Fractal Structures of the Carbon Nanotube System Arrays

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.

1. Introduction

Carbon nanotubes stand to be one of the wonder materials of the present century [1, 2, 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], large-amplitude 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 two-dimensional light bullet propagating without being dispersed and diffracted over long distance with respect to the wavelength.

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, 19, 20, 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 $\mathbf{E}=-\partial \mathbf{A}/\partial t$ , $\mathbf{E}$ and $\mathbf{A}$ 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

$$\mathrm{\Delta }\mathbf{A}-{\mathbf{A}}_{\mathit{tt}}/c^2=-{\mu }_0\mathbf{J},$$ E1

where subscripts denote the partial derivatives. Constants ${\mu }_0$ 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 $\mathbf{J}$ is directed along the axis of the nanotubes, i.e., $\mathbf{J}=J_z{\mathbf{e}}_z$ , where unitary vector ${\mathbf{e}}_z$ spans the $z$ -axis. Besides, we consider the case where the wave field is polarized in the same direction, and $\mathbf{A}=A{\mathbf{e}}_z$ .

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

$$f_t-{\mathit{qA}}_t{f}_p=\left(F_0-f\right)/\tau ,$$ E2

where constant $q$ stands for the electron charge. The relaxation time $\tau$ 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 $p\equiv p\left(p_{\phi },p_z\right)$ of the electron. The azimuthal component $p_{\phi }$ reads $p_{\phi }=s\mathrm{\Delta }{p}_{\phi }$ , and the axial component $p_z$ 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 $F_0$ is the equilibrium value of the distribution $f$ and is known as the Fermi-distribution function expressed as

$$F_0=1/\left[1+\text{exp}\left(E/k_B{T}_0\right)\right],$$ E3

in which quantities $k_B$ , $T_0$ , 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 $\pi$ -electron approximation as follows

$$E=\gamma \sqrt{1+4\text{cos}\left(\mathit{ap}\right)\text{cos}\left(\mathit{\pi s}/m\right)+4{\text{cos}}^2\left(\mathit{\pi s}/m\right)},$$ E4

with $\gamma =2.7\mathit{eV}$ and $a=3b/2\mathrm{\hslash }$ where constant $b=0.142\mathit{nm}$ 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 $J_s$ can hence be expressed as

$$J_s=\frac{2q}{2\pi \mathrm{\hslash }}\int \int {\mathit{\text{vfdp}}}_{\phi }\mathit{dp},$$ E5

where the velocity $v$ reads $v=\partial E/\partial p$ . The distribution function $f$ can be written as

$$f=\sum _s\mathrm{\Delta }{p}_{\phi }\delta \left(p_{\phi }-s\mathrm{\Delta }{p}_{\phi }\right)f_s\left(p,t\right),$$ E6

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

$$J=\frac{\mathit{Nq}}{\pi \mathrm{\hslash }}\sum _s\int {\mathit{vf}}_s\mathit{dp},$$ E7

where constant $N$ represents the surface density of nanotubes in the $\mathit{xy}$ -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 $\tau$ and the propagation length is very long with respect to the wavelength, we introduce the fast and slow variables

$$\theta =\left(t-x/V\right)/\epsilon ,\xi =\mathit{\epsilon x},$$ E8

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

$$f_s=f_0+\epsilon f_1+\cdots ,A=A_0+\epsilon A_1+\cdots .$$ E9

Thus, at leading order ${\epsilon }^{-1}$ , Eq. (2) yields

$$f_{0t}-{\mathit{qA}}_{0t}{f}_{0p}=0,$$ E10

in which solution $f_0$ reads $f_0=\phi \left(p+{\mathit{qA}}_0\right)$ with $\phi$ being an arbitrary function. However, at large $t$ , the wave $A$ vanishes and $f_0$ goes to its equilibrium value $F_0$ . Thus, from Eq. (7), we write

$$J_0=\frac{q}{\pi \mathrm{\hslash }}\sum _s\int v\left(p+\mathit{qA}\right)f_0\left(p\right)\mathit{dp}.$$ E11

