Fractal Structures of the Carbon Nanotube System Arrays

Raïssa S. Noule and Victor K. Kuetche

## Abstract

In this work, we investigate fractals in arrays of carbon nanotubes modeled by an evolution equation derived by using a rigorous application of the reductive perturbation formalism for the Maxwell equations and for the corresponding Boltzmann kinetic equation of the distribution function of electrons in such nanomaterials. We study the integrability properties of our dynamical system by using the Weiss-Tabor-Carnevale analysis. Actually, following the leading order analysis, we write the solution in the form of series of Laurent. We also use the Kruskal's simplification to find the solutions. Using the truncated Painlevé expansion, we construct the auto-Backlund transformation of the system. We take advantage of the above properties to construct a wide panel of structures with fractals properties. As a result, we unearth some typical features, namely the fractal dromion, the fractal lump, the stochastic and nonlocal fractal excitations. We also address some physical implications of the results obtained.

Keywords: carbon nanotubes, Weiss-Tabor-Carnevale analysis, Kruskal's simplification, auto-Backlund transformation, fractal excitations

### 1. Introduction

Carbon nanotubes stand to be one of the wonder materials of the present century [1–3] owing to their tremendous range of physical, mechanical, thermal, electronic, and optical properties. They are found in some flat panel displays, some field-effect transistors as emerging applications exploiting the good thermal and electronic conductivities of the above nanomaterials. The carbon nanotubes were synthesized previously in 1991 as graphitic carbon needles with diameter ranging from 4 to 30 nm and length up to 1 μm [4]. Large-scale synthesis [5] provided an impetus to research in the area of carbon fiber growth, as well as in the production and characterization of fullerene materials. Two years later [6], abundant single-shell tubes with diameters of about 1 nm were synthesized. In the past few years, some studies of various nonlinear effects in carbon nanotube arrays have been achieved. There are intrinsic localized modes in strongly nonlinear systems of anharmonic lattices [7, 8], largeamplitude oscillating modes with additional features of being nonlinear as well as discrete [9], spin-wave propagation [10], propagation of short optical pulses with dispersive nonmagnetic dielectric media [11], propagation of ultimately short optical pulses in coupled graphene waveguides [12, 13].

From the reductive perturbation method, Leblond and Mihalache [14, 15] investigated the formation of ultrashort spatiotemporal optical waveforms in arrays of carbon nanotubes while deriving a new coupled system. They actually used the multiscale analysis for the Maxwell equations and for the corresponding Boltzmann kinetic equation of the distribution function of electrons. The above authors [14] showed that a perturbed few-cycle plane-wave input evolves into a robust twodimensional light bullet propagating without being dispersed and diffracted over long distance with respect to the wavelength.

In order to determine the current, we use a semiclassical approximation [25] taking into account the dispersion law from the quantum-mechanical model and the evolution of the ensemble of particles by the classical Boltzmann kinetic equation in

where constant q stands for the electron charge. The relaxation time τ can be assessed according to Ref. [26]. The quantity f is the distribution function of electrons in the nanotubes depending upon the time t and the momentum

� � of the electron. The azimuthal component <sup>p</sup><sup>φ</sup> reads <sup>p</sup><sup>φ</sup> <sup>¼</sup> <sup>s</sup>Δpφ, and the axial component pz is merely denoted p below. It then appears that the integer s characterizes the momentum quantization transverse to the nanotube. We also mention that the function F<sup>0</sup> is the equilibrium value of the distribution f and is

in which quantities kB, T0, and E stand for the Boltzmann constant, the absolute temperature, and the energy in the conduction band, respectively. In account of the zigzag-type carbon nanotubes, the energy E is given by the Huckel π-

> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 4 cosð Þ ap cosð Þþ πs=m 4 cos <sup>2</sup>ð Þ πs=m

with γ ¼ 2:7eV and a ¼ 3b=2ℏ where constant b ¼ 0:142nm represents the distance between the adjacent carbon atoms. Constant m is the number of hexagons in the perimeter of a nanotube. The surface current density Js can hence be

where the velocity <sup>v</sup> reads <sup>v</sup> <sup>¼</sup> <sup>∂</sup>E=∂p. The distribution function <sup>f</sup> can be written as

Δpφδ p<sup>φ</sup> � sΔp<sup>φ</sup> � �fs

with quantity fs representing the longitudinal distribution function relative to the azimuthal quantum number s. The volume current density J is hence

where constant N represents the surface density of nanotubes in the xy-plane. We use the powerful reductive perturbation method in the short-wave approximation regime [13, 28]. Assuming that the typical duration of the pulse is very small with respect to τ and the propagation length is very long with respect to the

<sup>J</sup> <sup>¼</sup> Nq <sup>π</sup><sup>ℏ</sup> <sup>∑</sup> s ð vfs

Js <sup>¼</sup> <sup>2</sup><sup>q</sup> 2πℏ

f ¼ ∑ s

wavelength, we introduce the fast and slow variables

ft � qAt f <sup>p</sup> ¼ ð Þ F<sup>0</sup> � f =τ, (2)

F<sup>0</sup> ¼ 1=½ � 1 þ exp ð Þ E=kBT<sup>0</sup> , (3)

ð ð vfdpφdp, (5)

ð Þ p; t , (6)

dp, (7)

θ ¼ ð Þ t � x=V =ε, ξ ¼ εx, (8)

, (4)

the approximation of relaxation time. It comes

Fractal Structures of the Carbon Nanotube System Arrays

DOI: http://dx.doi.org/10.5772/intechopen.82306

known as the Fermi-distribution function expressed as

electron approximation as follows

E ¼ γ

q

p � p pφ; pz

expressed as

obtained as

85

In the present work, our motivation is to investigate whether other types of robust light bullets with different features can be supported by the previous arrays. Actually, from the governing system derived by Leblond and Mihalache [14], we need to tread into its structural properties of integrability while performing the Weiss-Tabor-Carnevale approach [15] to such a problem and discuss in detail the existence of fractal solutions to the system.

Weiss, Tabor and Carnevale [15] developed one of the most powerful methods known as the Painlevé analysis [16] which is very useful in proving the integrability of a model system. Such an analysis is helpful in generating some exact solutions, no matter the model is integrable or not. Also, if ones wants only to prove the Painlevé property of a model, the use of Kruskal's simplification [17] for the WTC approach is also addressable. Thus, if we need to find some more information from the model, it is better to use the original WTC approach or some extended forms [18–21]. In this work, we combine the standard WTC approach [15] with the Kruskal's simplification [17] in view of simplifying the proof of the Painlevé integrability.

We organize the work as follows: in Section 2, we briefly present the physical background of the system under investigation. In Section 3, we perform the WTC method to the governing equations under study. Next, in Section 4, we take advantage of the arbitrary functions generated by the previous analysis to discuss some higher dimensional pattern formations of light bullets, namely the fractals. In the last section, we end with a brief conclusion.

