# Quiescent Optical Solitons with Quadratic-Cubic and Generalized Quadratic-Cubic Nonlinearities
## 3 Quadratic-Cubic Nonlinearity
### Linear Temporal Evolution  
In this case, the NLSE shapes up as [3,4]  
\[iq_{t}+a\big{(}|q|^{n}q\big{)}_{xx}+\Big{(}b_{1}|q|+b_{2}|q|^{2}\Big{)}q=0, \tag{9}\]  
where \(q=q(x,t)\) purports the wave profile, while \(x\) and \(t\) stand for the spatial and temporal variables in sequence. The first term signifies the linear temporal evolution, while \(a\) comes from the nonlinear CD. Finally, \(b_{1}\) and \(b_{2}\) stem from the SPM, while \(n\) arises from the full nonlinearity. It must be noted that, if \(n=1\), we recover the linear CD that yields a mobile soliton. Equation (2) is the general integration algorithm for real-valued partial differential equations (PDEs). However, Equation (9) is a complex-valued PDE and consequently the phase component of the complex variable \(q(x,t)\) is split off so that the real part can be further analyzed as explained in (2). Therefore, model (9) permits the wave form  
\[q(x,t)=U(kx)e^{i(\omega t+\theta_{0})}, \tag{10}\]  
where \(\theta_{0}\) depicts the phase constant, and \(\omega\) denotes the wave number. Inserting (10) into (9) retrieves the governing equation  
\[ak^{2}(n+1)U^{n+1}U^{\prime\prime}+ak^{2}n(n+1)U^{n}U^{\prime}{}^{2}+b_{2}U^{4 }+b_{1}U^{3}-\omega U^{2}=0. \tag{11}\]  
Taking \(n=1\) simplifies (11) to  
\[2 ak^{2}UU^{\prime\prime}+2 ak^{2}{U^{\prime}}^{2}+b_{2}U^{3}+b_{1}U^{2}-\omega U =0. \tag{12}\]  
By the implementation of the balancing approach in (12), one reduces (4) to  
\[U(\xi)=\lambda_{0}+\lambda_{01}R(\xi)+\lambda_{10}Q(\xi)+\lambda_{11}R(\xi)Q( \xi)+\lambda_{02}R(\xi)^{2}+\lambda_{20}Q(\xi)^{2}. \tag{13}\]  
Inserting (13) along with (5) and (6) into (12) provides us with the results:  
Result-1:  
\[\lambda_{0}=-\frac{5b_{1}}{8b_{2}},\;\lambda_{01}=\lambda_{20}= \lambda_{11}=\lambda_{10}=0,\;\lambda_{02}=-\chi\lambda_{0},\] \[k=\pm\frac{1}{4}\sqrt{\frac{b_{1}}{2a}},\;\omega=-\frac{15b_{1}^ {2}}{64b_{2}}. \tag{14}\]  
Putting (14) along with (7) into (13) paves the way for the straddled soliton  
\[q(x,t)=-\frac{5b_{1}}{8b_{2}}\Bigg{[}1-\chi\left(\frac{4c}{4c^{2}\exp\left[\pm \frac{1}{4}\sqrt{\frac{b_{1}}{2a}}\;x\right]+\chi\exp\left[\mp\frac{1}{4}\sqrt {\frac{b_{1}}{2a}}\;x\right]}\right)^{2}\Bigg{]}e^{i\left(-\frac{15b_{1}^{2}}{ 8b_{2}}+\theta_{0}\right)}. \tag{15}\]If \(ab_{1}>0\) and \(\chi=\pm 4c^{2}\) in (15), the dark and singular solitons stick out as  
\[q(x,t)=-\frac{5b_{1}}{8b_{2}}\tanh^{2}\left[\frac{1}{4}\sqrt{\frac{b_{1}}{2a}}\ x \right]\ e^{i\left(-\frac{15b_{1}^{2}}{8b_{2}}t+\theta_{0}\right)}, \tag{16}\]  
and  
\[q(x,t)=-\frac{5b_{1}}{8b_{2}}\coth^{2}\left[\frac{1}{4}\sqrt{\frac{b_{1}}{2a}} \ x\right]\ e^{i\left(-\frac{15b_{1}^{2}}{8b_{2}}t+\theta_{0}\right)}. \tag{17}\]  
Figure 1 depicts the plots of dark and singular solitons (16) and (17), respectively. The parameter values chosen are : \(b_{1}=1\), \(b_{2}=1\) and \(a=1\). Result-2:  
\[q(x,t)=U(kx)e^{i(\omega t+\theta_{0})}, \tag{10}\]0 \[q(x,t)=U(kx)e^{i(\omega t+\theta_{0})}, \tag{10}\]1  
Putting (18) along with (8) into (13) provides us with the straddled soliton  
\[q(x,t)=U(kx)e^{i(\omega t+\theta_{0})}, \tag{10}\]2  
When \(ab_{1}>0\) and \(\eta=\pm b\) in (19), one retrieves the dark and singular solitons  
\[q(x,t)=U(kx)e^{i(\omega t+\theta_{0})}, \tag{10}\]3  
and  
\[q(x,t)=U(kx)e^{i(\omega t+\theta_{0})}, \tag{10}\]4  
Figure 2 depicts the plots of dark and singular solitons (20) and (21), respectively. The parameter values chosen are: \(b_{1}=1\), \(b_{2}=1\) and \(a=1\). \[q(x,t)=U(kx)e^{i(\omega t+\theta_{0})}, \tag{10}\]5 Profiles of stationary dark and singular solitons, respectively.