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

反应扩散方程早在20世纪初就作为描述群体遗传学、生态学等的数学模型出现，并在数学界进行了定性研究。20世纪后期，科学界出现了重大突破，即非线性非平衡科学和反应扩散方程作为描述各种非线性非平衡现象的数学模型出现在物理、化学、生物学等自然科学领域。因此，反应扩散系统的研究不仅在数学领域，而且在广泛的自然科学领域都得到了推进。本研究结果从数学和应用数学的角度建立了反应扩散方程时空格局的分析方法，对非线性-非平衡现象进行了数学解释。例如：(1)构建“无限维动力系统的不变流形理论”来处理表现为耗散结构和自组织的模式动力学；(2)构建“瞬态模式动力学解析理论”来理解从简单模式到复杂模式的转变过程，这是一种典型的非线性非平衡现象；(3)建立“奇异极限理论”来理解非线性非平衡系统的复杂动态模式和平稳形式。这些结果使我们能够从数学上理解非线性非平衡系统的时空模式。

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.