4. Discussion of some higher dimensional solutions

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 functions g expressed previously.

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

1. Nonlocal fractal pattern: we have the following

$$\Theta(\xi, t) = \sum\_{j=1}^{2} \lambda\_j \theta\_j |\theta\_j| \left\{ \left\{ a\_j \sin \left[ \ln \left( \theta\_j^2 \right) \right] + \beta\_j \cos \left[ \ln \left( \theta\_j^2 \right) \right] \right\} \right\}, \tag{47}$$

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 velocity of the j-wave component, respectively.

2. Fractal dromiom pattern: the dromion-like (lump-like) structure is exponentially (algebraically) localized on a large scale and possesses self-similar structure near the center of the pattern. The function Θ can be expressed as

$$\Theta(\xi, t) = \exp\left\{-|\theta|\overline{N}\{r + \mathfrak{s}\sin\left[\ln\left(\theta^2\right)\right] + w\cos\left[\ln\left(\theta^2\right)\right]\}\right\},\tag{48}$$

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, and s are arbitrary parameters. But also we can find

$$\Theta(\xi, t) = |\theta| \left\{ \overline{a} \sin \left[ \ln \left( \theta^2 \right) \right] + \overline{\beta} \cos \left[ \ln \left( \theta^2 \right) \right] \right\} \dot{N} / \left( 1 + \theta^4 \right), \tag{49}$$

for fractal lump solution. Constants α, β, and N~ are arbitrary parameters.

3. Stochastic fractal pattern: Such typical excitation is expressed through the differentiable Weierstrass function ℘ defined as

$$\otimes(\xi, t) = \sum\_{j=0}^{N} a^{-j/2} \sin \left( \beta^j \theta \right), N \to \infty,\tag{50}$$

with constants α and β being arbitrary parameters. A stochastic fractal excitation can be expressed as

$$\Theta(\xi,\mathfrak{t}) = \sum\_{i,j} R\_i(\theta\_i) R\_j(\theta\_j),\tag{51}$$

Depiction of Nonlocal fractal patterns at t ¼ 0. The parameters are chosen as a<sup>0</sup> ¼ 1, a<sup>1</sup> ¼ 1, a<sup>2</sup> ¼ 1, and a<sup>3</sup> ¼ 1 such that: For p xð Þ¼ ; t Θð Þ x; t , λ<sup>1</sup> ¼ 1=4, λ<sup>2</sup> ¼ 0, θ<sup>01</sup> ¼ 0, k<sup>1</sup> ¼ 1, and v<sup>1</sup> ¼ 1. For q y ð Þ¼ ; t Θð Þ y; t , λ<sup>1</sup> ¼ 1=4, λ<sup>2</sup> ¼ 0, θ<sup>02</sup> ¼ 0, k<sup>2</sup> ¼ 1, and v<sup>2</sup> ¼ 1. Note that α<sup>1</sup> ¼ 1 and β<sup>1</sup> ¼ 0. Panels (a) and (c) represent the pattern formations depicted in 3D-perspective, and the two others (b) and (d) are their corresponding densities represented within the square regions �3:<sup>6</sup> � <sup>10</sup>�<sup>2</sup>; <sup>3</sup>:<sup>6</sup> � <sup>10</sup>�<sup>2</sup> <sup>2</sup> and �2:<sup>32</sup> � <sup>10</sup>�<sup>10</sup>; <sup>2</sup>:<sup>32</sup> � <sup>10</sup>�<sup>10</sup> <sup>2</sup> , respectively.

Figure 3.

Figure 4.

93

�2:<sup>32</sup> � <sup>10</sup>�<sup>10</sup>; <sup>2</sup>:<sup>32</sup> � <sup>10</sup>�<sup>10</sup> <sup>2</sup>

Fractal lump excitations depicted at t ¼ 0 by the observable ∣B∣ � I which expression is given by Eq. (40). In this case, the parameters are selected as a<sup>0</sup> ¼ 1, a<sup>1</sup> ¼ 1, a<sup>2</sup> ¼ 1, and a<sup>3</sup> ¼ 2 such that: For p xð Þ¼ ; t Θð Þ x; t , θ<sup>01</sup> ¼ 0, k<sup>1</sup> ¼ 1, and v<sup>1</sup> ¼ 1. For q y ð Þ¼ ; t Θð Þ y; t , θ<sup>02</sup> ¼ 0, k<sup>2</sup> ¼ 1, and v<sup>2</sup> ¼ 1. Note that α ¼ 1, and β ¼ 0. Panels (a) and (c) represent the pattern formations depicted in 3D-perspective, and the two others (b) and (d)