Now, at leading order ${\epsilon }^{-2}$ , the Maxwell equations transform to

$$V=c.$$ E12

The order ${\epsilon }^0$ provides

$$\left(2/c\right){\partial }^2{A}_0/\partial \xi \partial \theta ={\mu }_0{J}_0,$$ E13

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

The energy $E$ is of the same order of magnitude as $\gamma$ . Calculating $\gamma /k_B=3.1\times {10}^{4}K$ shows that $E$ 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

$$E_{\mathit{min}}=\gamma \mid 1-2\mid \text{cos}\left(\mathit{\pi s}/m\right)\Vert ,$$ E14

for $p=\pm \pi /a$ when $\text{cos}\left(\mathit{\pi s}/m\right)>0$ and for $p=0$ when $\text{cos}\left(\mathit{\pi s}/m\right)<0$ . Thus, as $s$ varies, the minimum of $E_{\mathit{min}}$ is zero, i.e., $s/m=\pm 1/3$ or $s/m=\pm 2/3$ . Therefore, for $m=6$ , $s=2$ or $s=4$ . 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 $E_s\left(p\right)$ reads

$$E_s=2\gamma \mid \text{cos}\left(\mathit{ap}/2\right)\mid ,$$ E15

for $s/m=1/3$ and

$$E_s=2\gamma \mid \text{sin}\left(\mathit{ap}/2\right)\mid ,$$ E16

for $s/m=2/3$ . Calculating the velocity $v=\partial E/\partial p$ , where variable $p$ is substituted by $p-{\mathit{qA}}_0$ , and considering the value at which the minimum is reached, we get

$$v=-\mathit{a\gamma }\mathtt{sgn}\left(\text{sin}\left({\mathit{aqA}}_0/2\right)\right)\text{cos}\left({\mathit{aqA}}_0/2\right),$$ E17

where $\mathit{sgn}\left(X\right)$ denotes the sign of X in both cases.

The current $J_0$ can hence be expressed as

$$J_0=-Q\mathtt{sgn}\left(\text{sin}\left({\mathit{aqA}}_0/2\right)\right)\text{cos}\left({\mathit{aqA}}_0/2\right),$$ E18

where $Q=\left(4\mathit{Nq\gamma }/\pi \mathrm{\hslash }\right)\mathrm{\Phi }\left(\gamma /k_B{T}_0\right)$ . The function $\mathrm{\Phi }$ reads

$$\mathrm{\Phi }\left(X\right)={\int }_{-\pi /2}^{\pi /2}\mathit{dx}/\left[1+\text{exp}\left(2X|\text{sin}x|\right)\right].$$ E19

Inserting Eq. (18) into (13) yields the evolution equation. As a matter of illustration, we assume that $0<{\mathit{aqA}}_0/2<\pi$ and define $A_0^{\prime }$ such that $\mathit{aq}{A}_0^{\prime }/2={\mathit{aqA}}_0/2-\pi /2$ . Hence, we obtain

$${\partial }^2{A}_0^{\prime }/\partial \xi \partial \theta =-R\text{sin}\left(\mathit{aq}{A}_0^{\prime }/2\right),$$ E20

with $R=\left(2\mathit{Nq\gamma }/{\mathit{\pi \epsilon }}_0\mathrm{\hslash }c\right)\mathrm{\Phi }\left(\gamma /k_B{T}_0\right)$ . Assuming $\mid {\mathit{aqA}}_0/2\mid >\pi$ , we can set $A_0^{″}=A_0+\pi /\mathit{aq}$ . Therefore, Eq. (20) remains. This shows that Eq. (20) is valid for any $A_0$ . Retaining the second transverse derivative in the wave Eq. (13), we derive the following

$${\partial }^2{A}_0^{\prime }/\partial \xi \partial \theta =\left(c/2\right){\partial }^2{A}_0^{\prime }/\partial y^2-R\text{sin}\left(\mathit{aq}{A}_0^{\prime }/2\right),$$ E21

