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

The solitary wave solution of the cylindrical KdV equation is not generated by the typical initial profile often used, for example, in modem water-wave studies, namely, the familiar sech(2) 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 sech(2) initial profile, based on an appropriate small parameter (epsilon=1/initial radius, in non-dimensional variables). In terms of the limiting process epsilon --> 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. (C) 1999 Elsevier Science B.V. All rights reserved.