## Appendix A
Thus, the total pressure at the bubston surface \(p_{e}=p_{gas}+p_{Coul}\) can be presented as  
\[\theta^{-1}(I)\ =\ \frac{\langle v_{e}\rangle w_{a,e}}{R_{0}}\bigg{(}\frac{I}{I_ {0}}-1\bigg{)}. \tag{11}\]2  
The expansion rate \(u(t)\) can be found on the basis of Equations (A13), (A15) and (A17). Here, we should take into account that, according to Equation (A13), the pressure plays an essential role only at the time of the avalanche development \(t-\theta\times L_{\rm max}\), while \(\langle n_{e}\rangle\sim\langle n_{e}\rangle_{\rm max}\), \(x=x_{\rm max}\). This is why, when integrating in (A15), the quantity \(x\) (which depends on \(N_{e}\) only logarithmically) can be regarded as a constant, equal to \(x_{\rm max}\approx 8.4\). Thus, we arrive at  
\[\theta^{-1}(I)\ =\ \frac{\langle v_{e}\rangle w_{a,e}}{R_{0}}\bigg{(}\frac{I}{I_ {0}}-1\bigg{)}. \tag{11}\]3  
where  
\[\theta^{-1}(I)\ =\ \frac{\langle v_{e}\rangle w_{a,e}}{R_{0}}\bigg{(}\frac{I}{I_ {0}}-1\bigg{)}. \tag{11}\]4  
and parameter  
\[\theta^{-1}(I)\ =\ \frac{\langle v_{e}\rangle w_{a,e}}{R_{0}}\bigg{(}\frac{I}{I_ {0}}-1\bigg{)}. \tag{11}\]5  
This parameter stands for the contribution of the Coulomb repulsion of electrons in the process of bubston expansion. Substituting (111) in (A12) and taking into account that, in accordance with the definition of \(l_{thr}\) (see above), \(l_{0}=\theta(l_{thr})\times L_{\rm max}\), we obtain for the dependence \(t_{0}(\zeta)\)  
\[\theta^{-1}(I)\ =\ \frac{\langle v_{e}\rangle w_{a,e}}{R_{0}}\bigg{(}\frac{I}{I_ {0}}-1\bigg{)}. \tag{11}\]6  
The threshold intensity \(l_{thr}\) is determined from Equation (A3) after substituting either \(\theta(l_{thr})=t_{0}/L_{\rm max}\) at laser pulse duration \(\tau>t_{0}\), or \(\theta(l_{thr})\)= \(\tau/L_{\rm max}\) at \(\tau<t_{0}\). Thus, we have for the threshold intensity  
\[\theta^{-1}(I)\ =\ \frac{\langle v_{e}\rangle w_{a,e}}{R_{0}}\bigg{(}\frac{I}{I_ {0}}-1\bigg{)}. \tag{11}\]7  
In the case of \(w_{a,e}<<10^{-3}/\zeta^{1/2}\), the second term in (115) essentially exceeds unity, and we obtain  
\[\theta^{-1}(I)\ =\ \frac{\langle v_{e}\rangle w_{a,e}}{R_{0}}\bigg{(}\frac{I}{I_ {0}}-1\bigg{)}. \tag{11}\]8  
The case of \(\tau>t_{0}\) is more interesting. In this case, according to (115), we have  
\[\theta^{-1}(I)\ =\ \frac{\langle v_{e}\rangle w_{a,e}}{R_{0}}\bigg{(}\frac{I}{I_ {0}}-1\bigg{)}. \tag{11}\]9  
It is evident from Equations (115)-(117) that the threshold intensity \(I_{thr}\) does not depend on characteristics of molecular gas inside bubstons and is completely controlled by parameters of liquid, such as the energy of surface ionization \(\Delta\), the probability of electron adhesion \(w_{a,e}\), liquid density \(\rho\) and radius \(R_{0}\) of bubstons. In the case of \(\tau>t_{0}\), dependence on \(R_{0}\) is realized only due to the fact that \(u_{0}\sim(R_{0})^{-1/2}\) (see Equation (112)), and due to the dependence \(\mu(R_{0})\), see (113). It is clear that \(I_{thr}\) increases slowly while \(R_{0}\) drops. As shown in [29], \(R_{0}\) depends on the dielectric constant of the liquid, its temperature, surface tension coefficient, the adsorption energy of the surface ions of the dissolved electrolyte and its concentration. The characteristics of dissolved gas manifest themselves in the optical breakdown process only as a result of the ability of gas to be dissolved in liquid. These remarkable peculiarities allow us to speak about the optical breakdown of liquid but not about the breakdown of gas. Let us numerically estimate the time \(t_{0}\) and threshold intensity \(I_{thr}\). They depend on the extent (the measure) of inertial retention \(\zeta\). Assuming \(\zeta=0.1\), and bearing in mind that \(w_{a,e}<<10^{-3}\) and \(L_{\max}=11.8\), \(\mu=31\), we obtain \(t_{0}\approx 0.1\times 10^{-9}\) s, and the threshold intensity for \(\tau>t_{0}\), \(\lambda=1\)\(\mu\)m (\(n_{\text{ccr}}=3\times 10^{19}\) cm\({}^{-3}\)) and \(\Delta=6\) eV due to the fact (A24) is equal to \(I_{thr}=9.6\times 10^{6}\) W/cm\({}^{2}\).