Non-integrable lattice equations supporting kink and soliton solutions

2001 ◽  
Vol 12 (6) ◽  
pp. 709-718 ◽  
Author(s):  
A. K. COMMON ◽  
M. MUSETTE
Keyword(s):  
Solitary Waves ◽  
Difference Equations ◽  
Vries Equation ◽  
Soliton Solutions ◽  
Elastic Collision ◽  
Discrete Form ◽  
Integrable Lattice ◽  
De Vries ◽  
Burgers Hierarchy

Nonintegrable differential-difference equations are constructed which support two-kink and two-soliton solutions. These equations are related to the discrete Burgers hierarchy and a discrete form of the Korteweg-de Vries equation. In particular, discretisations of equations related to the Fitzhugh-Nagumo-Kolmogorov-Petrovskii-Piskunov, Satsuma-Burgers-Huxley equations are derived. Methods presented here can also be used to derive non-integrable differential-difference equations describing the elastic collision of more than two kinks or solitary waves.

Dynamics of Solitary-Waves in the Higher Order Korteweg –De Vries Equation Type (I)

2005 ◽  
Vol 60 (11-12) ◽  
pp. 757-767 ◽  
Author(s):  
Woo-Pyo Hong
Keyword(s):  
Solitary Waves ◽  
Wave Equations ◽  
Analytic Solution ◽  
Vries Equation ◽  
Soliton Solutions ◽  
Auxiliary Function ◽  
Higher Order ◽  
Type I ◽  
De Vries ◽  
52.35 Mw

We find new analytic solitary-wave solutions of the higher order wave equations of Korteweg - De Vries (KdV) type (I), using the auxiliary function method. We study the dynamical properties of the solitary-waves by numerical simulations. It is shown that the solitary-waves are stable for wide ranges of the model coefficients. We study the dynamics of the two solitary-waves by using the analytic solution as initial profiles and find that they interact elastically in the sense that the mass and energy of the system are conserved. This leads to the possibility of multi-soliton solutions of the higher order KdV type (I), which can not be found by current analytical methods. - PACS numbers: 03.40.Kf, 02.30.Jr, 47.20.Ky, 52.35.Mw

Stability and instability of some nonlinear dispersive solitary waves in higher dimension

1996 ◽  
Vol 126 (1) ◽  
pp. 89-112 ◽  
Author(s):  
Anne de Bouard
Keyword(s):  
Solitary Waves ◽  
Acoustic Waves ◽  
Vries Equation ◽  
Three Dimensional ◽  
Higher Dimension ◽  
Ion Acoustic Waves ◽  
Stability And Instability ◽  
Ion Acoustic ◽  
De Vries ◽  
The Stability

We study the stability of positive radially symmetric solitary waves for a three dimensional generalisation of the Korteweg de Vries equation, which describes nonlinear ion-acoustic waves in a magnetised plasma, and for a generalisation in dimension two of the Benjamin–Bona–Mahony equation.

THE SECOND-ORDER KdV EQUATION AND ITS SOLITON-LIKE SOLUTION

2009 ◽  
Vol 23 (14) ◽  
pp. 1771-1780 ◽  
Author(s):  
CHUN-TE LEE ◽  
JINN-LIANG LIU ◽  
CHUN-CHE LEE ◽  
YAW-HONG KANG
Keyword(s):  
Solitary Waves ◽  
Soliton Solution ◽  
Kdv Equation ◽  
Soliton Solutions ◽  
Second Order ◽  
Close Resemblance ◽  
De Vries

This paper presents both the theoretical and numerical explanations for the existence of a two-soliton solution for a second-order Korteweg-de Vries (KdV) equation. Our results show that there exists "quasi-soliton" solutions for the equation in which solitary waves almost retain their identities in a suitable physical regime after they interact, and bear a close resemblance to the pure KdV solitons.

Cnoidal Wave Trains and Solitary Waves in a Dissipation-Modified Korteweg—de Vries Equation

KdV ’95 ◽  
1995 ◽  
pp. 457-475
Author(s):  
A. Ye. Rednikov ◽  
M. G. Velarde ◽  
Yu. S. Ryazantsev ◽  
A. A. Nepomnyashchy ◽  
V. N. Kurdyumov
Keyword(s):  
Solitary Waves ◽  
Vries Equation ◽  
Cnoidal Wave ◽  
De Vries ◽  
Wave Trains

Analytical study of exact solutions of the nonlinear Korteweg–de Vries equation with space–time fractional derivatives

2018 ◽  
Vol 32 (02) ◽  
pp. 1850012 ◽  
Author(s):  
Jiangen Liu ◽  
Yufeng Zhang
Keyword(s):  
Exact Solutions ◽  
Fractional Derivatives ◽  
Analytical Study ◽  
Vries Equation ◽  
Soliton Solutions ◽  
Traveling Wave Solutions ◽  
Expansion Method ◽  
Space Time ◽  
Strong Basis ◽  
De Vries

This paper gives an analytical study of dynamic behavior of the exact solutions of nonlinear Korteweg–de Vries equation with space–time local fractional derivatives. By using the improved [Formula: see text]-expansion method, the explicit traveling wave solutions including periodic solutions, dark soliton solutions, soliton solutions and soliton-like solutions, are obtained for the first time. They can better help us further understand the physical phenomena and provide a strong basis. Meanwhile, some solutions are presented through 3D-graphs.

Decomposition, Darboux transformation and soliton solutions for the (2 + 1)-dimensional integrable nonlocal “breaking soliton” equation

2020 ◽  
Vol 34 (24) ◽  
pp. 2050251
Author(s):  
Xiaoming Zhu ◽  
Kelei Tian
Keyword(s):  
Darboux Transformation ◽  
Schrodinger Equation ◽  
Vries Equation ◽  
Soliton Solutions ◽  
Soliton Equation ◽  
Interaction Mechanisms ◽  
Dark Solitons ◽  
De Vries ◽  
Nonlocal Nonlinear Schrödinger Equation ◽  
Nonlocal Nonlinear

In this paper, we investigate an integrable nonlocal “breaking soliton” equation, which can be decomposed into the nonlocal nonlinear Schrödinger equation and the nonlocal complex modified Korteweg–de Vries equation. As an application, with the use of this decomposition and Darboux transformation, the dark solitons, antidark solitons, rational dark solitons and rational antidark solitons of the considered equation are given explicitly. In particular, the interaction mechanisms of these solutions are discussed and illustrated through some figures.

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