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 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 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.

New Types of Solitons with Fusion and Fission Properties in the (2+1)-Dimensional Generalized Broer-Kaup System

2006 ◽  
Vol 61 (10-11) ◽  
pp. 519-524
Author(s):  
Chao-Qing Dai ◽  
Jun-Lang Chen
Keyword(s):  
Arbitrary Number ◽  
Bäcklund Transformation ◽  
Variable Separation ◽  
Backlund Transformation ◽  
Linear Superposition ◽  
Soliton Fission ◽  
Superposition Theorem

In this paper, by virtue of a special Painlevé-Bäcklund transformation and the linear superposition theorem, the general variable separation solution with an arbitrary number of variable separated functions of the generalized Broer-Kaup (GBK) system is obtained. Based on the general variable separation solution with some suitable variable separated functions, new types of the V-shaped soliton fusion and Y-shaped soliton fission are firstly investigated. - PACS numbers: 05.45.Yv, 02.30.Jr, 02.03Ik

scholarly journals Solitary wave fission and fusion in the (2+1)-dimensional generalized Broer–Kaup system

2012 ◽  
Vol 17 (3) ◽  
pp. 271-279 ◽  
Author(s):  
Chaoqing Dai ◽  
Cuiyun Liu
Keyword(s):  
Solitary Wave ◽  
Bäcklund Transformation ◽  
Variable Separation ◽  
Backlund Transformation ◽  
Linear Superposition ◽  
Superposition Theorem

Via a special Painlevé–Bäcklund transformation and the linear superposition theorem, we derive the general variable separation solution of the (2 + 1)-dimensional generalized Broer–Kaup system. Based on the general variable separation solution and choosing some suitable variable separated functions, new types of V-shaped and A-shaped solitary wave fusion and Y-shaped solitary wave fission phenomena are reported.

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.

scholarly journals ON THE INTEGRABILITY, BÄCKLUND TRANSFORMATION AND SYMMETRY ASPECTS OF A GENERALIZED FISHER TYPE NONLINEAR REACTION–DIFFUSION EQUATION

2004 ◽  
Vol 14 (05) ◽  
pp. 1577-1600 ◽  
Author(s):  
P. S. BINDU ◽  
M. LAKSHMANAN ◽  
M. SENTHILVELAN
Keyword(s):  
Bäcklund Transformation ◽  
Reaction Diffusion ◽  
Lie Symmetry ◽  
Symmetry Analysis ◽  
Reaction Diffusion Equation ◽  
Theoretical Interpretation ◽  
Fisher Equation ◽  
Backlund Transformation ◽  
Lie Symmetry Analysis ◽  
Nonlinear Reaction

The dynamics of nonlinear reaction–diffusion systems is dominated by the onset of patterns, and Fisher equation is considered to be a prototype of such diffusive equations. Here we investigate the integrability properties of a generalized Fisher equation in both (1+1) and (2+1) dimensions. A Painlevé singularity structure analysis singles out a special case (m=2) as integrable. More interestingly, a Bäcklund transformation is shown to give rise to a linearizing transformation for the integrable case. A Lie symmetry analysis again separates out the same m=2 case as the integrable one and hence we report several physically interesting solutions via similarity reductions. Thus we give a group theoretical interpretation for the system under study. Explicit and numerical solutions for specific cases of nonintegrable systems are also given. In particular, the system is found to exhibit different types of traveling wave solutions and patterns, static structures and localized structures. Besides the Lie symmetry analysis, nonclassical and generalized conditional symmetry analysis are also carried out.

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