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 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 and interaction solutions of the Sawada–Kotera equation

2016 ◽  
Vol 30 (23) ◽  
pp. 1650293 ◽  
Author(s):  
Xiazhi Hao ◽  
Yinping Liu ◽  
Xiaoyan Tang ◽  
Zhibin Li
Keyword(s):  
Spectral Function ◽  
Darboux Transformation ◽  
Bäcklund Transformation ◽  
Original System ◽  
Lax Pair ◽  
Nonlocal Symmetries ◽  
Residual Symmetry ◽  
Interaction Solutions ◽  
Truncated Painlevé Expansion ◽  
Consistent Riccati Expansion

The nonlocal symmetries of the residual symmetry and the spectral function symmetry of Sawada–Kotera (SK) equation can be derived from the truncated Painlevé expansion and the Lax pair, respectively. By localizing the nonlocal symmetries of the original system to local ones of the prolonged system, the Bäcklund transformation and the Darboux transformation for both the original and the prolonged systems are obtained. Moreover, by the truncated Painlevé expansion, we further study the integrability of the SK quation in the sense of having a consistent Riccati expansion (CRE).

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.

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.

scholarly journals Symmetry Analysis, Bäcklund Transformations, and Interaction Solutions for an AB Modified KdV System

2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Man Jia
Keyword(s):  
Symmetry Reduction ◽  
Symmetry Analysis ◽  
Bäcklund Transformations ◽  
Cnoidal Waves ◽  
Nonlocal Symmetries ◽  
Interaction Solutions ◽  
Backlund Transformations ◽  
Akns System ◽  
Finite Transformation ◽  
Truncated Painlevé Expansion

An AB modified KdV (AB-mKdV) system which can be used to describe two-place event is studied in this manuscript. Because the AB-mKdV system is considered as a special reduction of the famous AKNS system, the properties of the AKNS system are first revealed by using symmetry analysis. The nonlocal symmetries related to truncated Painlevé expansion, the finite transformation, and the symmetry reduction solutions of the AKNS system are presented. The corresponding Bäcklund transformations and the interaction solutions of the AB-mKdV system are constructed based on the special reduction. The results demonstrate that the AB-mKdV system possesses many kinds of interaction solutions, such as the interactions between kink and soliton and kink and cnoidal waves. The soliton can be changed from bright to dark during propagation.

Residual Symmetries and Interaction Solutions for the Classical Korteweg-de Vries Equation

2017 ◽  
Vol 72 (3) ◽  
pp. 217-222 ◽  
Author(s):  
Jin-Xi Fei ◽  
Wei-Ping Cao ◽  
Zheng-Yi Ma
Keyword(s):  
Vries Equation ◽  
Expansion Method ◽  
Point Symmetry ◽  
Residual Symmetry ◽  
Interaction Solutions ◽  
Dependent Variables ◽  
Non Linear ◽  
De Vries ◽  
Finite Transformation ◽  
Non Local

AbstractThe non-local residual symmetry for the classical Korteweg-de Vries equation is derived by the truncated Painlevé analysis. This symmetry is first localised to the Lie point symmetry by introducing the auxiliary dependent variables. By using Lie’s first theorem, we then obtain the finite transformation for the localised residual symmetry. Based on the consistent tanh expansion method, some exact interaction solutions among different non-linear excitations are explicitly presented finally. Some special interaction solutions are investigated both in analytical and graphical ways at the same time.

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.

Nonlocal symmetries, solitary waves and cnoidal periodic waves of the (2+1)-dimensional breaking soliton equation

2017 ◽  
Vol 31 (36) ◽  
pp. 1750348
Author(s):  
Li Zou ◽  
Shou-Fu Tian ◽  
Lian-Li Feng
Keyword(s):  
Solitary Waves ◽  
Expansion Method ◽  
Periodic Waves ◽  
Nonlocal Symmetry ◽  
Soliton Equation ◽  
Nonlocal Symmetries ◽  
Interaction Solutions ◽  
Long Wave ◽  
Truncated Painlevé Expansion ◽  
Consistent Riccati Expansion

In this paper, we consider the (2[Formula: see text]+[Formula: see text]1)-dimensional breaking soliton equation, which describes the interaction of a Riemann wave propagating along the y-axis with a long wave along the x-axis. By virtue of the truncated Painlevé expansion method, we obtain the nonlocal symmetry, Bäcklund transformation and Schwarzian form of the equation. Furthermore, by using the consistent Riccati expansion (CRE), we prove that the breaking soliton equation is solvable. Based on the consistent tan-function expansion, we explicitly derive the interaction solutions between solitary waves and cnoidal periodic waves.

scholarly journals Resonant Soliton and Soliton-Cnoidal Wave Solutions for a (3+1)-Dimensional Korteweg-de Vries-Like Equation

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-11
Author(s):  
Zheng-Yi Ma ◽  
Jin-Xi Fei ◽  
Jun-Chao Chen ◽  
Quan-Yong Zhu
Keyword(s):  
Wave Interaction ◽  
Elliptic Functions ◽  
Cnoidal Wave ◽  
Linear Superposition ◽  
Jacobian Elliptic Functions ◽  
Multiple Wave ◽  
Residual Symmetry ◽  
Wave Solutions ◽  
De Vries ◽  
Truncated Painlevé Expansion

The residual symmetry of a (3+1)-dimensional Korteweg-de Vries (KdV)-like equation is constructed using the truncated Painlevé expansion. Such residual symmetry can be localized and the (3+1)-dimensional KdV-like equation is extended into an enlarged system by introducing some new variables. By using Lie’s first theorem, the finite transformation is obtained for this localized residual symmetry. Further, the linear superposition of multiple residual symmetries is localized and the n-th Bäcklund transformation in the form of the determinants is constructed for this equation. For illustration more detail, the first three multiple wave solutions-the collisions of resonant solitons are depicted. Finally, with the aid of the link between the consistent tanh expansion (CTE) method and the truncated Painlevé expansion, the explicit soliton-cnoidal wave interaction solution containing three kinds of Jacobian elliptic functions for this equation is derived.

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