Fractal stochastic nonlocal excitations depicted at t ¼ 0 by the observable ∣B∣ � I which expression is given by Eq. (40). In this case, the parameters are selected as a<sup>0</sup> ¼ 1, a<sup>1</sup> ¼ 1, a<sup>2</sup> ¼ 1, and a<sup>3</sup> ¼ 1 such that: For p xð Þ¼ ; t Θð Þ x; t , λ<sup>1</sup> ¼ 1=4, λ<sup>2</sup> ¼ 0, θ<sup>01</sup> ¼ 0, k<sup>1</sup> ¼ 1, and v<sup>1</sup> ¼ 1. For q y ð Þ¼ ; t Θð Þ y; t , λ<sup>1</sup> ¼ 1=4, λ<sup>2</sup> ¼ 0, θ<sup>02</sup> ¼ 0, k<sup>2</sup> ¼ 1, and v<sup>2</sup> ¼ 1. Note that α ¼ 3=2, β ¼ 3=2, and N ¼ 100. Panels (a) and (c) represent the pattern formations depicted in 3D-perspective, and the two others (b) and (d) are their corresponding densities

, respectively.

represented within the square regions �3:<sup>6</sup> � <sup>10</sup>�<sup>2</sup>; <sup>3</sup>:<sup>6</sup> � <sup>10</sup>�<sup>2</sup> <sup>2</sup> and �2:<sup>32</sup> � <sup>10</sup>�<sup>10</sup>; <sup>2</sup>:<sup>32</sup> � <sup>10</sup>�<sup>10</sup> <sup>2</sup>

are their corresponding densities represented within the square regions �3:<sup>6</sup> � <sup>10</sup>�<sup>2</sup>; <sup>3</sup>:<sup>6</sup> � <sup>10</sup>�<sup>2</sup> <sup>2</sup> and

, respectively.

Fractal Structures of the Carbon Nanotube System Arrays

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

#### Figure 2.

Fractal dromiom excitations depicted at t ¼ 0 by the observable ∣B∣ � I which expression is given by Eq. (40). In this case, the parameters are selected as a<sup>0</sup> ¼ 1, a<sup>1</sup> ¼ 1, a<sup>2</sup> ¼ 1, and a<sup>3</sup> ¼ 1 such that: For p xð Þ¼ ; t Θð Þ x; t , r ¼ 3=2, s ¼ 1, w ¼ 0, N ¼ 1, θ<sup>01</sup> ¼ 0, k<sup>1</sup> ¼ 1, and v<sup>1</sup> ¼ 1. For q y ð Þ¼ ; t Θð Þ y; t , θ<sup>02</sup> ¼ 0, k<sup>2</sup> ¼ 1, and v<sup>2</sup> ¼ 1. Panels (a) and (c) represent the pattern formations depicted in 3D-perspective, and the two others (b) and (d) are their corresponding densities represented within the square regions �<sup>7</sup> � <sup>10</sup>�<sup>3</sup>; <sup>7</sup> � <sup>10</sup>�<sup>3</sup> <sup>2</sup> and �2:<sup>8</sup> � <sup>10</sup>�<sup>8</sup>; <sup>2</sup>:<sup>8</sup> � <sup>10</sup>�<sup>8</sup> <sup>2</sup> , respectively.

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

#### Figure 3.

Figure 1.

Fractal Analysis

Figure 2.

92

�2:<sup>8</sup> � <sup>10</sup>�<sup>8</sup>; <sup>2</sup>:<sup>8</sup> � <sup>10</sup>�<sup>8</sup> <sup>2</sup>

, respectively.

Depiction of Nonlocal fractal patterns at t ¼ 0. The parameters are chosen as a<sup>0</sup> ¼ 1, a<sup>1</sup> ¼ 1, a<sup>2</sup> ¼ 1, and a<sup>3</sup> ¼ 1 such that: For p xð Þ¼ ; t Θð Þ x; t , λ<sup>1</sup> ¼ 1=4, λ<sup>2</sup> ¼ 0, θ<sup>01</sup> ¼ 0, k<sup>1</sup> ¼ 1, and v<sup>1</sup> ¼ 1. For q y ð Þ¼ ; t Θð Þ y; t , λ<sup>1</sup> ¼ 1=4, λ<sup>2</sup> ¼ 0, θ<sup>02</sup> ¼ 0, k<sup>2</sup> ¼ 1, and v<sup>2</sup> ¼ 1. Note that α<sup>1</sup> ¼ 1 and β<sup>1</sup> ¼ 0. Panels (a) and (c) represent the pattern formations depicted in 3D-perspective, and the two others (b) and (d) are their corresponding densities