## 2. Physical ground of light propagation within the carbon nanotube arrays

In a recent study, Belonenko et al. [22, 23] investigated both analytically and numerically the propagation of light bullets within an array of carbon nanotubes. They obtained an analytical function presenting some (2 + 1)-dimensional optical soliton with some diffraction displays in propagation. In view of suppressing the diffraction to obtain some robust light bullet waveform, the model is slightly modified [14] while deriving a new higher dimensional coupled system. Using the calibration <sup>E</sup> ¼ �∂A=∂t, <sup>E</sup> and <sup>A</sup> being the electric field and the potential vectors, respectively, and variable t being the time, taking into account of the dielectric and magnetic properties of carbon nanotubes [24], the Maxwell equations reduce to the following system

$$
\Delta \mathbf{A} - \mathbf{A}\_{tt}/c^2 = -\mu\_0 \mathbf{J},
\tag{1}
$$

where subscripts denote the partial derivatives. Constants μ<sup>0</sup> and c are magnetic permeability and light velocity in vacuum, respectively. We have neglected the diffraction blooming of the laser beam in the directions perpendicular to the propagation plane. The current J is directed along the axis of the nanotubes, i.e., J ¼ Jzez, where unitary vector e<sup>z</sup> spans the z-axis. Besides, we consider the case where the wave field is polarized in the same direction, and A ¼ Aez.

Fractal Structures of the Carbon Nanotube System Arrays DOI: http://dx.doi.org/10.5772/intechopen.82306

From the reductive perturbation method, Leblond and Mihalache [14, 15] investigated the formation of ultrashort spatiotemporal optical waveforms in arrays of carbon nanotubes while deriving a new coupled system. They actually used the multiscale analysis for the Maxwell equations and for the corresponding Boltzmann kinetic equation of the distribution function of electrons. The above authors [14] showed that a perturbed few-cycle plane-wave input evolves into a robust twodimensional light bullet propagating without being dispersed and diffracted over

In the present work, our motivation is to investigate whether other types of robust light bullets with different features can be supported by the previous arrays. Actually, from the governing system derived by Leblond and Mihalache [14], we need to tread into its structural properties of integrability while performing the Weiss-Tabor-Carnevale approach [15] to such a problem and discuss in detail the

Weiss, Tabor and Carnevale [15] developed one of the most powerful methods known as the Painlevé analysis [16] which is very useful in proving the integrability of a model system. Such an analysis is helpful in generating some exact solutions, no matter the model is integrable or not. Also, if ones wants only to prove the Painlevé property of a model, the use of Kruskal's simplification [17] for the WTC approach is also addressable. Thus, if we need to find some more information from the model, it is better to use the original WTC approach or some extended forms [18–21]. In this work, we combine the standard WTC approach [15] with the Kruskal's simpli-

We organize the work as follows: in Section 2, we briefly present the physical background of the system under investigation. In Section 3, we perform the WTC method to the governing equations under study. Next, in Section 4, we take advantage of the arbitrary functions generated by the previous analysis to discuss some higher dimensional pattern formations of light bullets, namely the fractals. In the

2. Physical ground of light propagation within the carbon nanotube

ΔA � Att=c

wave field is polarized in the same direction, and A ¼ Aez.

where subscripts denote the partial derivatives. Constants μ<sup>0</sup> and c are magnetic permeability and light velocity in vacuum, respectively. We have neglected the diffraction blooming of the laser beam in the directions perpendicular to the propagation plane. The current J is directed along the axis of the nanotubes, i.e., J ¼ Jzez, where unitary vector e<sup>z</sup> spans the z-axis. Besides, we consider the case where the

<sup>2</sup> ¼ �μ0J, (1)

In a recent study, Belonenko et al. [22, 23] investigated both analytically and numerically the propagation of light bullets within an array of carbon nanotubes. They obtained an analytical function presenting some (2 + 1)-dimensional optical soliton with some diffraction displays in propagation. In view of suppressing the diffraction to obtain some robust light bullet waveform, the model is slightly modified [14] while deriving a new higher dimensional coupled system. Using the calibration <sup>E</sup> ¼ �∂A=∂t, <sup>E</sup> and <sup>A</sup> being the electric field and the potential vectors, respectively, and variable t being the time, taking into account of the dielectric and magnetic properties of carbon nanotubes [24], the Maxwell equations reduce to the

fication [17] in view of simplifying the proof of the Painlevé integrability.

long distance with respect to the wavelength.

existence of fractal solutions to the system.

last section, we end with a brief conclusion.

arrays

Fractal Analysis

following system

84

In order to determine the current, we use a semiclassical approximation [25] taking into account the dispersion law from the quantum-mechanical model and the evolution of the ensemble of particles by the classical Boltzmann kinetic equation in the approximation of relaxation time. It comes

$$f\_t - qA\_t f\_p = (F\_0 - f) / \mathfrak{r},\tag{2}$$

where constant q stands for the electron charge. The relaxation time τ can be assessed according to Ref. [26]. The quantity f is the distribution function of electrons in the nanotubes depending upon the time t and the momentum p � p pφ; pz � � of the electron. The azimuthal component <sup>p</sup><sup>φ</sup> reads <sup>p</sup><sup>φ</sup> <sup>¼</sup> <sup>s</sup>Δpφ, and the axial component pz is merely denoted p below. It then appears that the integer s characterizes the momentum quantization transverse to the nanotube. We also mention that the function F<sup>0</sup> is the equilibrium value of the distribution f and is known as the Fermi-distribution function expressed as

$$F\_0 = \mathbf{1}/[\mathbf{1} + \exp\left(E/k\_B T\_0\right)],\tag{3}$$

in which quantities kB, T0, and E stand for the Boltzmann constant, the absolute temperature, and the energy in the conduction band, respectively. In account of the zigzag-type carbon nanotubes, the energy E is given by the Huckel πelectron approximation as follows

$$E = \gamma \sqrt{1 + 4 \cos(ap) \cos(\pi s/m) + 4 \cos^2(\pi s/m)},\tag{4}$$

with γ ¼ 2:7eV and a ¼ 3b=2ℏ where constant b ¼ 0:142nm represents the distance between the adjacent carbon atoms. Constant m is the number of hexagons in the perimeter of a nanotube. The surface current density Js can hence be expressed as

$$J\_s = \frac{2q}{2\pi\hbar} \int \left[vf dp\_\varphi dp,\right] \tag{5}$$

where the velocity <sup>v</sup> reads <sup>v</sup> <sup>¼</sup> <sup>∂</sup>E=∂p. The distribution function <sup>f</sup> can be written as

$$f = \sum\_{s} \Delta p\_{\rho} \delta \left( p\_{\rho} - s \Delta p\_{\rho} \right) f\_{s}(p, t), \tag{6}$$

with quantity fs representing the longitudinal distribution function relative to the azimuthal quantum number s. The volume current density J is hence obtained as

$$J = \frac{Nq}{\pi\hbar} \sum\_{s} \int v f\_s dp,\tag{7}$$

where constant N represents the surface density of nanotubes in the xy-plane.

