsolutions with arbitrary functions are derived for a (2+1)-dimensional modified dispersive water-wave system. The mainidea of this method is to express the solutions of this system as polynomials in the solution of the Riccati equation thatthe symmetrical Lucas functions satisfy. From the variable separation solution and by selecting appropriate functions,some novel Jacobian elliptic wave structure with variable modulus and their interactions with dromions and peakons are investigated.

PACS numbers: 02.30.Jr, 05.45.Yv, 03.65.GeKey words: Lucas functions, quasi-periodic structure, variable separation excitations, modified dispersive

water-wave system

1 Introduction

Due to the wide applications of soliton theory in math-ematics, physics, chemistry, biology, communications, as-

trophysics, and geophysics, the study of integrable models

has attracted much attention of many mathematicians and

physicists. To find some exact explicit soliton solutions

for integrable models is one of the most important and

significant tasks. There has been a great amount of activ-

ities aiming to find methods for exact solution of nonlin-

ear partial differential equations (NLPDEs). Such include

the Backlund transformation, Darboux transformation,[1]

various tanh methods,[2−6] various Jacobi elliptic function

methods,[7−8] multi-linear variable separation approach,[9]

Painleve property, homogeneous balance method, similar-ity reduction method and so on.[41−43]

For a given NPLDE with independent variables x =

(x0 = t, x1, x2, x3, . . . , xn) and dependent variable u,

P ( u, ut, uxi, uxixj , . . .) = 0 , (1)

where P is in general a polynomial function of its argu-

ment, and the subscripts denote the partial derivatives, by

using the traveling wave transformation, Eq. (1) possesses

the following ansatz,

u = u(ξ), ξ =m

i=0

ki xi , (2)

where ki, i = 0, 1, 2, . . . , m are all arbitrary constants.

Substituting Eq. (2) into Eq. (1) yields an ordinary differ-

ential equation (ODE):

O(u(ξ), u(ξ)ξ, u(ξ)ξξ , . . .) = 0. Then u(ξ) is expanded

into a polynomial in g(ξ)

u(ξ) = F (g(ξ)) =

ni=0

ai gi(ξ) , (3)

where ai are constants to be determined and n is fixed

by balancing the linear term of the highest order with thenonlinear term in Eq. (1). If we suppose g(ξ) = tanh ξ,

g(ξ) = sech ξ, and g(ξ) = s n ξ or g(ξ) = c n ξ respec-

tively, then the corresponding approach is usually called

the tanh-function method, the sech-function method, and

the Jacobian-function method. Although the Jacobian el-

liptic function method is more improved than the tanh-

function method and the sech-function method, the re-

peated calculations are often tedious since the different

function g(ξ) should be treated in a repeated way. The

main idea of the mapping approach is that, g(ξ) is not as-

sumed to be a specific function, such as tanh, sech, sn, and

cn, etc., but a solution of a mapping equation such as theRiccati equation (gξ = g2 + a0), or a solution of the cubic

non-linear Klein Gordon (g2ξ = a4g4 + a2g2 + a0), or a

solution of the general elliptic equation ( g2ξ =4

i=0 ai gi),

where ai (i = 0, 1, . . . , 4) are all arbitrary constants. Using

the mapping relations and the solutions of these mapping

equations, one can obtain many explicit and exact trav-

eling wave solutions of Eq. (1). Now an interesting or

important question is that whether the localized excita-

tions based on the former multilinear approach[9] can be

derived by the latter mapping approach, which is usually

used to search for traveling wave solutions. The crucial

technic is how to obtain some solutions of Eq. (1) withcertain arbitrary functions, also the quite rich localized ex-

citations, such as lumps, dromions, peakons, compactons,

foldons, ring solitons, fractal solitons, chaotic solitons and

so on[9−14,17−31,39] are obtained, and the novel interac-

tive behavior among the same types and various types of

soliton excitations are revealed.