Fractal dromiom excitations depicted at t ¼ 0 by the observable ∣B∣ � I which expression is given by Eq. (40). In this case, the parameters are selected as a<sup>0</sup> ¼ 1, a<sup>1</sup> ¼ 1, a<sup>2</sup> ¼ 1, and a<sup>3</sup> ¼ 1 such that: For p xð Þ¼ ; t Θð Þ x; t , r ¼ 3=2, s ¼ 1, w ¼ 0, N ¼ 1, θ<sup>01</sup> ¼ 0, k<sup>1</sup> ¼ 1, and v<sup>1</sup> ¼ 1. For q y ð Þ¼ ; t Θð Þ y; t , θ<sup>02</sup> ¼ 0, k<sup>2</sup> ¼ 1, and v<sup>2</sup> ¼ 1. Panels (a) and (c) represent the pattern formations depicted in 3D-perspective, and the two others (b) and (d) are their corresponding densities represented within the square regions �<sup>7</sup> � <sup>10</sup>�<sup>3</sup>; <sup>7</sup> � <sup>10</sup>�<sup>3</sup> <sup>2</sup> and

, respectively.

represented within the square regions �3:<sup>6</sup> � <sup>10</sup>�<sup>2</sup>; <sup>3</sup>:<sup>6</sup> � <sup>10</sup>�<sup>2</sup> <sup>2</sup> and �2:<sup>32</sup> � <sup>10</sup>�<sup>10</sup>; <sup>2</sup>:<sup>32</sup> � <sup>10</sup>�<sup>10</sup> <sup>2</sup>

Fractal lump excitations depicted at t ¼ 0 by the observable ∣B∣ � I which expression is given by Eq. (40). In this case, the parameters are selected as a<sup>0</sup> ¼ 1, a<sup>1</sup> ¼ 1, a<sup>2</sup> ¼ 1, and a<sup>3</sup> ¼ 2 such that: For p xð Þ¼ ; t Θð Þ x; t , θ<sup>01</sup> ¼ 0, k<sup>1</sup> ¼ 1, and v<sup>1</sup> ¼ 1. For q y ð Þ¼ ; t Θð Þ y; t , θ<sup>02</sup> ¼ 0, k<sup>2</sup> ¼ 1, and v<sup>2</sup> ¼ 1. Note that α ¼ 1, and β ¼ 0. Panels (a) and (c) represent the pattern formations depicted in 3D-perspective, and the two others (b) and (d) are their corresponding densities represented within the square regions �3:<sup>6</sup> � <sup>10</sup>�<sup>2</sup>; <sup>3</sup>:<sup>6</sup> � <sup>10</sup>�<sup>2</sup> <sup>2</sup> and �2:<sup>32</sup> � <sup>10</sup>�<sup>10</sup>; <sup>2</sup>:<sup>32</sup> � <sup>10</sup>�<sup>10</sup> <sup>2</sup> , respectively.

#### Figure 4.

Fractal stochastic nonlocal excitations depicted at t ¼ 0 by the observable ∣B∣ � I which expression is given by Eq. (40). In this case, the parameters are selected as a<sup>0</sup> ¼ 1, a<sup>1</sup> ¼ 1, a<sup>2</sup> ¼ 1, and a<sup>3</sup> ¼ 1 such that: For p xð Þ¼ ; t Θð Þ x; t , λ<sup>1</sup> ¼ 1=4, λ<sup>2</sup> ¼ 0, θ<sup>01</sup> ¼ 0, k<sup>1</sup> ¼ 1, and v<sup>1</sup> ¼ 1. For q y ð Þ¼ ; t Θð Þ y; t , λ<sup>1</sup> ¼ 1=4, λ<sup>2</sup> ¼ 0, θ<sup>02</sup> ¼ 0, k<sup>2</sup> ¼ 1, and v<sup>2</sup> ¼ 1. Note that α ¼ 3=2, β ¼ 3=2, and N ¼ 100. Panels (a) and (c) represent the pattern formations depicted in 3D-perspective, and the two others (b) and (d) are their corresponding densities represented within the square regions �3:<sup>6</sup> � <sup>10</sup>�<sup>2</sup>; <sup>3</sup>:<sup>6</sup> � <sup>10</sup>�<sup>2</sup> <sup>2</sup> and �2:<sup>32</sup> � <sup>10</sup>�<sup>10</sup>; <sup>2</sup>:<sup>32</sup> � <sup>10</sup>�<sup>10</sup> <sup>2</sup> , respectively.

where Ri <sup>¼</sup> <sup>℘</sup>ð Þþ <sup>θ</sup><sup>i</sup> <sup>θ</sup><sup>2</sup> <sup>i</sup> þ μi, with μ<sup>i</sup> standing for arbitrary parameter.

