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 anN-soliton solution which is discussed in detail. Individual solitons are hump-like in nature whereas the corresponding solution to the VE comprisesN-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 toN-soliton solutions andM-mode periodic solutions, respectively. Interactions between these types of solutions are investigated.
Highlights
The physical phenomena and processes that take place in nature generally have complicated nonlinear features
There are equations which possess solutions which are a nonlinear superposition of two solitary waves but which do not have all the properties enjoyed by soliton equations
We consider the two-loop soliton solution found in Section 5.6 in more detail
Summary
The physical phenomena and processes that take place in nature generally have complicated nonlinear features. This leads to nonlinear mathematical models for the real processes. There is much interest in the practical issues involved, as well as the development of methods to investigate the associated nonlinear mathematical problems including nonlinear wave propagation. An early example of the latter was the development of the inverse scattering method for the Korteweg-de Vries (KdV) equation [1] and the subsequent interest in soliton theory. Based on our experience of the study of the Vakhnenko equation (VE), we acquaint the reader with a series of methods and approaches which may be applied to certain nonlinear equations. We outline a way in which an uninitiated reader could investigate a new nonlinear equation
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