There is a well-known fact that two mathematical con-

stants of nature, the π- and e-numbers, play a great role

∗E-mail: emad [email protected]



No. 6 Quasi-Periodic Structures Based on Symmetrical Lucas Function of (2+1)-Dimensional · · · 1005

in mathematics and physics. Their importance consists

in the fact that they “generate” the main classes of so-called “elementary functions”: sin, cosine (the π-number),exponential, logarithmic and hyperbolic functions (the e-

number). It is impossible to imagine mathematics andphysics without these functions. For example, there is thewell-known greatest role of the classical hyperbolic func-

tions in geometry and in cosmological researches. How-ever, there is the one more mathematical constant play-ing a great role in modeling of processes in living nature

termed the Golden Section, Golden Proportion, GoldenRatio, Golden Mean.[32−36] However, we should certifythat a role of this mathematical constant is sometimes un-

deservedly humiliated in modern mathematics and math-ematical education. There is the well-known fact thatthe basic symbols of esoteric (pentagram, pentagonal star,

platonic solids etc.) are connected to the Golden Sec-tion closely. Moreover, the “materialistic” science to-gether with its “materialistic” education had decided to

“throw out” the Golden Section. However, in modern sci-

ence, an attitude towards the Golden Section and con-nected to its Fibonacci and Lucas numbers is changing

very quickly. The outstanding discoveries of modern sci-ence based on the Golden Section have a revolutionaryimportance for development of modern science. These

are enough convincing confirmation of the fact that hu-man science approaches to uncovering one of the mostcomplicated scientific notions, namely, the notion of Har-

mony, which is based on the Golden Section, Harmonywas opposed to Chaos and meant the organization of theuniverse. In Euclid’s the elements we find a geometric

problem called “the problem of division of a line segment

in the extreme and middle ratio”. Often this problemis called the golden section problem.[33−36] Solution of the

golden section problem reduces to the following. algebraicequation x2 = x + 1 this equation has two roots. We callthe positive root, α = (1 +

√5)/2, the golden proportion,

golden mean, or golden ratio. El Naschie’s works[34−38] de-velop the golden mean applications into modern physics.In the paper[38] devoted to the role of the Golden Meanin quantum physics El Naschie concludes the following:“In our opinion it is very worthwhile enterprise to followthe idea of Cantorian space-time with all its mathematical

and physical ramifications. The final version may well be a

synthesis between the results of quantum topology, quan-tum geometry and may be also Rossler’s endorphysics,which like Nottale’s latest work makes extensive use of

the ideas of Nelson’s stochastic mechanism”. Thus, inthe Shechtman’s, Butusov’s, Mauldin and Williams’, El-Naschie’s, Vladimirov’s works, the Golden Section occu-

pied a firm place in modern physics and it is impossible toimagine the future progress in physical researches withoutthe Golden Section.

In our present paper, we review symmetrical Lucasfunctions[39] and we find new solutions of the Riccati equa-

tion by using these functions. Also, we devise an algorithm

called Lucas–Riccati method to obtain new exact solutions

of NLPDEs. Along with the above line, i.e., in order to de-

rive some new solutions with certain arbitrary functions,

we assume that its solutions in the form,

u(x) =

ni=0

ai(x) F i(x) , (4)

with

F ′ = A + B F 2 , (5)

where x = (x0 = t, x1, x2, x3, . . . , xn) and A, B are con-

stants and the prime denotes differentiation with respect

to ξ. To determine u explicitly, one may take the following

steps: First, similar to the usual mapping approach, de-

termine n by balancing the highest non-linear terms and

the highest-order partial terms in the given NLPDE. Sec-

ond, substituting (4) and (5) into the given NLPDE and

collecting coefficients of polynomials of F , then eliminat-

ing each coefficient to derive a set of partial differential

equations of ai (i = 0, 1, 2, . . . , n) and ξ. Third, solv-ing the system of partial differential equations to obtainai and ξ. Substituting these results into (4), then a gen-

eral formula of solutions of equation (1) can be obtained.

Choose properly A and B in ODE (5) such that the cor-

responding solution F (ξ) is one of the symmetrical Lucas

function given bellow. Some definitions and properties of

the symmetrical Lucas function are given in Appendix.

Case 1 If A = ln α and B = − ln α, then (5) possesses

solutions

tLs(ξ), cotLs(ξ) .

Case 2 If A = ln α/2 and B = −ln α/2, then (5) pos-

sesses a solutiontLs(ξ)

1± secLs(ξ).

Case 3 If A = ln α and B = −4 ln α, then (5) possessesa solution

tLs(ξ)

1± tLs2(ξ).

In the following section we apply the Lucas–Riccati

method to obtain new localized excitations. Also, we payour attention to some novel Jacobian elliptic wave struc-

ture with variable modulus and their interactions withdromions and peakons.

