# Optical Bullets and Their Modulational Instability Analysis
## 1 Introduction  
Wave phenomena have become one of the mechanisms that helps to understand the nature of the medium in which they arise [1; 2; 3; 4; 5]. The structures of waves varies enormously and takes different forms based on the physical properties of the media. In addition, the continuous theoretical and experimental studies of wave features have led to considerable development and steady progress toward physical and industrial applications in different branches of science such as fluid dynamics, plasma physics, nonlinear optics, quantum electronics, signal processing, and many others [6; 7; 8; 9; 10]. Recently, the soliton has been considered one of the ubiquitous waves that has made a contribution to enhancing modern technology and applied sciences [11; 12; 13; 14; 15; 16; 17; 18; 19]. Shedding light on the field of nonlinear optics, solitons, which are stable and undistorted pulses arising due to the balance between the dispersion and nonlinearity, have been found to play a vital role in developing an electronic communication system where this type of wave efficiently transmits information through optical fiber over long distances. To examine the dynamic of solitons, various models that belong to the class of nonlinear Schrodinger equation (NLSE) have been developed and studied widely during the past decade, taking into account the distinct physical effects in the medium. For instance, the NLSE with different forms of nonlinear influences or in thepresence of higher-order dispersion has been investigated by many experts and scholars to observe soliton behaviors in optical fibers [20; 21; 22; 23]. The detection of ultrashort pulses, as an example of soliton propagation in optical fiber, that have a significant impact on developing photonic and optoelectronic devices can be carried out through studying the generalized NLSE of third order [24; 25]. There have been many studies conducted on the features and structures of optical solitons in fiber medium; see e.g., [26; 27; 28; 29; 30]. In multidimensional optical media, solitons are also known as optical bullets; this term was firstly introduced by [PERSON] in 1989 [31]. Mostly, the dominant model for generating optical bullets is the 3-dimensional NLSE. In contrast to self-focusing effects that tend to squeeze the pulse, both diffraction and dispersion cause spreading of the pulse. Subsequently, the balance between these two contradictory processes leads to the formation of this type of optical soliton. Optical bullets have different behaviors from solitons, where the former experience the loss of energy during collision events, and the latter survive collisions without losing energy [32]. In the literature, [PERSON] et al. [33] detected optical light bullets in nonlinear dissipative media that were governed by the (3+1)-dimensional complex cubic-quintic Ginzburg-Landau equation. They demonstrated that stable bullets had bell-shaped solitons, whereas unstable bullets were transformed into the shape of rockets. Furthermore, the formation of light bullets upon ultrashort laser pulse filamentation in dissipative media such as air, fused silica, fluorite, etc., has been studied in the presence of different physical effects, see [34; 35; 36; 37; 38]. For more examples of investigating the propagation characteristics of optical bullets, the reader is referred to the following studies [39; 40; 41; 42; 43; 44; 45; 46; 47]. Recently, several researchers have explored miscellaneous types of wave forms for an integrable system of coupled (2+1)-dimensional NLSE (2D-NLSS) given as  
\[ \begin{split}&\dot{\gamma}_{t}=\mathrm{Y}_{xy}+\mathrm{Y}\mathrm{T},\\ &\Gamma_{x}=2(|\mathrm{Y}|^{2})_{y},\end{split}  \tag{1}\]  
where Y and \(\Gamma\) represent complex and real functions, respectively, and both of them are functions of the variables \(x,y,\) and \(t\). On one side, Model (1) reduces to the NLSE as \(\partial_{x}=\partial_{y}\), and on the other side, it is converted into the complex sine-Gordon equation when \(\partial_{t}=0\).