# Quiescent Optical Solitons with Quadratic-Cubic and Generalized Quadratic-Cubic Nonlinearities
## 3 Quadratic-Cubic Nonlinearity
### Generalized Temporal Evolution  
Thus, the model evolves as  
\[i\Big{(}q^{l}\Big{)}_{t}+a\Big{(}|q|^{n}q^{l}\Big{)}_{xx}+\Big{(}b_{1}|q|+b_{2}|q| ^{2}\Big{)}q^{l}=0, \tag{22}\]  
where \(l\) stands for the generalized temporal evolution. If \(l=1\), Equation (22) collapses to (9) so that one falls back to the case of linear temporal evolution. Plugging (10) into (22) causes the main equation  
\[ak^{2}\Big{(}l^{2}+l(2n-1)+(n-1)n\Big{)}U^{n}{U^{\prime}}^{2}+ak^{2}(l+n)U^{n+1 }U^{\prime\prime}+b_{1}U^{3}+b_{2}U^{4}-l\omega U^{2}=0. \tag{23}\]  
Setting \(n=1\) condenses (23) to  
\[ak^{2}(l+1)UU^{\prime\prime}+ak^{2}l(l+1){U^{\prime}}^{2}+b_{2}U^{3}+b_{1}U^{2} -l\omega U=0. \tag{24}\]  
By the implementation of the balancing scheme in (24), one collapses (4) to  
\[U(\xi)=\lambda_{0}+\lambda_{01}R(\xi)+\lambda_{10}Q(\xi)+\lambda_{11}R(\xi)Q( \xi)+\lambda_{02}R(\xi)^{2}+\lambda_{20}Q(\xi)^{2}. \tag{25}\]  
Plugging (25) along with (5) and (6) into (24) provides us with the results:  
Result-1:  
\[\lambda_{0}=-\frac{b_{1}(2l+3)}{4b_{2}(l+1)},\;\lambda_{01}= \lambda_{20}=\lambda_{11}=\lambda_{10}=0,\;\lambda_{02}=-\chi\lambda_{0},\] \[k=\pm\frac{1}{2(l+1)}\sqrt{\frac{b_{1}}{2a}},\;\omega=-\frac{b_{ 1}^{2}(4l(l+2)+3)}{16b_{2}l(l+1)^{2}}. \tag{26}\]  
Inserting (26) along with (7) into (25) reveals the straddled soliton  
\[q(x,t)=-\frac{b_{1}(2l+3)}{4b_{2}(l+1)}\Bigg{\{}1-\chi\left( \frac{4c}{4c^{2}\exp\left[\pm\frac{1}{2(l+1)}\sqrt{\frac{b_{1}}{2a}}\;x\right] +\chi\exp\left[\mp\frac{1}{2(l+1)}\sqrt{\frac{b_{1}}{2a}}\;x\right]}\right)^{ 2}\Bigg{\}}\] \[\times e^{i\Big{(}-\frac{b_{1}^{2}(4l(l+2)+3)}{16b_{2}l(l+1)^{2}}t +\theta_{0}\Big{)}}. \tag{27}\]  
Taking \(ab_{1}>0\) and \(\chi=\pm 4c^{2}\) in (27) secures the dark and singular solitons  
\[q(x,t)=-\frac{b_{1}(2l+3)}{4b_{2}(l+1)}\tanh^{2}\left[\frac{1}{2(l+1)}\sqrt{ \frac{b_{1}}{2a}}\;x\right]\;e^{i\Big{(}-\frac{b_{1}^{2}(4l(l+2)+3)}{16b_{2}l( l+1)^{2}}t+\theta_{0}\Big{)}}, \tag{28}\]  
and  
\[q(x,t)=-\frac{b_{1}(2l+3)}{4b_{2}(l+1)}\coth^{2}\left[\frac{1}{2(l+1)}\sqrt{ \frac{b_{1}}{2a}}\;x\right]\;e^{i\Big{(}-\frac{b_{1}^{2}(4l(l+2)+3)}{16b_{2}l( l+1)^{2}}t+\theta_{0}\Big{)}}. \tag{29}\]  
Figure 2: Profiles of stationary dark and singular solitons, respectively.  
Figure 3 depicts the plots of dark and singular solitons (28) and (29), respectively. The parameter values chosen are : \(b_{1}=1\), \(b_{2}=1\) and \(a=1\).  
Result-2:  
\[\lambda_{0}=-\frac{b_{1}(2l+3)}{4b_{2}(l+1)},\;\lambda_{01}=\lambda_{11}= \lambda_{02}=0,\;\lambda_{20}=4\eta^{2}\lambda_{0},\]  
\[\lambda_{10}=-4\eta\lambda_{0},\;k=\pm\frac{1}{l+1}\sqrt{\frac{b_{1}}{2a}},\; \omega=-\frac{b_{1}^{2}(2l+1)(2l+3)}{16b_{2}l(l+1)^{2}}. \tag{30}\]  
Putting (30) along with (8) into (25) extracts the straddled soliton  
\[q(x,t)=-\frac{b_{1}(2l+3)}{4b_{2}(l+1)}\left\{1-\frac{4b\eta\exp\left[\frac{1} {l+1}\sqrt{\frac{b_{1}}{2a}}\;x\right]}{\left(b\exp\left[\frac{1}{l+1}\sqrt{ \frac{b_{1}}{2a}}\;x\right]+\eta\right)^{2}}\right\}e^{i\left(-\frac{b_{1}^{2 }(2l+1)(2l+3)}{16b_{2}l(l+1)^{2}}+\theta_{0}\right)}. \tag{31}\]  
Setting \(ab_{1}>0\) and \(\eta=\pm b\) in (31) recovers the dark and singular solitons  
\[q(x,t)=-\frac{b_{1}(2l+3)}{4b_{2}(l+1)}\tanh^{2}\left[\frac{1}{2(l+1)}\sqrt{ \frac{b_{1}}{2a}}\;x\right]e^{i\left(-\frac{b_{1}^{2}(2l+1)(2l+3)}{16b_{2}l(l+ 1)^{2}}+\theta_{0}\right)}, \tag{32}\]  
and  
\[q(x,t)=-\frac{b_{1}(2l+3)}{4b_{2}(l+1)}\coth^{2}\left[\frac{1}{2(l+1)}\sqrt{ \frac{b_{1}}{2a}}\;x\right]e^{i\left(-\frac{b_{1}^{2}(2l+1)(2l+3)}{16b_{2}l(l+ 1)^{2}}+\theta_{0}\right)}. \tag{33}\]  
Figure 4 depicts the plots of dark and singular solitons (32) and (33), respectively. The parameter values chosen are: \(b_{1}=1\), \(b_{2}=1\) and \(a=1\).