Chain of the Bäcklund transformation for the KdV equation

1981 ◽  
Vol 22 (8) ◽  
pp. 1608-1613 ◽  
Author(s):  
Akira Nakamura
Keyword(s):  
Bäcklund Transformation ◽  
Kdv Equation ◽  
Backlund Transformation ◽  
The Kdv Equation

scholarly journals The Painlevé Tests, Bäcklund Transformation and Bilinear Form for the KdV Equation with a Self-Consistent Source

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Yali Shen ◽  
Fengqin Zhang ◽  
Xiaomei Feng
Keyword(s):  
Soliton Solution ◽  
Bilinear Form ◽  
Bäcklund Transformation ◽  
Kdv Equation ◽  
Bilinear Operator ◽  
Backlund Transformation ◽  
Painlevé Property ◽  
Bilinear Bäcklund Transformation ◽  
The Kdv Equation ◽  
Self Consistent

The Painlevé property and Bäcklund transformation for the KdV equation with a self-consistent source are presented. By testing the equation, it is shown that the equation has the Painlevé property. In order to further prove its integrality, we give its bilinear form and construct its bilinear Bäcklund transformation by the Hirota's bilinear operator. And then the soliton solution of the equation is obtained, based on the proposed bilinear form.

scholarly journals A New Auto-Bäcklund Transformation of the KdV Equation with General Variable Coefficients and Its Application

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Chunping Liu
Keyword(s):  
Solitary Wave ◽  
Bäcklund Transformation ◽  
Kdv Equation ◽  
Variable Coefficients ◽  
Backlund Transformation ◽  
The Kdv Equation ◽  
Homogeneity Equation ◽  
Homogeneous Balance Method ◽  
Key Steps ◽  
Homogeneous Balance

First, by improving some key steps in the homogeneous balance method, a new auto-Bäcklund transformation (BT) to the KdV equation with general variable coefficients is derived. The new auto-BT in this paper does not require the coefficients of the equation to be linearly dependent. Then, based on the new auto-BT in which there is only one quadratic homogeneity equation to be solved, an exact soliton-like solution containing 2-solitary wave is given.

BILINEARIZATION AND SOLUTIONS OF THE KdV6 EQUATION

2008 ◽  
Vol 06 (04) ◽  
pp. 401-412 ◽  
Author(s):  
A. RAMANI ◽  
B. GRAMMATICOS ◽  
R. WILLOX
Keyword(s):  
Evolution Equation ◽  
Arbitrary Function ◽  
Bäcklund Transformation ◽  
Kdv Equation ◽  
Singularity Analysis ◽  
Backlund Transformation ◽  
Integrable Evolution Equation ◽  
The Kdv Equation ◽  
Integrable Evolution ◽  
Kdv6 Equation

We examine the recently proposed KdV6 integrable evolution equation. Starting from solutions suggested by singularity analysis and using the auto-Bäcklund transformation, we construct solutions of the KdV6 which involve one arbitrary function of time. Next, we proceed to bilinearize the equation and derive a new, simpler, auto-Bäcklund transformation. Starting from the solutions of the KdV equation we construct those of the KdV6 in the form of M kinks and N poles and which indeed involve an arbitrary function of time.

Multi-soliton solutions and Bäcklund transformation for a two-mode KdV equation in a fluid

2016 ◽  
Vol 27 (1) ◽  
pp. 1-14 ◽  
Author(s):  
Zi-Jian Xiao ◽  
Bo Tian ◽  
Hui-Ling Zhen ◽  
Jun Chai ◽  
Xiao-Yu Wu
Keyword(s):  
Bäcklund Transformation ◽  
Kdv Equation ◽  
Soliton Solutions ◽  
Backlund Transformation

scholarly journals Bäcklund Transformations between the KdV Equation and a New Nonlinear Evolution Equation

2017 ◽  
Vol 2017 ◽  
pp. 1-6
Author(s):  
Xifang Cao
Keyword(s):  
Evolution Equation ◽  
Nonlinear Evolution Equation ◽  
Nonlinear Evolution ◽  
Bäcklund Transformation ◽  
Kdv Equation ◽  
The Other ◽  
Bäcklund Transformations ◽  
Backlund Transformations ◽  
The Kdv Equation ◽  
New Equation

We first give a Bäcklund transformation from the KdV equation to a new nonlinear evolution equation. We then derive two Bäcklund transformations with two pseudopotentials, one of which is from the KdV equation to the new equation and the other from the new equation to itself. As applications, by applying our Bäcklund transformations to known solutions, we construct some novel solutions to the new equation.

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