## 3 Induced Cubic Nonlinear Processes Based on Cascading SHG
### Exact Solution of the Modified Problem  
In the framework of the long pulse duration approximation, the functions \(U,W\) depend only on \(z\) coordinate, and we can rewrite the problems (9) and (10) in the form:  
\[ \begin{split}&\frac{dU}{dz}-i\frac{\gamma^{2}}{\Delta_{21}k}(|U|^{ 2}U+3U^{*2}W+2U|W|^{2})=0,\\ &\frac{dW}{dz}-3i\frac{\gamma^{2}}{\Delta_{21}k}(U^{3}+2|U|^{2}W) +\Delta_{31}kW=0,\,0<z\leq L_{z}\\ & U(0)=1,\,W(0)=0.\end{split}  \tag{14}\]  
The conservation laws (12) and (13) are written as follows:  
\[ \begin{split}& I_{1}=|U|^{2}+|W|^{2},\\ & I_{3}=\frac{\gamma^{2}}{\Delta_{21}k}\Big{(}-12\operatorname{Re }(U^{3}W^{*})-3|U|^{4}-12|U|^{2}|W|^{2}\Big{)}+2\Delta_{31}k|W|^{2}.\end{split}  \tag{15}\]  
The energy conservation law of the problem (14) equals unity (\(I_{1}=1\)) due to the initial conditions. To write the intensity evolution for the interacting waves, we use a well-known representation for the complex amplitudes:  
\[U(z)=a_{1}e^{i\varphi_{1}(z)},\,W(z)=a_{3}e^{i\varphi_{3}(z)}, \tag{16}\]  
where \(a_{j},\,\varphi_{j}\) are the real functions describing amplitudes and phases of the waves. In the new notations, the equations in (14) are re-written as follows:  
\[ \begin{split}&\frac{da_{1}}{dz}=-\frac{3\gamma^{2}}{\Delta_{21}k}a_{ 1}^{2}a_{3}\sin\varphi,\\ &\frac{da_{3}}{dz}=\frac{3\gamma^{2}}{\Delta_{21}k}a_{1}^{3}\sin \varphi,\\ &\frac{d\varphi}{dz}-\frac{3\gamma^{2}}{\Delta_{21}k}\left(a_{1} \left(\frac{a_{1}^{2}}{a_{3}}-3a_{3}\right)\cos\varphi+a_{1}^{2}-2a_{3}^{2} \right)+\Delta_{31}k=0,\end{split}  \tag{17}\]  
where \(\varphi=\varphi_{3}-3\varphi_{1}\) is a phase difference between TH and FW. The invariants (15) are transformed to the following form:  
\[ \begin{split}& I_{1}=a_{1}^{2}+a_{3}^{2}=1,\\ & I_{3}=\frac{\gamma^{2}}{\Delta_{21}k}\Big{(}-12a_{1}^{3}a_{3} \cos\varphi-3a_{1}^{4}-12a_{1}^{2}a_{3}^{2}\Big{)}+\Delta_{31}ka_{3}^{2}.\end{split} \]  
To express \(\cos\varphi\), we modify the third invariant:  
\[\tilde{I}_{3}=I_{3}+3\frac{\gamma^{2}}{\Delta_{21}k}I_{1}^{2}=-12\frac{\gamma ^{2}}{\Delta_{21}k}a_{1}^{3}a_{3}\cos\varphi-6\frac{\gamma^{2}}{\Delta_{21}k} a_{1}^{2}a_{3}^{2}+3\frac{\gamma^{2}}{\Delta_{21}k}a_{3}^{4}+2\Delta_{31}ka_{3}^{2}=0,\]which equals zero because the TH incident intensity is equal to zero: \(a_{3}(0)=0\). Hence, we can express \(\cos\varphi\):  
\[\cos\varphi=\frac{a_{3}(-6a_{1}^{2}+3a_{3}^{2}+2q)}{12a_{1}^{3}}, \tag{18}\]  
where the parameter \(q\) is defined as:  
\[q=\frac{\Delta_{21}k\Delta_{31}k}{\gamma^{2}},\]  
which characterizes relation between phase incursions because of phase mismatching between SH and FW, and TH and FW, respectively, and because of the nonlinear coupling coefficient \(\gamma\). Obviously, the inequality  
\[|\cos\varphi|\leq 1\]  
must be valid. This leads to restriction on the intensities changing. Then, we find \(\sin\varphi\) from equality (18) and substitute it into the second equation of (17), which is preliminarily multiplied by \(a_{1}\). As a result, we write the following equation with respect to the TH intensity:  
\[\frac{da_{3}^{2}}{dz}=\pm 6\frac{\gamma^{2}}{\Delta_{21}k}a_{1}^{3}a_{3}\sqrt{1 -\frac{a_{3}^{2}(-6a_{1}^{2}+3a_{3}^{2}+2q)^{2}}{144a_{1}^{6}}}.\]  
Using the new variable \(p_{3}=a_{3}^{2}\) and expressing \(a_{1}^{2}\) from the first invariant \(a_{1}^{2}=1-p_{3}\), we finally obtain  
\[ \begin{split}&\frac{dp_{3}}{dz}=\pm 7.5\frac{\gamma^{2}}{\Delta_{21 }k}\sqrt{p_{3}f(p_{3})},\\ & f(p_{3})=\frac{16}{25}((1-p_{3})^{3}-\frac{1}{144}p_{3}(9p_{3}+ 2q-6)^{2}).\end{split}  \tag{19}\]  
where the sign in the right side of the equation corresponds to different processes--the plus sign corresponds to the conversion of the FW energy to TH energy and the minus sign to reverse energy conversion from the wave with tripled frequency to the FW. So, since TH incident intensity equals zero, we choose the plus sign. The intensity evolution is defined by the parameter \(q\) because it defines the number of real roots of the equation  
\[f(p_{3})=0,\]  
which can be found, for example, by using the Sturm theorem.