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.

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.

The integrability of the coupled Ramani equation with binary Bell polynomials

2020 ◽  
Vol 34 (32) ◽  
pp. 2050371
Author(s):  
Xue-Dong Chai ◽  
Chun-Xia Li
Keyword(s):  
Conservation Laws ◽  
Scale Invariance ◽  
Bäcklund Transformation ◽  
Lax Pair ◽  
Bell Polynomials ◽  
Backlund Transformation ◽  
Bell Polynomial ◽  
Bilinear Bäcklund Transformation ◽  
Binary Bell Polynomial ◽  
Binary Bell Polynomials

Binary Bell polynomial approach is applied to study the coupled Ramani equation, which is the generalization of the Ramani equation. Based on the concept of scale invariance, the coupled Ramani equation is written in terms of binary Bell polynomials of two dimensionless field variables, which leads to the bilinear coupled Ramani equation directly. As a consequence, the bilinear Bäcklund transformation, Lax pair and conservation laws are systematically constructed by virtue of binary Bell polynomials.

Bäcklund Transformation and Soliton Solutions for a (3+1)-Dimensional Variable-Coefficient Breaking Soliton Equation

2016 ◽  
Vol 71 (9) ◽  
pp. 797-805 ◽  
Author(s):  
Chen Zhao ◽  
Yi-Tian Gao ◽  
Zhong-Zhou Lan ◽  
Jin-Wei Yang
Keyword(s):  
Variable Coefficient ◽  
Bäcklund Transformation ◽  
Soliton Solutions ◽  
Variable Coefficients ◽  
Bell Polynomials ◽  
Propagation Characteristics ◽  
Backlund Transformation ◽  
Soliton Equation ◽  
Dimensional Variable ◽  
Interaction Behaviors

AbstractIn this article, a (3+1)-dimensional variable-coefficient breaking soliton equation is investigated. Based on the Bell polynomials and symbolic computation, the bilinear forms and Bäcklund transformation for the equation are derived. One-, two-, and three-soliton solutions are obtained via the Hirota method.N-soliton solutions are also constructed. Propagation characteristics and interaction behaviors of the solitons are discussed graphically: (i) solitonic direction and position depend on the sign of the wave numbers; (ii) shapes of the multisoliton interactions in the scaled space and time coordinates are affected by the variable coefficients; (iii) multisoliton interactions are elastic for that the velocity and amplitude of each soliton remain unchanged after each interaction except for a phase shift.

Solitons, Bäcklund transformation and Lax pair for a (2+1)-dimensional B-type Kadomtsev–Petviashvili equation in the fluid/plasma mechanics

2016 ◽  
Vol 30 (25) ◽  
pp. 1650265 ◽  
Author(s):  
Zhong-Zhou Lan ◽  
Yi-Tian Gao ◽  
Jin-Wei Yang ◽  
Chuan-Qi Su ◽  
Qi-Min Wang
Keyword(s):  
Bäcklund Transformation ◽  
Soliton Solutions ◽  
Lax Pair ◽  
Inelastic Interaction ◽  
Bell Polynomials ◽  
Backlund Transformation ◽  
Linear Superposition ◽  
Electrostatic Wave ◽  
Elastic Interactions ◽  
Petviashvili Equation

Under investigation in this paper is a (2[Formula: see text]+[Formula: see text]1)-dimensional B-type Kadomtsev–Petviashvili equation for the shallow water wave in a fluid or electrostatic wave potential in a plasma. Bilinear form, Bäcklund transformation and Lax pair are derived based on the binary Bell polynomials. Multi-soliton solutions are constructed via the Hirota’s method. Propagation and interaction of the solitons are illustrated graphically: (i) Through the asymptotic analysis, elastic and inelastic interactions between the two solitons are discussed analytically and graphically, respectively. The elastic interaction, amplitudes, velocities and shapes of the two solitons remain unchanged except for a phase shift. However, in the area of the inelastic interaction, amplitudes of the two solitons have a linear superposition. (ii) Elastic interactions among the three solitons indicate that the properties of the elastic interactions among the three solitons are similar to those between the two solitons. Moreover, oblique and overtaking interactions between the two solitons are displayed. Oblique interactions among the three solitons and interactions among the two parallel solitons and a single one are presented as well. (iii) Inelastic–elastic interactions imply that the interaction between the inelastic region and another one is elastic.

scholarly journals Soliton Solutions, Bäcklund Transformation and Lax Pair for Coupled Burgers System via Bell Polynomials

2015 ◽  
Vol 70 (5) ◽  
pp. 359-363 ◽  
Author(s):  
Ömer Ünsal ◽  
Filiz Taşcan
Keyword(s):  
Bilinear Form ◽  
Bäcklund Transformation ◽  
Soliton Solutions ◽  
Lax Pair ◽  
Polynomial Form ◽  
Bell Polynomials ◽  
Backlund Transformation ◽  
Bell Polynomial ◽  
Binary Bell Polynomial ◽  
Burgers System

AbstractIn this work, we apply the binary Bell polynomial approach to coupled Burgers system. In other words, we investigate possible integrability of referred system. Bilinear form and soliton solutions are obtained, some figures related to these solutions are given. We also get Bäcklund transformations in both binary Bell polynomial form and bilinear form. Based on the Bäcklund transformation, Lax pair is obtained. Namely, this is a study in which integrabilitiy of coupled burgers system is investigated.

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

Tunnelling Effects of Solitons in Optical Fibers with Higher-Order Effects

2012 ◽  
Vol 67 (6-7) ◽  
pp. 338-346
Author(s):  
Chao-Qing Dai ◽  
Hai-Ping Zhu ◽  
Chun-Long Zheng
Keyword(s):  
Optical Fibers ◽  
Schrodinger Equation ◽  
Soliton Solutions ◽  
Higher Order ◽  
Order Effects ◽  
Bright Solitons ◽  
Dark Solitons ◽  
Tunnelling Effect ◽  
Tunnelling Effects ◽  
Higher Order Effects

We construct four types of analytical soliton solutions for the higher-order nonlinear Schrödinger equation with distributed coefficients. These solutions include bright solitons, dark solitons, combined solitons, and M-shaped solitons. Moreover, the explicit functions which describe the evolution of the width, peak, and phase are discussed exactly.We finally discuss the nonlinear soliton tunnelling effect for four types of femtosecond solitons

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