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• W2300553681 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<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M1><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:math>-soliton solution which is discussed in detail. Individual solitons are hump-like in nature whereas the corresponding solution to the VE comprises<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M2><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:math>-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<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M3><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:math>-soliton solutions and<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M4><mml:mrow><mml:mi>M</mml:mi></mml:mrow></mml:math>-mode periodic solutions, respectively. Interactions between these types of solutions are investigated." @default.
• W2300553681 created "2016-06-24" @default.
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• W2300553681 date "2016-01-01" @default.
• W2300553681 modified "2023-10-10" @default.
• W2300553681 title "Approach in Theory of Nonlinear Evolution Equations: The Vakhnenko-Parkes Equation" @default.
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• W2300553681 doi "https://doi.org/10.1155/2016/2916582" @default.
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