We use the powerful reductive perturbation method in the short-wave approximation regime [13, 28]. Assuming that the typical duration of the pulse is very small with respect to τ and the propagation length is very long with respect to the wavelength, we introduce the fast and slow variables

$$
\theta = (\mathfrak{t} - \mathfrak{x}/\mathcal{V})/\mathfrak{e}, \quad \mathfrak{F} = \mathfrak{e}\mathfrak{x}, \tag{8}
$$

in which the quantities ε and V denote the small perturbative parameter and the wave velocity, respectively. Accordingly, we address the following expansions

$$f\_s = f\_0 + \epsilon f\_1 + \cdots, \quad A = A\_0 + \epsilon A\_1 + \cdots. \tag{9}$$

Thus, at leading order ε�1, Eq. (2) yields

$$f\_{0\iota} - qA\_{0\iota}f\_{0p} = 0,\tag{10}$$

<sup>J</sup><sup>0</sup> ¼ �<sup>Q</sup> sgn sin aqA0=<sup>2</sup> � � � � cos aqA0=<sup>2</sup> � �, (18)

dx=½ � 1 þ exp 2ð Þ Xjsin xj : (19)

<sup>0</sup> such that

<sup>0</sup>=<sup>2</sup> � �, (20)

<sup>0</sup>=<sup>2</sup> � �, (21)

BYYdT, (22)

where Q ¼ ð Þ 4Nqγ=πℏ Φð Þ γ=kBT<sup>0</sup> . The function Φ reads

ð<sup>π</sup>=<sup>2</sup> �π=2

Inserting Eq. (18) into (13) yields the evolution equation. As a matter of illus-

<sup>0</sup>=∂ξ∂<sup>θ</sup> ¼ �<sup>R</sup> sin aqA<sup>0</sup>

<sup>0</sup> ¼ A<sup>0</sup> þ π=aq. Therefore, Eq. (20) remains. This shows that Eq. (20) is valid for

which can be known as the two-dimensional sine-Gordon equation. Eq. (21) can

<sup>0</sup>=∂y<sup>2</sup> � <sup>R</sup> sin aqA<sup>0</sup>

ðT

ctrLr=<sup>2</sup> <sup>p</sup> . The assumption lim<sup>T</sup>!�∞<sup>A</sup> <sup>¼</sup> <sup>U</sup> is

with R ¼ ð Þ 2Nqγ=πε0ℏc Φð Þ γ=kBT<sup>0</sup> . Assuming ∣aqA0=2∣>π, we can set

any A0. Retaining the second transverse derivative in the wave Eq. (13), we

AT ¼ �BC, CT ¼ AB, BZ ¼ C þ

provided B ¼ E0=Er, Z ¼ x=Lr, T ¼ ð Þ t � x=c =tr, and Y ¼ y=wr, with

regarded. The system (22) has been investigated by means of a modified Euler scheme in Z in each substep of which the equations relative to the variable T are solved by a scheme of the same type [14]. Unlikely, we develop an analytical scheme known as the WTC formalism in view of studying the full integrability of the system above while unearthing other kinds of light bullet waveforms with

According to the standard WTC method [15], if equation Eq. (22) is Painlevé integrable, then all the possible solutions of the system can be written in the full

> ∞ k¼0

g ¼ g Yð Þ ; Z; T , Ak ¼ Akð Þ Y; Z; T , Bk ¼ Bkð Þ Y; Z; T , and Ck ¼ Ckð Þ Y; Z; T (k being nonzero integers) are analytical functions within the neighborhood of g ¼ 0. The

Bkg<sup>k</sup>þ<sup>β</sup>, C <sup>¼</sup> <sup>∑</sup>

∞ k¼0

α ¼ γ ¼ �2, β ¼ �1 (24)

Ckg<sup>k</sup>þ<sup>γ</sup>

, (23)

Akg<sup>k</sup>þ<sup>α</sup>, B <sup>¼</sup> <sup>∑</sup>

with sufficient arbitrary functions among Ak, Bk, Ck, and g, where

A0

Φð Þ¼ X

Fractal Structures of the Carbon Nanotube System Arrays

DOI: http://dx.doi.org/10.5772/intechopen.82306

tration, we assume that 0 < aqA0=2 < π and define A<sup>0</sup>

∂2 A0

<sup>0</sup>=∂ξ∂<sup>θ</sup> <sup>¼</sup> ð Þ <sup>c</sup>=<sup>2</sup> <sup>∂</sup><sup>2</sup>

� �, Lr ¼ �UEr=R, and wr <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>0</sup>=2 ¼ aqA0=2 � π=2. Hence, we obtain

∂2 A0

aqA0

A″

derive the following

be written as

Er ¼ 2= aqtr

compact supports.

3. Painlevé analysis

Laurent series as follows

87

A ¼ ∑ ∞ k¼0

constants α, β, and γ should all be negative integers. The leading order analysis provides the following

in which solution f <sup>0</sup> reads f <sup>0</sup> ¼ φ p þ qA<sup>0</sup> � � with φ being an arbitrary function. However, at large t, the wave A vanishes and f <sup>0</sup> goes to its equilibrium value F0. Thus, from Eq. (7), we write

$$J\_0 = \frac{q}{\pi\hbar} \sum\_{\varepsilon} \int v(p+qA)f\_0(p)dp. \tag{11}$$

Now, at leading order ε�2, the Maxwell equations transform to

$$V = \mathfrak{c}.\tag{12}$$

The order ε<sup>0</sup> provides

$$(2/c)\partial^2 A\_0/\partial\xi\partial\theta = \mu\_0 I\_0. \tag{13}$$

which, with Eq. (11), stands for the governing model system.

The energy E is of the same order of magnitude as γ. Calculating <sup>γ</sup>=kB <sup>¼</sup> <sup>3</sup>:<sup>1</sup> � <sup>10</sup><sup>4</sup><sup>K</sup> shows that <sup>E</sup> is very large with respect to room temperature. Thus, only the levels with the lowest energy are excited. Let us seek for this minimum for a given parameter s. Hence, we find that

$$E\_{\min} = \gamma |1 - 2|\cos\left(\pi s/m\right)|\,, \tag{14}$$

for p ¼ �π=a when cosð Þ πs=m >0 and for p ¼ 0 when cosð Þ πs=m < 0. Thus, as s varies, the minimum of Emin is zero, i.e., s=m ¼ �1=3 or s=m ¼ �2=3. Therefore, for m ¼ 6, s ¼ 2 or s ¼ 4. Besides, in other nanotubes, there is a nonzero gap between valence and conduction bands. The gap is so great that the conductivity is very low. Hence, only the nanotubes where m is multiple of 3 contributes. In this sense, the expression of Esð Þ p reads

$$E\_s = 2\chi |\cos\left(ap/2\right)|\,\tag{15}$$

for s=m ¼ 1=3 and

$$E\_s = 2\gamma |\sin\left(ap/2\right)|\,\tag{16}$$

for <sup>s</sup>=<sup>m</sup> <sup>¼</sup> <sup>2</sup>=3. Calculating the velocity <sup>v</sup> <sup>¼</sup> <sup>∂</sup>E=∂p, where variable <sup>p</sup> is substituted by p � qA0, and considering the value at which the minimum is reached, we get

$$v = -a\gamma \operatorname{sgn}\left(\sin\left(aqA\_0/2\right)\right)\cos\left(aqA\_0/2\right),\tag{17}$$

where sgn ð Þ X denotes the sign of X in both cases. The current J<sup>0</sup> can hence be expressed as

