Investigation of complex spatio-temporal regimes caused by the interaction of diffusion instabilities

Kuznetsov M.B.

P. N. Lebedev Physical Institute of the Russian Academy of Sciences, 53 Leninskiy Prospekt, Moscow, 119991, Russia

The discovery of new regimes in classical nonlinear models of autocatalytic reactions is of considerable fundamental interest. The results obtained in studies of model systems can be further applied in various branches of physics, chemistry, biology, as well as in remote areas. The most studied spatio-temporal structures in nonlinear nonequilibrium systems are stationary dissipative structures and autowaves. The former are formed as a result of Turing instability, the latter – as a result of wave instability, both of which are diffusion instabilities, leading to the excitation of a certain range of wave modes. Simultaneous fulfillment of the conditions of both instabilities can lead to more complex phenomena of various types. Some examples of them, described in literature, are waves on dissipative structures, modulated dissipative structures, modulated standing waves and self-completing solitary moving structures.

In this work we carried out an analytical and numerical study of the Brusselator model, which was extended by addition of a rapidly diffusing inhibitor, in a two-dimensional space in the area in parametric space where both indicated instabilities take place simultaneously. In particular, we considered a region with a very large ratio of diffusion coefficients of variables, the numerical study of the model in which is associated with significant computational difficulties. Methods for their overcoming are similar to those presented in [1] and are as well discussed within the scope of the given work. During the study of the system, a mixed regime was obtained, which consists of local undamped oscillations against the background of a stationary dissipative structure, that has not been previously described in the world literature. The conditions for the emergence of this regime are discussed.

The reported study was funded by RFBR according to the research project № 18-31-00411.


1. Kuznetsov M.B., Kolobov A.V., Polezhaev A.A. Pattern formation in a reaction-diffusion system of Fitzhugh-Nagumo type before the onset of subcritical Turing bifurcation // Physical Review E Vol. 95, No.5, Year 2017. Pp. 052208.

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