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

$$A_T=-\mathit{BC},C_T=\mathit{AB},B_Z=C+{\int }^T{B}_{\mathit{YY}}\mathit{dT},$$ E22

provided $B=E_0/E_r$ , $Z=x/L_r$ , $T=\left(t-x/c\right)/t_r$ , and $Y=y/w_r$ , with $E_r=2/\left({\mathit{aqt}}_r\right)$ , $L_r=-{\mathit{UE}}_r/R$ , and $w_r=\sqrt{{\mathit{ct}}_r{L}_r/2}$ . The assumption ${\text{lim}}_{T\to -\infty }A=U$ 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

$$A=\sum _{k=0}^{\infty }{A}_k{g}^{k+\alpha },B=\sum _{k=0}^{\infty }{B}_k{g}^{k+\beta },C=\sum _{k=0}^{\infty }{C}_k{g}^{k+\gamma },$$ E23

with sufficient arbitrary functions among $A_k$ , $B_k$ , $C_k$ , and $g$ , where $g=g\left(Y,Z,T\right)$ , $A_k=A_k\left(Y,Z,T\right)$ , $B_k=B_k\left(Y,Z,T\right)$ , and $C_k=C_k\left(Y,Z,T\right)$ ( $k$ being nonzero integers) are analytical functions within the neighborhood of $g=0$ . The constants $\alpha$ , $\beta$ , and $\gamma$ should all be negative integers.

The leading order analysis provides the following

$$\alpha =\gamma =-2,\beta =-1$$ E24

and

$$A_0=2\left(g_Z{g}_T-g_Y^2\right),B_0=2\mathit{i\epsilon }{g}_T,C_0=-\mathit{i\epsilon }{A}_0,$$ E25

where $\epsilon =\pm 1$ and $i^2=-1$ .

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

$${\mathcal{M}}_k{\mathcal{V}}_k={\mathcal{T}}_k,$$ E26

where ${\mathcal{M}}_k$ is a square matrix, ${\mathcal{V}}_k={\left(A_k,B_k,C_k\right)}^T$ , and ${\mathcal{T}}_k={\left({\mathcal{A}}_k,{\mathcal{B}}_k,{\mathcal{C}}_k\right)}^T$ with

$$\begin{array}{l}{\mathcal{A}}_k=-B_{k-2,\mathit{ZT}}+B_{k-2,\mathit{YY}}\\ -\left(k-2\right)\left[B_{k-1,Z}{g}_T+B_{k-1,T}{g}_Z-2B_{k-1,Y}{g}_Y+B_{k-1}\left(g_{\mathit{ZT}}-g_{\mathit{YY}}\right)\right]\\ +\sum _{j=1}^{k-1}{A}_j{B}_{k-j},\end{array}$$ E27

and

$${\mathcal{B}}_k=-A_{k-1,T}-\sum _{j=1}^{k-1}{C}_j{B}_{k-j},$$ E28

with

$${\mathcal{C}}_k=-C_{k-1,T}+\sum _{j=1}^{k-1}{A}_j{B}_{k-j},$$ E29

provided $A_k=B_k=C_k=0$ for $k<0$ . The matrix ${\mathcal{M}}_k$ is given by

$${\mathcal{M}}_k=\left(\begin{array}{lll}-B_0& k\left(k-3\right)A_0/2& 0\\ \left(k-2\right)g_T& C_0& B_0\\ -B_0& -A_0& \left(k-2\right)g_T\end{array}\right).$$ E30

Thus, the determinant ${\mathrm{\Delta }}_k$ of the matrix ${\mathcal{M}}_k$ is given by

$${\mathrm{\Delta }}_k=-\left(k+1\right)\left(k-2\right)\left(k-2\right)\left(k-4\right)\left(g_Z{g}_T-g_Y^2\right)g_T^2.$$ E31

If the determinant ${\mathrm{\Delta }}_k$ of the cœfficient matrix ${\mathcal{M}}_k$ is not equal to zero, then the functions $A_k$ , $B_k$ , and $C_k$ can be obtained from Eq. (26) straightforwardly as unique solutions. Nonetheless, when

$$k\in \left\{-\mathrm{1,2,2,4}\right\},$$ E32

resonances occur.

The resonance at $k=-1$ corresponds to the singularity manifold $g$ , which is an arbitrary function, and the case $k=0$ , 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 $k=2\mathrm{;}4$ , which are satisfied identically such that the other four arbitrary functions among $A_k$ , $B_k$ , and $C_k$ can be introduced into the general series expansion given by Eq. (23).