Fractal Structures of the Carbon Nanotube System Arrays DOI: http://dx.doi.org/10.5772/intechopen.82306

in which the quantities ε and V denote the small perturbative parameter and the

fs ¼ f <sup>0</sup> þ εf <sup>1</sup> þ ⋯, A ¼ A<sup>0</sup> þ εA<sup>1</sup> þ ⋯: (9)

f <sup>0</sup><sup>t</sup> � qA0<sup>t</sup> f <sup>0</sup><sup>p</sup> ¼ 0, (10)

� � with φ being an arbitrary function.

v pð Þ þ qA f <sup>0</sup>ð Þ p dp: (11)

V ¼ c: (12)

<sup>A</sup>0=∂ξ∂<sup>θ</sup> <sup>¼</sup> <sup>μ</sup>0J0, (13)

Emin ¼ γ∣1 � 2∣ cosð Þk πs=m , (14)

Es ¼ 2γ∣ cosð Þ ap=2 ∣, (15)

Es ¼ 2γ∣sin ð Þ ap=2 ∣, (16)

<sup>v</sup> ¼ �a<sup>γ</sup> sgn sin aqA0=<sup>2</sup> � � � � cos aqA0=<sup>2</sup> � �, (17)

wave velocity, respectively. Accordingly, we address the following expansions

However, at large t, the wave A vanishes and f <sup>0</sup> goes to its equilibrium value F0.

Thus, at leading order ε�1, Eq. (2) yields

in which solution f <sup>0</sup> reads f <sup>0</sup> ¼ φ p þ qA<sup>0</sup>

<sup>J</sup><sup>0</sup> <sup>¼</sup> <sup>q</sup> <sup>π</sup><sup>ℏ</sup> <sup>∑</sup> s ð

Now, at leading order ε�2, the Maxwell equations transform to

ð Þ <sup>2</sup>=<sup>c</sup> <sup>∂</sup><sup>2</sup>

which, with Eq. (11), stands for the governing model system. The energy E is of the same order of magnitude as γ. Calculating

minimum for a given parameter s. Hence, we find that

<sup>γ</sup>=kB <sup>¼</sup> <sup>3</sup>:<sup>1</sup> � <sup>10</sup><sup>4</sup><sup>K</sup> shows that <sup>E</sup> is very large with respect to room temperature. Thus, only the levels with the lowest energy are excited. Let us seek for this

for p ¼ �π=a when cosð Þ πs=m >0 and for p ¼ 0 when cosð Þ πs=m < 0. Thus, as s varies, the minimum of Emin is zero, i.e., s=m ¼ �1=3 or s=m ¼ �2=3. Therefore, for m ¼ 6, s ¼ 2 or s ¼ 4. Besides, in other nanotubes, there is a nonzero gap between valence and conduction bands. The gap is so great that the conductivity is very low. Hence, only the nanotubes where m is multiple of 3 contributes. In this sense, the

for <sup>s</sup>=<sup>m</sup> <sup>¼</sup> <sup>2</sup>=3. Calculating the velocity <sup>v</sup> <sup>¼</sup> <sup>∂</sup>E=∂p, where variable <sup>p</sup> is substituted by p � qA0, and considering the value at which the minimum is

where sgn ð Þ X denotes the sign of X in both cases.

The current J<sup>0</sup> can hence be expressed as

Thus, from Eq. (7), we write

Fractal Analysis

The order ε<sup>0</sup> provides

expression of Esð Þ p reads

for s=m ¼ 1=3 and

reached, we get

86

$$J\_0 = -Q \operatorname{sgn} \left( \sin \left( aqA\_0/2 \right) \right) \cos \left( aqA\_0/2 \right), \tag{18}$$

where Q ¼ ð Þ 4Nqγ=πℏ Φð Þ γ=kBT<sup>0</sup> . The function Φ reads

$$\Phi(\mathbf{X}) = \int\_{-\pi/2}^{\pi/2} d\mathbf{x} / [\mathbf{1} + \exp\left(2\mathbf{X}|\sin\mathbf{x}|\right)].\tag{19}$$

Inserting Eq. (18) into (13) yields the evolution equation. As a matter of illustration, we assume that 0 < aqA0=2 < π and define A<sup>0</sup> <sup>0</sup> such that aqA0 <sup>0</sup>=2 ¼ aqA0=2 � π=2. Hence, we obtain

$$
\partial^2 A\_0' / \partial \xi \partial \theta = -R \sin \left( a q A\_0' / 2 \right),
\tag{20}
$$

with R ¼ ð Þ 2Nqγ=πε0ℏc Φð Þ γ=kBT<sup>0</sup> . Assuming ∣aqA0=2∣>π, we can set A″ <sup>0</sup> ¼ A<sup>0</sup> þ π=aq. Therefore, Eq. (20) remains. This shows that Eq. (20) is valid for any A0. Retaining the second transverse derivative in the wave Eq. (13), we derive the following

$$
\partial^2 A\_0'/\partial \xi \partial \theta = (c/2)\partial^2 A\_0'/\partial \mathfrak{y}^2 - R\sin\left(aqA\_0'/2\right),
\tag{21}
$$

which can be known as the two-dimensional sine-Gordon equation. Eq. (21) can be written as

$$A\_T = -BC, \quad C\_T = AB, \quad B\_Z = C + \int^T B\_{YY} dT,\tag{22}$$

provided B ¼ E0=Er, Z ¼ x=Lr, T ¼ ð Þ t � x=c =tr, and Y ¼ y=wr, with Er ¼ 2= aqtr � �, Lr ¼ �UEr=R, and wr <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ctrLr=<sup>2</sup> <sup>p</sup> . The assumption lim<sup>T</sup>!�∞<sup>A</sup> <sup>¼</sup> <sup>U</sup> is regarded. The system (22) has been investigated by means of a modified Euler scheme in Z in each substep of which the equations relative to the variable T are solved by a scheme of the same type [14]. Unlikely, we develop an analytical scheme known as the WTC formalism in view of studying the full integrability of the system above while unearthing other kinds of light bullet waveforms with compact supports.

#### 3. Painlevé analysis

According to the standard WTC method [15], if equation Eq. (22) is Painlevé integrable, then all the possible solutions of the system can be written in the full Laurent series as follows

$$A = \sum\_{k=0}^{\infty} A\_k \mathbf{g}^{k+a}, \quad B = \sum\_{k=0}^{\infty} B\_k \mathbf{g}^{k+\beta}, \quad \mathbf{C} = \sum\_{k=0}^{\infty} \mathbf{C}\_k \mathbf{g}^{k+\gamma}, \tag{23}$$

with sufficient arbitrary functions among Ak, Bk, Ck, and g, where g ¼ g Yð Þ ; Z; T , Ak ¼ Akð Þ Y; Z; T , Bk ¼ Bkð Þ Y; Z; T , and Ck ¼ Ckð Þ Y; Z; T (k being nonzero integers) are analytical functions within the neighborhood of g ¼ 0. The constants α, β, and γ should all be negative integers.

The leading order analysis provides the following

$$a = \chi = -\mathfrak{Z}, \quad \beta = -\mathfrak{1} \tag{24}$$

and

$$A\_0 = 2(\mathbf{g}\_Z \mathbf{g}\_T - \mathbf{g}\_Y^2), \quad B\_0 = 2i \mathbf{eg}\_T, \quad C\_0 = -i\epsilon A\_0. \tag{25}$$

where ε ¼ �1 and i <sup>2</sup> ¼ �1.