Stochastic fractal dromion/solitoff excitations: such structures are obtained by including the Weierstrass function into the dromiom solution. Especially for solitoff excitations, we can try the following:

$$\Theta(\xi, t) = k + \sum\_{j=0} \eta\_j \otimes (\theta\_j) \tanh^{\mu\_j}(\theta\_j), \tag{52}$$

provided quantities k, ηj, and μ<sup>j</sup> being arbitrary parameters. Stochastic fractal lump pattern: Eq. (51) is reduced as

$$\Theta(\xi, t) = \sum\_{i, j} \rho\_j R\_j(\theta\_j), \tag{53}$$

Next, in Figure 3, we obtain the fractal lump which shows self-similar structures

In addition to the above self-similar regular fractal dromion and lump excitations, by using the lower dimensional stochastic fractal functions, we construct some other higher dimensional stochastic fractal patterns. Thus, in Figure 4, we generate a typical stochastic fractal nonlocal pattern with self-similarity in structure.

Besides, in Figure 5, with the selecting parameters and suitable choices of lower dimensional arbitrary stochastic fractal dromion function as presented in the captions of these figures, we obtain higher dimensional stochastic dromion excitations. The self-similarity in structure of the observable I � ∣B∣ shows how the peaks are

In Figure 6, we construct the fractal solitoff excitations. By reducing the region

In Figure 7 with the selecting parameters and suitable choices of lower dimensional arbitrary stochastic fractal lump function as presented in the captions of the figure, we obtain higher dimensional stochastic lump excitations. Through the panels (7(a) and 7(b)) depicting the variations of ∣B∣�observable, at t ¼ 0, the selfsimilarity in structure of this observable shows how the peaks are distributed

; <sup>7</sup> � <sup>10</sup>�<sup>3</sup> <sup>2</sup> and

Fractal stochastic solitoff excitations depicted at t ¼ 0 by the observable ∣B∣ � I which expression is given by Eq. (40). In this case, the parameters are selected as a<sup>0</sup> ¼ 1, a<sup>1</sup> ¼ 1, a<sup>2</sup> ¼ 1, and a<sup>3</sup> ¼ 1 such that: For p xð Þ¼ ; t Θð Þ x; t , κ ¼ 2, M ¼ 1, η<sup>0</sup> ¼ 0, η<sup>1</sup> ¼ 1=2, η<sup>m</sup> ¼ 0ð Þ m ≥2 , μ<sup>1</sup> ¼ 1, θ<sup>01</sup> ¼ �20, k<sup>1</sup> ¼ 4, and v<sup>1</sup> ¼ 1. For q y ð Þ¼ ; t Θð Þ y; t , κ ¼ 0, M ¼ 2, η<sup>0</sup> ¼ 0, η<sup>1</sup> ¼ 1=5, η<sup>2</sup> ¼ 1=4, η<sup>m</sup> ¼ 0ð Þ m ≥3 , μ<sup>1</sup> ¼ μ<sup>1</sup> ¼ 1, θ<sup>02</sup> ¼ �15, k<sup>2</sup> ¼ 2, and v<sup>2</sup> ¼ 2. Note that α ¼ 3=2, β ¼ 3=2, and N ¼ 100. Panels (a) and (c) represent the pattern formations depicted in 3D-perspective, and the two others (b) and (d) are their corresponding densities

represented within the square regions �1:<sup>2</sup> � <sup>10</sup>�<sup>2</sup>; <sup>1</sup>:<sup>2</sup> � <sup>10</sup>�<sup>2</sup> � �½ � <sup>5</sup>; <sup>10</sup> and

�1:<sup>5</sup> � <sup>10</sup>�<sup>8</sup>; <sup>1</sup>:<sup>5</sup> � <sup>10</sup>�<sup>8</sup> � �½ � <sup>5</sup>; <sup>10</sup> , respectively.

; <sup>1</sup>:<sup>2</sup> � <sup>10</sup>�<sup>2</sup> � �½ � <sup>5</sup>; <sup>10</sup> of panel 6ð Þ <sup>a</sup> to �1:<sup>5</sup> � <sup>10</sup>�8; <sup>1</sup>:<sup>5</sup> � <sup>10</sup>�<sup>8</sup> � �½ � <sup>5</sup>; <sup>10</sup> of panel 6ð Þc , we obtain a totally similar structure with density plots represented

; <sup>1</sup> � <sup>10</sup>�<sup>2</sup> <sup>2</sup> and

. In the above configurations, the stochastic fractal solitoff

distributed stochastically within the regions �<sup>1</sup> � <sup>10</sup>�<sup>2</sup>

Fractal Structures of the Carbon Nanotube System Arrays

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

�2:<sup>6</sup> � <sup>10</sup>�8; <sup>2</sup>:<sup>6</sup> � <sup>10</sup>�<sup>8</sup> <sup>2</sup> for stochastic fractal dromion.

in panels ð Þc and ð Þ d .

