scholarly journals Residual Symmetries and Bäcklund Transformations of Strongly Coupled Boussinesq–Burgers System

Symmetry ◽  
2019 ◽  
Vol 11 (11) ◽  
pp. 1365 ◽  
Author(s):  
Haifeng Wang ◽  
Yufeng Zhang
Keyword(s):  
Bäcklund Transformation ◽  
Point Symmetry ◽  
Backlund Transformation ◽  
Nonlocal Symmetry ◽  
Linear Superposition ◽  
Strongly Coupled ◽  
Residual Symmetry ◽  
Commutative Subalgebra ◽  
Truncated Painlevé Expansion ◽  
Burgers System

In this article, we construct a new strongly coupled Boussinesq–Burgers system taking values in a commutative subalgebra Z 2 . A residual symmetry of the strongly coupled Boussinesq–Burgers system is achieved by a given truncated Painlevé expansion. The residue symmetry with respect to the singularity manifold is a nonlocal symmetry. Then, we introduce a suitable enlarged system to localize the nonlocal residual symmetry. In addition, a Bäcklund transformation is obtained with the help of Lie’s first theorem. Further, the linear superposition of multiple residual symmetries is localized to a Lie point symmetry, and a N-th Bäcklund transformation is also obtained.

scholarly journals Residual Symmetries and Bäcklund Transformations of (2 + 1)-Dimensional Strongly Coupled Burgers System

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Haifeng Wang ◽  
Yufeng Zhang
Keyword(s):  
General Theory ◽  
Bäcklund Transformation ◽  
Symmetry Analysis ◽  
Point Symmetry ◽  
Backlund Transformation ◽  
Linear Superposition ◽  
Strongly Coupled ◽  
Residual Symmetry ◽  
Commutative Subalgebra ◽  
Burgers System

In this article, we mainly apply the nonlocal residual symmetry analysis to a (2 + 1)-dimensional strongly coupled Burgers system, which is defined by us through taking values in a commutative subalgebra. On the basis of the general theory of Painlevé analysis, we get a residual symmetry of the strongly coupled Burgers system. Then, we introduce a suitable enlarged system to localize the nonlocal residual symmetry. In addition, a Bäcklund transformation is derived by Lie’s first theorem. Further, the linear superposition of the multiple residual symmetries is localized to a Lie point symmetry, and an N-th Bäcklund transformation is also obtained.

scholarly journals Nonlocal Symmetry and Bäcklund Transformation of a Negative-Order Korteweg–de Vries Equation

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-10 ◽  
Author(s):  
Jinxi Fei ◽  
Weiping Cao ◽  
Zhengyi Ma
Keyword(s):  
Vries Equation ◽  
Bäcklund Transformation ◽  
Backlund Transformation ◽  
Nonlocal Symmetry ◽  
Linear Superposition ◽  
Residual Symmetry ◽  
Negative Order ◽  
De Vries ◽  
Finite Transformation ◽  
New Variables

The residual symmetry of a negative-order Korteweg–de Vries (nKdV) equation is derived through its Lax pair. Such residual symmetry can be localized, and the original nKdV equation is extended into an enlarged system by introducing four new variables. By using Lie’s first theorem, we obtain the finite transformation for the localized residual symmetry. Furthermore, we localize the linear superposition of multiple residual symmetries and construct n-th Bäcklund transformation for this nKdV equation in the form of the determinants.

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.

