Affiliation:
1. Department of Mathematics, University of Surrey, Guildford GU2 7XH, UK
Abstract
A new perspective on the emergence of the Korteweg–de Vries (KdV) equation is presented. The conventional view is that the KdV equation arises as a model when the dispersion relation of the linearization of some system of partial differential equations has the appropriate form, and the nonlinearity is quadratic. The assumptions of this paper imply that the usual spectral and nonlinearity assumptions for the derivation of the KdV equation are met. In addition to a new mechanism, the theory shows that the emergence of the KdV equation always takes a universal form, where the coefficients in the KdV equation are completely determined from the properties of the background state—even an apparently trivial background state. Moreover, the mechanism for the emergence of the KdV equation is simplified, reducing it to a single condition. Well-known examples, such as the KdV equation in shallow-water hydrodynamics and in the emergence of dark solitary waves, are predicted by the new theory for emergence of KdV.
Subject
General Physics and Astronomy,General Engineering,General Mathematics
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