Auto-Bäcklund Transformation and Analytic Solutions of (2+1)-Dimensional Boussinesq Equation

2008 ◽  
Vol 49 (2) ◽  
pp. 287-290
Author(s):  
Liu Guan-Ting
Keyword(s):  
Boussinesq Equation ◽  
Bäcklund Transformation ◽  
Analytic Solutions ◽  
Backlund Transformation

Nonlocal symmetries, Bäcklund transformation and interaction solutions for the integrable Boussinesq equation

2020 ◽  
Vol 34 (26) ◽  
pp. 2050288
Author(s):  
Jun Cai Pu ◽  
Yong Chen
Keyword(s):  
Boussinesq Equation ◽  
Bäcklund Transformation ◽  
Superposition Principle ◽  
Symmetry Transformation ◽  
Backlund Transformation ◽  
Nonlocal Symmetry ◽  
Linear Superposition ◽  
Nonlocal Symmetries ◽  
Interaction Solutions ◽  
Physical Phenomena

The nonlocal symmetry of the integrable Boussinesq equation is derived by the truncated Painlevé method. The nonlocal symmetry is localized to the Lie point symmetry by introducing auxiliary-dependent variables and the finite symmetry transformation related to the nonlocal symmetry is presented. The multiple nonlocal symmetries are obtained and localized base on the linear superposition principle, then the determinant representation of the [Formula: see text]th Bäcklund transformation is provided. The integrable Boussinesq equation is also proved to be consistent tanh expansion (CTE) form and accurate interaction solutions among solitons and other types of nonlinear waves are given out analytically and graphically by the CTE method. The associated structure may be related to large variety of real physical phenomena.

BÄCKLUND TRANSFORMATION AND ANALYTIC SOLUTIONS FOR A GENERALIZED VARIABLE-COEFFICIENT MODIFIED KORTEWEG–DE VRIES MODEL FROM FLUID DYNAMICS AND PLASMAS

2008 ◽  
Vol 19 (11) ◽  
pp. 1659-1671 ◽  
Author(s):  
FU-WEI SUN ◽  
YI-TIAN GAO ◽  
CHUN-YI ZHANG ◽  
XIAO-GE XU
Keyword(s):  
Fluid Dynamics ◽  
External Force ◽  
Variable Coefficient ◽  
Bäcklund Transformation ◽  
Solitary Wave Solution ◽  
Kdv Equation ◽  
Analytic Solutions ◽  
Mkdv Equation ◽  
Backlund Transformation ◽  
De Vries

We investigate a generalized variable-coefficient modified Korteweg–de Vries model with perturbed factor and external force (vc-GmKdV) describing fluid dynamics and space plasmas. In this paper, we propose an extended variable-coefficient balancing-act method (Evc-BAM), which is concise and straightforward, to obtain the generalized analytic solutions including solitary wave solution of the vc-GmKdV model with symbolic computation. Meanwhile, using the Evc-BAM, we obtain an auto-Bäcklund transformation for the vc-GmKdV model on the relevant constraint conditions of the coefficient functions. Using the given auto-Bäcklund transformation, the solutions of special equations for the vc-GmKdV model are also obtained as the variable-coefficient Korteweg–de Vries (vc-KdV) equation, the generalized KdV equation with perturbed factor and external force (GKdV), the variable-coefficient modified Korteweg–de Vries (vc-mKdV) equation, and the variable-coefficient cylindrical modified Korteweg–de Vries (vc-cmKdV) equation, respectively.

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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