Mathematical Theory of Nonlinear-Non-equilibrium Reaction-Diffusion Systems

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Reaction diffusion equations already appeared as mathematics models which described population genetics, ecology and so on from the early 20^<th> century, and the qualitative study was performed in the world of mathematics. In the late 20^<th> century, there was a great breakthrough in sciences, that is, nonlinear non-equilibrium science and reaction diffusion equations appeared in natural ecienc such as physics, chemistry, biology and other fields, as mathematics models describing various nonlinear non-equilibrium phenomena. Thus, the study of reaction diffusion system has been pushed forward not only in mathematics but also in widely natural sciences. The result in this study was able to establish the analytical technique of spatio-temporal patterns arising in reaction diffusion equations from viewpoints of mathematics and applied mathematics for mathematical elucidation of the nonlinear-non-equilibrium phenomenon. As examples, there are (1) the construction of "the invariant manifold theory of infinite dimension dynamical systems" to handle pattern dynamics appearing as dissipative structure and self-organization, (2) the construction of "analytical theory of transient pattern dynamics" to understand the transition process from a simple pattern to a complex one, which is a typical nonlinear-non-equilibrium phenomenon, and (3) the construction of "the singular limit theory" to understand complex dynamic patterns and stationary forms in nonlinear non-equilibrium systems. These results enable us to mathematically understand spatio-temporal patterns in nonlinear non-equilibrium systems.