# Quiescent Optical Solitons with Quadratic-Cubic and Generalized Quadratic-Cubic Nonlinearities
## 4 Generalized Quadratic-Cubic Nonlinearity
### Linear Temporal Evolution  
Therefore, the model stands as [8]  
\[iq_{t}+a\big{(}|q|^{n}q\big{)}_{xx}+\Big{(}b_{1}|q|^{m}+b_{2}|q|^{2m}\Big{)}q=0. \tag{34}\]  
Plugging (10) into (34) reveals the strategic equation  
\[ak^{2}(n+1)U^{n+1}U^{\prime\prime}+ak^{2}n(n+1)U^{n}{U^{\prime}}^{2}+b_{1}U^{m +2}+b_{2}U^{2m+2}-U^{2}\omega=0. \tag{35}\]  
Assume \(n=m\). Thus, Equation (35) shapes up as  
\[ak^{2}(m+1)U^{m+1}U^{\prime\prime}+ak^{2}m(m+1)U^{m}{U^{\prime}}^{2}+b_{1}U^{m +2}+b_{2}U^{2m+2}-U^{2}\omega=0. \tag{36}\]  
The relation  
\[U=V^{\frac{1}{m}} \tag{37}\]  
collapses Equation (36) to  
\[ak^{2}m(m+1)VV^{\prime\prime}+ak^{2}(m+1)V^{\prime 2}+m^{2}V\Big{(}b_{2}V^{2}+b_ {1}V-\omega\Big{)}=0. \tag{38}\]  
By the implementation of the balancing algorithm in (38), one simplifies (4) to  
\[V(\xi)=\lambda_{0}+\lambda_{01}R(\xi)+\lambda_{10}Q(\xi)+\lambda_{11}R(\xi)Q (\xi)+\lambda_{02}R(\xi)^{2}+\lambda_{20}Q(\xi)^{2}. \tag{39}\]  
Inserting (39) along with (5) and (6) into (38) retrieves the results:  
Result-1:  
\[\lambda_{0}=-\frac{b_{1}(2+3m)}{4b_{2}(1+m)},\ \lambda_{01}= \lambda_{20}=\lambda_{11}=\lambda_{10}=0,\ \lambda_{02}=-\chi\lambda_{0},\] \[k=\pm\frac{m}{2(1+m)}\sqrt{\frac{b_{1}}{2a}},\ \omega=-\frac{b_{1}^{2}(2+m)(2+3m)}{16b_{2}(1+m)^{2}}. \tag{40}\]  
Putting (40) along with (7) into (39) derives the straddled soliton  
\[q(x,t)=\left[-\frac{b_{1}(2+3m)}{4b_{2}(1+m)}\left\{1-\chi\left( \frac{4c}{4c^{2}\exp\left[\pm\frac{m}{2(1+m)}\sqrt{\frac{b_{1}}{2a}}\ x\right]+\chi\exp \left[\mp\frac{m}{2(1+m)}\sqrt{\frac{b_{1}}{2a}}\ x\right]}\right)^{2}\right\} \right]^{\frac{1}{m}}\] \[\times e^{i\left(-\frac{\frac{b_{1}^{2}(2+m)(2+3m)}{16b_{2}(1+m)^ {2}}+b_{0}}{16b_{2}(1+m)^{2}}+b_{0}\right)}. \tag{41}\]  
If \(ab_{1}>0\) and \(\chi=\pm 4c^{2}\) in (41), the dark and singular solitons evolve as  
\[q(x,t)=\left\{-\frac{b_{1}(2+3m)}{4b_{2}(1+m)}\tanh^{2}\left[\frac{m}{2(1+m)} \sqrt{\frac{b_{1}}{2a}}\ x\right]\right\}^{\frac{1}{m}}\ e^{i\left(-\frac{b_{1}^{2}(2+m)(2+3 m)}{16b_{2}(1+m)^{2}}+b_{0}\right)}, \tag{42}\]  
and  
\[q(x,t)=\left\{-\frac{b_{1}(2+3m)}{4b_{2}(1+m)}\coth^{2}\left[\frac{m}{2(1+m)} \sqrt{\frac{b_{1}}{2a}}\ x\right]\right\}^{\frac{1}{m}}\ e^{i\left(-\frac{b_{1}^{2}(2+m)(2+3 m)}{16b_{2}(1+m)^{2}}+b_{0}\right)}. \tag{43}\]  
Figure 5 depicts the plots of dark and singular solitons (42) and (43), respectively. The parameter values chosen are: \(b_{1}=1\), \(b_{2}=-1\) and \(a=1\).  
Result-2:  
\[\lambda_{0}=-\frac{b_{1}(2+3m)}{4b_{2}(1+m)},\;\lambda_{01}=\lambda_{11}=\lambda_ {02}=0,\;\lambda_{20}=4\eta^{2}\lambda_{0},\]  
\[\lambda_{10}=-4\eta\lambda_{0},\;k=\pm\frac{m}{1+m}\sqrt{\frac{b_{1}}{2a}},\; \omega=-\frac{b_{1}^{2}(2+m)(2+3m)}{16b_{2}(1+m)^{2}}. \tag{44}\]  
Inserting (44) along with (8) into (39) secures the straddled soliton  
\[q(x,t)=\left\{-\frac{b_{1}(2+3m)}{4b_{2}(1+m)}\left(1-\frac{4b\eta\exp\left[ \frac{m}{1+m}\sqrt{\frac{b_{1}}{2a}}\;x\right]}{\left(b\exp\left[\frac{m}{1+m} \sqrt{\frac{b_{1}}{2a}}\;x\right]+\eta\right)^{2}}\right)\right\}^{\frac{1}{m }}\;e^{i\left(-\frac{b_{1}^{2}(2+m)(2+3m)}{16b_{2}(1+m)^{2}}t+\theta_{0} \right)}. \tag{45}\]  
When \(ab_{1}>0\) and \(\eta=\pm b\) in (45), one acquires dark and singular solitons  
\[q(x,t)=\left\{-\frac{b_{1}(2+3m)}{4b_{2}(1+m)}\tanh^{2}\left[\frac{m}{2(1+m)} \sqrt{\frac{b_{1}}{2a}}\;x\right]\right\}^{\frac{1}{m}}\;e^{i\left(-\frac{b_{ 1}^{2}(2+m)(2+3m)}{16b_{2}(1+m)^{2}}t+\theta_{0}\right)}, \tag{46}\]  
and  
\[q(x,t)=\left\{-\frac{b_{1}(2+3m)}{4b_{2}(1+m)}\coth^{2}\left[\frac{m}{2(1+m)} \sqrt{\frac{b_{1}}{2a}}\;x\right]\right\}^{\frac{1}{m}}\;e^{i\left(-\frac{b_{ 1}^{2}(2+m)(2+3m)}{16b_{2}(1+m)^{2}}t+\theta_{0}\right)}. \tag{47}\]  
Figure 6 depicts the plots of dark and singular solitons (46) and (47), respectively. The parameter values chosen are: \(b_{1}=1\), \(b_{2}=-1\) and \(a=1\).