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).

Nonlocal Symmetries, Consistent Riccati Expansion, and Analytical Solutions of the Variant Boussinesq System

2017 ◽  
Vol 72 (7) ◽  
pp. 655-663 ◽  
Author(s):  
Lian-Li Feng ◽  
Shou-Fu Tian ◽  
Tian-Tian Zhang ◽  
Jun Zhou
Keyword(s):  
Periodic Wave ◽  
Boussinesq System ◽  
Nonlocal Symmetry ◽  
Nonlocal Symmetries ◽  
Interaction Solutions ◽  
Long Wave ◽  
Symmetry Transformations ◽  
Truncated Painlevé Expansion ◽  
Consistent Riccati Expansion ◽  
Similarity Reductions

AbstractUnder investigation in this paper is the variant Boussinesq system, which describes the propagation of surface long wave towards two directions in a certain deep trough. With the help of the truncated Painlevé expansion, we construct its nonlocal symmetry, Bäcklund transformation, and Schwarzian form, respectively. The nonlocal symmetries can be localised to provide the corresponding nonlocal group, and finite symmetry transformations and similarity reductions are computed. Furthermore, we verify that the variant Boussinesq system is solvable via the consistent Riccati expansion (CRE). By considering the consistent tan-function expansion (CTE), which is a special form of CRE, the interaction solutions between soliton and cnoidal periodic wave are explicitly studied.

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.

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.

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.

scholarly journals Residual Symmetry Reduction and Consistent Riccati Expansion to a Nonlinear Evolution Equation

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-9 ◽  
Author(s):  
Lamine Thiam ◽  
Xi-zhong Liu
Keyword(s):  
Evolution Equation ◽  
Nonlinear Evolution Equation ◽  
Nonlinear Evolution ◽  
Symmetry Reduction ◽  
Lie Symmetry ◽  
Residual Symmetry ◽  
Interaction Solutions ◽  
Lie Symmetry Method ◽  
Consistent Riccati Expansion ◽  
Symmetry Method

The residual symmetry of a (1 + 1)-dimensional nonlinear evolution equation (NLEE) ut+uxxx−6u2ux+6λux=0 is obtained through Painlevé expansion. By introducing a new dependent variable, the residual symmetry is localized into Lie point symmetry in an enlarged system, and the related symmetry reduction solutions are obtained using the standard Lie symmetry method. Furthermore, the (1 + 1)-dimensional NLEE equation is proved to be integrable in the sense of having a consistent Riccati expansion (CRE), and some new Bäcklund transformations (BTs) are given. In addition, some explicitly expressed solutions including interaction solutions between soliton and cnoidal waves are derived from these BTs.

Nonlocal Symmetries, Conservation Laws and Interaction Solutions of the Generalised Dispersive Modified Benjamin–Bona–Mahony Equation

2018 ◽  
Vol 73 (5) ◽  
pp. 399-405 ◽  
Author(s):  
Xue-Wei Yan ◽  
Shou-Fu Tian ◽  
Min-Jie Dong ◽  
Xiu-Bin Wang ◽  
Tian-Tian Zhang
Keyword(s):  
Direct Method ◽  
Symmetry Transformation ◽  
Expansion Method ◽  
Local Symmetry ◽  
Dispersive Media ◽  
Jacobi Elliptic Function ◽  
Nonlocal Symmetries ◽  
Interaction Solutions ◽  
Non Local ◽  
Truncated Painlevé Expansion

AbstractWe consider the generalised dispersive modified Benjamin–Bona–Mahony equation, which describes an approximation status for long surface wave existed in the non-linear dispersive media. By employing the truncated Painlevé expansion method, we derive its non-local symmetry and Bäcklund transformation. The non-local symmetry is localised by a new variable, which provides the corresponding non-local symmetry group and similarity reductions. Moreover, a direct method can be provided to construct a kind of finite symmetry transformation via the classic Lie point symmetry of the normal prolonged system. Finally, we find that the equation is a consistent Riccati expansion solvable system. With the help of the Jacobi elliptic function, we get its interaction solutions between solitary waves and cnoidal periodic waves.

Nonlocal Symmetry, CRE Solvability and Exact Interaction Solutions of the Asymmetric Nizhnik–Novikov–Veselov System

2015 ◽  
Vol 70 (9) ◽  
pp. 729-737 ◽  
Author(s):  
Xiaorui Hu ◽  
Yong Chen
Keyword(s):  
Symmetry Group ◽  
Soliton Solution ◽  
Special Form ◽  
Dark Soliton ◽  
Complex Interaction ◽  
Nonlocal Symmetry ◽  
Interaction Solutions ◽  
Function Expansion ◽  
Truncated Painlevé Expansion ◽  
Consistent Riccati Expansion

AbstractApplying the truncated Painlevé expansion to the (2+1)-dimensional asymmetric Nizhnik–Novikov–Veselov (ANNV) system, some Bäcklund transformations (BTs) including auto BT and non-auto BT are obtained. The auto BT leads to a nonlocal symmetry which corresponds to the residual of the truncated Painlevé expansion and the related nonlocal symmetry group is presented with the help of the localization procedure. Further, it is shown that the ANNV system has a consistent Riccati expansion (CRE). Stemming from the consistent tan-function expansion (CTE), which is a special form of CRE, some complex interaction solutions between soliton and arbitrary other seed waves of the ANNV system are readily constructed, such as bight-dark soliton solution, dark-dark soliton solution, soliton-cnoidal interaction solutions, solitoff solutions and so on.

scholarly journals CRE Solvability, Exact Soliton-Cnoidal Wave Interaction Solutions, and Nonlocal Symmetry for the Modified Boussinesq Equation

2016 ◽  
Vol 2016 ◽  
pp. 1-7 ◽  
Author(s):  
Wenguang Cheng ◽  
Biao Li
Keyword(s):  
Boussinesq Equation ◽  
Wave Interaction ◽  
Original System ◽  
Cnoidal Wave ◽  
Nonlocal Symmetry ◽  
Residual Symmetry ◽  
Symmetry Approach ◽  
Modified Boussinesq Equation ◽  
The Relationship ◽  
Truncated Painlevé Expansion

It is proved that the modified Boussinesq equation is consistent Riccati expansion (CRE) solvable; two types of special soliton-cnoidal wave interaction solution of the equation are explicitly given, which is difficult to be found by other traditional methods. Moreover, the nonlocal symmetry related to the consistent tanh expansion (CTE) and the residual symmetry from the truncated Painlevé expansion, as well as the relationship between them, are obtained. The residual symmetry is localized after embedding the original system in an enlarged one. The symmetry group transformation of the enlarged system is derived by applying the Lie point symmetry approach.

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