Glasgow Mathematical Journal

Glasgow Mathematical Journal (2001), 43:65-90 Cambridge University Press
Copyright © 2001 Glasgow Mathematical Journal Trust

Additional volume of selected papers from a Conference on Integrable Systems, Islay 1999

The N-soliton solution of a generalised Vakhnenko equation

A. J.  Morrison  a1 and E. J.  Parkes  a1
a1 Department of Mathematics, University of Strathclyde, Glasgow G1 1XH, UK e-mail: and


The N-soliton solution of a generalised Vakhnenko equation is found, where N is an arbitrary positive integer. The solution, which is obtained by using a blend of transformations of the independent variables and Hirota's method, is expressed in terms of a Moloney & Hodnett (1989) type decomposition. Different types of soliton are possible, namely loops, humps or cusps. Details of the different types of interactions between solitons, including resonant soliton interactions, are discussed in detail for the case N=2. A proof of the ‘N-soliton condition’ is given in the Appendix.

(Received February 23 2000)