## 2 Problem Statement
Starting from [PERSON]'s equations, one can derive the following set of dimensionless Schrodinger equations with respect to slowly varying envelopes of the wave packets:  
\[\frac{\partial A_{1}}{\partial z}+iD_{1}\frac{\partial^{2}A_{1}} {\partial t^{2}}+i\Big{(}\gamma_{12}A_{1}^{*}A_{2}e^{-i\Delta_{21}kz}+\gamma_{ 23}A_{2}^{*}A_{3}e^{-i(\Delta_{31}k-\Delta_{21}k)z}\Big{)}=0,\] \[\frac{\partial A_{2}}{\partial z}+\
u_{21}\frac{\partial A_{2}}{ \partial t}+iD_{2}\frac{\partial^{2}A_{2}}{\partial t^{2}}+i\Big{(}\gamma_{11 }A_{1}^{2}e^{i\Delta_{21}kz}+2\gamma_{13}A_{1}^{*}A_{3}e^{-i(\Delta_{31}k- \Delta_{21}k)z}\Big{)}=0, \tag{1}\] \[\frac{\partial A_{3}}{\partial z}+\
u_{31}\frac{\partial A_{3}}{ \partial t}+iD_{3}\frac{\partial^{2}A_{3}}{\partial t^{2}}+3i\gamma_{21}A_{1} A_{2}e^{i(\Delta_{31}k-\Delta_{21}k)z}=0,\,0<z\leq L_{z},0<t<L_{t}\]  
which describes the optical frequency tripling in a medium with combined nonlinear response in the framework of the plane wave approximation (without taking into account a laser beam diffraction and absorption of the crystal). Here, \(A_{1}\), \(A_{2}\) and \(A_{3}\) are the complex amplitudes of waves with the FF (\(\omega\)), the doubled frequency (\(2\omega\)), and the tripled frequency (\(3\omega\)), respectively, normalized at \(A_{01}=\sqrt{I_{01}}\) (a square root of the incident pulse maximum intensity at the FF). Parameter \(\gamma_{jk}\) is the interacting waves' coupling coefficient due to the quadratic nonlinear response at the corresponding frequencies. In general case, this coefficient may be different for various generation processes. However, for simplicity, we choose the same value (\(\gamma_{jk}=\gamma\)) for all these processes involved in the set of equations (1) Parameter \(\Delta_{21}k=k_{2}-2k_{1}\) characterizes the phase mismatching on the length \(Z_{n}=1\) mm, in which units a longitudinal coordinate \(z\) is measured. \(k_{1}\) and \(k_{2}\) denote dimensionless wave numbers for waves with the FF and doubled frequency. Parameter \(\Delta_{31}k=k_{3}-3k_{1}\) characterizes the phase mismatching on \(Z_{n}\) at THG. The symbol \(k_{3}\) denotes dimensionless TH wave number. Coordinate \(t\) is a time, measured in units of the pulse duration \(\tau_{P}\) at the FF. Parameters \(\
u_{21}\) and \(\
u_{31}\) characterize a dimensionless GVM between SH and FW, and TH and FW, respectively. Parameters \(D_{j},j=1,2,3\) characterize dimensionless second-order dispersion (SOD) of the corresponding wave packet. Parameters \(L_{t}\) and \(L_{z}\) define the time interval, during which the laser pulse interaction and the maximum distance of the pulse propagation are analysed, respectively. The initial distributions of the complex amplitudes are the following:  
\[A_{1}(0,t)=A_{10}(t),\,A_{2}(0,t)=0,\,A_{3}(0,t)=0,\,t\in[0,L_{t}], \tag{2}\]  
and corresponds to a finite distribution of the laser pulses. Let us discuss the terms in the set of equations (1). The term \(\gamma A_{1}^{2}e^{i\Delta_{21}kz}\) (\(\chi^{(2)}(2\omega;\,\omega,\omega)\)) in the second equation corresponds to the generation of a wave with doubled frequency for the process (\(\omega+\omega\to 2\omega\)). The term \(\gamma A_{1}^{*}A_{2}e^{-i\Delta_{21}kz}\) in the first equation is responsible for the opposite process (\(2\omega\rightarrow\omega+\omega\)). A process of the THG via the SFG is described by the term \(\gamma A_{1}A_{2}e^{i(\Delta_{31}k-\Delta_{21}k)z}\) (\(\chi^{(2)}(3\omega;\,\omega,2\omega)\)) in the third equation. In turn, the term \(\gamma A_{2}^{*}A_{3}e^{-i(\Delta_{31}k-\Delta_{21}k)z}\) in the first equation and the term \(\gamma A_{1}^{*}A_{3}e^{-i(\Delta_{31}k-\Delta_{21}k)z}\) in the second equation correspond to the reverse processes (\(3\omega\to 2\omega+\omega\)).