2 New Variable Separation Solutions of (2+1)-Dimensional Modified Dispersive Water-Wave System

We consider here the (2+1)-dimensional modified dis-

persive water-wave (MDWW) system

uyt + uxxy − 2 vxx − (u2)xy = 0 ,

vt − vxx − 2(vu)x = 0 . (6)



1006 Emad A-B. ABDEL-SALAM Vol. 52

The (2+1)-dimensional MDWW system was used to

model nonlinear and dispersive long gravity waves travel-

ing in two horizontal directions on shallow waters of uni-

form depth, and can also be derived from the well-known

Kadomtsev–Petviashvili (KP) equation using the symme-

try constraint.[15−16] Abundant propagating localized ex-

citations were derived by Tang et al .[9] with the help of

Painleve–Backlund transformation and a multilinear vari-able separation approach. It is worth mentioning that

this system has been widely applied in many branches of

physics, such as plasma physics, fluid dynamics, nonlinear

optics, etc. So, a good understanding of more solutions of

the (2+1)-dimensional MDWW system (6) is very help-

ful, especially for coastal and civil engineers in applying

the nonlinear water model in harbor and coastal design.

Meanwhile, finding more types of solutions to system (6)

is of fundamental interest in fluid dynamics.

Now we apply the Lucas–Riccati method to Eqs. (6).

First, let us make a transformation of the system (6):

v = uy. Substituting this transformation into system (6),yields

uty − uxxy − (u2)xy = 0 . (7)

Balancing the highest order derivative term with the non-

linear term in Eq. (7), gives n = 1, we have the ansatz

u(x, y, t) = a0(x, y, t) + a1(x, y, t)F (ϕ(x, y, t)) , (8)

where a0(x, y, t) ≡ a0, a1(x,y,t) ≡ a1, and ϕ(x, y, t) ≡ ϕ

are arbitrary functions of x, y, t to be determined. Sub-

stituting (8) with (5) into (6), and equating each of the

coefficients of F (ϕ) to zero, we obtain system of PDEs.

Solving this system of PDEs, with the help of Maple, we

obtain the following solution:

a0(x, y, t) = −ϕxx(x, y, t) − ϕt(x, y, t)

2 ϕx(x, y, t), (9)

a1(x, y, t) = −B ϕx(x, y, t) , (10)

ϕ(x, y, t) = f (x, t) + g(y) , (11)

where f (x, t) ≡ f and g(y) = g are two arbitrary functions

of x, t, and y, respectively.

Now based on the solutions of (5), one can obtain new

types of localized excitations of the (2+1)-dimensional

MDWW system. We obtain the general formulae of the

solutions of the (2+1)-dimensional MDWW system

u = −f xx − f t2f x

−Bf x F (f + g) , (12)

v = −ABf xgy −B2f x gyF 2(f + g) . (13)

By selecting the special values of the A, B and the cor-

responding function F we have the following solutions of

(2+1)-dimensional MDWW system:

u1 = −f xx − f t2f x

+ f xtLs(f + g) ln α , (14)

v1 = f xgy ln α2 − f xgytLs2( f + g) ln α2 , (15)


+ f xcotLs(f + g) ln α , (16)

v2 = f xgy ln α2 − f x gycotLs2(f + g) ln α2 , (17)


+f xtLs(f + g) ln α

2[1± secLs(f + g)], (18)

v3 =f x gy ln α2

4− f x gy ln α2

4

tLs(f + g)

1± secLs(f + g)

2, (19)


+ 4f x tLs(f + g) ln α1 + tLs2(f + g)

, (20)

v4 = 4f x gy ln α2 − 16f x gy ln α2 tLs(f + g)

1 + tLs2(f + g)

2, (21)

where f (x, t) and g(y) are two arbitrary variable separa-

tion functions. Especially, for the potential U = u1y has

the following form

U = 4 f x gy sec Ls2(f + g) ln α2 . (22)

3 Periodic and Quasi-Periodic and Waves of (2+1)-Dimensional MDWW System

All rich localized coherent structures, such as non-

propagating solitons, dromions, peakons, compactons,

foldons, instantons, ghostons, ring solitons, and the in-

teractions between these solitons,[9−14,17−31,39] can be

derived by the quantity U expressed by (22). It is

known that for the (2+1)-dimensional integrable mod-

els, there are many more abundant localized structures

than in (1+1)-dimensional case because some types of

arbitrary functions can be included in the explicit solu-

tion expression.[9] Moreover, the periodic waves also have

been studied by some authors. However, the quasi- and

non-periodic wave solutions of the integrable systems havebeen given limited attention with a fewer publications in

the literature. In the present paper, we pay our attention

to some periodic, quasi-periodic, and non-periodic wave

evolutional behaviors for the field U in (2+1) dimensions.

