## 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, 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

where subscripts denote the partial derivatives. Constants

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

where constant

in which quantities

with

where the velocity

with quantity

where constant

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

in which the quantities

Thus, at leading order

in which solution

Now, at leading order

The order

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

The energy

for

for

for

where

The current

where

Inserting Eq. (18) into (13) yields the evolution equation. As a matter of illustration, we assume that

with

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

provided

## 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

with sufficient arbitrary functions among

The leading order analysis provides the following

and

where

In order to obtain the recursion relations to determine the functions

where

and

with

provided

Thus, the determinant

If the determinant

resonances occur.

The resonance at

For

On the other hand, solving the case for

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),

where one of the quantities among

For

Let us emphasize that

For

where

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

Then, for

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

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

After vanishing

From Eq. (41), it follows that

with parameter

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

provided

with

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

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

where the parameter

## 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

1. *Nonlocal fractal pattern:* we have the following

provided quantities

*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

with

for fractal lump solution. Constants

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

with constants

where

*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:

provided quantities

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

where quantities

Now, let us analyze different figures with respect to the previous classifications. Thus, in Figure 1, we depict the variations of the

In a

Following the above figure, in Figure 2, we generate the fractal dromiom depicting self-similar structure with density plots represented in panels

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

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

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

From a physical viewpoint, the observable B which has the meaning of the dimensionless electric field shows that its intensity

## 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.