Lie symmetry analysis and invariant solutions of (3+1)-dimensional Date–Jimbo–Kashiwara–Miwa equation

2022 ◽  
Author(s):  
Hengchun Hu ◽  
Runlan Sun
Keyword(s):  
Vector Fields ◽  
Periodic Wave ◽  
Lie Symmetry ◽  
Periodic Wave Solution ◽  
Lie Symmetry Analysis ◽  
Similarity Reduction ◽  
Infinitesimal Generators ◽  
New Exact Solutions ◽  
Trigonometric Function Solutions ◽  
Function Selection

In this paper, the (3+1)-dimensional constant coefficient of Date–Jimbo–Kashiwara–Miwa (CCDJKM) equation is studied. All of the vector fields, infinitesimal generators, Lie symmetry reductions and different similarity reduction solutions are constructed. Due to the arbitrary functions in the infinitesimal generators, the (3+1)-dimensional CCDJKM equation can further be reduced to many (2+1)-dimensional partial differential equations. The explicit solutions of the similarity reduction equations, which include the quasi-periodic wave solution, the interaction solution between the periodic wave and a kink soliton and the trigonometric function solutions, are constructed with proper arbitrary function selection, and these new exact solutions are given out analytically and graphically.

Lie symmetry analysis, explicit solutions and conservation laws of the time fractional Clannish Random Walker’s Parabolic equation

2020 ◽  
pp. 2150074
Author(s):  
Panpan Wang ◽  
Wenrui Shan ◽  
Ying Wang ◽  
Qianqian Li
Keyword(s):  
Power Series ◽  
Conservation Laws ◽  
Vector Fields ◽  
Lie Symmetry ◽  
Symmetry Analysis ◽  
Series Solutions ◽  
Lie Symmetry Analysis ◽  
Similarity Reduction ◽  
Power Series Solutions ◽  
Conservation Theorem

In this paper, we mainly study the symmetry analysis and conservation laws of the time fractional Clannish Random Walker’s Parabolic (CRWP) equation. The vector fields and similarity reduction of the time fractional CRWP equation are obtained. In addition, based on the power series theory, a simple and effective approach for constructing explicit power series solutions is proposed. Finally, by use of the new conservation theorem, the conservation laws of the time fractional CRWP equation are constructed.

scholarly journals Lie Symmetry Analysis and Exact Solutions of Generalized Fractional Zakharov-Kuznetsov Equations

Symmetry ◽  
2019 ◽  
Vol 11 (5) ◽  
pp. 601 ◽  
Author(s):  
Changzhao Li ◽  
Juan Zhang
Keyword(s):  
Exact Solutions ◽  
Vector Fields ◽  
Lie Symmetry ◽  
Symmetry Analysis ◽  
Lie Point Symmetries ◽  
Lie Symmetry Analysis ◽  
Symmetry Reductions ◽  
New Exact Solutions ◽  
Symmetry Properties

This paper considers the Lie symmetry analysis of a class of fractional Zakharov-Kuznetsov equations. We systematically show the procedure to obtain the Lie point symmetries for the equation. Accordingly, we study the vector fields of this equation. Meantime, the symmetry reductions of this equation are performed. Finally, by employing the obtained symmetry properties, we can get some new exact solutions to this fractional Zakharov-Kuznetsov equation.

scholarly journals Some new exact solutions of $(3+1)$-dimensional Burgers system via Lie symmetry analysis

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Elnaz Alimirzaluo ◽  
Mehdi Nadjafikhah ◽  
Jalil Manafian
Keyword(s):  
Vector Fields ◽  
Nonlinear Differential Equations ◽  
Three Dimensional ◽  
Lie Symmetry ◽  
Symmetry Analysis ◽  
Optimal System ◽  
Lie Symmetry Analysis ◽  
3 Dimensional ◽  
New Exact Solutions ◽  
Burgers System

AbstractIn this paper, by using the Lie symmetry analysis, all of the geometric vector fields of the $(3+1)$ ( 3 + 1 ) -Burgers system are obtained. We find the 1, 2, and 3-dimensional optimal system of the Burger system and then by applying the 3-dimensional optimal system reduce the order of the system. Also the nonclassical symmetries of the $(3+1)$ ( 3 + 1 ) -Burgers system will be found by employing nonclassical methods. Finally, the ansatz solutions of BS equations with the aid of the tanh method has been presented. The achieved solutions are investigated through two- and three-dimensional plots for different values of parameters. The analytical simulations are presented to ensure the efficiency of the considered technique. The behavior of the obtained results for multiple cases of symmetries is captured in the present framework. The outcomes of the present investigation show that the considered scheme is efficient and powerful to solve nonlinear differential equations that arise in the sciences and technology.

scholarly journals Symmetry Reduction, Exact Solutions, and Conservation Laws of the ZK-BBM Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-9
Author(s):  
Wenbin Zhang ◽  
Jiangbo Zhou ◽  
Sunil Kumar
Keyword(s):  
Conservation Laws ◽  
Exact Solutions ◽  
Invariant Solution ◽  
Lie Symmetry ◽  
Similarity Reduction ◽  
Group Invariant ◽  
New Exact Solutions ◽  
Reduction Group ◽  
The Given ◽  
Bbm Equation

Employing the classical Lie method, we obtain the symmetries of the ZK-BBM equation. Applying the given Lie symmetry, we obtain the similarity reduction, group invariant solution, and new exact solutions. We also obtain the conservation laws of ZK-BBM equation of the corresponding Lie symmetry.