�1:<sup>2</sup> � <sup>10</sup>�<sup>2</sup>

Figure 6.

95

in panels (b) and (d), respectively.

�2:<sup>1</sup> � <sup>10</sup>�<sup>10</sup>; <sup>2</sup>:<sup>1</sup> � <sup>10</sup>�<sup>10</sup> <sup>2</sup>

stochastically within the regions �<sup>7</sup> � <sup>10</sup>�<sup>3</sup>

where quantities η<sup>j</sup> and ρ<sup>j</sup> being arbitrary parameter.

Now, let us analyze different figures with respect to the previous classifications. Thus, in Figure 1, we depict the variations of the ∣B∣-observable with space at t ¼ 0.

In a 3D�representation, the features presented in panel 1ð Þ a within the space region �3:<sup>6</sup> � <sup>10</sup>�<sup>2</sup> ; <sup>3</sup>:<sup>6</sup> � <sup>10</sup>�<sup>2</sup> <sup>2</sup> � ∣B∣ and those depicted in ð Þc within region �2:<sup>32</sup> � <sup>10</sup>�<sup>10</sup>; <sup>2</sup>:<sup>32</sup> � <sup>10</sup>�<sup>10</sup> <sup>2</sup> � ∣B∣ are self-similar nonlocal. Such a similarity in the profiles is clearly shown in panels ð Þ b and ð Þ d standing for their density plots, respectively.

Following the above figure, in Figure 2, we generate the fractal dromiom depicting self-similar structure with density plots represented in panels ð Þ b and ð Þ d , respectively. In comparison to the previous nonlocal fractal patterns, it appears that the fractal dromions have relatively high amplitudes.

#### Figure 5.

Fractal stochastic dromion excitations depicted at t ¼ 0 by the observable ∣B∣ � I which expression is given by Eq. (40). In this case, the parameters are selected as a<sup>0</sup> ¼ 1, a<sup>1</sup> ¼ 1, a<sup>2</sup> ¼ 1, and a<sup>3</sup> ¼ 1 such that: For p xð Þ¼ ; t Θð Þ x; t , r ¼ 3=2, s ¼ 1, w ¼ 0, N ¼ 1, θ<sup>01</sup> ¼ 0, k<sup>1</sup> ¼ 1, and v<sup>1</sup> ¼ 1. For q y ð Þ¼ ; t Θð Þ y; t , θ<sup>02</sup> ¼ 0, k<sup>2</sup> ¼ 1, and v<sup>2</sup> ¼ 1. Note that α ¼ 3=2, β ¼ 3=2, and N ¼ 100. Panels (a) and (c) represent the pattern formations depicted in 3D-perspective, and the two others (b) and (d) are their corresponding densities represented within the square regions �<sup>1</sup> � <sup>10</sup>�<sup>2</sup>; <sup>1</sup> � <sup>10</sup>�<sup>2</sup> <sup>2</sup> and �2:<sup>6</sup> � <sup>10</sup>�<sup>8</sup>; <sup>2</sup>:<sup>6</sup> � <sup>10</sup>�<sup>8</sup> <sup>2</sup> , respectively.

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

where Ri <sup>¼</sup> <sup>℘</sup>ð Þþ <sup>θ</sup><sup>i</sup> <sup>θ</sup><sup>2</sup>

Fractal Analysis

region �3:<sup>6</sup> � <sup>10</sup>�<sup>2</sup>

respectively.

Figure 5.

94

�2:<sup>32</sup> � <sup>10</sup>�<sup>10</sup>; <sup>2</sup>:<sup>32</sup> � <sup>10</sup>�<sup>10</sup> <sup>2</sup>

excitations, we can try the following:

; <sup>3</sup>:<sup>6</sup> � <sup>10</sup>�<sup>2</sup> <sup>2</sup>

<sup>i</sup> þ μi, with μ<sup>i</sup> standing for arbitrary parameter.

tanh<sup>μ</sup><sup>j</sup> θ<sup>j</sup>

� ∣B∣ and those depicted in ð Þc within region

� ∣B∣ are self-similar nonlocal. Such a similarity in the

, (52)

, respectively.

