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and stochastic fractal lump excitations appear to be the waves with greater ampli-

, respectively.

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>

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

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.

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

tudes in comparison to the previous ones.

5. Summary

96

Figure 7.

Fractal Analysis

Raïssa S. Noule and Victor K. Kuetche\* National Advanced School of Engineering, University of Yaounde I, Cameroon

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

© 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, provided the original work is properly cited.