In order to obtain the recursion relations to determine the functions Ak, Bk, and Ck, we substitute Eqs. (23)–(25) into (22). This leads us to the following algebraic system

$$
\mathcal{M}\_k \mathcal{V}\_k = \mathcal{T}\_k. \tag{26}
$$

For k ¼ 1, we can easily obtain from Eq. (26)

Fractal Structures of the Carbon Nanotube System Arrays

DOI: http://dx.doi.org/10.5772/intechopen.82306

C<sup>1</sup> ¼ � ιεA1:

For k ¼ 3, Eq. (26) gives us

<sup>þ</sup> ιε

2A<sup>0</sup>

þ 1 2A<sup>0</sup>

Let us emphasize that <sup>∂</sup>g=∂<sup>T</sup> � gT, and so on.

A<sup>4</sup> þ 2C<sup>4</sup> � 4gTC<sup>4</sup>

, B<sup>4</sup> <sup>¼</sup> <sup>1</sup>

C<sup>3</sup> ¼ � ιεA3:

For k ¼ 4, we can obtain

where C<sup>4</sup> is an arbitrary function.

bilinearization [27–30] are derived.

<sup>A</sup><sup>4</sup> <sup>¼</sup> ιε 6gT

ing relations are derived:

89

<sup>A</sup><sup>3</sup> <sup>¼</sup> ιε

<sup>B</sup><sup>3</sup> <sup>¼</sup> <sup>1</sup>

is derived

Eq. (26),

<sup>A</sup><sup>1</sup> <sup>¼</sup> ιε gTB0, <sup>Z</sup> <sup>þ</sup> gZB0,T � <sup>2</sup>gYB0, <sup>Y</sup> <sup>þ</sup> ð Þ gZT�gYY B0 <sup>þ</sup> <sup>C</sup>0,T

<sup>B</sup><sup>1</sup> <sup>¼</sup> gTB0, <sup>Z</sup> <sup>þ</sup> gZB0,T � <sup>2</sup>gYB0, <sup>Y</sup> <sup>þ</sup> ð Þ gZT�gYY B0 <sup>þ</sup> <sup>2</sup>C0,T

<sup>B</sup><sup>2</sup> <sup>¼</sup> <sup>C</sup>1,T � <sup>A</sup>1B<sup>1</sup> � <sup>B</sup>0A<sup>2</sup> A<sup>0</sup>

where one of the quantities among A<sup>2</sup> and C<sup>2</sup> is arbitrary.

<sup>2</sup>gT �ð Þ gZT�gYY <sup>B</sup><sup>2</sup> <sup>þ</sup> <sup>A</sup>1B<sup>2</sup> <sup>þ</sup> <sup>A</sup>2B<sup>1</sup> 

A<sup>0</sup>

It is straightforward to see that the resonance condition given by Eq. (34) is satisfied identically because of Eqs. (25) and (33). Then, we have, after solving

> <sup>2</sup>gT �B1,ZT <sup>þ</sup> <sup>B</sup>1,YY � gTB2,Z � gZB2, <sup>T</sup> <sup>þ</sup> <sup>2</sup>gYB2,Y

�3B1,ZT þ 3B1,YY � 3gTB2,Z � 3gZB2,T þ 6gYB2,Y 

3A<sup>0</sup>

Nevertheless, let us make a remark that throughout the above study, the follow-

Then, for k ¼ 4, Eq. (38) verifies the resonance condition. All of the resonance conditions with four arbitrary functions are satisfied identically. Hence, the system (22) is Painlevé integrable. Its complete integrability will be established if some other essential properties such as the Bäcklund transformation (BT) and the Hirota

The Painlevé analysis can also be used to obtain other interesting properties [15] of the (2 + 1)-dimensional coupled system above. In this work, we use the standard

A<sup>4</sup> � C<sup>4</sup> þ 2gTC<sup>4</sup>

B<sup>k</sup> � ιεC<sup>k</sup> ¼ 0, kð Þ ¼ 1; 2; 3; 4 : (38)

, (37)

�3ð Þ gZT�gYY B<sup>2</sup> þ 2C2,T þ A1B<sup>2</sup> þ A2B<sup>1</sup> ,

On the other hand, solving the case for k ¼ 2, the following resonance condition

gT ,

�B0,ZT þ B0,YY þ C1,T ¼ 0: (34)

,

, C<sup>2</sup> ¼ �ιεA2, (35)

(33)

(36)

where M<sup>k</sup> is a square matrix, V<sup>k</sup> ¼ ð Þ Ak; Bk;Ck <sup>T</sup>, and <sup>T</sup> <sup>k</sup> <sup>¼</sup> ð Þ <sup>A</sup>k; <sup>B</sup>k; <sup>C</sup><sup>k</sup> <sup>T</sup> with

$$\begin{aligned} \mathcal{A}\_k &= -B\_{k-2,ZT} + B\_{k-2,YY} \\ &- (k-2) \left[ B\_{k-1,Z} \mathbf{g}\_T + B\_{k-1,T} \mathbf{g}\_Z - 2B\_{k-1,Y} \mathbf{g}\_Y + B\_{k-1} (\mathbf{g}\_{ZT} - \mathbf{g}\_{YY}) \right] \\ &+ \sum\_{j=1}^{k-1} A\_j B\_{k-j} \end{aligned} \tag{27}$$

and

$$\mathcal{B}\_k = -A\_{k-1,T} - \sum\_{j=1}^{k-1} \mathcal{C}\_j B\_{k-j},\tag{28}$$

with

$$\mathcal{C}\_{k} = -\mathbf{C}\_{k-1,T} + \sum\_{j=1}^{k-1} A\_{j} B\_{k-j},\tag{29}$$

provided Ak ¼ Bk ¼ Ck ¼ 0 for k < 0. The matrix M<sup>k</sup> is given by

$$\mathcal{M}\_{k} = \begin{pmatrix} -B\_{0} & k(k-3)A\_{0}/2 & 0\\ (k-2)\mathbf{g}\_{T} & C\_{0} & B\_{0} \\ -B\_{0} & -A\_{0} & (k-2)\mathbf{g}\_{T} \end{pmatrix}. \tag{30}$$

Thus, the determinant Δ<sup>k</sup> of the matrix M<sup>k</sup> is given by

$$
\Delta\_k = -(k+1)(k-2)(k-2)(k-4)\left(\mathbf{g}\_Z \mathbf{g}\_T - \mathbf{g}\_Y^2\right) \mathbf{g}\_T^2. \tag{31}
$$

If the determinant Δ<sup>k</sup> of the cœfficient matrix M<sup>k</sup> is not equal to zero, then the functions Ak, Bk, and Ck can be obtained from Eq. (26) straightforwardly as unique solutions. Nonetheless, when

$$k \in \{-1, 2, 2, 4\},\tag{32}$$

resonances occur.

The resonance at k ¼ �1 corresponds to the singularity manifold g, which is an arbitrary function, and the case k ¼ 0, which is then satisfied identically by the leading order analysis provided by Eq. (25). If the model is Painlevé integrable, we require two resonance conditions at k ¼ 2;4, which are satisfied identically such that the other four arbitrary functions among Ak, Bk, and Ck can be introduced into the general series expansion given by Eq. (23).

