## Appendix A  
Below, we derive the equation set (9) that describes the frequency conversion process at large phase-mismatching between IFW and LFW: \(|\Delta_{21}k|>>\gamma\). In this case, the process of the optical pulse interactions possess various spatial scales: in particular, a small scale, defined by large-phase mismatching \(|\Delta_{21}k|\), and a long spatial scale defined by the smallest dispersion length of the interacting pulses. Thus, we introduce a small parameter \(\mu=\frac{1}{\Delta_{21}k}\) and introduce various scales along the \(z\) coordinate: a small scale equal to the inverse phase-mismatching length: \(\xi=\frac{z}{\mu}\) and large longitudinal scales \(z_{l}=\mu^{l}z\), \(l=0,1,2\ldots\). Therefore, the complex amplitudes are expanded in a power series of \(\mu\):\[ \begin{split} A_{1}&=U+\mu U_{1}+\mu^{2}U_{2}+\dots,\\ A_{2}&=V+\mu V_{1}+\mu^{2}V_{2}+\dots,\\ A_{3}&=W+\mu W_{1}+\mu^{2}W_{2}+\dots.\end{split}  \tag{10}\]  
Obviously, the functions in (10) depend on all variables \((t,\zeta,z_{i}|l\geq 0)\). Then, we transform the differential operators using new variables:  
\[ \begin{split} L_{j}&=\frac{\partial}{\partial z}+ \
u_{j1}\frac{\partial}{\partial t}+iD_{j}\frac{\partial^{2}}{\partial t^{2}}= \frac{\partial\xi}{\partial z}\frac{\partial}{\partial\xi}+\sum_{l=0}^{\infty }\frac{\partial z_{l}}{\partial z}\frac{\partial}{\partial z_{l}}+iD_{j}\frac{ \partial^{2}}{\partial t^{2}}=\frac{1}{\mu}\frac{\partial}{\partial\xi}+\sum _{l=0}^{\infty}\mu^{l}\frac{\partial}{\partial z_{l}}+iD_{j}\frac{\partial^{ 2}}{\partial t^{2}}\\ &=\frac{1}{\mu}\frac{\partial}{\partial\xi}+L_{j}^{0}+\mu\frac{ \partial}{\partial z_{1}}+\mu^{2}\frac{\partial}{\partial z_{2}}+\dots,j=1,2, 3.\end{split}  \tag{11}\]  
Here, the operator \(L_{j}\) is defined by  
\[L_{j}^{(0)}=\frac{\partial}{\partial z_{0}}+\
u_{j1}\frac{\partial}{\partial t }+iD_{j}\frac{\partial^{2}}{\partial t^{2}}.\]  
Then, we substitute the representation (10) of the complex amplitudes into the equation set (1), and keep terms with an order, which is greater than \(\mu^{2}\):  
\[ \begin{split}&\frac{1}{\mu}\frac{\partial U}{\partial\xi}+L_{1}^{ (0)}U+\mu\frac{\partial U}{\partial z_{1}}+\frac{\partial U_{1}}{\partial\xi} +\mu L_{1}^{(0)}U_{1}+\mu\frac{\partial U_{2}}{\partial\xi}+\\ &+i\gamma\Big{(}U^{*}Ve^{-i\xi}+V^{*}We^{i\xi}+\mu((U^{*}V_{1}+U_ {1}^{*}V)e^{-i\xi}+(V^{*}W_{1}+V_{1}^{*}W)e^{i\xi})\Big{)}+O(\mu^{2})=0,\\ &\frac{1}{\mu}\frac{\partial V}{\partial\xi}+L_{2}^{(0)}V+\mu \frac{\partial V}{\partial z_{1}}+\frac{\partial V_{1}}{\partial\xi}+\mu L_{1 }^{(0)}V_{1}+\mu\frac{\partial V_{2}}{\partial\xi}+\\ &+i\gamma\Big{(}U^{2}e^{i\xi}+2U^{*}We^{i\xi}+\mu(2 UI_{1}e^{i\xi} +(U^{*}W_{1}+U_{1}^{*}W)e^{i\xi})\Big{)}+O(\mu^{2})=0,\\ &\frac{1}{\mu}\frac{\partial W}{\partial\xi}+L_{3}^{(0)}W+\mu \frac{\partial W}{\partial z_{1}}+\frac{\partial W_{1}}{\partial\xi}+\mu L_{1 }^{(0)}W_{1}+\mu\frac{\partial W_{2}}{\partial\xi}+\\ &+3i\gamma\Big{(}UVe^{-i\xi}+\mu(UV_{1}+U_{1}V)e^{-i\xi}\Big{)}+ O(\mu^{2})=0.\end{split}  \tag{12}\]  
Grouping the terms with respect to the power of \(\mu\), we obtain the equations:  
\[\frac{\partial U}{\partial\xi}=\frac{\partial V}{\partial\xi}=\frac{\partial W }{\partial\xi}=0,\]  
corresponding to \(\frac{1}{\mu}\) of the power series. Consequently, the functions \(U\), \(V\) and \(W\) do not depend on the fast changing coordinate \(\xi\). Therefore, these functions do not change at the small scale. For the next order \(O(1)\) of the power series on \(\mu\), we obtain the following set of equations:  
\[ \begin{split}& L_{1}^{(0)}U+\frac{\partial U_{1}}{\partial\xi}+i \gamma(U^{*}Ve^{-i\xi}+V^{*}We^{i\xi})=0,\\ & L_{2}^{(0)}V+\frac{\partial V_{1}}{\partial\xi}+i\gamma(U^{2}e ^{i\xi}+2U^{*}We^{i\xi})=0,\\ & L_{3}^{(0)}W+\frac{\partial W_{1}}{\partial\xi}+3i\gamma UVe^{- i\xi}=0.\end{split}  \tag{13}\]  
The first terms in these equations do not depend on \(\xi\); however, as other terms depend on this variable, we can separate equations into two parts.