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

$$\begin{array}{l}{A}_1=\frac{\mathit{\iota \epsilon }\left[{}_{\mathit{gT}}{B}_0{,}_{\mathit{Z}}+{}_{\mathit{gZ}}{B}_{0,\mathit{T}}-2{}_{\mathit{gY}}{B}_0{,}_{\mathit{Y}}+{}_{\mathrm{}\left(\mathit{gZT}-\mathit{gYY}\right)}{\text{B}}_{\mathit{0}}+C_{0,T}\right]}{\mathit{gT}},\\ B_1=\frac{{}_{\mathit{gT}}{B}_0{,}_{\mathit{Z}}+{}_{\mathit{gZ}}{B}_{0,\mathit{T}}-2{}_{\mathit{gY}}{B}_0{,}_{\mathit{Y}}+{}_{\mathrm{}\left(\mathit{gZT}-\mathit{gYY}\right)}{\text{B}}_{\mathit{0}}+2C_{0,T}}{A_0},\\ C_1=-\mathit{\iota \epsilon }{A}_1.\end{array}$$ E33

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

$$-B_{0,\mathit{ZT}}+B_{0,\mathit{YY}}+C_{1,T}=0.$$ E34

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),

$$B_2=\frac{C_{1,T}-A_1{B}_1-B_0{A}_2}{A_0},C_2=-\mathit{\iota \epsilon }{A}_2,$$ E35

where one of the quantities among $A_2$ and $C_2$ is arbitrary.

For $k=3$ , Eq. (26) gives us

$$\begin{array}{l}{A}_3=\frac{\mathit{\iota \epsilon }}{2\mathit{gT}}\left[-B_{1,\mathit{ZT}}+B_{1,\mathit{YY}}-{}_{\mathit{gT}}{B}_{2,\mathit{Z}}-{}_{\mathit{gZ}}{B}_2{,}_{\mathit{T}}+2{}_{\mathit{gY}}{B}_{2,\mathit{Y}}\right]\\ +\frac{\mathit{\iota \epsilon }}{2\mathit{gT}}\left[-,{}_{\mathrm{}\left(\mathit{gZT}-\mathit{gYY}\right)}{B}_2+A_1{B}_2+A_2{B}_1\right]\\ B_3=\frac{1}{2A_0}\left[-3B_{1,\mathit{ZT}}+3B_{1,\mathit{YY}}-3{}_{\mathit{gT}}{B}_{2,\mathit{Z}}-3{}_{\mathit{gZ}}{B}_{2,\mathit{T}}+6{}_{\mathit{gY}}{B}_{2,\mathit{Y}}\right]\\ +\frac{1}{2A_0}\left[-3{}_{\mathrm{}\left(\mathit{gZT}-\mathit{gYY}\right)}{B}_2+2C_{2,T}+A_1{B}_2+A_2{B}_1\right],\\ C_3=-\mathit{\iota \epsilon }{A}_3.\end{array}$$ E36

Let us emphasize that ${\partial }_g/{\partial }_T\equiv gT$ , and so on.

For $k=4$ , we can obtain

$$A_4=\frac{\mathit{\iota \epsilon }}{6g_T}\left({\mathcal{A}}_4+2{\mathcal{C}}_4-4g_T{C}_4\right),B_4=\frac{1}{3A_0}\left({\mathcal{A}}_4-{\mathcal{C}}_4+2g_T{C}_4\right),$$ E37

where $C_4$ is an arbitrary function.