, (53)

Stochastic fractal dromion/solitoff excitations: such structures are obtained by including the Weierstrass function into the dromiom solution. Especially for solitoff

> j¼0 ηj ℘ θ<sup>j</sup>

Θð Þ¼ ξ; t Σ

profiles is clearly shown in panels ð Þ b and ð Þ d standing for their density plots,

Following the above figure, in Figure 2, we generate the fractal dromiom depicting self-similar structure with density plots represented in panels ð Þ b and ð Þ d , respectively. In comparison to the previous nonlocal fractal patterns, it appears that

Fractal stochastic dromion excitations depicted at t ¼ 0 by the observable ∣B∣ � I which expression is given by Eq. (40). In this case, the parameters are selected as a<sup>0</sup> ¼ 1, a<sup>1</sup> ¼ 1, a<sup>2</sup> ¼ 1, and a<sup>3</sup> ¼ 1 such that: For p xð Þ¼ ; t Θð Þ x; t , r ¼ 3=2, s ¼ 1, w ¼ 0, N ¼ 1, θ<sup>01</sup> ¼ 0, k<sup>1</sup> ¼ 1, and v<sup>1</sup> ¼ 1. For q y ð Þ¼ ; t Θð Þ y; t , θ<sup>02</sup> ¼ 0, k<sup>2</sup> ¼ 1, and v<sup>2</sup> ¼ 1. Note that α ¼ 3=2, β ¼ 3=2, and N ¼ 100. Panels (a) and (c) represent the pattern formations depicted in 3D-perspective, and the two others (b) and (d) are their corresponding densities

represented within the square regions �<sup>1</sup> � <sup>10</sup>�<sup>2</sup>; <sup>1</sup> � <sup>10</sup>�<sup>2</sup> <sup>2</sup> and �2:<sup>6</sup> � <sup>10</sup>�<sup>8</sup>; <sup>2</sup>:<sup>6</sup> � <sup>10</sup>�<sup>8</sup> <sup>2</sup>

i,j

Now, let us analyze different figures with respect to the previous classifications. Thus, in Figure 1, we depict the variations of the ∣B∣-observable with space at t ¼ 0. In a 3D�representation, the features presented in panel 1ð Þ a within the space

ρjRj θ<sup>j</sup>

Θð Þ¼ ξ; t k þ Σ

provided quantities k, ηj, and μ<sup>j</sup> being arbitrary parameters.

Stochastic fractal lump pattern: Eq. (51) is reduced as

where quantities η<sup>j</sup> and ρ<sup>j</sup> being arbitrary parameter.

the fractal dromions have relatively high amplitudes.

Next, in Figure 3, we obtain the fractal lump which shows self-similar structures in panels ð Þc and ð Þ d .

In addition to the above self-similar regular fractal dromion and lump excitations, by using the lower dimensional stochastic fractal functions, we construct some other higher dimensional stochastic fractal patterns. Thus, in Figure 4, we generate a typical stochastic fractal nonlocal pattern with self-similarity in structure.

Besides, in Figure 5, with the selecting parameters and suitable choices of lower dimensional arbitrary stochastic fractal dromion function as presented in the captions of these figures, we obtain higher dimensional stochastic dromion excitations. The self-similarity in structure of the observable I � ∣B∣ shows how the peaks are distributed stochastically within the regions �<sup>1</sup> � <sup>10</sup>�<sup>2</sup> ; <sup>1</sup> � <sup>10</sup>�<sup>2</sup> <sup>2</sup> and

�2:<sup>6</sup> � <sup>10</sup>�8; <sup>2</sup>:<sup>6</sup> � <sup>10</sup>�<sup>8</sup> <sup>2</sup> for stochastic fractal dromion.

In Figure 6, we construct the fractal solitoff excitations. By reducing the region �1:<sup>2</sup> � <sup>10</sup>�<sup>2</sup> ; <sup>1</sup>:<sup>2</sup> � <sup>10</sup>�<sup>2</sup> � �½ � <sup>5</sup>; <sup>10</sup> of panel 6ð Þ <sup>a</sup> to �1:<sup>5</sup> � <sup>10</sup>�8; <sup>1</sup>:<sup>5</sup> � <sup>10</sup>�<sup>8</sup> � �½ � <sup>5</sup>; <sup>10</sup> of panel 6ð Þc , we obtain a totally similar structure with density plots represented in panels (b) and (d), respectively.

In Figure 7 with the selecting parameters and suitable choices of lower dimensional arbitrary stochastic fractal lump function as presented in the captions of the figure, we obtain higher dimensional stochastic lump excitations. Through the panels (7(a) and 7(b)) depicting the variations of ∣B∣�observable, at t ¼ 0, the selfsimilarity in structure of this observable shows how the peaks are distributed stochastically within the regions �<sup>7</sup> � <sup>10</sup>�<sup>3</sup> ; <sup>7</sup> � <sup>10</sup>�<sup>3</sup> <sup>2</sup> and

�2:<sup>1</sup> � <sup>10</sup>�<sup>10</sup>; <sup>2</sup>:<sup>1</sup> � <sup>10</sup>�<sup>10</sup> <sup>2</sup> . In the above configurations, the stochastic fractal solitoff

