tokens: determined by the resonant conditions ( 52 ) and ( 53 ) are of order unity (see ( 42 ). In the opposite short-length limit | k | ≫ 1, the eigenvalue relation ( 30 ) yields: , k, n ≠ 0, n = 1, 2, ..., i.e. the linear growthrates are also small in the radially short-length limit, , e.g. | k | ≫ 1 for n ∼ 1 and | q | ∼ 1. In the latter case the eigenfrequency is diminished together with the growthrate. Now, let us discuss the three-wave resonant interaction in the presence of the short-length modes. First, note that a resonant triad cannot be formed by three short-length modes because of the frequency resonant condition ( 53 ) that is degenerated with all three small frequencies which can be included in the mismatch value, δω. Second, the resonant condition ( 53 ) is also degenerated for two short-length modes with | k | ≫ 1 interacting with the single long-length mode with | k | ≪ 1, since all they have zero frequencies. Finally, evidently that the radial wavenumber resonant condition ( 52 ) cannot be satisfied by the single short-length term with | k | ≪ 1 and two long-length terms with | k | ≫ 1 (at axial wavenumber n ∼ 1). Thus, resonant triads cannot be formed by interacting short- and long-length modes. In particular, this means that energy transfer from very short- to very long-length modes and vice versa cannot be realized with three-wave resonant interaction. Rigorous analytical solutions of the reduced weakly non-linear VSI problem discovered in the present study can serve as a test for numerical modelling of the full non-linear hydrodynamic problem, since the intermediate non-linear stage of VSI should be inevitably crossed before saturation. Predictions of the reduced VSI model allow to identify and discover physical mechanisms that are driving the VSI and its transition to turbulence, as well as identify tuning parameters of the physical and numerical models of the full problem. Figure 1. Amplitudes α ( j ) , j = 1, 2, 3 versus slow time θ for the triad of class I. The two-mode regime of the resonant triad evolution. n (1) = 1, n (2) = 1, n (3) = 2 (see Table 1 ); R = +1; S = +1; , φ 0 = π/4, δω = +1. The scaled resonant frequencies and radial wavenumbers are in Table 1 with and , , ( j = 1, 2). Dashed and solid lines are solutions of the linear and non-linear systems ( 71 )–( 74 ), respectively, for triad amplitudes. Figure 2. Amplitudes α ( j ) , j = 1, 2, 3 versus slow time θ for the triad of class II. The two-mode regime of the resonant triad evolution as in Fig. 1 , but for S = −1. Figure 3. Triad amplitudes α ( j ) , j = 1, 2, 3 versus slow time θ for the triad of class III. The three-mode regime of bounded oscillations of all interacting modes as in Fig. 1 , but for R = −1. Figure 4. Amplitudes α ( j ) versus slow time θ for the triad of class I as in Fig. 1 ( S = +1, R = +1), but at small initial amplitudes . Figure 5. Amplitudes α ( j ) versus slow time θ for the triad of class II as in Fig. 2 ( S = −1, R = +1), but
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