## Appendix A  
For demonstrating appearance of the induced cubic-like response of a medium due to cascading SHG, we use the multi-scale method [57]. This method was early applied in [51] for a demonstration of appearing induced nonlinearity similar to Kerr effect and responsible for the pulse self-compression or its decompression because of the process under consideration. Because we consider a large phase mismatching between SH and FW: \(|\Delta_{21}k|>>1\), the process of wave interaction possesses various space scales: in particular, a small scale, defined by large phase mismatching \(|\Delta_{21}k|\), and a long space scale defined by the dispersion lengths of the interacting pulses. Therefore, let us introduce a small parameter \(\mu=\frac{1}{\Delta_{21}k}\) (for simplicity, we suppose that the phase mismatching has a positive sign) and introduce various scales along \(z\) coordinate: small scale equal to the inverse phase mismatching length: \(\xi=\frac{z}{\mu}\), and big longitudinal scales \(z_{l}=\mu^{l}z\), \(l=0,1,2\ldots\). Thus, the complex amplitudes are expanded in a power series of \(\mu\):  
\[ \begin{split} A_{1}&=U+\mu U_{1}+\mu^{2}U_{2}+...,\\ A_{2}&=V+\mu V_{1}+\mu^{2}V_{2}+...,\\ A_{3}&=W+\mu W_{1}+\mu^{2}W_{2}+....\end{split}  \tag{1}\]  
Evidently, the functions in (1) depend on all the variables \((t,\xi,z_{l}|l\geq 0)\). The differential operators in terms of new variables take a form  
\[ \begin{split} L_{j}&=\frac{\partial}{\partial z}+ \
u_{l1}\frac{\partial}{\partial t}+iD_{j}\frac{\partial^{2}}{\partial t^{2}}= \frac{\partial^{2}_{\xi}}{\partial z}\frac{\partial}{\partial\xi}+\sum_{l=0} ^{\infty}\frac{\partial z_{l}}{\partial z}\frac{\partial}{\partial z_{l}}+iD _{j}\frac{\partial^{2}}{\partial t^{2}}=\frac{1}{\mu}\frac{\partial}{\partial \xi}+\sum_{l=0}^{\infty}\mu^{l}\frac{\partial}{\partial z_{l}}+iD_{j}\frac{ \partial^{2}}{\partial t^{2}}\\ &=\frac{1}{\mu}\frac{\partial}{\partial\xi}+D_{j}^{0}+\mu\frac{ \partial}{\partial z_{1}}+\mu^{2}\frac{\partial}{\partial z_{2}}+...j=1,2,3. \end{split}  \tag{2}\]  
Here, operator \(L_{j}^{(0)}\) is defined as  
\[L_{j}^{(0)}=\frac{\partial}{\partial z_{0}}+\
u_{l1}\frac{\partial}{\partial t }+iD_{j}\frac{\partial^{2}}{\partial t^{2}}.\]  
Then, substituting expansion (1) into the equation set (7) and collecting terms, which orders are bigger than \(\mu^{2}\), we obtain the equations:\[ \begin{split}&\frac{1}{\mu}\frac{\partial U}{\partial\zeta}+L_{1}^{(0)}U +\mu\frac{\partial U}{\partial z_{1}}+\frac{\partial U_{1}}{\partial\zeta}+\mu L _{1}^{(0)}U_{1}+\mu\frac{\partial U_{2}}{\partial\zeta}+\\ +i\gamma\Big{(}U^{*}Ve^{-i\xi}+V^{*}We^{i\xi}+\mu((U^{*}V_{1}+U_ {1}^{*}V)e^{-i\xi}+(V^{*}W_{1}+V_{1}^{*}W)e^{i\xi}\Big{)}\Big{)}+O(\mu^{2})=0,\\ &\frac{1}{\mu}\frac{\partial V}{\partial\zeta}+L_{2}^{(0)}V+\mu \frac{\partial V}{\partial z_{1}}+\frac{\partial V_{1}}{\partial\xi}+\mu L_{1}^ {(0)}V_{1}+\mu\frac{\partial V_{2}}{\partial\xi}+\\ +i\gamma\Big{(}U^{2}e^{i\xi}+2U^{*}We^{i\xi}+\mu(2 UU_{1}e^{i\xi} +(U^{*}W_{1}+U_{1}^{*}W)e^{i\xi})\Big{)}+O(\mu^{2})=0,\\ &\frac{1}{\mu}\frac{\partial W}{\partial\zeta}+L_{3}^{(0)}W+\mu \frac{\partial W}{\partial z_{1}}+\frac{\partial W_{1}}{\partial\xi}+\mu L_{1} ^{(0)}W_{1}+\mu\frac{\partial W_{2}}{\partial\xi}+\\ +3i\gamma\Big{(}UVe^{-i\xi}+\mu(UV_{1}+U_{1}V)e^{-i\xi}\Big{)}+i \Delta_{31}k(W+\mu W_{1})+O(\mu^{2})=0.\end{split}  \tag{10}\]  
The terms with respect to power of \(\frac{1}{\mu}\) allow us to write the equations:  
\[\frac{\partial U}{\partial\xi}=\frac{\partial V}{\partial\xi}=\frac{\partial W }{\partial\xi}=0.\]  
Consequently, the functions \(U\), \(V\) and \(W\) do not depend on fast changing coordinate \(\xi\), and they do not change at the small scale. Next order \(O(1)\) of power \(\mu\) leads to the following equations:  
\[