## 3 Induced Cubic Nonlinear Processes Based on Cascading SHG  
### Derivation of the Modified Equations  
For simplicity of deriving modified equations describing THG, let us make a substitution of the TH complex amplitude as follows:  
\[A_{3}=\bar{A}_{3}e^{i\Delta_{31}kz}\]  
(for brevity, the bar is omitted below). The Equations' set (1) is transformed to the form  
\[\frac{\partial A_{1}}{\partial z}+iD_{1}\frac{\partial^{2}A_{1}}{ \partial t^{2}}+i\gamma\Big{(}A_{1}^{*}A_{2}e^{-i\Delta_{21}kz}+A_{2}^{*}A_{3} e^{i\Delta_{21}kz}\Big{)}=0,\] \[\frac{\partial A_{2}}{\partial z}+v_{21}\frac{\partial A_{2}}{ \partial t}+iD_{2}\frac{\partial^{2}A_{2}}{\partial t^{2}}+i\gamma\Big{(}A_{ 1}^{2}e^{i\Delta_{21}kz}+2A_{1}^{*}A_{3}e^{i\Delta_{21}kz}\Big{)}=0, \tag{7}\] \[\frac{\partial A_{3}}{\partial z}+v_{31}\frac{\partial A_{3}}{ \partial t}+iD_{3}\frac{\partial^{2}A_{3}}{\partial t^{2}}+3i\gamma A_{1}A_{2 }e^{-i\Delta_{21}kz}+i\Delta_{31}kA_{3}=0,\,0<z\leq L_{z},0<t<L_{t}.\]  
As we mentioned above, the three-wave interaction in a medium with quadratic susceptibility at cascading SHG leads to a response of a medium similar to the response of a medium with the cubic susceptibility. It is easy to show by using the multi-scale method [57]. Because the derivation is rather complicated, we place it in Appendix A. Here, we present the final result, in which the complex amplitudes \(A_{j}\), \(j=1,\,2,\,3\) are written as:  
\[A_{1}=U,A_{2}=\frac{1}{\Delta_{21}k}(-\gamma(U^{2}+2U^{*}W)e^{i\Delta_{21}kz}+ v_{1}),A_{3}=We^{i\Delta_{31}kz}, \tag{8}\]  
accounting for the terms of the order of \(O(1/\Delta_{21}k)\). The functions \(U,\,W,\,v_{1}\) are governed by the set of nonlinear equations  
\[\frac{\partial U}{\partial z}+iD_{1}\frac{\partial^{2}U}{\partial t ^{2}}-i\frac{\gamma^{2}}{\Delta_{21}k}(|U|^{2}U+3U^{*2}W+2U|W|^{2})=0,\] \[\frac{\partial W}{\partial z}+v_{31}\frac{\partial W}{\partial t }+iD_{3}\frac{\partial^{2}W}{\partial t^{2}}-3i\frac{\gamma^{2}}{\Delta_{21} k}(U^{3}+2|U|^{2}W)+i\Delta_{31}kW=0, \tag{9}\] \[\frac{\partial v_{1}}{\partial t}+v_{21}\frac{\partial v_{1}}{ \partial t}+iD_{2}\frac{\partial^{2}v_{1}}{\partial t^{2}}=0\]  
with the initial conditions  
\[U|_{z=0}=A_{10}(t),\,W|_{z=0}=0,\,v_{1}|_{z=0}=\gamma A_{10}^{2}(t). \tag{10}\]  
and zero-valued boundary conditions, because all pulses have a finite initial distribution and we consider the bounded domain in the \(z\)-coordinate. One can see that a term proportional to the third power of the FW complex amplitude is present in the second equation of the problem (9), and this term is responsible for THG. We call the below problems (9) and (10) the modified problem. It should be emphasized that the first two equations in (9) can be obtained in much simpler way. For this aim, we propose to use the following representation of the complex amplitude \(A_{2}\):  
\[A_{2}=(A_{2}^{(0)}+A_{2}^{(1)}+A_{2}^{(2)}+...)e^{i\Delta_{21}kz}.\]  
Substituting this series into the second equation of the set (7), we obtain the following equation in the first order of approximation on parameter \((1/\Delta_{21}k)\):  
\[\frac{\partial A_{1}}{\partial z}+iD_{1}\frac{\partial^{2}A_{1}}{ \partial t^{2}}+i\gamma\Big{(}A_{1}^{*}A_{2}e^{-i\Delta_{21}kz}+A_{2}^{*}A_{3} e^{i\Delta_{21}kz}\Big{)}=0,\]0that can be rewritten as  
\[\frac{\partial A_{1}}{\partial z}+iD_{1}\frac{\partial^{2}A_{1}}{ \partial t^{2}}+i\gamma\Big{(}A_{1}^{*}A_{2}e^{-i\Delta_{21}kz}+A_{2}^{*}A_{3} e^{i\Delta_{21}kz}\Big{)}=0,\]1  
The last three terms contain both complex amplitude \(A_{2}^{(0)}\) and \(1/\Delta_{21}k\). Therefore, they are much smaller than the two first terms, and they can be neglected if we account for the terms of order \(1/\Delta_{21}k\) only.