## Appendix A
\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}+i\Delta_{31}kW=0.\end{split}  \tag{11}\]  
Because the first terms in these equations do not depend on \(\xi\) meanwhile other terms do depend on this variable, then one can separate equations into two parts. The first of them is written as  
\[L_{1}^{(0)}U=L_{2}^{(0)}V=L_{3}^{(0)}W+i\Delta_{31}kW=0. \tag{12}\]  
The functions \(U_{1},\,V_{1},\,W_{1}\) can be found from the other parts by integrating (11) with respect to \(\xi\):  
\[ \begin{split}& U_{1}=\gamma(U^{*}Ve^{-i\xi}-V^{*}We^{i\xi})+u_{1}(t,z_{0},\,z_{1}...),\\ & V_{1}=\gamma(-U^{2}e^{i\xi}-2U^{*}We^{i\xi})+v_{1}(t,z_{0},\,z _{1}...),\\ & W_{1}=3\gamma UVe^{-i\xi}+w_{1}(t,z_{0},\,z_{1}...).\end{split}  \tag{13}\]  
where \(u_{1},v_{1},w_{1}\) are the function of integration: they do not depend on \(\xi\). The equations with respect these functions are written further. The order \(O(\mu)\) of the power series leads to the equations:  
\[ \begin{split}&\frac{\partial U_{2}}{\partial\xi}+L_{1}^{(0)}U_{1}+ \frac{\partial U}{\partial z_{1}}+i\gamma((U^{*}V_{1}+U_{1}^{*}V)e^{-i\xi}+(V^{ *}W_{1}+V_{1}^{*}W)e^{i\xi})=0,\\ &\frac{\partial V_{2}}{\partial\xi}+L_{2}^{(0)}V_{1}+\frac{ \partial V}{\partial z_{1}}+i\gamma(2 UU_{1}e^{i\xi}+(U^{*}W_{1}+U_{1}^{*}W)e^ {i\xi})=0,\\ &\frac{\partial W_{2}}{\partial\xi}+L_{3}^{(0)}W_{1}+\frac{ \partial W}{\partial z_{1}}+3i\gamma(UV_{1}+U_{1}V)e^{-i\xi}+\Delta_{31}kW_{1}= 0.\end{split} \]  
Using the representation (13), this set transforms into the form:\[\frac{\partial U_{2}}{\partial\xi}+\gamma(L_{1}^{(0)}(U^{*}V)e^{-i \xi}-L_{1}^{(0)}(V^{*}W)e^{i\xi})+i\gamma^{2}(U^{*}v_{1}e^{-i\xi}-V^{2}W^{*}e^{-2 i\xi}+u_{1}^{*}Ve^{-i\xi}+V^{*}w_{1}e^{i\xi}+v_{1}^{*}We^{i\xi})=\] \[-\left(\frac{\partial U}{\partial z_{1}}+i\gamma^{2}(-|U|^{2}U-3U^ {*2}W+4U|V|^{2}-2U|W|^{2})\right)-L_{1}^{(0)}u_{1},\] \[\frac{\partial V_{2}}{\partial\xi}-\gamma(L_{2}^{(0)}(U^{2})e^{i \xi}+2L_{2}^{(0)}(U^{*}W)e^{i\xi})+i\gamma^{2}(-2 UV^{*}We^{2i\xi}+2 Uu_{1}e^{i \xi}+2 UV^{*}We^{i\xi}+2u_{1}^{*}We^{i\xi}+2U^{*}we^{i\xi})=\] \[-\left(\frac{\partial V}{\partial z_{1}}+2i\gamma^{2}(4|U|-|W|^{2 })V\right)-L_{2}^{(0)}v_{1},\] \[\frac{\partial W_{2}}{\partial\xi}+3\gamma L_{3}^{(0)}(UV)e^{-i \xi}+3i\gamma^{2}(Uv_{1}e^{-i\xi}+U^{*}V^{2}*e^{-2i\xi}+u_{1}V_{1}e^{-i\xi})+3 i\Delta_{31}k\gamma UVe^{-i\xi}=\] \[-\left(\frac{\partial W}{\partial z_{1}}-3i\gamma^{2}(U^{3}+2|U|^ {2}W+|V|^{2}W)\right)-L_{1}^{(0)}w_{1}-i\Delta_{31}kw_{1}.\]  
As above, we can state that the right-hand sides of the equations must be 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  
\[\frac{\partial U}{\partial z_{1}}+i\gamma^{2}(-|U|^{2}U-3U^{*2}W+4U| V|^{2}-2U|W|^{2})=-L_{1}^{(0)}u_{1},\] \[\frac{\partial V}{\partial z_{1}}+2i\gamma^{2}(4|U|^{2}-|W|^{2})V =L_{2}^{(0)}v_{1},\] \[\frac{\partial W}{\partial z_{1}}-3i\gamma^{2}(U^{3}+2|U|^{2}W+| V|^{2}W)=L_{3}^{(0)}w_{1}+i\Delta_{31}kw_{1}.\]  
Because the functions \(u_{1}\), \(v_{1}\), \(w_{1}\) are of order \(O(\mu)\) (see the representation (A6)) meanwhile the functions \(U,\,V,\,W\) are of order \(O(1)\), then we can once again separate the obtained equation into two parts:  
\[\frac{\partial U}{\partial z_{1}}+i\gamma^{2}(-|U|^{2}U-3U^{*2}W+4 U|V|^{2}-2U|W|^{2})=0,\] \[-\left(\frac{\partial U}{\partial z_{1}}+i\gamma^{2}(-|U|^{2}U-3U^ {*2}W+4U|V|^{2}-2U|W|^{2})\right)-L_{1}^{(0)}u_{1},\]0 (A7) \[-\left(\frac{\partial U}{\partial z_{1}}+i\gamma^{2}(-|U|^{2}U-3U^ {*2}W+4U|V|^{2}-2U|W|^{2})\right)-L_{1}^{(0)}u_{1},\]1  
Consequently, the equations  
\[-\left(\frac{\partial U}{\partial z_{1}}+i\gamma^{2}(-|U|^{2}U-3U^ {*2}W+4U|V|^{2}-2U|W|^{2})\right)-L_{1}^{(0)}u_{1},\]2 (A8)  
are valid.