On occurrence conditions of regular structures in condensed media under external radiation

Background. The authors have established a method for calculating long-distance asymptotic solutions of nonlinear diffusion equations and their systems. The main purpose of this work is to obtain the growth criteria for space-periodic disturbances and formation of regular long-wave structures in conditions of a space- and time-constant external source of defects in non-linear diffusion systems. Materials and methods. The proposed method for calculating long-wave asymptotic solutions is based on one of the variants of the method of multiscale expansions with correction of convergence of series by resonances elimination. The ratio of the diffusion transfer rate to the relaxation rate of local concentration fluctuations is considered as a series expansion parameter. The approach makes it possible to link the diffusion parameters for relaxation process amplitudes with the characteristics of external components of the environment. As a result, it is possible to obtain the growth criteria for long-wave periodic components of the diffusion process of environmental components transfer, linking them with the value of external sources. The method applies to the case of multi-media. The procedure of deriving the equations for the transfer process amplitudes was carried out for a general class of nonlinear diffusion equations in a general form by nonlinear diffusion coefficients and non-linear springs. The authors investigated the importance of nonlinear relaxation in formation of periodic structures in asymptotics of long periods of time. Results. On the basis of the proposed approach the authors obtained the criteria of asymptotic growth of long-wave perturbations for the total functional dependence of the diffusion coefficient on concentration of defects and small nonlinear sources in the right part of the equation. Similar criteria were obtained for systems of diffusion equations that describe the migration process in multi-component media. The researchers have discovered an effect of suppressing the growth of long-wave asymptotic space-periodic fluctuations in the case of a quadratic nonlinear source in the zero order perturbation. Conclusions. The obtained growth criteria for spatially periodic long-wave perturbations provide an opportunity to explain occurrence of regular structures in environments, subjected to constant external radiation. This approach can be used for both problems of laser irradiation of materials and material flows for irradiating neutrons or other types of radiation.