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

$${\mathcal{B}}_k-\mathit{\iota \epsilon }{\mathcal{C}}_k=0,\left(k=\mathrm{1,2,3,4}\right).$$ E38

Then, for $k=4$ , 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, 28, 29, 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, 28, 29, 30] of the system (22). By setting

$$A_{k+1}=B_k=C_{k+1}=0,\mathit{\text{for}}k\ge 2,$$ E39

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

$$A=A_0/g^2+A_1/g+A_2,B=B_0/g+B_1,C=C_0/g^2+C_1/g+C_2.$$ E40

After vanishing $A_3$ and $B_3$ , and using Eq. (38), we can reduce the system (36) to

$$B_{1,\mathit{ZT}}=A_2{B}_1+B_{1,\mathit{YY}},A_{2,T}=-B_1{C}_2,C_{2,T}=A_2{B}_1.$$ E41

From Eq. (41), it follows that $A_2,B_1,C_2$ 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 $A_2$ , $B_1$ , and $C_2$ . For convenience, we fix the original seed solution as

$$A_2=\nu ,B_1=0,C_2=-\mathit{\iota \epsilon \nu },$$ E42

with parameter $\nu$ 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

$$\begin{array}{l}{D}_Z{D}_{T}H\cdot F=v_1\mathit{HF},D_Y{D}_{T}H\cdot F=-v_2\mathit{HF},D_Y^{2}H\cdot F=-v_2\mathit{HF},\\ D_Y{D}_{T}F\cdot F=H^2/2,D_T^{2}F\cdot F=H^2/2,D_Y^{2}F\cdot F=H^2/2,\end{array}$$ E43

provided $A=D_Z+E_Y$ and $C\propto \left(B_Z-B_Y\right)$ so as to express

$$B=H/F,D={\nu }_{1}Z-2{\partial }_T\text{ln}\left(F\right),E={\nu }_{2}Y+2{\partial }_Y\text{ln}\left(F\right),$$ E44

with $\nu ={\nu }_1+{\nu }_2$ . The symbols $D_Y$ , $D_Z$ , and $D_T$ refer to the Hirota operators [29, 30, 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 $\left(2+1\right)$ -dimensional system above is completely integrable.

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

$$A=\nu +\frac{\left(D_Y^2-D_Z{D}_T\right)g\cdot g}{g^2},B=2\mathit{\iota \epsilon }{\partial }_T\left(\text{ln}|g|\right),C=-\mathit{\iota \epsilon A}.$$ E45

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

$$g=a_0+a_{1}P+a_{2}Q+a_3\mathit{PQ},$$ E46

where the parameter $a_k\left(k=\mathrm{0,1,2,3}\right)$ is an arbitrary constant and $P=P\left(Z,T\right)$ and $Q=Q\left(Y,T\right)$ are arbitrary functions of $\left(Z,T\right)$ and $\left(Y,T\right)$ , respectively.

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 $\mathrm{\Theta }$ of two generalized coordinates $\left(\xi ,t\right)$ as defined by [33].

1. Nonlocal fractal pattern: we have the following

$$\mathrm{\Theta }\left(\xi ,t\right)=\sum _{j=1}^2{\lambda }_j{\theta }_j\mid {\theta }_j\mid \left\{\left\{{\alpha }_j\text{sin}\left[\text{ln}\left({\theta }_j^2\right)\right]+{\beta }_j\text{cos}\left[\text{ln}\left({\theta }_j^2\right)\right]\right\}\right\},$$ E47

provided quantities ${\theta }_{0j}$ , ${\lambda }_j$ , ${\alpha }_j$ , and ${\beta }_j$ being arbitrary parameters. Also, ${\theta }_j\equiv k_j\xi -v_{j}t+{\theta }_{0j}$ . Variables $\xi$ , $k_j$ , and $v_j$ 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 $\mathrm{\Theta }$ can be expressed as

$$\mathrm{\Theta }\left(\xi ,t\right)=\text{exp}\left\{-{\left|\theta \right|}^{\overline{N}}\left\{r+s\text{sin}\left[\text{ln}\left({\theta }^2\right)\right]+w\text{cos}\left[\text{ln}\left({\theta }^2\right)\right]\right\}\right\},$$ E48

with ${\theta }_j\equiv \mathit{k\xi }-\mathit{vt}+{\theta }_{0j}$ , ${\theta }_0$ being an arbitrary parameter, and constants $\overline{N}$ , $r$ , $w$ , and $s$ are arbitrary parameters. But also we can find

$$\mathrm{\Theta }\left(\xi ,t\right)=\mid \theta \mid {\left\{\overline{\alpha }\text{sin}\left[\text{ln}\left({\theta }^2\right)\right]+\overline{\beta }\text{cos}\left[\text{ln}\left({\theta }^2\right)\right]\right\}}^{\tilde{N}}/\left(1+{\theta }^4\right),$$ E49

for fractal lump solution. Constants $\overline{\alpha }$ , $\overline{\beta }$ , and $\tilde{N}$ are arbitrary parameters.

