## Appendix A
Thus, the solution to Equation (A25) has the form  
\[\langle Y_{e}\rangle\ =\ T_{e}\ =\ \int_{0}^{\Delta}Y_{e}f(Y_{e})dY_{e}\ =\ \Delta/3,\]2 (A27)  
where \(\eta\ =\ u^{2}(R_{0})/\left[2(1+2\mu)u_{0}^{2}\right]\). The maximum velocity \(u(R_{0})\), according to Equation (A18), is \(u(R_{0})\ =\ (1+\mu)u_{0}^{2}\theta(I_{thr})/R_{0}\), and therefore, taking Equations (A19) and (A20) into account, we obtain \(\eta\approx\zeta/2<1\). At a sufficiently large expansion of bubstons in the cluster, they merge together into one large bubble; that is, the coalescence is completed. As a quantitative criterion of completing the coalescence, we impose the condition that the maximum radius of the expansion is \(R_{1}=l/2\), where \(l\) is the distance between the centers of the bubstons in the cluster. According to [29], \(l\approx 2a_{i}\), where \(a_{i}=\sqrt{\epsilon T/(8\pi e^{2}n_{thr})}\) is the Debye screening radius in aqueous electrolyte solution, see., e.g., [50]; here, \(T\) is temperature, and \(\varepsilon\) is the dielectric constant. According to the results of our preliminary measurements, the pH value for the water samples is pH \(\approx 5.5\). Such a value of pH is due to the dissolution of CO\({}_{2}\) in water, followed by its hydrolysis and the formation of carbonic acid H\({}_{2}\)CO\({}_{3}\). After estimating \(n_{i0}=3\times 10^{15}\) cm\({}^{-3}\), we obtain \(a_{i}=2\times 10^{-5}\) cm. Under these conditions (these can be regarded as typical ones), the criterion of the coalescence has the form: \(R_{1}/R_{0}=a_{i}/R_{0}\approx 2\). As follows from Equation (A27), the time duration necessary for such expansion (we shall refer to it as the coalescence time \(t_{coal}\)) is equal to  
\[\langle Y_{e}\rangle\ =\ T_{e}\ =\ \int_{0}^{\Delta}Y_{e}f(Y_{e})dY_{e}\ =\ \Delta/3,\]3  
Since \(0<\eta<1\), this integral does not diverge in the vicinity of \(z=1\). Assuming \(\eta\approx 0.01\), this integral at \(a_{i}/R_{0}\approx 2\) is equal to \(3.5\). Thus, for the coalescence time we have an estimate \(t_{coal}=10^{-10}\) s \(\approx t_{0}\); i.e., the stimulated optical coalescence (SOC) of a bubston cluster occurs at the stage of inertial retention. Thus, it follows from here that if the pulse duration \(\tau>t_{coal}\) (it means that \(\tau>t_{0}=0.1\times 10^{-9}\) s) and its intensity \(I>I_{thr}=9.6\times 10^{6}\) W/cm\({}^{2}\), such a pulse should result in the SOC effect. Since the walls of the macroscopic bubble formed as a result of SOC remain positively charged, the free electrons inside the bubble must stick to these walls; that is, the effective charge of such a bubble must be close to zero. However, due to the presence of anions adsorbed on the inner surface of the bubston cluster, the charge of a macroscopic bubble cannot be exactly equal to zero. It can be shown, see Ref. [29], that despite the presence of an adsorbed charge, such a macroscopic bubble will not be ion-stabilized; i.e., such a bubble will rapidly collapse according to the kinetics described in the work [44]. Summarizing, we can say that stimulated optical coalescence sets in at laser intensities of about \(10^{7}\) W/cm\({}^{2}\). If the intensity of laser radiation significantly exceeds this threshold value, and if the laser pulse has not yet finished by the time the coalescence process is completed, the electrons continue to swing by the electromagnetic wave and collide with heavy particles (\"fragments\" of the liquid walls of the bubston cluster). However, these collisions are no longer elastic, the electrons lose their energy in these collisions, which leads to the effect of bremsstrahlung and an optical breakdown flash. It is the presence of a flash of breakdown that is the main feature of breakdown in classical experimental works on this subject, see review [13].