## 3 Induced Cubic Nonlinear Processes Based on Cascading SHG
### Derivation of the Modified Equations
Thus, we obtain the following equality:  
\[\frac{\partial A_{1}}{\partial z}+iD_{1}\frac{\partial^{2}A_{1}}{ \partial t^{2}}+i\gamma\Big{(}A_{1}^{*}A_{2}e^{-i\Delta_{21}kz}+A_{2}^{*}A_{3} e^{i\Delta_{21}kz}\Big{)}=0,\]2  
Evidently, the terms in the right part describe phase gratings, induced at doubled frequency, where \(A_{1}^{2}\) will lead to THG and FW self-modulation. The term \(2A_{1}^{*}A_{3}\) will lead to cross-modulation of the waves and to reverse energy conversion from TH to FW. Substituting representation (11) into the first equation and the third equation of set (7)  
\[\frac{\partial A_{1}}{\partial z}+iD_{1}\frac{\partial^{2}A_{1}}{ \partial t^{2}}+i\gamma\Big{(}A_{1}^{*}A_{2}e^{-i\Delta_{21}kz}+A_{2}^{*}A_{3} e^{i\Delta_{21}kz}\Big{)}=0,\]3 \[\frac{\partial A_{1}}{\partial z}+iD_{1}\frac{\partial^{2}A_{1}}{ \partial t^{2}}+i\gamma\Big{(}A_{1}^{*}A_{2}e^{-i\Delta_{21}kz}+A_{2}^{*}A_{3} e^{i\Delta_{21}kz}\Big{)}=0,\]4  
coincides with the first two equations of set (9). The equations in (9) possess the energy conservation law:  
\[\frac{\partial A_{1}}{\partial z}+iD_{1}\frac{\partial^{2}A_{1}}{ \partial t^{2}}+i\gamma\Big{(}A_{1}^{*}A_{2}e^{-i\Delta_{21}kz}+A_{2}^{*}A_{3} e^{i\Delta_{21}kz}\Big{)}=0,\]5  
The values of the energy's invariant (12) and (4) coincide with each other: they are equal to \(\int\limits_{0}^{L_{t}}|A_{10}|^{2}dt\). However, because the invariant for the set of the modified equations does not contain the SH wave, the total energy of two others waves will be greater than those computed using the original problem. The Hamiltonian of the Equations' set (9) is written in the form:  
\[ \begin{split}& I_{3}=\int\limits_{0}^{L_{t}}\Big{(}2v_{31}\, \text{Im}\left(W^{*}\frac{\partial W}{\partial t}\right)-6D_{1}\Big{|}\frac{ \partial U}{\partial t}\Big{|}^{2}-2D_{3}\Big{|}\frac{\partial W}{\partial t }\Big{|}^{2}-\\ &\frac{3\gamma^{2}}{\Delta_{21}k}\Big{(}4\,\text{Re}(U^{3}W^{*}) +|U|^{4}+4|U|^{2}|W|^{2}\Big{)}+2\Delta_{31}k|W|^{2}\Big{)}dt=const.\end{split}  \tag{13}\]  
It should be mentioned that the physical mechanism of a response, which inherent cubic nonlinearity, at the laser pulse propagation in a medium with the quadratic susceptibility under the cascading SHG is discussed in [51]. The authors stress that the self-lensing term \(|A|^{2}A\), which describes the nonlinear phase shift, appeared because of sequences of cascaded photon processes, each involving up-conversion to \(2\omega\) followed by back-conversion to \(\omega\). This two-step process leads to an effective cubic susceptibility \(\chi^{(2)}(\omega;2\omega,2\omega)\chi^{(2)}(2\omega;\omega,\omega)\to\chi^{(3 )}_{eff}(\omega;\omega,2\omega,\omega)\) that is characteristic of the effective four-photon interaction \(\omega=\omega-\omega+\omega\). However, in that paper as well as in most others papers, the authors consider only the first two equations of the set (7) with respect to FW and SH. We believe that the physical mechanism of this phenomenon can also be explained on the base of the analysis of the optical phase gratings induced by the interacting waves. So, if the phase mismatching \(\Delta_{21}k\) is large, then the SH complex amplitude is proportional to \(A_{1}^{2}\) (see above). This means that the response of a medium contains the phase grating at the doubled frequency. The optical wave at FF scatters on the grating and the optical wave at the tripled frequency appears. If the phase matching between the TH and FW occurs then the frequency conversion will be efficient. However, this mechanism of the frequency conversion has not been discussed. Mathematically, we can obtain the term responsible for THG in the equation if we analyse all three equations in (7) and use the term \(A_{1}^{2}\) instead of a small amplitude \(A_{2}\) in the third equation. This is rough mathematical reasoning. Proper mathematical expression is derived on the basis of the multi-scale method (see Appendix A) or using our approach proposed above, which results in (11).