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

$$\wp \left(\xi ,t\right)=\sum _{j=0}^N{\alpha }^{-j/2}\text{sin}\left({\beta }^j\theta \right),N\to \infty ,$$ E50

with constants $\alpha$ and $\beta$ being arbitrary parameters. A stochastic fractal excitation can be expressed as

$$\mathrm{\Theta }\left(\xi ,t\right)=\underset{i,j}{\mathrm{\Sigma }}{R}_i\left({\theta }_i\right)R_j\left({\theta }_j\right),$$ E51

where $R_i=\wp \left({\theta }_i\right)+{\theta }_i^2+{\mu }_i$ , with ${\mu }_i$ 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:

$$\mathrm{\Theta }\left(\xi ,t\right)=k+\underset{j=0}{\mathrm{\Sigma }}{\eta }_j\wp \left({\theta }_j\right){\text{tanh}}^{{\mu }_j}\left({\theta }_j\right),$$ E52

provided quantities $k$ , ${\eta }_j$ , and ${\mu }_j$ being arbitrary parameters.

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

$$\mathrm{\Theta }\left(\xi ,t\right)=\underset{i,j}{\mathrm{\Sigma }}{\rho }_j{R}_j\left({\theta }_j\right),$$ E53

where quantities ${\eta }_j$ and ${\rho }_j$ 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 $\mid \mathbf{B}\mid$ -observable with space at $t=0$ .

Figure 1.

Depiction of Nonlocal fractal patterns at $t=0$ . The parameters are chosen as $a_0=1$ , $a_1=1$ , $a_2=1$ , and $a_3=1$ such that: For $p\left(x,t\right)=\mathrm{\Theta }\left(x,t\right)$ , ${\lambda }_1=1/4$ , ${\lambda }_2=0$ , ${\theta }_{01}=0$ , $k_1=1$ , and $v_1=1$ . For $q\left(y,t\right)=\mathrm{\Theta }\left(y,t\right)$ , ${\lambda }_1=1/4$ , ${\lambda }_2=0$ , ${\theta }_{02}=0$ , $k_2=1$ , and $v_2=1$ . Note that ${\alpha }_1=1$ and ${\beta }_1=0$ . 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 ${\left[-3.6\cdot {10}^{-2}\mathrm{,3.6}\cdot {10}^{-2}\right]}^2$ and ${\left[-2.32\cdot {10}^{-10}\mathrm{,2.32}\cdot {10}^{-10}\right]}^2$ , respectively.

In a $3D-$ representation, the features presented in panel 1 $\left(a\right)$ within the space region ${\left[-3.6\cdot {10}^{-2}\mathrm{,3.6}\cdot {10}^{-2}\right]}^2\times \mid \mathbf{B}\mid$ and those depicted in $\left(c\right)$ within region ${\left[-2.32\cdot {10}^{-10}\mathrm{,2.32}\cdot {10}^{-10}\right]}^2\times \mid \mathbf{B}\mid$ are self-similar nonlocal. Such a similarity in the profiles is clearly shown in panels $\left(b\right)$ and $\left(d\right)$ 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 $\left(b\right)$ and $\left(d\right)$ , respectively. In comparison to the previous nonlocal fractal patterns, it appears that the fractal dromions have relatively high amplitudes.

Figure 2.

Fractal dromiom excitations depicted at $t=0$ by the observable $\mid \mathbf{B}\mid \equiv I$ which expression is given by Eq. (40). In this case, the parameters are selected as $a_0=1$ , $a_1=1$ , $a_2=1$ , and $a_3=1$ such that: For $p\left(x,t\right)=\mathrm{\Theta }\left(x,t\right)$ , $r=3/2$ , $s=1$ , $w=0$ , $N=1$ , ${\theta }_{01}=0$ , $k_1=1$ , and $v_1=1$ . For $q\left(y,t\right)=\mathrm{\Theta }\left(y,t\right)$ , ${\theta }_{02}=0$ , $k_2=1$ , and $v_2=1$ . 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 ${\left[-7\cdot {10}^{-3},7\cdot {10}^{-3}\right]}^2$ and ${\left[-2.8\cdot {10}^{-8}\mathrm{,2.8}\cdot {10}^{-8}\right]}^2$ , respectively.

