## 3 Modified Equations Derived in the Framework of the Multi-Scale Method  
### Set of Modified Equations  
If the phase mismatching between LFW and IFW is much larger than the nonlinear coefficient (\(|\Delta_{21}k|>>\gamma\)), we can derive the qualitative approximation of the problem (1), (2) solution using the multi-scale method:  
\[\begin{split} A_{1}&=U+\frac{1}{\Delta_{21}k}\Big{(} \gamma(U^{*}Ve^{-i\Delta_{21}kz}-V^{*}We^{i\Delta_{21}kz})+u_{1}\Big{)},\\ A_{2}&=V+\frac{1}{\Delta_{21}k}\Big{(}-\gamma(U^{2 }+2U^{*}W)e^{i\Delta_{21}kz}+v_{1}\Big{)},\\ A_{3}&=W+\frac{1}{\Delta_{21}k}\Big{(}3\gamma UVe^{ -i\Delta_{21}kz}+w_{1}\Big{)},\end{split} \tag{4}\]  
with accuracy of \(O(\Delta_{21}k^{-2})\). Here, the functions \(U,\,V,\mathrm{W}\) are governed by the following set of equations:  
\[\begin{split}&\frac{\partial U}{\partial z}+iD_{1}\frac{\partial^{2} U}{\partial t^{2}}-i\tilde{\alpha}(|U|^{2}U+3U^{*2}W-4U|V|^{2}+2U|W|^{2})=0,\\ &\frac{\partial V}{\partial z}+v_{21}\frac{\partial V}{\partial t }+iD_{2}\frac{\partial^{2}V}{\partial t^{2}}+2i\tilde{\alpha}(4|U|^{2}-|W|^{2 })V=0,\\ &\frac{\partial W}{\partial z}+v_{31}\frac{\partial W}{\partial t }+iD_{3}\frac{\partial^{2}W}{\partial t^{2}}-3i\tilde{\alpha}(U^{3}+2|U|^{2}W+ |V|^{2}W)=0,\end{split} \tag{5}\]  
where \(\tilde{\alpha}=\gamma^{2}/\Delta_{21}k\). The functions \(u_{1},\,v_{1},\,w_{1}\) are the solutions of the following linear equations:  
\[\begin{split}&\frac{\partial u_{1}}{\partial z}+iD_{1}\frac{ \partial^{2}u_{1}}{\partial t^{2}}=0,\\ &\frac{\partial v_{1}}{\partial z}+v_{21}\frac{\partial v_{1}}{ \partial t}+iD_{2}\frac{\partial^{2}v_{1}}{\partial t^{2}}=0,\\ &\frac{\partial w_{1}}{\partial z}+v_{31}\frac{\partial w_{1}}{ \partial t}+iD_{3}\frac{\partial^{2}w_{1}}{\partial t^{2}}=0.\end{split} \tag{6}\]  
Let us note that the derivation of these equations is made in A. The distributions for the new functions corresponding to the complex amplitudes of the incident pulses are the following:  
\[\begin{split}& U(0,t)=A_{10}(t),\,V(0,t)=A_{20}(t),\,W(0,t)=A_{30} (t),\\ & u_{1}(0,t)=\gamma(A_{20}^{*}(t)A_{30}(t)-A_{10}^{*}(t)A_{20}(t)), \,v_{1}(0,t)=\gamma\Big{(}A_{10}^{2}(t)+2A_{10}^{*}(t)A_{30}(t)\Big{)},\\ & w_{1}(0,t)=-3\gamma A_{10}(t)A_{20}(t),\,t\in[0,L_{t}],\\ & U(z,0)=V(z,0)=W(z,0)=U(z,L_{t})=V(z,L_{t})=W(z,L_{t})=0,\\ & u_{1}(z,0)=v_{1}(z,0)=w_{1}(z,0)=u_{1}(z,L_{t})=v_{1}(z,L_{t})= w_{1}(z,L_{t})=0,\,z\in[0,L_{z}].\end{split} \tag{7}\]  
The set of Equation (5) possesses some conservation laws (invariants):\[I_{1 UW}=\int\limits_{0}^{L_{t}}\Big{(}|U|^{2}+|W|^{2}\Big{)}dt=const, \tag{8}\] \[I_{1V}=\int\limits_{0}^{L_{t}}|V|^{2}dt=const,\] \[I_{3}=\int\limits_{0}^{L_{t}}\Big{(}3\
u_{21}Im\bigg{(}V^{*} \frac{\partial V}{\partial t}\bigg{)}+2\
u_{31}Im\bigg{(}W^{*}\frac{\partial W }{\partial t}\bigg{)}-6D_{1}\bigg{|}\frac{\partial U}{\partial t}\bigg{|}^{2} -3D_{2}\bigg{|}\frac{\partial V}{\partial t}\bigg{|}^{2}-2D_{3}\bigg{|}\frac{ \partial W}{\partial t}\bigg{|}^{2}\] \[-3\tilde{\kappa}\Big{(}4 Re(U^{3}W^{*})+|U|^{4}+4|U|^{2}|W|^{2}-8| U|^{2}|V|^{2}+2|V|^{2}|W|^{2}\Big{)}\Big{)}dt=const.\]  
These invariants show that, in the framework of the modified equations, the energy exchanges occur only between LFW and HFW. Meanwhile, the IFW energy remains constant. This fact explains why we choose the normalization value \(A_{0}\) in (3) as the square root from the sum of the intensities of LFW and HFW achieved at the pulse center.