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.

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.

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.

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.

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 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 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.

Symmetry reduction and new interaction solutions for the negative-order potential KdV equation

2020 ◽  
pp. 2150108
Author(s):  
Hengchun Hu ◽  
Zhenya Zhang
Keyword(s):  
Kdv Equation ◽  
Symmetry Transformation ◽  
Expansion Method ◽  
Local Symmetry ◽  
Similarity Solutions ◽  
Seed Solution ◽  
Interaction Solutions ◽  
New Exact Solutions ◽  
Negative Order ◽  
Non Local

New soliton–cnoidal interaction solutions for the negative-order potential KdV equation are studied with the help of the consistent tanh expansion method. The non-local symmetry for the negative-order potential KdV equation is derived from the truncated Painlevé expansion method. The non-local symmetry is transformed into the standard Lie point symmetry by introducing new dependent variables and the finite symmetry transformation is also presented to construct new exact solutions with non-zero seed solution. The similarity solutions of the enlarged negative-order potential KdV system are obtained through different constant selections.

Nonlocal symmetries and the nth finite symmetry transformation or AKNS system

2018 ◽  
Vol 32 (27) ◽  
pp. 1850332
Author(s):  
Xiazhi Hao ◽  
Yinping Liu ◽  
Xiaoyan Tang ◽  
Zhibin Li ◽  
Wen-Xiu Ma
Keyword(s):  
Exact Solutions ◽  
Symmetry Transformation ◽  
Original System ◽  
Point Symmetry ◽  
Auxiliary Variables ◽  
Nonlocal Symmetry ◽  
Nonlocal Symmetries ◽  
Akns System ◽  
Lie Point Symmetry ◽  
Physical Phenomena

In this paper, by introduction of pseudopotentials, the nonlocal symmetry is obtained for the Ablowitz–Kaup–Newell–Segur system, which is used to describe many physical phenomena in different applications. Together with some auxiliary variables, this kind of nonlocal symmetry can be localized to Lie point symmetry and the corresponding once finite symmetry transformation is calculated for both the original system and the prolonged system. Furthermore, the nth finite symmetry transformation represented in terms of determinant and exact solutions are derived.

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