# Optical Bullets and Their Modulational Instability Analysis
## 5 Stability Analysis  
The emergence of instability in a nonlinear wave medium that is known to occur because of the interaction between the dispersion and nonlinear influences results in the modulation of the steady-state. Thus, the technique of linear stability analysis can be exploited to study the modulation instability (MI) of the stationary solutions of 3D-NLSS (2). Therefore, we assume that the stationary solutions are defined as  
\[\begin{split}\mathrm{Y}(x,y,z,t)&=\rho e^{i\theta t },\\ \Gamma(x,y,z,t)&=\mu,\end{split} \tag{63}\]where \(\rho\) and \(\mu\) are real constants. Inserting (63) into 3D-NLSS (2), one can deduce \(\vartheta=-\mu\). To apply the standard linear stability analysis, perturbed fields are introduced in the amplitudes of stationary waves to find an expression given as  
\[\begin{split}\mathrm{Y}(x,y,z,t)&=(\rho+V)e^{-i\mu t },\\ \Gamma(x,y,z,t)&=(\mu+W),\end{split} \tag{64}\]  
where \(V=V(x,y,z,t)\) and \(W=W(x,y,z,t)\) are complex and real functions, respectively. Substituting (64) into 3D-NLSS (2), one can acquire the linearized disturbance equations of the form  
\[\begin{split} iV_{t}&=V_{xy}+V_{xx}+\rho W,\\ W_{x}&=2\rho[(V_{y}+V_{y}^{*})+(V_{z}+V_{z}^{*})], \end{split} \tag{65}\]  
where * indicates the complex conjugate. Consider that the solutions of Equation (65) are written as  
\[\begin{split}& V=\alpha_{1}e^{i(\kappa_{1}x+\kappa_{2}y+\kappa_{3}z- \alpha t)}\ +i\alpha_{2}e^{-i(\kappa_{1}x+\kappa_{2}y+\kappa_{3}z-\alpha t)},\\ & W=(\beta_{1}+i\beta_{2})e^{i(\kappa_{1}x+\kappa_{2}y+\kappa_{3} z-\alpha t)}+(\beta_{1}-i\beta_{2})e^{-i(\kappa_{1}x+\kappa_{2}y+\kappa_{3}z- \alpha t)},\end{split} \tag{66}\]  
where \(\kappa_{j}\,(j=1,2,3)\) and \(\varpi\) are the normalized wave numbers and frequency of perturbation, respectively, while \(\alpha_{l}\) and \(\beta_{l}\,(l=1,2)\) are the coefficients of linear combination. Upon applying the ansatz (66) to Equation (65), we reach the following four homogeneous equations as  
\[\begin{split}&-(\kappa_{1}(\kappa_{2}+\kappa_{3})+\varpi) \alpha_{1}+\rho\beta_{1}+i\rho\beta_{2}=0,\\ &-i(\kappa_{1}(\kappa_{2}+\kappa_{3})-\varpi)\alpha_{2}+\rho \beta_{1}-i\rho\beta_{2}=0,\\ &-2i\rho(\kappa_{2}+\kappa_{3})\alpha_{1}-2\rho(\kappa_{2}+ \kappa_{3})\alpha_{2}+i\kappa_{1}\beta_{1}-\kappa_{1}\beta_{2}=0,\\ & 2i\rho(\kappa_{2}+\kappa_{3})\alpha_{1}-2\rho(\kappa_{2}+\kappa_{3}) \alpha_{2}-i\kappa_{1}\beta_{1}-\kappa_{1}\beta_{2}=0.\end{split} \tag{67}\]  
From the above system, we can construct a matrix for the coefficients of \(\alpha_{l}\) and \(\beta_{l}\,(l=1,2)\). In order to assure nontrivial solutions for the system (67), the determinant of the matrix must be zero. Accordingly, the dispersion relation is produced as  
\[\kappa_{1}^{4}(\kappa_{2}+\kappa_{3})^{2}-\kappa_{1}^{2}\Big{(}\varpi^{2}+4 \rho^{2}(\kappa_{2}+\kappa_{3})^{2}\Big{)}+8\rho^{4}(\kappa_{2}+\kappa_{3})^{ 2}=0, \tag{68}\]  
which has the solutions for \(\varpi\) in the form  
\[\varpi=\pm\frac{(\kappa_{2}+\kappa_{3})}{\kappa_{1}}\sqrt{\big{(}\kappa_{1}^{ 2}-2\rho^{2}\big{)}^{2}+4\rho^{4}}. \tag{69}\]  
Based on this expression, we can determine the situation of the MI for the 3D-NLSS (2). It is clearly seen that \((\kappa_{1}^{2}-2\rho^{2})^{2}+4\rho^{4}\) is always \(\geq 0\), meaning that the MI does not exist because \(\mathrm{Im}\,\varpi=0\). Eventually, the steady state is stable against wave number perturbations.