## Appendix A
The first of them is written as  
\[L_{1}^{(0)}U=L_{2}^{(0)}V=L_{3}^{(0)}W=0. \tag{14}\]The functions \(U_{1},\ V_{1},\ W_{1}\) can be written solving the second system by integrating (A4), with respect to \(\xi\) variable:  
\[ \begin{split} U_{1}&=\gamma(U^{*}Ve^{-i\xi}-V^{*}We^{ i\xi})+u_{1}(t,z_{0},\ z_{1}\dots),\\ V_{1}&=\gamma(-U^{2}e^{i\xi}-2U^{*}We^{i\xi})+v_{1} (t,z_{0},\ z_{1}\dots),\\ W_{1}&=3\gamma UVe^{-i\xi}+w_{1}(t,z_{0},\ z_{1} \dots).\end{split}  \tag{10}\]  
Here \(u_{1},v_{1},w_{1}\) are the functions of integration: they do not depend on the \(\xi\)-coordinate. The equations with respect to these functions are derived below. For the functions at the order \(O(\mu)\) of the power series, we obtain the following equations:  
\[ \begin{split}&\frac{\partial U_{2}}{\partial\xi}+L_{1}^{(0)}U_{1}+ \frac{\partial U}{\partial z_{1}}+i\gamma((U^{*}V_{1}+U_{1}^{*}V)e^{-i\xi}+(V ^{*}W_{1}+V_{1}^{*}W)e^{i\xi})=0,\\ &\frac{\partial V_{2}}{\partial\xi}+L_{2}^{(0)}V_{1}+\frac{ \partial V}{\partial z_{1}}+i\gamma(2 UI_{1}e^{i\xi}+(U^{*}W_{1}+U_{1}^{*}W)e^ {i\xi})=0,\\ &\frac{\partial W_{2}}{\partial\xi}+L_{3}^{(0)}W_{1}+\frac{ \partial W}{\partial z_{1}}+3i\gamma(UV_{1}+U_{1}V)e^{-i\xi}=0.\end{split} \]  
Using the representation (10), this set of equations transforms into the form:  
\[ \begin{split}&\frac{\partial U_{2}}{\partial\xi}+\gamma(L_{1}^{(0)}(U ^{*}V)e^{-i\xi}-L_{1}^{(0)}(V^{*}W)e^{i\xi})+i\gamma^{2}(U^{*}v_{1}e^{-i\xi}-V ^{2}W^{*}e^{-2i\xi}+u_{1}^{*}Ve^{-i\xi}+V^{*}w_{1}e^{i\xi}+v_{1}^{*}We^{i\xi} )=\\ &-\left(\frac{\partial U}{\partial z_{1}}+i\gamma^{2}(-|U|^{2}U- 3U^{*2}W+4 UI|V|^{2}-2U|W|^{2})\right)-L_{1}^{(0)}u_{1},\\ &\frac{\partial V_{2}}{\partial\xi}-\gamma(L_{2}^{(0)}(U^{2})e^{ i\xi}+2L_{2}^{(0)}(U^{*}W)e^{i\xi})+i\gamma^{2}(-2 UV^{*}We^{2i\xi}+2 LU_{1}e^{i\xi}+2 UV^{*}We^{i\xi}+2u_{1}^{*}We^{i\xi}+2U^{*}we^{i\xi})=\\ &-\left(\frac{\partial V}{\partial z_{1}}+2i\gamma^{2}(4|U|-|W|^ {2})V\right)-L_{2}^{(0)}v_{1},\\ &\frac{\partial W_{2}}{\partial\xi}+3\gamma L_{3}^{(0)}(UV)e^{- i\xi}+3i\gamma^{2}(Uv_{1}e^{-i\xi}+U^{*}V^{2}*e^{-2i\xi}+u_{1}V_{1}e^{-i\xi})=\\ &-\left(\frac{\partial W}{\partial z_{1}}-3i\gamma^{2}(U^{3}+2|U| ^{2}W+|V|^{2}W)\right)-L_{1}^{(0)}w_{1}.\end{split} \]  
As before, we can state that the right-hand sides of the equations are equal to zero because they do not depend on \(\xi\), in contrast to the left-hand sides of the equations. Thus, we write the equations  
\[ \begin{split}&\frac{\partial U}{\partial z_{1}}+i\gamma^{2}(-|U|^{2}U- 3U^{*2}W+4U|V|^{2}-2U|W|^{2})=-L_{1}^{(0)}u_{1},\\ &\frac{\partial V}{\partial z_{1}}+2i\gamma^{2}(4|U|^{2}-|W|^{2}) V=L_{2}^{(0)}v_{1},\\ &\frac{\partial W}{\partial z_{1}}-3i\gamma^{2}(U^{3}+2|U|^{2}W+| V|^{2}W)=L_{3}^{(0)}w_{1}.\end{split} \]  
Above, we separate terms that contain \(u_{1}\), \(v_{1}\), \(w_{1}\). They belong to order \(O(\mu)\) in the representation (11); meanwhile, \(U\), \(V\), \(W\) are aof order \(O(1)\), then one can once again separate the obtained equation into two parts:  
\[ \begin{split}&\frac{\partial U}{\partial z_{1}}+i\gamma^{2}(-|U|^{2}U- 3U^{*2}W+4U|V|^{2}-2U|W|^{2})=0,\\ &\frac{\partial V}{\partial z_{1}}+2i\gamma^{2}(4|U|^{2}-|W|^{2}) V=0,\\ &\frac{\partial W}{\partial z_{1}}-3i\gamma^{2}(U^{3}+2|U|^{2}W+| V|^{2}W)=0.\end{split}  \tag{11}\]Consequently, the equations  
\[L_{1}^{(0)}u_{1} =0,\] \[L_{2}^{(0)}v_{1} =0,\] (10) \[L_{3}^{(0)}w_{1} =0\] take place. After returning to the original variables, (\(\zeta=\Delta_{21}kz\), \(z_{0}=z\), \(z_{1}=z/\Delta_{21}k\), \(\frac{\partial}{\partial z}=\Delta_{21}k\frac{\partial}{\partial z}+\frac{ \partial}{\partial z_{0}}+\frac{1}{\Delta_{21}k}\frac{\partial}{\partial z_{1 }}+O((\Delta_{21}k)^{-2})\)), we obtain the sets of Equations (5) and (6). In turn, the expansion series (10) transforms into the form (4).