Stability of finite perturbations of the phase transition interface for one problem of water evaporation in a porous medium

We study global dynamics of phase transition evaporation interfaces in horizontally extended domains of porous layers where a water located over a vapor. The derivation of the model equation describing the secondary structures, which bifurcate from the ground state in a small neighborhood of the instability threshold in the case of a quasi-stationary approach to the description of the diffusion process, is presented. The resulting equation is reduced to the equation in the form of Kolmogorov-Petrovsky-Piskounov. The obtained approximate equation predicts the existence of stationary solutions in the full problem. To verify the obtained results, the numerical solution of the problem of the motion of the phase transition interface is performed using the original program code developed by the authors. The results of numerical simulation are used to verify the possibility of using stationary solutions obtained in the weakly nonlinear approximation to determine the scenario for the development of the initial localized finite amplitude perturbation. It is shown that the obtained approximate stationary solutions accurately predict the behavior of the perturbation in the vicinity of the turning point of the bifurcation diagram. A modification of the formulas describing an approximate stationary soliton-like solution is proposed in the case when the perturbation amplitude is comparable with the height of a low-permeable layer of a porous medium in which the phase transition interface is located. By numerical simulation it is shown that this modified approximate solution is in good agreement with the results of numerical calculation for the full problem.