## 3 Induced Cubic Nonlinear Processes Based on Cascading SHG
### Exact Solution of the Modified Problem
We denote the roots as \(P_{31}\), \(P_{32}\), \(P_{33}\), where \(P_{31}\) is always real. If two other roots are also real, then there is an inequality \(P_{31}\leq P_{32}\leq P_{33}\) between them. Obviously, this assumption does not restrict our consideration. It should be also emphasized that a solution of the Equation (19) is expressed through the elliptical functions or, in certain cases, through elementary functions. Below, we briefly discuss the problem's solution corresponding to various values of the parameter \(q\). Moreover, we develop also the TH phase evolution. Because of the complexity of the obtained formulas, we do not write them here. However, using the formulas with respect to the phase and the intensity, it is possible to approximate the solution of problems (1) and (2) with good accuracy. We show it below using the computer simulation results. **Single frequency conversion mode: domain \(q<-1.5\), \(q>3+\frac{9}{\sqrt[3]{2}}-9\sqrt[3]{2}\); two complex roots \(P_{32},P_{33}\). **  
Only root \(P_{31}\) is positive. As a result, the TH intensity \(p_{3}(z)\) changes between zero and \(P_{31}\) in accordance to the following formula:\[p_{3}(z)=|A_{3}(z)|^{2}=\frac{cP_{31}(1-\text{cn}(7.5\frac{\gamma^{2}}{\Delta_{21} }\kappa z,m))}{(d-c)\,\text{cn}(7.5\frac{\gamma^{2}}{\Delta_{21}}\kappa z,m)+(c+ d)}, \tag{20}\]  
where  
\[c=\sqrt{r^{2}+s^{2}},\,d=\sqrt{(p_{31}-r)^{2}+s^{2}},\kappa=\sqrt{cd},m=\frac{ \kappa^{2}+(r+P_{31})r-s^{2}}{2\kappa^{2}},\]  
\[r=\text{Re}(P_{32}),\,s=\text{Im}(P_{32}).\]  
This domain of the parameter \(q\) contains an important case of THG corresponding to the phase matching between FW and TH (\(\Delta_{31}k=0\)). Under this condition, there is only a unique branch of the TH intensity changing between 0 and \(P_{31}=0.8\). Two other roots are defined by a solution of the equation \(5p_{3}^{2}-8p_{3}+4=0\), and they are the complex ones (\(P_{32,33}=0.8\pm 0.4i\)). As a result, the TH intensity evolution is described by:  
\[p_{3}(z)=\frac{4(1-\text{cn}(\frac{3\sqrt{5}\gamma^{2}}{\Delta_{21}}\kappa z; 0.5(1-\frac{1}{\sqrt{5}})))}{(5+\sqrt{5})-(5-\sqrt{5})\,\text{cn}(\frac{3 \sqrt{5}\gamma^{2}}{\Delta_{21}}\kappa z;0.5(1-\frac{1}{\sqrt{5}}))}. \tag{21}\]  
As follows from (21), the coupling coefficient \(\gamma\) and the phase mismatching between FW and SH (\(\Delta_{21}k\)) only influence the intensity oscillation period. Let us note that in all other modes (at \(\Delta_{31}k\
eq 0\)) of THG, there is more complicated dependencies of the TH maximum intensity and period on \(\Delta_{21}k\). Therefore, we do not discuss this below. **Bistable frequency conversion mode: domain \(-1.5<q<3+\frac{9}{\sqrt[3]{2}}-9\sqrt[3]{2}\); three real roots. **  
In this case, all three roots \(P_{3j}\) are positive. Therefore, there are two regimes of the wave interaction: the low-efficiency mode and the high-efficiency mode. In the low-efficiency regime, the TH intensity changes between 0 and \(P_{31}\) as follows:  
\[p_{3}(z)=\frac{P_{31}P_{33}\,\text{sn}^{2}(3.75\frac{\gamma^{2}}{\Delta_{21} }\sqrt{P_{32}(P_{33}-P_{31})z},m)}{P_{31}\,\text{sn}^{2}(3.75\frac{\gamma^{2} }{\Delta_{21}}\sqrt{(P_{32})(P_{33}-P_{31})z},m)+(P_{33}-P_{31})}, \tag{22}\]  
\[m=\sqrt{\frac{P_{31}(P_{33}-P_{32})}{P_{32}(P_{33}-P_{31})}}.\]  
In the high-efficiency regime, the TH intensity changes between values of the roots \(P_{32}\) and \(P_{33}\) according to the rule:  
\[p_{3}(z)=\frac{P_{32}P_{33}}{(P_{33}-P_{32})\,\text{sn}^{2}(3.75\frac{\gamma^{ 2}}{\Delta_{21}k}\sqrt{P_{32}(P_{33}-P_{31})z},m)+P_{32}} \tag{23}\]  
with the same coefficient \(m\) as in (22). **Single frequency conversion mode and mode of preserving TH maximum intensity (\(P_{3}=1\)): the first special case \(q=-1.5;0<P_{31}<P_{32}=P_{33}=1\). **  
For this parameter value \(q\), the TH intensity \(p_{3}(z)\) changes again between 0 and \(P_{31}=0.64\).