3.1 Doubly Periodic and Line Periodic Solitary

Waves

It is known that for the nonlinear system, the peri-

odic (and doubly periodic) wave solutions can usually be

expressed by means of the Jacobi elliptic functions with

constant modulus. If we takef = cn(k1x + ω1t + x01; m1) ,

g = cn(K 1y + y01; n1) , (23)

of (22) leads to a periodic solution for the potential U . In

(23), cn(k1x+ ω1t+ x01; m1), cn(K 1y + y01; n1) are the Ja-

cobian elliptic cn functions with the modulus m1, n1, and

x01, y01, k1, K 1 are arbitrary constants. Figure 1 shows

the detailed structures of (22) with (23) and

k1 = ω1 = K 1 = 1, x01 = y01 = 12 , (24)




when t = 0. Figure 1(a) is related to the modulus of the

Jacobian cn functions being taken as m1 = n1 = 0.6. Fig-

ure 1(b) shows the density plot of 1(a). If we select the

functions f and g such that one of them possesses a lo-

calized structure while the other has a periodic structure,

then the wave solution (22) becomes a line-periodic wave

excitation. For example, the selection

f = cn(k1x + ω1t + x01; m1) ,

g = tanh(K 1y + y01) , (25)

makes the wave solution (22) to be line-periodic solitary

wave. Figure 2 displays the structure of equation (22)

with the condition (25) and the parameter selections as

k1 = K 1 = 0.6, ω1 = 1, x01 = y01 = 8 , (26)

at time t = 0. Figure 2(a) is related to the modulus of the

Jacobian cn functions being taken as m1 = 0.4.

Fig. 1 The structures of doubly-periodic solution (22) with (23) and (24): (a) m1 = n1 = 0.6; (b) the density plot of (a) when t = 0.

Fig. 2 The structures of line-periodic solution (22) with (23) and (24): (a) m1 = 0.4; (b) the density plot of (a) whent = 0.

3.2 Quasi-Periodic Wave

In this subsection, we give doubly quasi-periodic(quasi-periodic in both x and y directions) wave solutions

by selecting the arbitrary functions as Jacobi elliptic func-

tions with variable modulus. To see more concretely, wetake some concrete selections for the arbitrary functionsf and g for the (2+1)-dimensional MDWW system. Thefirst example, we take f and g as

f = cn(ξ1; m(ξ2)) ,

m = 0.3 + 0.2 tanh(ξ2) + 0.4 tanh2(ξ2) ,

g = cn(ζ 1; n(ζ 2)) ,

n = 0.3 + 0.2 tanh(ζ 2) + 0.4 tanh

2

(ζ 2) ,ξi = kix + ωit + x0i, ζ i = K iy + y0i, i = 1, 2 , (27)

then Eq. (22) denotes a doubly quasi-periodic wave solu-

tion. Figure 3 displays the structure of Eq. (22) with the

condition (27) and the parameter selection as

k1 = k2 = 0.2, K 1 = K 2 = 0.4, ω1 = ω2 = 1 ,

x01 = y01 = x02 = y02 = 6 , (28)

at time t = 0.




Fig. 3 (a) The structures of quasi-periodic solution (22) with (27) and (28) (b) the density plot of (a) when t = 0.

Fig. 4 (a) The structures of quasi-periodic solution (22) with (29) and (30); (b) the density plot of (a) when t = 0.

Fig. 5 (a) The structures of quasi-periodic solution (22) with (31) and (32); (b) the density plot of (a) when t = 0.

