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A note on an asymptotic solution of the cylindrical Korteweg-de Vries equation

Lookup NU author(s): Professor Robin Johnson


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The solitary wave solution of the cylindrical KdV equation is not generated by the typical initial profile often used, for example, in modern water-wave studies, namely, the familiar sech2 profile. One reason is that this solution carries zero mass and therefore cannot, alone, describe the evolution of a wave of elevation. This paper describes an alternative approach; this is an asymptotic solution of the cylindrical KdV equation, given a sech2 initial profile, based on an appropriate small parameter (ε=1/initial radius, in non-dimensional variables). In terms of the limiting process ε→0, the various components of the resulting wave are described: the leading wave (a pulse), the trailing shelf and the oscillatory transition back to undisturbed conditions. The solution that is obtained takes a very simple form (and is therefore likely to be useful in more complicated scenarios), it satisfies mass conservation and each of the three elements of the solution satisfy the matching principle. The resulting evolution of the leading wave, and the complete structure of the asymptotic solution, are compared with numerical solutions of the cylindrical KdV equation; the agreement is exceptionally good, for both outward and inward propagation. © 1999 Elsevier Science B.V.

Publication metadata

Author(s): Johnson RS

Publication type: Article

Publication status: Published

Journal: Wave Motion

Year: 1999

Volume: 30

Issue: 1

Pages: 1-16

Print publication date: 03/12/1998

ISSN (print): 0165-2125

Publisher: Elsevier BV


DOI: 10.1016/S0165-2125(98)00052-3


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