For k ¼ 1, we can easily obtain from Eq. (26)

and

Fractal Analysis

system

where ε ¼ �1 and i

A<sup>k</sup> ¼ �Bk�2,ZT þ Bk�2,YY

AjBk�j,

M<sup>k</sup> ¼

solutions. Nonetheless, when

resonances occur.

88

0

B@

the general series expansion given by Eq. (23).

Thus, the determinant Δ<sup>k</sup> of the matrix M<sup>k</sup> is given by

þ ∑ k�1 j¼1

and

with

<sup>A</sup><sup>0</sup> <sup>¼</sup> <sup>2</sup> gZ gT � <sup>g</sup><sup>2</sup>

<sup>2</sup> ¼ �1.

where M<sup>k</sup> is a square matrix, V<sup>k</sup> ¼ ð Þ Ak; Bk;Ck

Y

In order to obtain the recursion relations to determine the functions Ak, Bk, and Ck, we substitute Eqs. (23)–(25) into (22). This leads us to the following algebraic

� ð Þ k � 2 Bk�1,Z gT þ Bk�1,T gZ � 2Bk�1,Y gY þ Bk�<sup>1</sup> gZT � gYY

B<sup>k</sup> ¼ �Ak�1,T � ∑

C<sup>k</sup> ¼ �Ck�1,T þ ∑

<sup>Δ</sup><sup>k</sup> ¼ �ð Þ <sup>k</sup> <sup>þ</sup> <sup>1</sup> ð Þ <sup>k</sup> � <sup>2</sup> ð Þ <sup>k</sup> � <sup>2</sup> ð Þ <sup>k</sup> � <sup>4</sup> gZ gT � <sup>g</sup><sup>2</sup>

If the determinant Δ<sup>k</sup> of the cœfficient matrix M<sup>k</sup> is not equal to zero, then the functions Ak, Bk, and Ck can be obtained from Eq. (26) straightforwardly as unique

The resonance at k ¼ �1 corresponds to the singularity manifold g, which is an arbitrary function, and the case k ¼ 0, which is then satisfied identically by the leading order analysis provided by Eq. (25). If the model is Painlevé integrable, we require two resonance conditions at k ¼ 2;4, which are satisfied identically such that the other four arbitrary functions among Ak, Bk, and Ck can be introduced into

provided Ak ¼ Bk ¼ Ck ¼ 0 for k < 0. The matrix M<sup>k</sup> is given by

� � � �

k�1 j¼1

k�1 j¼1

�B<sup>0</sup> k kð Þ � 3 A0=2 0 ð Þ k � 2 gT C<sup>0</sup> B<sup>0</sup> �B<sup>0</sup> �A<sup>0</sup> ð Þ k � 2 gT

� �, B<sup>0</sup> <sup>¼</sup> <sup>2</sup>iεgT, C<sup>0</sup> ¼ �iεA0, (25)

MkV<sup>k</sup> ¼ T k, (26)

<sup>T</sup>, and <sup>T</sup> <sup>k</sup> <sup>¼</sup> ð Þ <sup>A</sup>k; <sup>B</sup>k; <sup>C</sup><sup>k</sup>

CjBk�j, (28)

AjBk�j, (29)

1

Y � �g<sup>2</sup>

k∈ f g �1; 2; 2; 4 , (32)

CA: (30)

<sup>T</sup>: (31)

<sup>T</sup> with

(27)

$$\begin{array}{rclcrcl} A\_{1} &=& \frac{\iota \varepsilon \left[ \, \_{\varrho T} \mathcal{B}\_{0, Z} + \, \_{\varrho Z} \mathcal{B}\_{0, T} - \, \_{\varrho Y} \mathcal{B}\_{0, Y} + \, \_{\langle \varrho Z T - \varrho Y \rangle} \mathcal{B}\_{0} + \mathcal{C}\_{0, T} \right]}{\mathcal{g}T}, \\ B\_{1} &=& \frac{\, \_{\varrho T} \mathcal{B}\_{0, Z} + \, \_{\varrho Z} \mathcal{B}\_{0, T} - \, \_{\varrho Y} \mathcal{B}\_{0, Y} + \, \_{\langle \varrho Z T - \varrho Y \rangle} \mathcal{B}\_{0} + \mathcal{2} \mathcal{C}\_{0, T}}{\mathcal{A}\_{0}}, \\ C\_{1} &=& \, -\iota \epsilon A\_{1}. \end{array} \tag{33}$$

On the other hand, solving the case for k ¼ 2, the following resonance condition is derived

$$-B\mathbf{0}\_{\mathbf{0}}\mathbf{z}\mathbf{r} + B\mathbf{0}\_{\mathbf{0}}\mathbf{y}\mathbf{y} + \mathbf{C}\_{\mathbf{I}}\mathbf{r} = \mathbf{0}.\tag{34}$$

It is straightforward to see that the resonance condition given by Eq. (34) is satisfied identically because of Eqs. (25) and (33). Then, we have, after solving Eq. (26),

$$B\_2 = \frac{C\_{1,T} - A\_1 B\_1 - B\_0 A\_2}{A\_0}, \quad C\_2 = -\iota \varepsilon A\_2. \tag{35}$$

where one of the quantities among A<sup>2</sup> and C<sup>2</sup> is arbitrary. For k ¼ 3, Eq. (26) gives us

$$\begin{split} A\_{3} &= \; \frac{\imath\epsilon}{2\overline{g}T} \left[ -B\_{1,ZT} + B\_{1,YY} - \varrho\_{T}B\_{2,Z} - \varrho\_{Z}B\_{2,T} + 2\_{\overline{g}Y}B\_{2,Y} \right] \\ &+ \; \frac{\imath\epsilon}{2\overline{g}T} \left[ -\left( \varrho\_{ZT-\overline{g}YY} \right)B\_{2} + A\_{1}B\_{2} + A\_{2}B\_{1} \right] \\ B\_{3} &= \; \frac{1}{2A\_{0}} \left[ -3B\_{1,ZT} + 3B\_{1,YY} - 3\_{\overline{g}T}B\_{2,Z} - 3\_{\overline{g}Z}B\_{2,T} + 6\_{\overline{g}Y}B\_{2,Y} \right] \\ &+ \; \frac{1}{2A\_{0}} \left[ -3\_{(\overline{g}ZT-\overline{g}YY)}B\_{2} + 2C\_{2,T} + A\_{1}B\_{2} + A\_{2}B\_{1} \right], \\ C\_{3} &= \; -\imath\epsilon A\_{3}. \end{split} \tag{36}$$

Let us emphasize that <sup>∂</sup>g=∂<sup>T</sup> � gT, and so on. For k ¼ 4, we can obtain

$$A\_4 = \frac{\imath\varepsilon}{6\mathfrak{g}\_T} \left(\mathcal{A}\_4 + 2\mathcal{C}\_4 - 4\mathfrak{g}\_T \mathcal{C}\_4\right), \quad B\_4 = \frac{1}{3A\_0} \left(\mathcal{A}\_4 - \mathcal{C}\_4 + 2\mathfrak{g}\_T \mathcal{C}\_4\right), \tag{37}$$

where C<sup>4</sup> is an arbitrary function.

