Autowaves are self-supporting non-linear waves in active media (i.e. those that provide distributed energy sources). The term is generally used in processes where the waves carry relatively low energy, which is necessary for synchronization or switching the active medium.
Introduction
Relevance and significance
In 1980, the Soviet scientists G.R. Ivanitsky, V.I. Krinsky, A.N. Zaikin, A.M. Zhabotinsky, B.P. Belousov became winners of the highest state award of the USSR, Lenin Prize "for the discovery of a new class of autowave processes and the study of them in disturbance of stability of the distributed excitable systems."
A brief history of autowave researches
The first who studied actively the self-oscillations was Academician AA Andronov, and the term "auto-oscillations" in Russian terminology was introduced by AA Andronov in 1928. His followers from Lobachevsky University further contributed greatly to the development of autowave theory.
The simplest autowave equations describing combustion processes have been studied by A.N. Kolmogorov, I.E. Petrovsky, N.S. Piskunov in 1937., as well as by Ya.B. Zel'dovich и D.A. Frank-Kamenetsky in 1938.
The classical axiomatic model with autowaves in myocardium was published in 1946 by Norbert Wiener and Arturo Rosenblueth.
During 1970-80, major efforts to study autowaves were concentrated in the Institute of Biological Physics of the USSR Academy of Sciences, located in the suburban town Pushchino, near Moscow. It was here, under the guidance of V.I.Krinsky, such world-famous now experts in the field of the autowave researches as A.V.Panfilov, I.R.Efimov, R.R.Aliev, K.I. Agladze, O.A.Mornev, M.A.Tsyganov were educated and trained. V.V.Biktashev, Yu.E. Elkin, A.V. Moskalenko gained their experience with the autowave theory also in Pushchino, in the neighboring Institute of Mathematical Problems of Biology, under the guidance of E.E.Shnoll.
The term "autowaves" was proposed, probably, on the analogy of previously "auto-oscillations".
Almost immediately after the Dissolution of the Soviet Union, many of these Russian scientists left their native country for working in foreign institutions, where they still continue their studies of autowaves. In particular, E.R.Efimov is developing the theory of virtual electrode, which describes some effects occurring during defibrillation.
Among other notable scientists, who are engaged in these investigation, there are A.N. Zaikin and E.E.Shnoll (autowaves and bifurcation memory in the blood coagulation system); A.Yu. Loskutov (general autowave theory as well as dynamic chaos in autowaves); V.G. Yakhno (general autowave theory as well as connections between autowaves and process of thinking); K.I. Agladze (autowaves in chemical media); V.N.Biktashev (general autowave theory as well as different sorts of autowave drift); O.A.Mornev (general autowave theory); M.A.Tsyganov (the role of autowave in population dynamics); Yu.E. Elkin, A.V. Moskalenko, (bifurcation memory in a model of cardiac tissue).
A huge role in the study of autowave models of cardiac tissue belongs to Denis Noble and members of his team from the University of Oxford.
The basic definitions
One of the first definitions of autowaves was as follows:
Unlike linear waves — such as sound waves, electromagnetic waves and other, which are inherent in conservative systems and mathematically described by linear second order hyperbolic equations (wave equations), — dynamics of an autowave in terms of differential equations can be described by parabolic equation with nonlinear free member of a special form.
The concrete form of the free member is extremely important, because:
Commonly, have the form of -shaped dependence on . In this sense, the system of equations, known as the Aliev–Panfilov model, is a very exotic example, because has in it a very complex form of two intersecting parabolas, besides more crossed with two straight lines, resulting in a more pronounced nonlinear properties of this model.
Autowaves is an example of a self-sustaining wave process in extensive nonlinear systems containing distributed energy sources. It is correct for simple autowaves, that period, wavelength, propagation speed, amplitude, and some other characteristics of an autowave are determined solely by local properties of the medium. However, in the 21st century, researchers began to discover a growing number of examples of self-wave solutions when the "classical" principle is violated.
(See also general information in literature, for example, in).