scholarly journals A vector asymmetrical NNV equation: Soliton solutions, bilinear Bäcklund transformation and Lax pair

2008 ◽  
Vol 344 (2) ◽  
pp. 593-600 ◽  
Author(s):  
Guo-Fu Yu ◽  
Hon-Wah Tam
Keyword(s):  
Bäcklund Transformation ◽  
Soliton Solutions ◽  
Lax Pair ◽  
Backlund Transformation ◽  
Bilinear Bäcklund Transformation

Bäcklund transformation and soliton solutions for two (3+1)-dimensional nonlinear evolution equations

2016 ◽  
Vol 30 (24) ◽  
pp. 1650309
Author(s):  
Lin Wang ◽  
Qixing Qu ◽  
Liangjuan Qin
Keyword(s):  
Bilinear Form ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Bäcklund Transformation ◽  
Soliton Solutions ◽  
Nonlinear Evolution Equations ◽  
Backlund Transformation ◽  
Wronskian Determinant ◽  
Bilinear Bäcklund Transformation ◽  
Bilinear Equations

In this paper, two (3[Formula: see text]+[Formula: see text]1)-dimensional nonlinear evolution equations (NLEEs) are under investigation by employing the Hirota’s method and symbolic computation. We derive the bilinear form and bilinear Bäcklund transformation (BT) for the two NLEEs. Based on the bilinear form, we obtain the multi-soliton solutions for them. Furthermore, multi-soliton solutions in terms of Wronskian determinant for the first NLEE are constructed, whose validity is verified through direct substitution into the bilinear equations.

N-SOLITON-LIKE SOLUTIONS AND BÄCKLUND TRANSFORMATIONS FOR A NON-ISOSPECTRAL AND VARIABLE-COEFFICIENT MODIFIED KORTEWEG-DE VRIES EQUATION

2011 ◽  
Vol 25 (05) ◽  
pp. 723-733 ◽  
Author(s):  
QIAN FENG ◽  
YI-TIAN GAO ◽  
XIANG-HUA MENG ◽  
XIN YU ◽  
ZHI-YUAN SUN ◽  
...  
Keyword(s):  
Variable Coefficient ◽  
Vries Equation ◽  
Bäcklund Transformation ◽  
Lax Pair ◽  
Mkdv Equation ◽  
Wronskian Technique ◽  
Backlund Transformation ◽  
Bilinear Bäcklund Transformation ◽  
De Vries ◽  
Bilinear Equations

A non-isospectral and variable-coefficient modified Korteweg–de Vries (mKdV) equation is investigated in this paper. Starting from the Ablowitz–Kaup–Newell–Segur procedure, the Lax pair is established and the Bäcklund transformation in original variables is also derived. By a dependent variable transformation, the non-isospectral and variable-coefficient mKdV equation is transformed into bilinear equations, by virtue of which the N-soliton-like solution is obtained. In addition, the bilinear Bäcklund transformation gives a one-soliton-like solution from a vacuum one. Furthermore, the N-soliton-like solution in the Wronskian form is constructed and verified via the Wronskian technique.

Lump soliton solutions and Bäcklund transformation for the (3+1)-dimensional Boussinesq equation with Bell polynomials

2018 ◽  
Vol 32 (21) ◽  
pp. 1850244
Author(s):  
Xiao-Ge Xu ◽  
Xiang-Hua Meng ◽  
Qi-Xing Qu
Keyword(s):  
Boussinesq Equation ◽  
Bäcklund Transformation ◽  
Soliton Solutions ◽  
Quadratic Function ◽  
Bell Polynomials ◽  
Backlund Transformation ◽  
Temporal Variables ◽  
Bilinear Bäcklund Transformation ◽  
The Relationship ◽  
Hirota Bilinear

In this paper, the (3+1)-dimensional Boussinesq equation which can describe the propagation of gravity waves on the surface of water is investigated. Using the Bell polynomials, the bilinear form of the (3+1)-dimensional Boussinesq equation is obtained and the lump soliton solutions for the equation are derived by means of the quadratic function method. As an important integrable property, the Bäcklund transformation for the (3+1)-dimensional Boussinesq equation is constructed by the Bell polynomials considering the constraints on the derivatives with respect to spatial and temporal variables. Through the relationship between the Bell polynomials and the Hirota bilinear operators, the bilinear Bäcklund transformation for the (3+1)-dimensional Boussinesq equation is given.

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