scholarly journals Symmetry Classification of Third-Order Nonlinear Evolution Equations. Part I: Semi-simple Algebras

2012 ◽  
Vol 124 (1) ◽  
pp. 123-170 ◽  
Author(s):  
P. Basarab-Horwath ◽  
F. Güngör ◽  
V. Lahno
Keyword(s):  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Nonlinear Evolution Equations ◽  
Third Order ◽  
Symmetry Classification ◽  
Simple Algebras

scholarly journals Symmetry classification of KdV-type nonlinear evolution equations

2004 ◽  
Vol 45 (6) ◽  
pp. 2280-2313 ◽  
Author(s):  
F. Güngör ◽  
V. I. Lahno ◽  
R. Z. Zhdanov
Keyword(s):  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Nonlinear Evolution Equations ◽  
Symmetry Classification

Lie group classification of the N-th-order nonlinear evolution equations

2011 ◽  
Vol 54 (12) ◽  
pp. 2553-2572 ◽  
Author(s):  
ShouFeng Shen ◽  
ChangZheng Qu ◽  
Qing Huang ◽  
YongYang Jin
Keyword(s):  
Lie Group ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Group Classification ◽  
Nonlinear Evolution Equations ◽  
Lie Group Classification

scholarly journals Approach in Theory of Nonlinear Evolution Equations: The Vakhnenko-Parkes Equation

2016 ◽  
Vol 2016 ◽  
pp. 1-39 ◽  
Author(s):  
V. O. Vakhnenko ◽  
E. J. Parkes
Keyword(s):  
Bound States ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Kdv Equation ◽  
Soliton Solutions ◽  
Traveling Wave Solutions ◽  
Nonlinear Evolution Equations ◽  
Alternative Form ◽  
Third Order ◽  
Spectral Equation

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.

Formint — A program for the classification of integrable nonlinear evolution equations

1985 ◽  
Vol 34 (3) ◽  
pp. 303-311 ◽  
Author(s):  
V.P. Gerdt ◽  
A.B. Shvachka ◽  
A.Yu. Zharkov
Keyword(s):  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Nonlinear Evolution Equations

On classification of soliton solutions of multicomponent nonlinear evolution equations

2008 ◽  
Vol 41 (31) ◽  
pp. 315213 ◽  
Author(s):  
V S Gerdjikov ◽  
D J Kaup ◽  
N A Kostov ◽  
T I Valchev
Keyword(s):  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Soliton Solutions ◽  
Nonlinear Evolution Equations

scholarly journals A class of third-order nonlinear evolution equations admitting invariant subspaces and associated reductions

2014 ◽  
Vol 21 (1) ◽  
pp. 132-148 ◽  
Author(s):  
Yujian Ye ◽  
Wen-Xiu Ma ◽  
Shoufeng Shen ◽  
Danda Zhang
Keyword(s):  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Invariant Subspaces ◽  
Nonlinear Evolution Equations ◽  
Third Order

scholarly journals Classification of local and nonlocal symmetries of fourth-order nonlinear evolution equations

2010 ◽  
Vol 65 (3) ◽  
pp. 337-366 ◽  
Author(s):  
Qing Huang ◽  
C.Z. Qu ◽  
R. Zhdanov
Keyword(s):  
Fourth Order ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Nonlinear Evolution Equations ◽  
Nonlocal Symmetries
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