## 3 Modified Equations Derived in the Framework of the Multi-Scale Method
### Frequency Down-Conversion in the Framework of the Long Pulse Duration Approximation  
Although our primary goal was to analyze a process of the frequency down-conversion, accounting for SOD, we will briefly present an analysis of this process using the long pulse duration approximation. It allowed us to estimate maximal conversion efficiency. Let us suppose that the SOD coefficients are equal to zero: \(D_{j}=0\), \(j=1\), \(2\), \(3\). Thus, the functions \(U\), \(V\), W depend only on \(z\)--coordinate. In this case, the problem (5), (7) can be rewritten as follows:  
\[\frac{dU}{dz}-i\tilde{\kappa}(|U|^{2}U+3U^{*2}W-4U|V|^{2}+2U|W|^{ 2})=0, \tag{9}\] \[\frac{dV}{dz}+2i\tilde{\kappa}(4|U|^{2}-|W|^{2})V=0,\] \[\frac{dW}{dz}-3i\tilde{\kappa}(U^{3}+2|U|^{2}W+|V|^{2}W)=0,\] \[U(0)=A_{10},\;V(0)=A_{20},\;W(0)=A_{30}.\]  
The conservation laws (8) transform in the following way:  
\[I_{1 UW}=|U|^{2}+|W|^{2}=1, \tag{10}\] \[I_{1V}=|V|^{2}=|A_{20}|^{2},\] \[I_{3}=-3\tilde{\kappa}\Big{(}4 Re(U^{3}W^{*})+|U|^{4}+4|U|^{2}|W|^ {2}-8|U|^{2}|V|^{2}+2|V|^{2}|W|^{2}\Big{)}=const.\]  
The energy invariant \(I_{1 UW}\) is chosen equal to unity, for definiteness. This invariant's choice leads to more simple formulas for LFW generation/amplification efficiency. In general, this value is easy to realize by renormalization of the complex amplitudes. The Equation (9) can be reduced to a single differential equation. For this aim, we represent the complex amplitudes in a standard way:  
\[U(z)=a_{1}(z)\exp{i\varphi_{1}(z)},\;V(z)=a_{20}\exp(i\varphi_{2}(z)),\;W(z)= a_{3}(z)\exp(i\varphi_{3}(z)), \tag{11}\]  
here, the functions \(a_{j},\varphi_{j}\), \(j=1\), \(2\), \(3\) are real-valued functions. We notice that an equality of \(a_{20}=|A_{20}|\) takes place because of the IFW energy preservation. Thus, the problem (9) takes the form \[ \begin{split}\frac{da_{1}}{dz}&=-3\dot{x}a_{1}^{2}a_{3} \sin\varphi,\\ \frac{da_{3}}{dz}&=3\dot{x}a_{1}^{3}\sin\varphi,\\ \frac{d\varphi}{dz}&-\ddot{x}\left(3\left(\frac{a_{ 1}^{3}}{a_{3}}-3a_{1}a_{3}\right)\cos\varphi+3a_{1}^{2}-6a_{3}^{2}+15a_{20} ^{2}\right)=0,\\ \frac{d\varphi_{2}}{dz}&+2i\dot{x}(4a_{1}^{2}-a_{ 3}^{2})=0,\\ a_{1}(0)&=|A_{10}|,\,a_{3}(0)=|A_{30}|,\,\varphi(0)= \varphi_{0},\,\varphi_{2}(0)=\arg(A_{20}).\end{split}  \tag{12}\]  
Here, \(\varphi=\varphi_{3}-3\varphi_{1}\) is a phase difference between HFW and LFW, \(\varphi_{0}\) is its value in the input section of a medium. Obviously, the invariants (10) are transformed to a form:  
\[ \begin{split}& I_{1_{a_{1}a_{3}}}=a_{1}^{2}+a_{3}^{2}=1,\\ & I_{1_{a_{2}}}=a_{2}^{2}=a_{20}^{2},\\ & I_{3}=3\dot{x}\left(-4a_{1}^{3}a_{3}\cos\varphi-a_{1}^{4}-4a_{1 }^{2}a_{3}^{2}+8a_{20}a_{1}^{2}-2a_{20}^{2}a_{3}^{2}\right).\end{split} \]  
To write the algebraic equation with respect to the phase difference, we modify Hamiltonian by dividing it on \(3\dot{x}\):  
\[\tilde{I}_{3}=-4a_{1}^{3}a_{3}\cos\varphi-a_{1}^{4}-4a_{1}^{2}a_{3}^{2}+8a_{20 }a_{1}^{2}-2a_{20}^{2}a_{3}^{2}=\tilde{I}_{30} \tag{13}\]  
Thus, we can express \(\cos\varphi\) from the Hamiltonian (13):  
\[\cos\varphi=\frac{2a_{20}^{2}(4a_{1}^{2}-a_{3}^{2})-\tilde{I}_{30}-a_{1}^{4}- 4a_{1}^{2}a_{3}^{2}}{4a_{1}^{2}a_{3}^{2}} \tag{14}\]  
Substituting the expression (14) into the first equation of the set (12), we obtain the equation with respect to the LFW amplitude \(a_{1}\):  
\[\frac{da_{1}}{dz}=\mp\dot{x}a_{1}^{2}a_{3}\sqrt{1-\frac{(2a_{20}^{2}(4a_{1}^{ 2}-a_{3}^{2})-\tilde{I}_{30}-a_{1}^{4}-4a_{1}^{2}a_{3}^{2})}{16a_{1}^{6}a_{3} ^{2}}}.\]  
Multiplying both parts of the equation on \(a_{1}\), and using a new notation \(p_{1}=a_{1}^{2}\) (it means the LFW intensity), we write the differential equation with respect to the intensity of the LFW \(p_{1}\), accounting for an equality \(a_{3}^{2}=1-p_{1}\):  
\[ \begin{split}&\frac{dp_{1}}{dz}=\mp 7.5\dot{x}\sqrt{f(p_{1})},\\ & f(p_{1})=-p_{1}^{4}+(0.64+0.48B)p_{1}^{3}+(0.24A-0.16B^{2})p_{1 }^{2}-0.16 ABp_{1}-0.04A^{2},\\ & A=\tilde{I}_{30}+2a_{20}^{2},\,\,B=2-5a_{20}^{2}.\end{split}