## 1 Introduction
The appearance of a spatio-temporal soliton was experimentally demonstrated in [39]. A comprehensive review of cascading \(\chi^{(2)}\) solitons and their practical applications can be found in [40]. Recently, a special type of solitons--Akhmediev breathers--in a medium with quadratic susceptibility accounting for the dispersion of a pulse or optical beam diffraction was studied in [41] and [42], respectively. One of the cascading SHG applications is soliton compression. By choosing an appropriate sign of phase mismatching, pulse compression at the FF can be reached. Soliton compression at a wavelength of 1030 nm at its propagating in a LiNbO\({}_{3}\) crystal with a length equal to 10 cm has been discussed in [43]. Few-cycle pulse's formation at the wavelength of 1250 nm via soliton compression of the pulse with a duration of 100 fs and a power density of 30 GW/cm\({}^{2}\) to the pulse with a duration of 6 fs using a defocusing quadratic medium was presented in [44]. In [45], such a pulse compression was observed in a BBO crystal with length of about 3  cm. Limits to pulse compression in cascaded quadratic soliton compressors were discussed theoretically in [46] using the nonlocal temporal response of a medium. Nonlinear phase shift [47], induced by quadratic cascaded processes, can also be used for compensation of self-phase modulation, caused by Kerr nonlinearity, as was shown numerically and experimentally in [48]. Spectral broadening, induced by intracavity self-phase modulation, in a Nd:YLF regenerative amplifier was overcome [49] with correct choice of phase mismatching between FW and SH waves. Cascaded SHG may lead to self-focusing or defocusing of the beam at the FF, as was shown experimentally in [50], and then theoretically in [51]. In the axial-symmetric case, the beam intensity may increase about 70 times [52]. Another important application of the cascading SHG is the suppression of random fluctuations of both duration and maximum power density of the sub-pulses generated by laser operating in the so-called free-generation mode [53; 54; 55; 56]. Only about 2-3% of the incident energy is lost when suppressing those fluctuations because of the frequency conversion to SH. Thus, the phenomena inherent to the Kerr effect in a medium with quadratic susceptibility at the cascaded SHG (for brevity, we will name it as induced cubic nonlinearity or susceptibility) was observed in many papers. Herewith, those authors who studied various manifestations of the self-focusing effect accounted only for the phase self-modulation of the beam or pulse at the FF. As mentioned above, THG can be provided via the cascaded processes (SHG and then SFG) at the phase matching between FW and TH. However, in this case, there is a large phase-mismatching between FW and SH as well as between SH and TH. Note that this scheme of the frequency conversion refers to cascading processes similar to the cascading SHG discussed above. One of the first papers accounting for a contribution of the cascaded THG at frequency tripling due to cubic susceptibility, was the paper [25], in which a measurement of the cubic susceptibility was made. This contribution was estimated in the framework of the energy non-depletion approximation for waves with FF and SH and without accounting for phase self- and cross-modulation of interacting waves. A few studies should be mentioned that refer to cascaded THG in the BBO crystal. A BBO crystal with a length of 7 mm was used for the frequency tripling of the pulse with a duration of 5 ps and a power density varying between 1 and 50 GW/cm\({}^{2}\)[26]. Absolute value of phase mismatching between SH and FW varies between 187 and 600 mm\({}^{-1}\). The reached frequency conversion efficiency was about 1%. For a longer pulse (45 ps) with a smaller incident power density (1.5 GW/cm\({}^{2}\)), the frequency conversion efficiency was only 0.007% [27].