Next, in Figure 3, we obtain the fractal lump which shows self-similar structures in panels $\left(c\right)$ and $\left(d\right)$ .

Figure 3.

Fractal lump excitations depicted at $t=0$ by the observable $\mid \mathbf{B}\mid \equiv I$ which expression is given by Eq. (40). In this case, the parameters are selected as $a_0=1$ , $a_1=1$ , $a_2=1$ , and $a_3=2$ such that: For $p\left(x,t\right)=\mathrm{\Theta }\left(x,t\right)$ , ${\theta }_{01}=0$ , $k_1=1$ , and $v_1=1$ . For $q\left(y,t\right)=\mathrm{\Theta }\left(y,t\right)$ , ${\theta }_{02}=0$ , $k_2=1$ , and $v_2=1$ . Note that $\overline{\alpha }=1$ , and $\overline{\beta }=0$ . 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 ${\left[-3.6\cdot {10}^{-2}\mathrm{,3.6}\cdot {10}^{-2}\right]}^2$ and ${\left[-2.32\cdot {10}^{-10}\mathrm{,2.32}\cdot {10}^{-10}\right]}^2$ , respectively.

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.

Figure 4.

Fractal stochastic nonlocal excitations depicted at $t=0$ by the observable $\mid \mathbf{B}\mid \equiv I$ which expression is given by Eq. (40). In this case, the parameters are selected as $a_0=1$ , $a_1=1$ , $a_2=1$ , and $a_3=1$ such that: For $p\left(x,t\right)=\mathrm{\Theta }\left(x,t\right)$ , ${\lambda }_1=1/4$ , ${\lambda }_2=0$ , ${\theta }_{01}=0$ , $k_1=1$ , and $v_1=1$ . For $q\left(y,t\right)=\mathrm{\Theta }\left(y,t\right)$ , ${\lambda }_1=1/4$ , ${\lambda }_2=0$ , ${\theta }_{02}=0$ , $k_2=1$ , and $v_2=1$ . Note that $\alpha =3/2$ , $\beta =3/2$ , and $N=100$ . 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 ${\left[-3.6\cdot {10}^{-2}\mathrm{,3.6}\cdot {10}^{-2}\right]}^2$ and ${\left[-2.32\cdot {10}^{-10}\mathrm{,2.32}\cdot {10}^{-10}\right]}^2$ , respectively.

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 $I\equiv \mid \mathbf{B}\mid$ shows how the peaks are distributed stochastically within the regions ${\left[-1\cdot {10}^{-2},1\cdot {10}^{-2}\right]}^2$ and ${\left[-2.6\cdot {10}^{-8}\mathrm{,2.6}\cdot {10}^{-8}\right]}^2$ for stochastic fractal dromion.

Figure 5.

Fractal stochastic dromion excitations depicted at $t=0$ by the observable $\mid \mathbf{B}\mid \equiv I$ which expression is given by Eq. (40). In this case, the parameters are selected as $a_0=1$ , $a_1=1$ , $a_2=1$ , and $a_3=1$ such that: For $p\left(x,t\right)=\mathrm{\Theta }\left(x,t\right)$ , $r=3/2$ , $s=1$ , $w=0$ , $\overline{N}=1$ , ${\theta }_{01}=0$ , $k_1=1$ , and $v_1=1$ . For $q\left(y,t\right)=\mathrm{\Theta }\left(y,t\right)$ , ${\theta }_{02}=0$ , $k_2=1$ , and $v_2=1$ . Note that $\alpha =3/2$ , $\beta =3/2$ , and $N=100$ . 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 ${\left[-1\cdot {10}^{-2},1\cdot {10}^{-2}\right]}^2$ and ${\left[-2.6\cdot {10}^{-8}\mathrm{,2.6}\cdot {10}^{-8}\right]}^2$ , respectively.

