The asymptotic form of the N soliton solution of the Korteweg-de Vries equation

1972 ◽  
Vol 5 (12) ◽  
pp. L132-L135 ◽  
Author(s):  
J D Gibbon ◽  
J C Eilbeck
Keyword(s):  
Soliton Solution ◽  
Asymptotic Form ◽  
Vries Equation ◽  
De Vries

scholarly journals Quasi-soliton solution of Korteweg-de Vries equation and its application in ion acoustic waves

2018 ◽  
Vol 67 (11) ◽  
pp. 110201
Author(s):  
Wang Jian-Yong ◽  
Cheng Xue-Ping ◽  
Zeng Ying ◽  
Zhang Yuan-Xiang ◽  
Ge Ning-Yi
Keyword(s):  
Soliton Solution ◽  
Acoustic Waves ◽  
Vries Equation ◽  
Ion Acoustic Waves ◽  
Ion Acoustic ◽  
De Vries

Asymptotic analysis of the Korteweg-de Vries equation by the nonlinear WKB technique

2021 ◽  
Vol 8 (3) ◽  
pp. 368-378
Author(s):  
S. I. Lyashko ◽  
◽  
V. H. Samoilenko ◽  
Yu. I. Samoilenko ◽  
N. I. Lyashko ◽  
...  
Keyword(s):  
Asymptotic Solution ◽  
Soliton Solution ◽  
Vries Equation ◽  
Wave Process ◽  
Singular Part ◽  
Variable Coefficients ◽  
Singularly Perturbed ◽  
Main Term ◽  
Phase Variable ◽  
De Vries

The paper deals with the Korteweg-de Vries equation with variable coefficients and a small parameter at the highest derivative. The non-linear WKB technique has been used to construct the asymptotic step-like solution to the equation. Such a solution contains regular and singular parts of the asymptotics. The regular part of the solution describes the background of the wave process, while its singular part reflects specific features associated with soliton properties. The singular part of the searched asymp\-totic solution has the main term that, like the soliton solution, is the quickly decreasing function of the phase variable $\tau$. In contrast, other terms do not possess this property. An algorithm of constructing asymptotic step-like solutions to the singularly perturbed Korteweg--de Vries equation with variable coefficients is presented. In some sense, the constructed asymptotic solution is similar to the soliton solution to the Korteweg-de Vries equation $u_t+uu_x+u_{xxx}=0$. Statement on the accuracy of the main term of the asymptotic solution is proven.

Non-local symmetries and the nth finite symmetry transformations for the (2+1)-dimensional Korteweg–de Vries equation

2020 ◽  
pp. 2050432
Author(s):  
Xiazhi Hao ◽  
Xiaoyan Li
Keyword(s):  
Spectral Function ◽  
Soliton Solution ◽  
Vries Equation ◽  
Kdv Equation ◽  
Symmetry Transformation ◽  
De Vries ◽  
Symmetry Transformations ◽  
Non Local ◽  
Local Symmetries

Non-local symmetries in forms of square spectral function and residue over the (2+1)-dimensional Korteweg–de Vries (KdV) equation are studied in some detail. Then, we present [Formula: see text]-soliton solution to this equation with the help of symmetry transformation.

N -Soliton Solution in Wronskian Form for a Generalized Variable-Coefficient Korteweg–de Vries Equation

2008 ◽  
Vol 25 (11) ◽  
pp. 3890-3893 ◽  
Author(s):  
Xu Xiao-Ge ◽  
Meng Xiang-Hua ◽  
Gao Yi-Tian
Keyword(s):  
Soliton Solution ◽  
Variable Coefficient ◽  
Vries Equation ◽  
De Vries

Interpretation of three-soliton interactions in terms of resonant triads

1978 ◽  
Vol 87 (1) ◽  
pp. 17-31 ◽  
Author(s):  
D. Anker ◽  
N. C. Freeman
Keyword(s):  
Soliton Solution ◽  
Analytic Solution ◽  
Vries Equation ◽  
Two Dimensional ◽  
Soliton Interactions ◽  
Resonant Interactions ◽  
De Vries ◽  
Computer Calculations ◽  
Resonant Triads

The three-soliton solution of the two-dimensional Korteweg-de Vries equation is analysed to show that the structure of the interaction can be represented in terms of the motion of two-soliton resonant interactions (resonant triads) as described by Miles (1977). The schematic development of the interaction with time is obtained and shown to approximate closely to computer calculations of the analytic solution. Similar results follow for interactions of more solitons and other equations.

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