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.

ON IMPROVED HOMOGENEOUS BALANCE METHOD, AUTO-BÄCKLUND TRANSFORMATION AND MULTI-SOLITONIC SOLUTIONS OF A VARIABLE-COEFFICIENT BURGERS EQUATION

2008 ◽  
Vol 19 (12) ◽  
pp. 1821-1827 ◽  
Author(s):  
R. SABRY
Keyword(s):  
Solitary Wave ◽  
Solitary Waves ◽  
Variable Coefficient ◽  
Burgers Equation ◽  
Bäcklund Transformation ◽  
Backlund Transformation ◽  
Homogeneous Balance Method ◽  
Ihb Method ◽  
Improved Homogeneous Balance Method ◽  
Homogeneous Balance

An improved homogeneous balance (IHB) method is introduced. On using the IHB method, a new auto-Bäcklund transformation and multi-solitonic solutions were obtained for a generalized variable-coefficient Burgers equation. The obtained solitary waves were found to propagate with a variable propagating speed which depends on the coefficients of the studied model. Also, fusion of two single solitary waves into a one-resonant solitary wave is pointed out.

Abundant New Multiple Soliton-like Solutions and Rational Solutions of the (2+1)-Dimensional Broer-Kaup Equation

2001 ◽  
Vol 56 (12) ◽  
pp. 816-824 ◽  
Author(s):  
Zhenya Yan
Keyword(s):  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Bäcklund Transformation ◽  
Soliton Solutions ◽  
Nonlinear Evolution Equations ◽  
Heat Equations ◽  
Backlund Transformation ◽  
Rational Solutions ◽  
Homogeneous Balance Method ◽  
Homogeneous Balance

Abstract In this paper we firstly improve the homogeneous balance method due to Wang, which was only used to obtain single soliton solutions of nonlinear evolution equations, and apply it to (2 + 1)-dimensional Broer-Kaup (BK) equations such that a Backlund transformation is found again. Considering further the obtained Backlund transformation, the relations are deduced among BK equations, well-known Burgers equations and linear heat equations. Finally, abundant multiple soliton-like solutions and infinite rational solutions are obtained from the relations.

Auto-B¨Acklund Transformations And Exact Solutions For The Generalized Two-Dimensional Korteweg-De Vries-Burgers-Type Equations And Burgers-Type Equations

2003 ◽  
Vol 58 (7-8) ◽  
pp. 464-472
Author(s):  
Biao Li ◽  
Yong Chen ◽  
Hongqing Zhang
Keyword(s):  
Exact Solutions ◽  
Bäcklund Transformation ◽  
Type Equation ◽  
Balance Method ◽  
Two Dimensional ◽  
Backlund Transformation ◽  
De Vries ◽  
Homogeneous Balance Method ◽  
Burgers Type Equations ◽  
Homogeneous Balance

In this paper, based on the idea of the homogeneous balance method and with the help of Mathematica, we obtain a new auto-Bäcklund transformation for the generalized two-dimensional Kortewegde Vries-Burgers-type equation and a new auto-Bäcklund transformation for the generalized twodimensional Burgers-type equation by introducing two appropriate transformations. Then, based on these two auto-Bäcklund transformation, some exact solutions for these equations are derived. Some figures are given to show the properties of the solutions.

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.

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