# Optical Bullets and Their Modulational Instability Analysis
## 4 Optical Bullet Solutions
### Solving by PREM  
The series form based on the solutions of projective Riccati equations is exploited here to derive the solutions of Equation (29), which are mutated by (19) to optical wave solutions for the model (2). Balancing between the terms \(u^{\prime\prime}\) and \(u^{3}\) in Equation (29) leads to \(m=1\). According to (6), the general solution form of Equation (29) reads  
\[u(\xi)=a_{0}+a_{1}f+b_{1}g. \tag{33}\]  
Substituting (33) along with (7) into (29), combining all the terms having the same exponent of \(f^{i}g^{j}\) together, and equating each coefficient to zero leads to a system of algebraic equations. Solving the resulting system gives rise to three sets of solutions that reveal distinct cases of solutions for the model (2). **Set I. **  
\[a_{0}=0,\quad a_{1}=\frac{A_{0}}{2}\sqrt{-\frac{r^{2}+\delta}{A_{1}A_{3}}}, \quad b_{1}=\sqrt{-\frac{A_{0}}{2A_{3}}},\quad R=-\frac{2 pA_{1}}{A_{0}},\quad p =\pm 1,\quad\delta=\pm 1. \tag{34}\]  
**Case II. ** If \(p=-1\), from (34) along with (8)-(10), one can find optical soliton solutions for Equation (2) as  
\[ \begin{split}\mathrm{Y}(x,y,z,t)&=\pm\sqrt{-\frac{ A_{1}}{A_{3}}}\left[\frac{A\sqrt{r^{2}+\delta}+B\sinh(\sqrt{\frac{2A_{1}}{A_{0}}} \xi)+C\cosh(\sqrt{\frac{2A_{1}}{A_{0}}}\xi)}{B\cosh(\sqrt{\frac{2A_{1}}{A_{0} }}\xi)+C\sinh(\sqrt{\frac{2A_{1}}{A_{0}}}\xi)+Ar}\right]e^{i\Phi(x,y,z,t)},\\ \Gamma(x,y,z,t)&=-A_{1}\left[\frac{A\sqrt{r^{2}+ \delta}+B\sinh(\sqrt{\frac{2A_{1}}{A_{0}}}\xi)+C\cosh(\sqrt{\frac{2A_{1}}{A_{0 }}}\xi)}{B\cosh(\sqrt{\frac{2A_{1}}{A_{0}}}\xi)+C\sinh(\sqrt{\frac{2A_{1}}{A_{ 0}}}\xi)+Ar}\right]^{2},\end{split}  \tag{35}\]  
where \(C^{2}=A^{2}+B^{2}\) when \(\delta=1\), and \(B^{2}=A^{2}+C^{2}\) when \(\delta=-1\). \[ \begin{split}\mathrm{Y}(x,y,z,t)&=\pm\sqrt{-\frac{ A_{1}}{A_{3}}}\left[\frac{\sqrt{r^{2}+1}+\cosh(\sqrt{\frac{2A_{1}}{A_{0}}} \xi)}{\sinh(\sqrt{\frac{2A_{1}}{A_{0}}}\xi)+r}\right]e^{i\Phi(x,y,z,t)},\\ \Gamma(x,y,z,t)&=-A_{1}\left[\frac{\sqrt{r^{2}+1}+ \cosh(\sqrt{\frac{2A_{1}}{A_{0}}}\xi)}{\sinh(\sqrt{\frac{2A_{1}}{A_{0}}}\xi)+ r}\right]^{2},\end{split}  \tag{36}\]  
where solution (36) demands \(\delta=1\). \[ \begin{split}\mathrm{Y}(x,y,z,t)&=\pm\sqrt{-\frac{ A_{1}}{A_{3}}}\left[\frac{\sqrt{r^{2}-1}+\sinh(\sqrt{\frac{2A_{1}}{A_{0}}} \xi)}{\cosh(\sqrt{\frac{2A_{1}}{A_{0}}}\xi)+r}\right]e^{i\Phi(x,y,z,t)},\\ \Gamma(x,y,z,t)&=-A_{1}\left[\frac{\sqrt{r^{2}-1}+ \sinh(\sqrt{\frac{2A_{1}}{A_{0}}}\xi)}{\cosh(\sqrt{\frac{2A_{1}}{A_{0}}}\xi)+ r}\right]^{2},\end{split}  \tag{37}\]  
where solution (37) is obtained when \(\delta=-1\). **Case I2. ** If \(p=1\), from (34) in company with (8)-(10), we arrive at periodic wave solutions for Equation (2) presented by  
where solution (43) demands \(\delta=1\). \[ \begin{split}\mathrm{Y}(x,y,z,t)&=\pm\sqrt{-\frac{2A_ {1}}{A_{3}}}\Bigg{[}\frac{1}{\cosh(\sqrt{-\frac{A_{1}}{A_{0}}\xi})}\Bigg{]}e^{i \Phi(x,y,z,t)},\\ \Gamma(x,y,z,t)&=-2A_{1}\Bigg{[}\frac{1}{\cosh( \sqrt{-\frac{A_{1}}{A_{0}}\xi})}\Bigg{]}^{2},\end{split}  \tag{44}\]  
where solution (44) implies \(\delta=-1\). **Case II2. ** If \(p=1\), from (41) together with (8)-(10), we procure periodic wave solutions for Equation (2) as  
\[ \begin{split}\mathrm{Y}(x,y,z,t)&=\pm\sqrt{\frac{2 \delta A_{1}}{A_{3}}}\Bigg{[}\frac{A}{B\cos(\sqrt{\frac{A_{1}}{A_{0}}\xi})+C \sin(\sqrt{\frac{A_{1}}{A_{0}}\xi})}\Bigg{]}e^{i\Phi(x,y,z,t)},\\ \Gamma(x,y,z,t)&=2\delta A_{1}\Bigg{[}\frac{A}{B \cos(\sqrt{\frac{A_{1}}{A_{0}}\xi})+C\sin(\sqrt{\frac{A_{1}}{A_{0}}\xi})} \Bigg{]}^{2},\end{split}  \tag{45}\]  
where \(A^{2}=-(B^{2}+C^{2})\) when \(\delta=1\), and \(A^{2}=B^{2}+C^{2}\) when \(\delta=-1\). \[ \begin{split}\mathrm{Y}(x,y,z,t)&=\pm\sqrt{-\frac{2 A_{1}}{A_{3}}}\Bigg{[}\frac{1}{\sin(\sqrt{\frac{A_{1}}{A_{0}}\xi})}\Bigg{]}e^{i \Phi(x,y,z,t)},\\ \Gamma(x,y,z,t)&=-2A_{1}\Bigg{[}\frac{1}{\sin(\sqrt {\frac{A_{1}}{A_{0}}\xi})}\Bigg{]}^{2},\end{split}  \tag{46}\]  
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