Nevertheless, let us make a remark that throughout the above study, the following relations are derived:

$$\mathcal{B}\_k - \iota \epsilon \mathcal{C}\_k = \mathbf{0}, (k = 1, 2, 3, 4). \tag{38}$$

Then, for k ¼ 4, Eq. (38) verifies the resonance condition. All of the resonance conditions with four arbitrary functions are satisfied identically. Hence, the system (22) is Painlevé integrable. Its complete integrability will be established if some other essential properties such as the Bäcklund transformation (BT) and the Hirota bilinearization [27–30] are derived.

The Painlevé analysis can also be used to obtain other interesting properties [15] of the (2 + 1)-dimensional coupled system above. In this work, we use the standard truncation of the WTC expansion to obtain the BT and the Hirota bilinearization [27–30] of the system (22). By setting

$$A\_{k+1} = B\_k = \mathbf{C}\_{k+1} = \mathbf{0}, \quad \text{for} \quad k \ge 2,\tag{39}$$

In the next section, because of the arbitrariness of some functions derived from the Painlevé analysis, we aim at focusing our interest to solutions for which the quantities A and B are expressed in the reduction form Eq. (45). In order to express some exact solutions of our initial coupled evolution system, we consider the gen-

where the parameter akð Þ k ¼ 0; 1; 2; 3 is an arbitrary constant and P ¼ P Zð Þ ; T

With this aim, we follow the method developed by Tang and Lou [32] for generating some families of diverse pattern formations while using the arbitrary

Let us mention that for some convenience, we rewrite the variables X, Y, and T into their lower cases. Paying particular attention to fractal pattern formations, based upon the previous works carried out on the subject, we classify the above waves according to the different expressions of the generic lower dimensional

j

2. Fractal dromiom pattern: the dromion-like (lump-like) structure is exponentially (algebraically) localized on a large scale and possesses self-similar structure

<sup>Θ</sup>ð Þ¼ <sup>ξ</sup>; <sup>t</sup> exp �j j <sup>θ</sup> N r <sup>þ</sup> <sup>s</sup>sin ln <sup>θ</sup><sup>2</sup> � � � � <sup>þ</sup> <sup>w</sup> cos ln <sup>θ</sup><sup>2</sup> � � � � � � � � , (48)

<sup>Θ</sup>ð Þ¼ <sup>ξ</sup>; <sup>t</sup> <sup>∣</sup>θ<sup>∣</sup> <sup>α</sup> sin ln <sup>θ</sup><sup>2</sup> � � � � <sup>þ</sup> <sup>β</sup> cos ln <sup>θ</sup><sup>2</sup> � � � � � � <sup>N</sup><sup>~</sup> <sup>=</sup> <sup>1</sup> <sup>þ</sup> <sup>θ</sup><sup>4</sup> � �, (49)

with θ<sup>j</sup> � kξ � vt þ θ0<sup>j</sup>, θ<sup>0</sup> being an arbitrary parameter, and constants N, r, w,

for fractal lump solution. Constants α, β, and N~ are arbitrary parameters. 3. Stochastic fractal pattern: Such typical excitation is expressed through the

α�j=<sup>2</sup> sin β<sup>j</sup>

with constants α and β being arbitrary parameters. A stochastic fractal excitation

Rið Þ θ<sup>i</sup> Rj θ<sup>j</sup>

provided quantities θ0<sup>j</sup>, λj, αj, and β<sup>j</sup> being arbitrary parameters. Also, θ<sup>j</sup> � kjξ � vjt þ θ0<sup>j</sup>. Variables ξ, kj, and vj are spacelike-defined, wave number, and

h i � � <sup>þ</sup> <sup>β</sup><sup>j</sup> cos ln <sup>θ</sup><sup>2</sup>

n o n o h i � � , (47)

j

<sup>θ</sup> � �, N ! <sup>∞</sup>, (50)

� �, (51)

and Q ¼ Q Yð Þ ; T are arbitrary functions of ð Þ Z; T and ð Þ Y; T , respectively.

4. Discussion of some higher dimensional solutions

function Θ of two generalized coordinates ð Þ ξ; t as defined by [33].

λjθj∣θj∣ α<sup>j</sup> sin ln θ<sup>2</sup>

near the center of the pattern. The function Θ can be expressed as

1. Nonlocal fractal pattern: we have the following

g ¼ a<sup>0</sup> þ a1P þ a2Q þ a3PQ , (46)

eral ansatz for the function g in the form

DOI: http://dx.doi.org/10.5772/intechopen.82306

Fractal Structures of the Carbon Nanotube System Arrays

functions g expressed previously.

Θð Þ¼ ξ; t ∑

2 j¼1

velocity of the j-wave component, respectively.

and s are arbitrary parameters. But also we can find

differentiable Weierstrass function ℘ defined as

can be expressed as

91

℘ð Þ¼ ξ; t ∑

N j¼0

Θð Þ¼ ξ; t Σ

i,j

Eq. (23) with (24) becomes a standard truncated expansion

$$\mathbf{A} = \mathbf{A}\_0/\mathbf{g}^2 + \mathbf{A}\_1/\mathbf{g} + \mathbf{A}\_2, \quad \mathbf{B} = \mathbf{B}\_0/\mathbf{g} + \mathbf{B}\_1, \quad \mathbf{C} = \mathbf{C}\_0/\mathbf{g}^2 + \mathbf{C}\_1/\mathbf{g} + \mathbf{C}\_2. \tag{40}$$

After vanishing A<sup>3</sup> and B3, and using Eq. (38), we can reduce the system (36) to

$$B\_{1,ZT} = A\_2 B\_1 + B\_{1,YY}, \quad A\_{2,T} = -B\_1 C\_2, \quad C\_{2,T} = A\_2 B\_1. \tag{41}$$

From Eq. (41), it follows that A2, B1, C<sup>2</sup> is a solution of the system (22). Besides, the truncated expansion Eq. (40) actually stands for a BT. Generally, in order to construct a typical family of solution to Eq. (22) in a simple manner, it is useful to consider very simple expressions of A2, B1, and C2. For convenience, we fix the original seed solution as

$$A\_2 = \nu, \quad B\_1 = \mathbf{0}, \quad C\_2 = -\iota \varepsilon \nu,\tag{42}$$

with parameter ν being an arbitrary constant. The seed solution is actually used for constructing many other solutions. However, many other classes of solutions are obtained for other existing seed solutions. It is that property of the Painlevé approach for constructing various kinds of solutions by means of arbitrary functions that makes it potential and powerfully underlying. The solutions are given by Eq. (40) expressed in a truncated form. Many solutions are constructed in a straightforward way due to the arbitrariness of these functions, provided to solve analytically or numerically some nonlinear partial differential constraint equations.