In Figure 6, we construct the fractal solitoff excitations. By reducing the region $\left[-1.2\cdot {10}^{-2}\mathrm{,1.2}\cdot {10}^{-2}\right]\times \left[-5,10\right]$ of panel 6 $\left(a\right)$ to $\left[-1.5\cdot {10}^{-8}\mathrm{,1.5}\cdot {10}^{-8}\right]\times \left[-5,10\right]$ of panel 6 $\left(c\right)$ , we obtain a totally similar structure with density plots represented in panels (b) and (d), respectively.

Figure 6.

Fractal stochastic solitoff excitations depicted at $t=0$ by the observable $\mid \mathbf{B}\mid \equiv I$ which expression is given by Eq. (40). In this case, the parameters are selected as $a_0=1$ , $a_1=1$ , $a_2=1$ , and $a_3=1$ such that: For $p\left(x,t\right)=\mathrm{\Theta }\left(x,t\right)$ , $\kappa =2$ , $M=1$ , ${\eta }_0=0$ , ${\eta }_1=1/2$ , ${\eta }_m=0\left(m\ge 2\right)$ , ${\mu }_1=1$ , ${\theta }_{01}=-20$ , $k_1=4$ , and $v_1=1$ . For $q\left(y,t\right)=\mathrm{\Theta }\left(y,t\right)$ , $\kappa =0$ , $M=2$ , ${\eta }_0=0$ , ${\eta }_1=1/5$ , ${\eta }_2=1/4$ , ${\eta }_m=0\left(m\ge 3\right)$ , ${\mu }_1={\mu }_1=1$ , ${\theta }_{02}=-15$ , $k_2=2$ , and $v_2=2$ . Note that $\alpha =3/2$ , $\beta =3/2$ , and $N=100$ . 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 $\left[-1.2\cdot {10}^{-2}\mathrm{,1.2}\cdot {10}^{-2}\right]\times \left[-5,10\right]$ and $\left[-1.5\cdot {10}^{-8}\mathrm{,1.5}\cdot {10}^{-8}\right]\times \left[-5,10\right]$ , 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 $\mid \mathbf{B}\mid -$ observable, at $t=0$ , the self-similarity in structure of this observable shows how the peaks are distributed stochastically within the regions ${\left[-7\cdot {10}^{-3},7\cdot {10}^{-3}\right]}^2$ and ${\left[-2.1\cdot {10}^{-10}\mathrm{,2.1}\cdot {10}^{-10}\right]}^2$ . In the above configurations, the stochastic fractal solitoff and stochastic fractal lump excitations appear to be the waves with greater amplitudes in comparison to the previous ones.

Figure 7.

Fractal stochastic lump excitations depicted at $t=0$ by the observable $\mid \mathbf{B}\mid \equiv I$ which expression is given by Eq. (40). In this case, the parameters are selected as $a_0=1$ , $a_1=1$ , $a_2=1$ , and $a_3=2$ such that: For $p\left(x,t\right)=\mathrm{\Theta }\left(x,t\right)$ , ${\theta }_{01}=0$ , $k_1=1$ , and $v_1=1$ . For $q\left(y,t\right)=\mathrm{\Theta }\left(y,t\right)$ , ${\theta }_{02}=0$ , $k_2=1$ , and $v_2=1$ . Note that $\overline{\alpha }=1$ , $\overline{\beta }=0$ , $\tilde{N}=2$ associated to $\alpha =3/2$ , $\beta =3/2$ , and $N=100$ . 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 ${\left[-7\cdot {10}^{-3},7\cdot {10}^{-3}\right]}^2$ and ${\left[-2.1\cdot {10}^{-10}\mathrm{,2.1}\cdot {10}^{-10}\right]}^2$ , respectively.

From a physical viewpoint, the observable B which has the meaning of the dimensionless electric field shows that its intensity $\mid B\mid$ can be nonlocal or rather self-confined. 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 system is Painlevé integrable [15]. We derived another important properties, namely the Bäcklund transformation and the Hirota bilinearization [27, 28, 29, 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, 32, 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, 35, 36, 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.