New interaction solutions of the similarity reduction for the integrable (2+1)-dimensional Boussinesq equation

2021 ◽  
Author(s):  
Hengchun Hu ◽  
Xiaodan Li
Keyword(s):  
Boussinesq Equation ◽  
Rational Solution ◽  
Symmetry Transformation ◽  
Nonlocal Symmetry ◽  
Similarity Reduction ◽  
Interaction Solution ◽  
Interaction Solutions ◽  
Dependent Variables ◽  
New Formula ◽  
Truncated Painlevé Expansion

The nonlocal symmetry of the new integrable [Formula: see text]-dimensional Boussinesq equation is studied by the standard truncated Painlevé expansion. This nonlocal symmetry can be localized to the Lie point symmetry of the prolonged system by introducing two auxiliary dependent variables. The corresponding finite symmetry transformation and similarity reduction related to the nonlocal symmetry of the new integrable [Formula: see text]-dimensional Boussinesq equation are studied. The rational solution, the triangle solution, two solitoff-interaction solution and the soliton–cnoidal interaction solutions for the new [Formula: see text]-dimensional Boussinesq equation are presented analytically and graphically by selecting the proper arbitrary constants.

scholarly journals Nonlocal Symmetry, Painlevé Integrable and Interaction Solutions for CKdV Equations

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1268
Author(s):  
Yarong Xia ◽  
Ruoxia Yao ◽  
Xiangpeng Xin ◽  
Yan Li
Keyword(s):  
Differential Equation ◽  
Nonlinear Partial Differential Equation ◽  
Three Dimensional ◽  
Dynamical Evolution ◽  
Symmetry Transformation ◽  
Nonlocal Symmetry ◽  
Similarity Reduction ◽  
Nonlocal Symmetries ◽  
Interaction Solutions ◽  
Dependent Variables

In this paper, we provide a method to construct nonlocal symmetry of nonlinear partial differential equation (PDE), and apply it to the CKdV (CKdV) equations. In order to localize the nonlocal symmetry of the CKdV equations, we introduce two suitable auxiliary dependent variables. Then the nonlocal symmetries are localized to Lie point symmetries and the CKdV equations are extended to a closed enlarged system with auxiliary dependent variables. Via solving initial-value problems, a finite symmetry transformation for the closed system is derived. Furthermore, by applying similarity reduction method to the enlarged system, the Painlevé integral property of the CKdV equations are proved by the Painlevé analysis of the reduced ODE (Ordinary differential equation), and the new interaction solutions between kink, bright soliton and cnoidal waves are given. The corresponding dynamical evolution graphs are depicted to present the property of interaction solutions. Moreover, With the help of Maple, we obtain the numerical analysis of the CKdV equations. combining with the two and three-dimensional graphs, we further analyze the shapes and properties of solutions u and v.

scholarly journals Nonlocal symmetry and interaction solutions for the new (3+1)-dimensional integrable Boussinesq equation

2022 ◽  
Author(s):  
Hengchun Hu ◽  
Xiaodan Li
Keyword(s):  
Soliton Solution ◽  
Boussinesq Equation ◽  
Symmetry Transformation ◽  
Expansion Method ◽  
Point Symmetry ◽  
Periodic Waves ◽  
Nonlocal Symmetry ◽  
Interaction Solutions ◽  
Nonlinear Excitations ◽  
Dependent Variables

The nonlocal symmetry of the new (3+1)-dimensional Boussinesq equation is obtained with the truncated Painlev\'{e} method. The nonlocal symmetry can be localized to the Lie point symmetry for the prolonged system by introducing auxiliary dependent variables. The finite symmetry transformation related to the nonlocal symmetry of the integrable (3+1)-dimensional Boussinesq equation is studied. Meanwhile, the new (3+1)-dimensional Boussinesq equation is proved by the consistent tanh expansion method and many interaction solutions among solitons and other types of nonlinear excitations such as cnoidal periodic waves and resonant soliton solution are given.

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.

Nonlocal Symmetry and its Applications in Perturbed mKdV Equation

2016 ◽  
Vol 71 (6) ◽  
pp. 557-564 ◽  
Author(s):  
Bo Ren ◽  
Ji Lin
Keyword(s):  
Constant Coefficient ◽  
Direct Method ◽  
Symmetry Transformation ◽  
Mkdv Equation ◽  
Nonlocal Symmetry ◽  
Rational Solutions ◽  
Initial Value ◽  
Interaction Solutions ◽  
Different Types ◽  
Painlevé Ii

AbstractBased on the modified direct method, the variable-coefficient perturbed mKdV equation is changed to the constant-coefficient perturbed mKdV equation. The truncated Painlevé method is applied to obtain the nonlocal symmetry of the constant-coefficient perturbed mKdV equation. By introducing one new dependent variable, the nonlocal symmetry can be localized to the Lie point symmetry. Thanks to the localization procedure, the finite symmetry transformation is presented by solving the initial value problem of the prolonged systems. Furthermore, many explicit interaction solutions among different types of solutions such as solitary waves, rational solutions, and Painlevé II solutions are obtained using the symmetry reduction method to the enlarged systems. Two special concrete soliton-cnoidal interaction solutions are studied in both analytical and graphical ways.

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 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.

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.

scholarly journals Rational Solutions and Their Interaction Solutions of the (3+1)-Dimensional Jimbo-Miwa Equation

2020 ◽  
Vol 2020 ◽  
pp. 1-18 ◽  
Author(s):  
Xiaomin Wang ◽  
Sudao Bilige ◽  
Jing Pang
Keyword(s):  
Nonlinear Evolution ◽  
Rational Solution ◽  
Three Dimensional ◽  
High Order ◽  
Lump Solution ◽  
Rational Solutions ◽  
Bilinear Method ◽  
Interaction Solution ◽  
Interaction Solutions ◽  
Contour Plots

In this paper, we gave a form of rational solution and their interaction solution to a nonlinear evolution equation. The rational solution contained lump solution, general lump solution, high-order lump solution, lump-type solution, etc. Their interaction solution contained the classical interaction solution, such as the lump-kink solution and the lump-soliton solution. As the example, by using the generalized bilinear method and symbolic computation Maple, we obtained abundant high-order lump-type solutions and their interaction solutions between lumps and other function solutions under certain constraints of the (3+1)-dimensional Jimbo-Miwa equation. Via three-dimensional plots, contour plots and density plots with the help of Maple, the physical characteristics and structures of these waves are described very well. These solutions have greatly enriched the exact solutions of the (3+1)-dimensional Jimbo-Miwa equation on the existing literature.

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.

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