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.

scholarly journals Bilinear Form and Two Bäcklund Transformations for the (3+1)-Dimensional Jimbo-Miwa Equation

2015 ◽  
Vol 2015 ◽  
pp. 1-5 ◽  
Author(s):  
He Li ◽  
Yi-Tian Gao
Keyword(s):  
Symbolic Computation ◽  
Bilinear Form ◽  
Integrable Systems ◽  
Bäcklund Transformation ◽  
Soliton Solutions ◽  
Bäcklund Transformations ◽  
Bell Polynomials ◽  
Backlund Transformation ◽  
Bell Polynomial ◽  
Bilinear Bäcklund Transformation

With Bell polynomials and symbolic computation, this paper investigates the (3+1)-dimensional Jimbo-Miwa equation, which is one of the equations in the Kadomtsev-Petviashvili hierarchy of integrable systems. We derive a bilinear form and construct a bilinear Bäcklund transformation (BT) for the (3+1)-dimensional Jimbo-Miwa equation, by virtue of which the soliton solutions are obtained. Bell-polynomial-typed BT is also constructed and cast into the bilinear BT.

Discrete Solitons and Bäcklund Transformation for the Coupled Ablowitz–Ladik Equations

2017 ◽  
Vol 72 (10) ◽  
pp. 963-972
Author(s):  
Xiao-Yu Wu ◽  
Bo Tian ◽  
Lei Liu ◽  
Yan Sun
Keyword(s):  
Lattice Spacing ◽  
Bäcklund Transformation ◽  
Soliton Solutions ◽  
Bound State ◽  
Bell Polynomials ◽  
Backlund Transformation ◽  
Bright Solitons ◽  
Dark Solitons ◽  
Spacing Parameter ◽  
Binary Bell Polynomials

AbstractUnder investigation in this paper are the coupled Ablowitz–Ladik equations, which are linked to the optical fibres, waveguide arrays, and optical lattices. Binary Bell polynomials are applied to construct the bilinear forms and bilinear Bäcklund transformation. Bright/dark one- and two-soliton solutions are also obtained. Asymptotic analysis indicates that the interactions between the bright/dark two solitons are elastic. Amplitudes and velocities of the bright solitons increase as the value of the lattice spacing increases. Increasing value of the lattice spacing can lead to the increase of both the bright solitons’ amplitudes and velocities, and the decrease of the velocities of the dark solitons. The lattice spacing parameter has no effect on the amplitudes of the dark solitons. Overtaking interaction between the unidirectional bright two solitons and a bound state of the two equal-velocity solitons is presented. Overtaking interaction between the unidirectional dark two solitons and the two parallel dark solitons is also plotted.

INTEGRABLE PROPERTIES FOR A GENERALIZED NON-ISOSPECTRAL AND VARIABLE-COEFFICIENT KORTEWEG–DE VRIES MODEL

2010 ◽  
Vol 24 (10) ◽  
pp. 1023-1032 ◽  
Author(s):  
XIAO-GE XU ◽  
XIANG-HUA MENG ◽  
FU-WEI SUN ◽  
YI-TIAN GAO
Keyword(s):  
Bilinear Form ◽  
Variable Coefficient ◽  
Bäcklund Transformation ◽  
Backlund Transformation ◽  
Bilinear Method ◽  
Solitonic Solution ◽  
Bilinear Bäcklund Transformation ◽  
De Vries ◽  
The One ◽  
Hirota Bilinear

Applicable in fluid dynamics and plasmas, a generalized variable-coefficient Korteweg–de Vries (vcKdV) equation is investigated analytically employing the Hirota bilinear method in this paper. The bilinear form for such a model is derived through a dependent variable transformation. Based on the bilinear form, the integrable properties such as the N-solitonic solution, the Bäcklund transformation and the Lax pair for the vcKdV equation are obtained. Additionally, it is shown that the bilinear Bäcklund transformation can turn into the one denoted in the original variables.

scholarly journals Bell Polynomials Approach Applied to (2 + 1)-Dimensional Variable-Coefficient Caudrey-Dodd-Gibbon-Kotera-Sawada Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Wen-guang Cheng ◽  
Biao Li ◽  
Yong Chen
Keyword(s):  
Variable Coefficient ◽  
Soliton Solutions ◽  
Lax Pair ◽  
Variable Coefficients ◽  
Bell Polynomials ◽  
Bilinear Method ◽  
Bilinear Bäcklund Transformation ◽  
Dimensional Variable ◽  
Binary Bell Polynomials ◽  
Hirota Bilinear

The bilinear form, bilinear Bäcklund transformation, and Lax pair of a (2 + 1)-dimensional variable-coefficient Caudrey-Dodd-Gibbon-Kotera-Sawada equation are derived through Bell polynomials. The integrable constraint conditions on variable coefficients can be naturally obtained in the procedure of applying the Bell polynomials approach. Moreover, theN-soliton solutions of the equation are constructed with the help of the Hirota bilinear method. Finally, the infinite conservation laws of this equation are obtained by decoupling binary Bell polynomials. All conserved densities and fluxes are illustrated with explicit recursion formulae.

Grammian solutions for a (2+1)-dimensional integrable coupled modified Date–Jimbo–Kashiwara–Miwa equation

2019 ◽  
Vol 33 (10) ◽  
pp. 1950119 ◽  
Author(s):  
Sheng-Nan Wang ◽  
Juan Hu
Keyword(s):  
Bäcklund Transformation ◽  
Perturbation Expansion ◽  
Hirota Bilinear Method ◽  
Backlund Transformation ◽  
Bilinear Method ◽  
Bilinear Bäcklund Transformation ◽  
Hirota Bilinear ◽  
Djkm Equation ◽  
Generation Procedure

In this paper, Grammian solutions to a (2[Formula: see text]+[Formula: see text]1)-dimensional modified Date–Jimbo–Kashiwara–Miwa (mDJKM) equation are presented by using Hirota bilinear method and perturbation expansion. Starting from the Grammian solutions, an integrable coupled mDJKM equation is then obtained and the corresponding Grammian solutions are first derived by utilizing the source generation procedure. Besides, we also construct and solve a coupled DJKM equation via source generation procedure. It is interesting that the coupled mDJKM system constitute a bilinear Bäcklund transformation for the coupled DJKM system. This means that the commutativity of source generation procedure and Bäcklund transformation is valid for the (2[Formula: see text]+[Formula: see text]1)-dimensional DJKM equation.

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.

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