## 2 Problem Statement  
The three-wave interaction process with multiple frequencies in a medium with quadratic nonlinear responses can be described by the set of nonlinear Schrodinger equations [40]:  
\[\frac{\partial A_{1}}{\partial z}+iD_{1}\frac{\partial^{2}A_{1}}{ \partial t^{2}}+i\Big{(}\gamma_{12}A_{1}^{*}A_{2}e^{-i\Delta_{21}kz}+\gamma_{2 3}A_{2}^{*}A_{3}e^{-i(\Delta_{31}k-\Delta_{21}k)z}\Big{)}=0,\] \[\frac{\partial A_{2}}{\partial z}+\gamma_{21}\frac{\partial A_{2} }{\partial t}+iD_{2}\frac{\partial^{2}A_{2}}{\partial t^{2}}+i\Big{(}\gamma_{ 11}A_{1}^{2}e^{i\Delta_{21}kz}+2\gamma_{13}A_{1}^{*}A_{3}e^{-i(\Delta_{31}k- \Delta_{21}k)z}\Big{)}=0, \tag{1}\] \[\frac{\partial A_{3}}{\partial z}+\gamma_{31}\frac{\partial A_{3} }{\partial t}+iD_{3}\frac{\partial^{2}A_{3}}{\partial t^{2}}+3i\gamma_{21}A_{1 }A_{2}e^{i(\Delta_{31}k-\Delta_{21}k)z}=0,\,0<z\leq L_{z}.\,0<t<L_{t}.\]  
The initial condition and boundary conditions (BCs) are the following:  
\[A_{1}(0,t)=A_{10}(t),\,A_{2}(0,t)=A_{20}(t),\,A_{3}(0,t)=A_{30}(t), \,t\in[0,L_{t}], \tag{2}\] \[A_{1}(z,0)=A_{2}(z,0)=A_{3}(z,0)=A_{1}(z,L_{t})=A_{2}(z,L_{t})=A _{3}(z,L_{t})=0,\,z\in[0,L_{z}],\]  
which correspond to the absence of the laser radiation of all optical waves out of a certain time interval. Here, \(A_{j}\) are dimensionless complex amplitudes of LFW, IFW, and HFW. Coefficients \(\gamma_{jk}\)--of nonlinear coupling of interacting waves--characterize a second-order nonlinear response at the corresponding frequencies. Below, for simplicity, we omit a dependence of this coefficient on the optical frequency. Therefore, these coefficients are equal to the same value: \(\gamma_{jk}=\gamma\). We emphasize that this is not a strong restriction for our analysis. Parameters \(\Delta_{21}k\) and \(\Delta_{31}k\) denote dimensionless phases mismatching between IFW and LFW, and HFW and LFW, respectively. Coefficients \(\
u_{21}\) and \(\
u_{31}\) are dimensionless mismatching of the group velocities between pulses of IFW and LFW, and HFW and LFW, respectively. Parameters \(D_{1},\ D_{2}\) and \(D_{3}\) are dimensionless coefficients of the second order dispersion (SOD) at the corresponding frequencies, characterizing GVD of the propagating pulses. \(t\) and \(z\) are dimensionless time and space coordinates, correspondingly. Let us note that the diffraction is not taken into account in Equation (1). This is valid if the radii of the incident beams are large enough to satisfy the following conditions: the diffraction length is greater than the crystal's length and the nonlinear length, which is defined by the incident beam's maximal intensity. Such conditions often occur in physical experiments (see, for example, [41; 42; 43]). Writing Equation (1), we neglected the cubic nonlinear response of a crystal. It can be done if the maximal intensity of the incident pulse is less than the certain value, depending on the used crystal [40]. The dimensionless variables and parameters are expressed through the physical ones as follows:  
\[ \begin{split}&\
u_{j1}=\left(\frac{\partial\bar{k}}{\partial \bar{\omega}}|_{\bar{\omega}}-\frac{\partial\bar{k}}{\partial\bar{\omega}}|_{ \bar{\omega}_{1}}\right)\frac{Z_{n}}{\tau_{p}},\ j=2,\,3,\,D_{j}=-\frac{1}{2} \frac{\partial^{2}\bar{k}}{\partial\bar{\omega}^{2}}|_{\bar{\omega}}\frac{Z _{n}}{\tau_{p}^{2}},\ A_{j}=\frac{\bar{A}_{l}}{A_{0}},\ j=1,2,3,\\ &\gamma=\frac{2\pi\chi^{(2)}\bar{k}_{1}A_{01}}{n^{2}}Z_{n},\ \Delta_{21}k=\Delta_{21}\bar{k}Z_{n},\ \Delta_{31}k=\Delta_{31}\bar{k}Z_{n}.\end{split}  \tag{3}\]  
where \(\tau_{p}\) is the incident pulse duration of the LFW (\(\bar{\omega}_{1}=\bar{\omega}\)), which is chosen to be equal to 100 fs, if we do not specify otherwise. Time coordinates are measured in \(\tau_{p}\). Parameter \(Z_{n}\) is a normalization length chosen equal 1 mm. The propagation distance of the optical pulse is measured in this units. Coefficients \(\chi^{(2)}(\bar{\omega}_{j},\bar{\omega}_{l})\) are the quadratic susceptibilities of a medium at the frequencies \(\bar{\omega}_{j},\bar{\omega}_{l}\).