Approach in theory of nonlinear evolution equations : the VakhnenkoParkes equation
Vakhnenko, V.O. and Parkes, E. J. (2016) Approach in theory of nonlinear evolution equations : the VakhnenkoParkes equation. Advances in Mathematical Physics, 2016. 2916582. ISSN 16879139

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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. It is shown how the equation arises in modelling the propagation of highfrequency waves in a relaxing medium. The VE is related to a particular form of the Whitham equation. Periodic and solitary traveling wave solutions are found by direct integration. Some of these solutions are looplike in nature. The VE can be written in an alternative form, now known as the VakhnenkoParkes equation (VPE), by a change of independent variables. The VPE can be written in Hirota bilinear form. It is then possible to show that the VPE satisfies the ‘Nsoliton condition’, in other words that the equation has an Nsoliton solution. This solution is found by using a blend of the Hirota method and ideas originally proposed by Moloney & Hodnett. This solution is discussed in detail, including the derivation of phase shifts due to interaction between solitons. Individual solitons are humplike in nature. However, when transformed back into the original variables, the corresponding solution to the VE comprises N looplike solitons. It is shown how aspects of the inverse scattering transform (IST) method, as applied originally to the KdV equation, can be used to find one and twosoliton solutions to the VPE even though, in contrast to the KdV equation, the VPE’s spectral equation is not secondorder (the isospectral Schr¨odinger equation). A B¨acklund transformation is found for the VPE and this is used to construct conservation laws. It is shown that the specral equation for the VPE is actually thirdorder. Then, based on ideas of Kaup and Caudrey, the standard IST method for thirdorder spectral problems is used to investigate solutions corresponding to bound states of the spectrum and to a continuous spectrum. This leads to Nsoliton solutions andMmode periodic solutions respectively. Interactions between these types of solutions are investigated.
Creators(s):  Vakhnenko, V.O. and Parkes, E. J.; 

Item type:  Article 
ID code:  55729 
Keywords:  nonlinear equations, VakhnenkoParkes equation, travelling wave solutions, Mathematics, Mathematics(all) 
Subjects:  Science > Mathematics 
Department:  Faculty of Science > Mathematics and Statistics 
Depositing user:  Pure Administrator 
Date deposited:  29 Feb 2016 17:01 
Last modified:  23 Oct 2020 04:38 
Related URLs:  
URI:  https://strathprints.strath.ac.uk/id/eprint/55729 
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