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Slowly Varying Waves and Shock Structures in Reaction‐Diffusion Equations
Author(s) -
Howard L. N.,
Kopell N.
Publication year - 1977
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1002/sapm197756295
Subject(s) - shock wave , reaction–diffusion system , plane (geometry) , simple (philosophy) , diffusion , mathematical analysis , initial value problem , ideal gas , space (punctuation) , shock (circulatory) , basis (linear algebra) , plane wave , mathematics , classical mechanics , physics , mechanics , geometry , thermodynamics , quantum mechanics , computer science , medicine , operating system , philosophy , epistemology
Solutions of reaction‐diffusion equations which, locally in space and time, are close to plane wave trains are investigated. Apart from certain exceptional (and localized) regions, such solutions are approximately described by solutions of a Hamilton‐Jacobi equation obtained from the properties of ideal plane waves. This provides a fairly simple description of many features of the initial value problem, but implies that transition layers in which the approximate description is invalid will generally arise. The structure of these layers, which are mathematically analogous to gas‐dynamic shocks, is investigated on the basis of the full equations.