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.

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.

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.

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.

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 N-Soliton Solutions for the Cylindrical Nonlinear Schrodinger Equation from the Diverging Quasi-Plane Envelope Waves

2012 ◽  
Vol 67 (8-9) ◽  
pp. 441-450
Author(s):  
Pan Wang ◽  
Bo Tian ◽  
Wen-Jun Liu ◽  
Yan Jiang
Keyword(s):  
Bilinear Form ◽  
Bäcklund Transformation ◽  
Soliton Solutions ◽  
Nonlinearity Parameter ◽  
Backlund Transformation ◽  
Nonlinear Schrödinger ◽  
Soliton Dynamics ◽  
Bilinear Bäcklund Transformation ◽  
Double Wronskian ◽  
Physical Quantities

This paper investigates a cylindrical nonlinear Schrodinger (cNLS) equation, which describes the cylindrically diverging quasi-plane envelope waves in a nonlinear medium. With the Hirota method and symbolic computation, bilinear form and N-soliton solutions in the form of an Nth-order polynomial in N exponentials are obtained for the cNLS equation. By means of the properties of double Wronskian, the N-soliton solutions in terms of the double Wronskian is testified through the direct substitution into the bilinear form. Based on the bilinear form and exchange formulae, the bilinear Backlund transformation is also given. Those solutions are graphically depicted to understand the soliton dynamics of the cylindrically diverging quasi-plane envelope waves. Soliton properties are discussed and physical quantities are also analyzed. Dispersion parameter has the effect that it may extend (or shorten) the periodic time of soliton interaction and change the direction of soliton propagation. Amplitudes of solitons are related to the cubic nonlinearity parameter.

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 Bäcklund transformations of the first kind associated with Monge-Ampère’s equations

1973 ◽  
Vol 51 ◽  
pp. 161-184
Author(s):  
Michihiko Matsuda
Keyword(s):  
Integrable Systems ◽  
Bäcklund Transformation ◽  
Bäcklund Transformations ◽  
Contact Transformation ◽  
Backlund Transformation ◽  
Backlund Transformations ◽  
The Given

Due to Clairin and Goursat, a Bäcklund transformation of the first kind can be associated with Monge-Ampère’s equation. We shall consider Monge-Ampère’s equation of the form s + f(x, y, z, p, q) + g(x, y, z, p, q) t = 0, where p = ∂z/∂x, q = ∂z/∂y, s = ∂2z/∂x∂y, t = ∂2z/∂y2. The following theorems will be obtained:1. The transformed equation takes on the same form s′ + f′ + g′t′ = 0 if and only if the given equation can be transformed to a Teixeira equation s + L(x, y, z, q)t + M(x, y, z, q)p + N(x, y, z, q) = 0 by a contact transformation.2. Teixeira equation s + tL + pM + N = 0 is solved by integrable systems of order n if and only if the transformed equation is solved by integrable systems of order n — 1.

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