#### Figure 6.

Fractal stochastic solitoff excitations depicted at t ¼ 0 by the observable ∣B∣ � I which expression is given by Eq. (40). In this case, the parameters are selected as a<sup>0</sup> ¼ 1, a<sup>1</sup> ¼ 1, a<sup>2</sup> ¼ 1, and a<sup>3</sup> ¼ 1 such that: For p xð Þ¼ ; t Θð Þ x; t , κ ¼ 2, M ¼ 1, η<sup>0</sup> ¼ 0, η<sup>1</sup> ¼ 1=2, η<sup>m</sup> ¼ 0ð Þ m ≥2 , μ<sup>1</sup> ¼ 1, θ<sup>01</sup> ¼ �20, k<sup>1</sup> ¼ 4, and v<sup>1</sup> ¼ 1. For q y ð Þ¼ ; t Θð Þ y; t , κ ¼ 0, M ¼ 2, η<sup>0</sup> ¼ 0, η<sup>1</sup> ¼ 1=5, η<sup>2</sup> ¼ 1=4, η<sup>m</sup> ¼ 0ð Þ m ≥3 , μ<sup>1</sup> ¼ μ<sup>1</sup> ¼ 1, θ<sup>02</sup> ¼ �15, k<sup>2</sup> ¼ 2, and v<sup>2</sup> ¼ 2. Note that α ¼ 3=2, β ¼ 3=2, and N ¼ 100. Panels (a) and (c) represent the pattern formations depicted in 3D-perspective, and the two others (b) and (d) are their corresponding densities represented within the square regions �1:<sup>2</sup> � <sup>10</sup>�<sup>2</sup>; <sup>1</sup>:<sup>2</sup> � <sup>10</sup>�<sup>2</sup> � �½ � <sup>5</sup>; <sup>10</sup> and �1:<sup>5</sup> � <sup>10</sup>�<sup>8</sup>; <sup>1</sup>:<sup>5</sup> � <sup>10</sup>�<sup>8</sup> � �½ � <sup>5</sup>; <sup>10</sup> , respectively.

system is Painlevé integrable [15]. We derived another important properties, namely the Bäcklund transformation and the Hirota bilinearization [27–30] while

took advantage of the existence of some arbitrary functions to construct some interesting solutions such as fractals. Actually, following the investigation of fractals in many physical systems [31–33], we constructed some localized nonlinear excitations with some fractal support. As a result, we found the following typical features: the fractal dromion, the fractal lump, the stochastic and nonlocal fractal excitations. One of the advantages of the WTC method discussed in this work is the generation of arbitrary functions useful in constructing many kinds and different solutions to the governing system. From such property endowing the method with the powerfulness, it would be rather interesting again to construct other types of nonlinear excitations such as the bubbles, the solitoffs, the dromions, the peakons, the fractals, among others [34–37]. These typical excitations would be useful in the understanding, more deeply, of the interaction between light incident excitations and carbon nanotubes for some practical issues in nanomechanical, nanoelectronic, and nanophotonic devices, alongside some emerging applications exploiting the good thermal and electronic conductivities of carbon nanotubes in some flat panel

In the wake of the result obtained from the WTC approach of integrability, we

Also, we intend using the WTC method in order to discover more other interesting properties still unknown in the carbon nanotube arrays. Previously, we discovered the properties of compactons in CNT [38]. The different properties will allow us in the future to improve the different uses of carbon nanotube in different areas of life. Moreover, because of these electrical and mechanical properties (very resistant, flexible, and lightweight), they are very suitable for the design of pressure sensors. These could be used by engineers to prevent structural collapses in civil engineering. They will have to measure either the pressure or the shear. Similarly, these sensors can be used in medicine while incorporating the system in textiles for better follow-up of patients or in a shoe sole. In another view, the sensors will have to be able to perform a good measure of the desired size. So to refine the design of these sensors, it will be essential to get even more information about this material. By discovering more properties of the material, we will know more how to exploit it in a safe way in all the various disciplines combining research and innovation.

National Advanced School of Engineering, University of Yaounde I, Cameroon

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

establishing the complete integrability of the system.

Fractal Structures of the Carbon Nanotube System Arrays

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

displays and field-effect transistors, among others.

Author details

97

Raïssa S. Noule and Victor K. Kuetche\*

provided the original work is properly cited.

\*Address all correspondence to: vkuetche@yahoo.fr

Figure 7.