The Residual Symmetry and Consistent Tanh Expansion for the Benney System

2017 ◽  
Vol 72 (9) ◽  
pp. 863-871 ◽  
Author(s):  
Zheng-Yi Ma ◽  
Jin-Xi Fei ◽  
Jun-Chao Chen
Keyword(s):  
Bäcklund Transformation ◽  
Error Function ◽  
Periodic Waves ◽  
Linear Superposition ◽  
Residual Symmetry ◽  
Interaction Solutions ◽  
Finite Transformation ◽  
Benney Equation ◽  
New Variables ◽  
Truncated Painlevé Expansion

AbstractThe residual symmetry of the (2+1)-dimensional Benney system is derived from the truncated Painlevé expansion. Such residual symmetry is localised and the original Benney equation is extended into an enlarged system by introducing four new variables. By using Lies first theorem, we obtain the finite transformation for the localised residual symmetry. More importantly, we further localise the linear superposition of multiple residual symmetries and construct the nth Bäcklund transformation for the Benney system in the form of the determinant. Moreover, it is proved that the (2+1)-dimensional Benney system is consistent tanh expansion (CTE) solvable. The exact interaction solutions between solitons and any other types of potential Burgers waves are also obtained, which include soliton-error function waves, soliton-periodic waves, and so on.

A Truncated Painlevé Expansion and Exact Analytical Solutions for the Nonlinear Schr¨Odinger Equation with Variable Coefficients

2005 ◽  
Vol 60 (11-12) ◽  
pp. 768-774 ◽  
Author(s):  
Biao Li ◽  
Yong Chen
Keyword(s):  
Analytical Solutions ◽  
Bäcklund Transformation ◽  
Variable Coefficients ◽  
Backlund Transformation ◽  
Exact Analytical Solutions ◽  
Interaction Of Solitons ◽  
Varying Dispersion ◽  
Physical Phenomena ◽  
Truncated Painlevé Expansion ◽  
Singular Soliton

By using the truncated Painlevé expansion analysis an auto-Bäcklund transformation is found for the nonlinear Schrödinger equation with varying dispersion, nonlinearity, and gain or absorption. Then, based on the obtained auto-Bäcklund transformation and symbolic computation, we explore some explicit exact solutions including soliton-like solutions, singular soliton-like solutions, which may be useful to explain the corresponding physical phenomena. Further, the formation and interaction of solitons are simulated by computer. - PACS Nos.: 05.45.Yv, 02.30.Jr, 42.65.Tg

Multisoliton Excitations for the Kadomtsev-Petviashvili Equation

2006 ◽  
Vol 61 (1-2) ◽  
pp. 32-38 ◽  
Author(s):  
Zheng-Yi Ma ◽  
Guo-Sheng Hua ◽  
Chun-Long Zheng
Keyword(s):  
Bäcklund Transformation ◽  
Backlund Transformation ◽  
Electrons And Positrons ◽  
Straight Line ◽  
Petviashvili Equation ◽  
Truncated Painlevé Expansion ◽  
Multisoliton Solutions

By means of the standard truncated Painlevé expansion and a special Bäcklund transformation, some exact multisoliton solutions are derived for the Kadomtsev-Petviashvili equation. The evolution properties of the multisoliton excitations are investigated and some novel features or interesting behaviors are revealed. The results show that four straight-line solitons are annihilated or produced with the time increases, which is very similar to the completely nonelastic collision among electrons and positrons.

Truncated Painlevé Expansion with Symbolic Computation for a General Kadomtsev-Petviashvili Equation with Variable Coefficients

1996 ◽  
Vol 51 (3) ◽  
pp. 175-178
Author(s):  
Bo Tian ◽  
Yi-Tian Gao
Keyword(s):  
Symbolic Computation ◽  
Variable Coefficient ◽  
Bäcklund Transformation ◽  
Variable Coefficients ◽  
Explicit Solutions ◽  
Current Interest ◽  
Backlund Transformation ◽  
Petviashvili Equation ◽  
Mathematical Sciences ◽  
Truncated Painlevé Expansion

Able to realistically model various physical situations, the variable-coefficient generalizations of the celebrated Kadmotsev-Petviashvili equation are of current interest in physical and mathematical sciences. In this paper, we make use of both the truncated Painleve expansion and symbolic computation to obtain an auto-Bäcklund transformation and certain soliton-typed explicit solutions for a general Kadomtsev-Petviashvili equation with variable coefficients.

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