The second selection of the functions f and g reads

f = sn(ξ1; m(ξ2)) ,

m = 0.3 + 0.2sech(ξ2) + 0.4 sech2(ξ2) ,

g = sn(ζ 1; n(ζ 2)) ,

n = 0.3 + 0.2sech(ζ 2) + 0.4 sech2(ζ 2) ,

ξi = kix + ωit + x0i ,

ζ i = K iy + y0i, i = 1, 2 , (29)

then Eq. (22) denotes another doubly quasi-periodic wavesolution. Figure 4 displays the structure of Eq. (22) withthe condition (29) and the parameter selections as

k1 = k2 = 0.1, K 1 = K 2 = 0.2, ω1 = ω2 = 1 ,

x01 = y01 = x02 = y02 = 4, (30)

at time t = 0. The last selection of the functions f and greads

f = cn(ξ1; m(ξ2)) ,




m = 0.6 + 0.3 tanh(ξ2) ,

g = cn(ζ 1; n(ζ 2)) ,

n = 0.6 + 0.3 tanh(ζ 2) ,

ξi = kix + ωit + x0i ,

ζ i = K iy + y0i, i = 1, 2 , (31)

then Eq. (20) denotes another doubly quasi-periodic wave

solution. Figure 5 displays the structure of Eq. (22) with

the condition (31) and the parameter selections as

k1 = k2 = 0.2, K 1 = K 2 = 0.3, ω1 = ω2 = 1 ,

x01 = y01 = x02 = y02 = 3 , (32)

at time t = 0. The idea to introduce the variable modulus

into the conoidal wave solutions is simple but important

and never been bethought. The introduction of the vari-

able modulus in the Jacobi elliptic functions is essential

and it results completely different types of waves solutions.

3.3 Interaction of Quasi-Periodic Waves and Dromions

Fig. 6 (a) The interaction of dromion and quasi-periodic solution (22) with (33) and (34); (b) the density plot of (a)when t = 0.

Fig. 7 (a) The interaction of dromion and line quasi-periodic solution (22) with (33) and (34) when m = 1; (b) thedensity plot of (a) when t = 0.

The dromion solutions which are localized in all di-

rections are driven by multiple straight-line ghost soli-

tons with some suitable dispersion relation. Also, multiple

dromion solutions are driven by curved line and straight

line solitons. Here, we study the interaction between this

localized structure in the background of Jacobian elliptic

wave and the Jacobian elliptic waves with variable modu-

lus. If we take

f = cn(k1x + ω1t + x01; m) + tanh(k1x + ω1t + x01) ,

g = sn(ζ 1; n(ζ 2)), n = 0.3 + 0.2 tanh(ζ 2) ,

ζ i = K iy + y0i, i = 1, 2 , (33)

from (22) we can obtain the interaction between the Jaco-

bian elliptic waves with variable modulus and dromions.

In order to understand the properties of the interaction

between the periodic and quasi-periodic waves, we illus-

trate it by several figures. Figure 5 shows the detail on

the interaction properties of the quasi-periodic wave and




dromions with the parameter values

K 1 = k1 = 1, K 2 = k2 = 2, ω1 = 3 ,

x01 = y01 = y02 = 0, m = 0.3 , (34)

and t = 0, respectively. It is very interesting to see the

interaction properties of the Jacobian elliptic waves and

dromions in the limit case. As m is close to 1, from (22) we

can obtain the interaction of dromions and line-quasi pe-riodic solution depicted in Fig. 7 with the same parameter

values as Fig. 6.

3.4 Interaction of Quasi-Periodic Waves and Peakons

Fig. 8 (a) The interaction of peakon and quasi-periodic solution (22) with (35) and (36) when m = 0.4; (b) the densityplot of (a) when t = 0.

Fig. 9 (a) The interaction of peakon and quasi-line periodic solution (22) with (35) and (36) when m = 1; (b) thedensity plot of (a) when t = 0.

The celebrated (1+1)-dimensional Camassa–Holm equation[40]

ut + 2kux − uxxt + 3uux = 2uxuxx + uuxxx ,

possesses a special type of novel solution, namely the peakon solution (weak continuous solution) which is discontinuous

at its crest.u(x, t) = −k + c e(−|x−ct|), k → 0 .

Moreover, we can also discuss the interaction between Jacobian elliptic waves with variable modulus and peakon with

the background of Jacobian elliptic wave. When we consider

f = cn(kx + ωt + x0; m) +

M i=1

F i(kix + ωit), kix + ωit ≤ 0 ,

−M i=1

F i(kix + ωit) + 2F i(0), kix + ωit > 0 ,

g = cn(ζ 1; n(ζ 2)), n = 0.3 + 0.2 tanh(ζ 2), ζ i = K iy + y0i, i = 1, 2 , (35)




the interaction between the Jacobian elliptic cn wave withvariable modulus and peakon with the background of Ja-

cobian elliptic wave can be constructed. A simple exampleof this interaction is depicted in Fig. 8 when choosing

F 1 = 0.3 ex− t, M = 1, k = 0.3, k1 = 1 ,

K 1 = K 2 = 0.2, m = 0.5, ω = 0 ,

ω1 =

−1, x0 = y01 = y02 = 0 , (36)

and t = 0, respectively. Figure 8 shows the interaction be-tween the Jacobian elliptic waves with variable modulusand peakon. It is very interesting to see the interactionproperties of the Jacobian elliptic waves and peakon in thelimit case. As m is close to 1, from (22) we can obtain theinteraction of peakon and line quasi-periodic solution de-picted in Fig. 9 with the same parameter values as Fig. 8.

