On the Reduction of Evolution Equations in Extended Systems

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Abstract
A simple geometrical picture incorporates a number of important notions such as adiabatic elimination, removal of secularity, solvability condition, normal form and functional ansatz, i.e., notions associated with the reduction of evolution equations to simpler forms. Various theories of reduction are reinterpreted from this viewpoint. We begin with finite-dimensional dynamical systems, and then proceed to partial differential equations. In the latter case, we are especially concerned with the type of theories in which the reduced equations, too, are partial differential equations or related ones. Three existing theories of this kind, i.e., the Newell-Whitehead theory, phasedynamics theory and the Chapman-Enskog theory, are reexamined, and a universal structure underlying these theories is pointed out.

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