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Approach in Theory of Nonlinear Evolution Equations: The Vakhnenko-Parkes Equation
Joint Authors
Source
Advances in Mathematical Physics
Issue
Vol. 2016, Issue 2016 (31 Dec. 2016), pp.1-39, 39 p.
Publisher
Hindawi Publishing Corporation
Publication Date
2016-02-22
Country of Publication
Egypt
No. of Pages
39
Main Subjects
Abstract EN
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 N -soliton solution which is discussed in detail.
Individual solitons are hump-like in nature whereas the corresponding solution to the VE comprises N -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 N -soliton solutions and M -mode periodic solutions, respectively.
Interactions between these types of solutions are investigated.
American Psychological Association (APA)
Vakhnenko, V. O.& Parkes, E. J.. 2016. Approach in Theory of Nonlinear Evolution Equations: The Vakhnenko-Parkes Equation. Advances in Mathematical Physics،Vol. 2016, no. 2016, pp.1-39.
https://search.emarefa.net/detail/BIM-1095795
Modern Language Association (MLA)
Vakhnenko, V. O.& Parkes, E. J.. Approach in Theory of Nonlinear Evolution Equations: The Vakhnenko-Parkes Equation. Advances in Mathematical Physics No. 2016 (2016), pp.1-39.
https://search.emarefa.net/detail/BIM-1095795
American Medical Association (AMA)
Vakhnenko, V. O.& Parkes, E. J.. Approach in Theory of Nonlinear Evolution Equations: The Vakhnenko-Parkes Equation. Advances in Mathematical Physics. 2016. Vol. 2016, no. 2016, pp.1-39.
https://search.emarefa.net/detail/BIM-1095795
Data Type
Journal Articles
Language
English
Notes
Includes bibliographical references
Record ID
BIM-1095795