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
2018 ◽  
Vol 67 (11) ◽  
pp. 110201
Author(s):  
Wang Jian-Yong ◽  
Cheng Xue-Ping ◽  
Zeng Ying ◽  
Zhang Yuan-Xiang ◽  
Ge Ning-Yi

2021 ◽  
Vol 8 (3) ◽  
pp. 368-378
Author(s):  
S. I. Lyashko ◽  
◽  
V. H. Samoilenko ◽  
Yu. I. Samoilenko ◽  
N. I. Lyashko ◽  
...  

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.


2020 ◽  
pp. 2050432
Author(s):  
Xiazhi Hao ◽  
Xiaoyan Li

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.


2008 ◽  
Vol 25 (11) ◽  
pp. 3890-3893 ◽  
Author(s):  
Xu Xiao-Ge ◽  
Meng Xiang-Hua ◽  
Gao Yi-Tian

1978 ◽  
Vol 87 (1) ◽  
pp. 17-31 ◽  
Author(s):  
D. Anker ◽  
N. C. Freeman

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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