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.

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.

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 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 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.

Direct Similarity Reduction and New Exact Solutions for the Variable-Coefficient Kadomtsev–Petviashvili Equation

2015 ◽  
Vol 70 (6) ◽  
pp. 445-450 ◽  
Author(s):  
Rehab M. El-Shiekh
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Variable Coefficient ◽  
Nonlinear Partial Differential Equation ◽  
Direct Methods ◽  
Periodic Wave ◽  
Variable Coefficients ◽  
Nonlinear Phenomena ◽  
Similarity Reduction ◽  
Petviashvili Equation

AbstractIn this paper, the generalized (3+1)-dimensional variable-coefficient Kadomtsev–Petviashvili equation (VCKPE), which can describe nonlinear phenomena in fluids or plasmas, is studied by using two different Clarkson and Kruskal (CK) direct methods, namely, the classical CK and the modified enlarged CK method. A similarity reduction to a (2+1)-dimensional nonlinear partial differential equation and a direct similarity reduction to a nonlinear ordinary differential equation are obtained, respectively. By solving the reduced ordinary differential equation, new solitary, periodic, and singular solutions for the VCKPE are obtained. Some figures for the soliton and periodic wave solutions are given to reflect the effect of the variable coefficients on the solution propagation. Finally, the comparison between the two different CK techniques indicates that the modified enlarged CK technique is clearly more powerful and simple than the classical CK technique.

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.

Characteristics of Abundant Lumps and Interaction Solutions in the (4+1)-Dimensional Nonlinear Partial Differential Equation

2020 ◽  
Vol 21 (3-4) ◽  
pp. 283-289
Author(s):  
Xiu-Bin Wang ◽  
Bo Han
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Rogue Wave ◽  
Nonlinear Partial Differential Equation ◽  
Soliton Solutions ◽  
Nonlinear Phenomena ◽  
Rational Solutions ◽  
Interaction Solutions ◽  
Rogue Wave Solution ◽  
Physics Model

AbstractIn this work, the (4+1)-dimensional Fokas equation, which is an important physics model, is under investigation. Based on the obtained soliton solutions, the new rational solutions are successfully constructed. Moreover, based on its bilinear formalism, a concise method is employed to explicitly construct its rogue-wave solution and interaction solution with an ansätz function. Finally, the main characteristics of these solutions are graphically discussed. Our results can be helpful for explaining some related nonlinear phenomena.

scholarly journals Breather Wave Solutions and Interaction Solutions for Two Mixed Calogero-Bogoyavlenskii-Schiff and Bogoyavlensky-Konopelchenko Equations

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Hongcai Ma ◽  
Caoyin Zhang ◽  
Aiping Deng
Keyword(s):  
Differential Equation ◽  
Three Dimensional ◽  
Test Method ◽  
Dynamic Behaviors ◽  
Interaction Solutions ◽  
Wave Solutions

In this paper, based on a bilinear differential equation, we study the breather wave solutions by employing the extended homoclinic test method. By constructing the different forms, we also consider the interaction solutions. Furthermore, it is natural to analyse dynamic behaviors of three-dimensional plots.

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