Substituting the BT from Eq. (40) and using the Eq. (42) into Eq. (22), we derive some bilinear equations which can be decoupled as

$$\begin{aligned} D\_2 \mathbf{D}\_T \mathbf{H} \cdot \mathbf{F} &= \nu\_1 \mathbf{H} \mathbf{F}, \quad D\_Y \mathbf{D}\_T \mathbf{H} \cdot \mathbf{F} = -\nu\_2 \mathbf{H} \mathbf{F}, \quad D\_Y^2 \mathbf{H} \cdot \mathbf{F} = -\nu\_2 \mathbf{H} \mathbf{F}, \\ D\_Y \mathbf{D}\_T \mathbf{F} \cdot \mathbf{F} &= \mathbf{H}^2 / 2, \quad D\_T^2 \mathbf{F} \cdot \mathbf{F} = \mathbf{H}^2 / 2, \quad D\_Y^2 \mathbf{F} \cdot \mathbf{F} = \mathbf{H}^2 / 2, \end{aligned} \tag{43}$$

provided A ¼ DZ þ EY and C∝ð Þ BZ � BY so as to express

$$B = H/F, \quad D = \nu\_1 Z - 2\partial\_T \ln \left( F \right), \quad E = \nu\_2 Y + 2\partial\_Y \ln \left( F \right), \tag{44}$$

with ν ¼ ν<sup>1</sup> þ ν2. The symbols DY, DZ, and DT refer to the Hirota operators [29–31] with respect to the variables Y, Z, and T, respectively. According to the usual procedure, the dependent function is expanded into suitable power series of a perturbation parameter and using them in Eq. (43), we can straightforwardly construct the one-, two- and N-soliton solutions (N being an integer) to Eq. (22). Nevertheless, the investigation of these solutions will be studied in detail in a separate paper. Now, knowing the BT and the related Hirota bilinearization of Eq. (22), we can conclude that the 2ð Þ þ 1 -dimensional system above is completely integrable.

After substitution Eqs. (25) and (33) into (40), we find

$$A = \nu + \frac{(D\_Y^2 - D\_Z D\_T)\mathbf{g} \cdot \mathbf{g}}{\mathbf{g}^2}, \quad B = 2\nu\epsilon\vartheta\_T(\ln|\mathbf{g}|), \quad \mathbf{C} = -\nu\epsilon A. \tag{45}$$

Fractal Structures of the Carbon Nanotube System Arrays DOI: http://dx.doi.org/10.5772/intechopen.82306

truncation of the WTC expansion to obtain the BT and the Hirota bilinearization

<sup>A</sup> <sup>¼</sup> <sup>A</sup>0=g<sup>2</sup> <sup>þ</sup> <sup>A</sup>1=<sup>g</sup> <sup>þ</sup> <sup>A</sup>2, B <sup>¼</sup> <sup>B</sup>0=<sup>g</sup> <sup>þ</sup> <sup>B</sup>1, C <sup>¼</sup> <sup>C</sup>0=g<sup>2</sup> <sup>þ</sup> <sup>C</sup>1=<sup>g</sup> <sup>þ</sup> <sup>C</sup>2: (40)

After vanishing A<sup>3</sup> and B3, and using Eq. (38), we can reduce the system (36) to

From Eq. (41), it follows that A2, B1, C<sup>2</sup> is a solution of the system (22). Besides, the truncated expansion Eq. (40) actually stands for a BT. Generally, in order to construct a typical family of solution to Eq. (22) in a simple manner, it is useful to consider very simple expressions of A2, B1, and C2. For convenience, we fix the

with parameter ν being an arbitrary constant. The seed solution is actually used for constructing many other solutions. However, many other classes of solutions are

=2, D<sup>2</sup>

<sup>B</sup> <sup>¼</sup> <sup>H</sup>=F, D <sup>¼</sup> <sup>ν</sup>1<sup>Z</sup> � <sup>2</sup>∂Tln ð Þ <sup>F</sup> , E <sup>¼</sup> <sup>ν</sup>2<sup>Y</sup> <sup>þ</sup> <sup>2</sup>∂Yln ð Þ <sup>F</sup> , (44)

with ν ¼ ν<sup>1</sup> þ ν2. The symbols DY, DZ, and DT refer to the Hirota operators [29–31] with respect to the variables Y, Z, and T, respectively. According to the usual procedure, the dependent function is expanded into suitable power series of a perturbation parameter and using them in Eq. (43), we can straightforwardly construct the one-, two- and N-soliton solutions (N being an integer) to Eq. (22). Nevertheless, the investigation of these solutions will be studied in detail in a separate paper. Now, knowing the BT and the related Hirota bilinearization of Eq. (22), we can conclude that the 2ð Þ þ 1 -dimensional system above is completely

obtained for other existing seed solutions. It is that property of the Painlevé approach for constructing various kinds of solutions by means of arbitrary functions that makes it potential and powerfully underlying. The solutions are given by Eq. (40) expressed in a truncated form. Many solutions are constructed in a straightforward way due to the arbitrariness of these functions, provided to solve analytically or numerically some nonlinear partial differential constraint equations. Substituting the BT from Eq. (40) and using the Eq. (42) into Eq. (22), we

derive some bilinear equations which can be decoupled as

DZDTH � <sup>F</sup> <sup>¼</sup> <sup>v</sup>1HF, DYDTH � <sup>F</sup> ¼ �v2HF, D<sup>2</sup>

provided A ¼ DZ þ EY and C∝ð Þ BZ � BY so as to express

After substitution Eqs. (25) and (33) into (40), we find

TF � <sup>F</sup> <sup>¼</sup> <sup>H</sup><sup>2</sup>

=2, D<sup>2</sup>

B1,ZT ¼ A2B<sup>1</sup> þ B1,YY, A2,T ¼ �B1C2, C2,T ¼ A2B1: (41)

Eq. (23) with (24) becomes a standard truncated expansion

Akþ<sup>1</sup> ¼ Bk ¼ Ckþ<sup>1</sup> ¼ 0, for k≥2, (39)

A<sup>2</sup> ¼ ν, B<sup>1</sup> ¼ 0, C<sup>2</sup> ¼ �ιεν, (42)

YF � <sup>F</sup> <sup>¼</sup> <sup>H</sup><sup>2</sup>

<sup>g</sup><sup>2</sup> , B <sup>¼</sup> <sup>2</sup>ιε∂Tð Þ ln <sup>j</sup>g<sup>j</sup> , C ¼ �ιεA: (45)

YH � F ¼ �v2HF,

<sup>=</sup>2, (43)

[27–30] of the system (22). By setting

Fractal Analysis

original seed solution as

DYDTF � <sup>F</sup> <sup>¼</sup> <sup>H</sup><sup>2</sup>

integrable.

90

A ¼ ν þ

D2

<sup>Y</sup> � DZDT <sup>g</sup> � <sup>g</sup>

In the next section, because of the arbitrariness of some functions derived from the Painlevé analysis, we aim at focusing our interest to solutions for which the quantities A and B are expressed in the reduction form Eq. (45). In order to express some exact solutions of our initial coupled evolution system, we consider the general ansatz for the function g in the form

$$\mathbf{g} = \mathbf{a}\_0 + \mathbf{a}\_1 \mathbf{P} + \mathbf{a}\_2 \mathbf{Q} + \mathbf{a}\_3 \mathbf{P} \mathbf{Q},\tag{46}$$

where the parameter akð Þ k ¼ 0; 1; 2; 3 is an arbitrary constant and P ¼ P Zð Þ ; T and Q ¼ Q Yð Þ ; T are arbitrary functions of ð Þ Z; T and ð Þ Y; T , respectively.