4 Summary and Discussion

In conclusion, the Lucas Riccati method is applied toobtain variable separation solutions of (2+1)-dimensionalMDWW system. Based on the quantity (22), some

novel Jacobian elliptic wave with variable modulus, quasi-periodic wave evolutional, and their interaction behaviorsare found. We hope that in future experimental stud-ies these novel structures and quasi-periodic wave evolu-tional behaviors obtained here can be realized in somefields. Actually, our present short paper is merely a be-

ginning work, more application to other nonlinear physicalsystems should be concerned and deserve further investi-gation. In our future work, on the one hand, we devoteto generalizing this method to other (2+1)-dimensionalnonlinear systems such as the ANNV system and BKKsystem, Boiti–Leon–Pempinelle system, etc. On the other

hand, we will look for more interesting localized excita-tions. The solutions obtained here be useful to some phys-ical problems in fluid dynamics.

AppendixStakhov and Rozin in [32] introduced a new class

of hyperbolic functions that unite the characteristics of the classical hyperbolic functions and the recurring Fi-bonacci and Lucas series. The hyperbolic Fibonacciand Lucas functions, which are the being extension of Binet’s formulas for the Fibonacci and Lucas numbers

in continuous domain, transform the Fibonacci numberstheory into “continuous” theory because every identityfor the hyperbolic Fibonacci and Lucas functions hasits discrete analogy in the framework of the Fibonacci

and Lucas numbers. Taking into consideration a great

role played by the hyperbolic functions in geometry and

physics, (“Lobatchevski’s hyperbolic geometry”, “Four-

dimensional Minkowski’s world”, etc.), it is possible to

expect that the new theory of the hyperbolic functions will

bring to new results and interpretations on mathematics,

biology, physics, and cosmology. In particular, the result

is vital for understanding the relation between transfinit-

ness i.e. fractal geometry and the hyperbolic symmetrical

character of the disintegration of the neural vacuum, as

pointed out by El Naschie.

The definition and properties of the symmetrical Lu-

cas functions, the symmetrical Lucas sine function (sLs),

the symmetrical Lucas cosine function (cLs) and the sym-

metrical Lucas tangent function (tLs) are defined[32−35]

as

sLs(x) = αx − α−x, cLs(x) = αx + α−x ,

tLs(x) =αx − α−x

αx + α−x. (A1)

They are introduced to consider so-called symmetrical rep-resentation of the hyperbolic Lucas functions and they

may present a certain interest for modern theoretical

physics taking into consideration a great role played by the

Golden Section, Golden Proportion, Golden ratio, Golden

Mean in modern physical researches.[32−33] The symmet-

rical Lucas cotangent function (cotLs) is cotLs(x) =

1/tLs(x), the symmetrical Lucas secant function (secLs)

is secLs(x) = 1/cLs(x), the symmetrical Lucas cosecant

function (cscLs) is cscLs(x) = 1/sLs(x). These functions

satisfy the following relations[32−33]

cLs2

(x)−sLs2

(x) = 4 , 1−tLs2

(x) = 4secLs2

(x) ,cotLs2(x) − 1 = 4cscLs2(x) . (A2)

Also, from the above definition, we give the derivative for-

mulas of the symmetrical Lucas functions as follows:

d sLs(x)

dx= cLs(x) ln α ,

d cLs(x)

dx= sLs(x) ln α ,

d tLs(x)

dx= 4secLs2(x) ln α . (A3)

The above symmetrical hyperbolic Lucas functions are

connected with the classical hyperbolic functions by the

following simple correlations:

sLs(x) = 2 sinh(x ln α) , cFs(x) = 2 cosh(x ln α) ,

tLs(x) = tanh(x ln α) . (A4)

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Solitons, Springer-Verlag, Berlin (1991).

[2] W. Malfliet, Am. J. Phys. 60 (1992) 650.

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