Approach in theory of nonlinear evolution equations : the Vakhnenko-Parkes equation

Vakhnenko, V.O. and Parkes, E. J. (2016) Approach in theory of nonlinear evolution equations : the Vakhnenko-Parkes equation. Advances in Mathematical Physics, 2016. ISSN 1687-9139

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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 high-frequency 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 loop-like in nature. The VE can be written in an alternative form, now known as the Vakhnenko-Parkes 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 ‘N-soliton condition’, in other words that the equation has an N-soliton 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 hump-like in nature. However, when transformed back into the original variables, the corresponding solution to the VE comprises N loop-like 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 two-soliton solutions to the VPE even though, in contrast to the KdV equation, the VPE’s spectral equation is not second-order (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 third-order. Then, based on ideas of Kaup and Caudrey, 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 N-soliton solutions andM-mode periodic solutions respectively. Interactions between these types of solutions are investigated.