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Advances in Mathematical Physics
Volume 2016 (2016), Article ID 2916582, 39 pages
http://dx.doi.org/10.1155/2016/2916582
Review Article

Approach in Theory of Nonlinear Evolution Equations: The Vakhnenko-Parkes Equation

1Institute of Geophysics, National Academy of Sciences of Ukraine, Kyiv 01054, Ukraine
2Department of Mathematics & Statistics, University of Strathclyde, Glasgow G1 1XH, UK

Received 13 September 2015; Accepted 12 November 2015

Academic Editor: Andrei D. Mironov

Copyright © 2016 V. O. Vakhnenko and E. J. Parkes. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A variety of methods for examining the properties and solutions of nonlinear evolution equations are explored by using the Vakhnenko equation (VE) as an example. The VE, which arises in modelling the propagation of high-frequency waves in a relaxing medium, has periodic and solitary traveling wave solutions some of which are loop-like in nature. The VE can be written in an alternative form, known as the Vakhnenko-Parkes equation (VPE), by a change of independent variables. The VPE has an -soliton solution which is discussed in detail. Individual solitons are hump-like in nature whereas the corresponding solution to the VE comprises -loop-like solitons. Aspects of the inverse scattering transform (IST) method, as applied originally to the KdV equation, are used to find one- and two-soliton solutions to the VPE even though the VPE’s spectral equation is third-order and not second-order. A Bäcklund transformation for the VPE is used to construct conservation laws. The standard IST method for third-order spectral problems is used to investigate solutions corresponding to bound states of the spectrum and to a continuous spectrum. This leads to -soliton solutions and -mode periodic solutions, respectively. Interactions between these types of solutions are investigated.