RENORMALIZATION-GROUP AND SINGULAR PERTURBATIONS - MULTIPLE SCALES, BOUNDARY-LAYERS, AND REDUCTIVE PERTURBATION-THEORY

Perturbative renormalization group theory is developed as a unified to ol for global asymptotic analysis. With numerous examples, we illustra te its application to ordinary differential equation problems involvin g multiple scales, boundary layers with technically difficult asymptot ic matching, and WKB analysis. In contrast to conventional methods, th e renormalization group approach requires neither nd hoc assumptions a bout the structure of perturbation series nor the use of asymptotic ma tching. Our renormalization group approach provides approximate soluti ons which an practically superior to those obtained conventionally, al though the latter can be reproduced, if desired, by appropriate expans ion of the renormalization group approximant. We show that the renorma lization group equation may be interpreted as an amplitude equation, a nd from this point of view develop reductive perturbation theory for p artial differential equations describing spatially extended Systems ne ar bifurcation points, deriving both amplitude equations and the cente r manifold.