Fractal stochastic lump excitations depicted at t ¼ 0 by the observable ∣B∣ � I which expression is given by Eq. (40). In this case, the parameters are selected as a<sup>0</sup> ¼ 1, a<sup>1</sup> ¼ 1, a<sup>2</sup> ¼ 1, and a<sup>3</sup> ¼ 2 such that: For p xð Þ¼ ; t Θð Þ x; t , θ<sup>01</sup> ¼ 0, k<sup>1</sup> ¼ 1, and v<sup>1</sup> ¼ 1. For q y ð Þ¼ ; t Θð Þ y; t , θ<sup>02</sup> ¼ 0, k<sup>2</sup> ¼ 1, and v<sup>2</sup> ¼ 1. Note that <sup>α</sup> <sup>¼</sup> <sup>1</sup>, <sup>β</sup> <sup>¼</sup> <sup>0</sup>, <sup>N</sup><sup>~</sup> <sup>¼</sup> <sup>2</sup> associated to <sup>α</sup> <sup>¼</sup> <sup>3</sup>=2, <sup>β</sup> <sup>¼</sup> <sup>3</sup>=2, and <sup>N</sup> <sup>¼</sup> <sup>100</sup>. Panels (a) and (c) represent the pattern formations depicted in 3D-perspective, and the two others (b) and (d) are their corresponding densities represented within the square regions �<sup>7</sup> � <sup>10</sup>�<sup>3</sup>; <sup>7</sup> � <sup>10</sup>�<sup>3</sup> <sup>2</sup> and �2:<sup>1</sup> � <sup>10</sup>�<sup>10</sup>; <sup>2</sup>:<sup>1</sup> � <sup>10</sup>�<sup>10</sup> <sup>2</sup> , respectively.

and stochastic fractal lump excitations appear to be the waves with greater amplitudes in comparison to the previous ones.

From a physical viewpoint, the observable B which has the meaning of the dimensionless electric field shows that its intensity ∣B∣ can be nonlocal or rather selfconfined. Actually, the intensity of the electromagnetic wave propagating along the carbon nanotube arrays is compact within the arrays. The previous study has revealed that the light bullet intensity describes a fractal-like excitation which provides more insights into the structural dynamics of the system under investigation.

#### 5. Summary

Throughout the present work, we investigated the formation of fractal ultrashort spatiotemporal optical waveforms in arrays of carbon nanotubes. We followed the short-wave approximation to derive a generic (2+1)-dimensional coupled system. Such a coupled system was constructed via the use of the reductive perturbation analysis for the Maxwell equations and for the corresponding Boltzmann kinetic equation of the distribution function of electrons in the carbon nanotubes. Prior to the construction of different solutions to the previous coupled equations, we first studied the integrability of the governing system within the viewpoint of WTC formalism [15]. Thus, we investigated the singularity structure of the system. In this analysis, we expanded the different observables in the form of the Laurent series. Therefore, we found the leading order terms useful to solve the recurrent system. Solving this last system, we unearthed the different resonances of the governing equations. At the end, we found that the number of resonances balances seemingly the number of arbitrary functions in such a way that the governing system has sufficient and enough arbitrary functions. Hence, we derived that the

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

system is Painlevé integrable [15]. We derived another important properties, namely the Bäcklund transformation and the Hirota bilinearization [27–30] while establishing the complete integrability of the system.

In the wake of the result obtained from the WTC approach of integrability, we took advantage of the existence of some arbitrary functions to construct some interesting solutions such as fractals. Actually, following the investigation of fractals in many physical systems [31–33], we constructed some localized nonlinear excitations with some fractal support. As a result, we found the following typical features: the fractal dromion, the fractal lump, the stochastic and nonlocal fractal excitations.

One of the advantages of the WTC method discussed in this work is the generation of arbitrary functions useful in constructing many kinds and different solutions to the governing system. From such property endowing the method with the powerfulness, it would be rather interesting again to construct other types of nonlinear excitations such as the bubbles, the solitoffs, the dromions, the peakons, the fractals, among others [34–37]. These typical excitations would be useful in the understanding, more deeply, of the interaction between light incident excitations and carbon nanotubes for some practical issues in nanomechanical, nanoelectronic, and nanophotonic devices, alongside some emerging applications exploiting the good thermal and electronic conductivities of carbon nanotubes in some flat panel displays and field-effect transistors, among others.

Also, we intend using the WTC method in order to discover more other interesting properties still unknown in the carbon nanotube arrays. Previously, we discovered the properties of compactons in CNT [38]. The different properties will allow us in the future to improve the different uses of carbon nanotube in different areas of life. Moreover, because of these electrical and mechanical properties (very resistant, flexible, and lightweight), they are very suitable for the design of pressure sensors. These could be used by engineers to prevent structural collapses in civil engineering. They will have to measure either the pressure or the shear. Similarly, these sensors can be used in medicine while incorporating the system in textiles for better follow-up of patients or in a shoe sole. In another view, the sensors will have to be able to perform a good measure of the desired size. So to refine the design of these sensors, it will be essential to get even more information about this material. By discovering more properties of the material, we will know more how to exploit it in a safe way in all the various disciplines combining research and innovation.
