# Optical Bullets and Their Modulational Instability Analysis
## 4 Optical Bullet Solutions
### Solving by PREM
\begin{split}\mathrm{Y}(x,y,z,t)&\pm\sqrt{-\frac{2 A_{1}}{A_{3}}}\Bigg{[}\frac{1}{\cos(\sqrt{\frac{A_{1}}{A_{0}}\xi})} \Bigg{]}e^{i\Phi(x,y,z,t)},\\ \Gamma(x,y,z,t)&=-2A_{1}\Bigg{[}\frac{1}{\cos(\sqrt {\frac{A_{1}}{A_{0}}\xi})}\Bigg{]}^{2},\end{split}  \tag{47}\]  
where solutions (46) and (47) need \(\delta=-1\). **Set III. **  
\[a_{0}=a_{1}=0,\quad b_{1}=\sqrt{-\frac{2A_{0}}{A_{3}}},\quad R=-\frac{pA_{1} }{2A_{0}},\quad r=0,\quad p=\pm 1,\quad\delta=\pm 1. \tag{48}\]  
**Case III. ** If \(p=-1\), from (48) along with (8)-(10), this provides optical soliton solutions for Equation (2) given by  
\[ \begin{split}\mathrm{Y}(x,y,z,t)&=\pm\sqrt{-\frac{ A_{1}}{A_{3}}}\Bigg{[}\frac{B\sinh(\sqrt{\frac{A_{1}}{2A_{0}}\xi})+C \cosh(\sqrt{\frac{A_{1}}{2A_{0}}\xi})}{B\cosh(\sqrt{\frac{A_{1}}{2A_{0}}\xi}) +C\sinh(\sqrt{\frac{A_{1}}{2A_{0}}\xi})}\Bigg{]}e^{i\Phi(x,y,z,t)},\\ \Gamma(x,y,z,t)&=-A_{1}\Bigg{[}\frac{B\sinh(\sqrt{ \frac{A_{1}}{2A_{0}}\xi})+C\cosh(\sqrt{\frac{A_{1}}{2A_{0}}\xi})}{B\cosh( \sqrt{\frac{A_{1}}{2A_{0}}\xi})+C\sinh(\sqrt{\frac{A_{1}}{2A_{0}}\xi})} \Bigg{]}^{2},\end{split}  \tag{49}\]  
The balance between the terms \(u^{\prime\prime}\) and \(u^{3}\) in Equation (29) results in a relation between \(n\) and \(k\) given as  
\[n=k-1. \tag{57}\]  
Taking \(k=3\) in Equation (57), this yields \(n=2\), and as a result the general form of the solution is given by  
\[u(\xi)=l_{0}+l_{1}H+l_{2}H^{2}. \tag{58}\]  
Plugging (58) together with (16) into Equation (29), a polynomial of degree six in \(H\) is obtained. Gathering like terms with the same degree of \(H\) and setting each coefficient to zero, a system of algebraic equations is created. The solutions of this system give rise to the following cases of solutions for the 3D-NLSS (2). **Case 1. **: For \(b\
eq d\), it is found that  
\[l_{0}=\sqrt{-\frac{A_{1}}{A_{3}}},\quad l_{1}=0,\quad l_{2}=2d\sqrt{-\frac{2A _{0}}{A_{3}}},\quad b=\sqrt{-\frac{A_{1}}{2A_{0}}}. \tag{59}\]  
These findings generate an optical soliton solution for Equation (2) as  
\[ \begin{split}\mathrm{Y}(x,y,z,t)&=\pm\sqrt{-\frac{ A_{1}}{A_{3}}}\frac{h\sqrt{\frac{A_{1}}{A_{0}}}+d\sqrt{2}\,e^{\sqrt{\frac{2A_{1}}{A_{ 0}}}\xi}}{h\sqrt{\frac{A_{1}}{A_{0}}}-d\sqrt{2}\,e^{\sqrt{\frac{2A_{1}}{A_{0}} }\xi}}e^{i\Phi(x,y,z,t)},\\ \Gamma(x,y,z,t)&=-A_{1}\left[\frac{h\sqrt{\frac{A_{1 }}{A_{0}}}+d\sqrt{2}\,e^{\sqrt{\frac{2A_{1}}{A_{0}}}\xi}}{h\sqrt{\frac{A_{1}} {A_{0}}}-d\sqrt{2}\,e^{\sqrt{\frac{2A_{1}}{A_{0}}}\xi}}\right].\end{split}  \tag{60}\]
**Case 2. **: For \(b=d\), we reach  
\[l_{0}=\sqrt{-\frac{A_{1}}{A_{3}}},\quad l_{1}=0,\quad l_{2}=2\sqrt{-\frac{A_{ 1}}{A_{3}}},\quad d=\sqrt{\frac{A_{1}}{2A_{0}}}. \tag{61}\]  
From this consequence, we retrieve an optical soliton solution for Equation (2) as  
\[ \begin{split}\mathrm{Y}(x,y,z,t)&=\pm\sqrt{-\frac{ A_{1}}{A_{3}}}\frac{(h+1)-(h-1)\tanh\Bigl{(}\sqrt{\frac{A_{1}}{2A_{0}}}\xi \Bigr{)}}{(h-1)-(h+1)\tanh\Bigl{(}\sqrt{\frac{A_{1}}{2A_{0}}}\xi\Bigr{)}}e^{i \Phi(x,y,z,t)},\\ \Gamma(x,y,z,t)&=-A_{1}\left[\frac{(h+1)-(h-1)\tanh \Bigl{(}\sqrt{\frac{A_{1}}{2A_{0}}}\xi\Bigr{)}}{(h-1)-(h+1)\tanh\Bigl{(}\sqrt{ \frac{A_{1}}{2A_{0}}}\xi\Bigr{)}}\right]^{2}.\end{split}  \tag{62}\]