Lie symmetry analysis for obtaining exact soliton solutions of generalized Camassa–Holm–Kadomtsev–Petviashvili equation

2020 ◽  
pp. 2150028
Author(s):  
Sachin Kumar ◽  
Monika Niwas ◽  
Ihsanullah Hamid
Keyword(s):  
Rational Function ◽  
Vector Fields ◽  
Soliton Solutions ◽  
Lie Symmetry ◽  
Trigonometric Function ◽  
Symmetry Analysis ◽  
Lie Symmetry Analysis ◽  
Nonlinear Odes ◽  
Petviashvili Equation ◽  
Exact Soliton Solutions

The prime objective of this paper is to obtain the exact soliton solutions by applying the two mathematical techniques, namely, Lie symmetry analysis and generalized exponential rational function (GERF) method to the (2+1)-dimensional generalized Camassa–Holm–Kadomtsev–Petviashvili (g-CHKP) equation. First, we obtain Lie infinitesimals, possible vector fields, and commutative product of vectors for the g-CHKP equation. By the means of symmetry reductions, the g-CHKP equation reduced to various nonlinear ODEs. Subsequently, we implement the GERF method to the reduced ODEs with the help of computerized symbolic computation in Mathematica. Some abundant exact soliton solutions are obtained in the shapes of different dynamical structures of multiple-solitons like one-soliton, two-soliton, three-soliton, four-soliton, bell-shaped solitons, lump-type soliton, kink-type soliton, periodic solitary wave solutions, trigonometric function, hyperbolic trigonometric function, exponential function, and rational function solutions. Consequently, the dynamical structures of attained exact analytical solutions are discussed through 3D-plots via numerical simulation. A comparison with other results is also presented.

Lie Symmetry Analysis, Analytical Solutions, and Conservation Laws of the Generalised Whitham–Broer–Kaup–Like Equations

2017 ◽  
Vol 72 (3) ◽  
pp. 269-279 ◽  
Author(s):  
Xiu-Bin Wang ◽  
Shou-Fu Tian ◽  
Chun-Yan Qin ◽  
Tian-Tian Zhang
Keyword(s):  
Power Series ◽  
Conservation Laws ◽  
Vector Fields ◽  
Wave Equations ◽  
Boussinesq Equations ◽  
Lie Symmetry ◽  
Symmetry Analysis ◽  
Optimal System ◽  
Lie Symmetry Analysis ◽  
Bidirectional Propagation

AbstractIn this article, a generalised Whitham–Broer–Kaup–Like (WBKL) equations is investigated, which can describe the bidirectional propagation of long waves in shallow water. The equations can be reduced to the dispersive long wave equations, variant Boussinesq equations, Whitham–Broer–Kaup–Like equations, etc. The Lie symmetry analysis method is used to consider the vector fields and optimal system of the equations. The similarity reductions are given on the basic of the optimal system. Furthermore, the power series solutions are derived by using the power series theory. Finally, based on a new theorem of conservation laws, the conservation laws associated with symmetries of this equations are constructed with a detailed derivation.

scholarly journals Lie Symmetry Analysis for Soliton Solutions of Generalised Kadomtsev-Petviashvili-Boussinesq Equation in (3+1)-dimensions

2021 ◽  
Author(s):  
VISHAKHA JADAUN ◽  
Navnit Jha ◽  
Sachin Ramola
Keyword(s):  
Vector Fields ◽  
Evolution Equations ◽  
Invariant Solution ◽  
Soliton Solutions ◽  
Dark Soliton ◽  
Lie Symmetry ◽  
Similarity Reduction ◽  
Closed Form Solutions ◽  
Exact Invariant ◽  
Infinitesimal Transformations

The Lie group of infinitesimal transformations technique and similarity reduction is performed for obtaining an exact invariant solution to generalized Kadomstev-Petviashvili-Boussinesq (gKPB) equation in (3+1)-dimensions. We obtain generators of infinitesimal transformations, which provide us a set of Lie algebras. In addition, we get geometric vector fields, a commutator table of Lie algebra, and a group of symmetries. It is observed that the analytic solution (closed-form solutions) to the nonlinear gKPB evolution equations can easily be treated employing the Lie symmetry technique. A detailed geometrical framework related to the nature of the solutions possessing traveling wave, bright and dark soliton, standing wave with multiple breathers, and one-dimensional kink, for the appropriate values of the parameters involved.

Lie symmetry analysis and some exact solutions for modified Zakharov–Kuznetsov equation

2018 ◽  
Vol 32 (31) ◽  
pp. 1850383 ◽  
Author(s):  
Xuan Zhou ◽  
Wenrui Shan ◽  
Zhilei Niu ◽  
Pengcheng Xiao ◽  
Ying Wang
Keyword(s):  
Exact Solutions ◽  
Detailed Analysis ◽  
Symmetry Group ◽  
Nonlinear Waves ◽  
Lie Symmetry ◽  
Lie Symmetry Analysis ◽  
One Dimensional ◽  
Infinitesimal Generators ◽  
Lie Symmetry Method ◽  
Symmetry Method

In this study, the Lie symmetry method is used to perform detailed analysis on the modified Zakharov–Kuznetsov equation. We have obtained the infinitesimal generators, commutator table of Lie algebra and symmetry group. In addition to that, optimal system of one-dimensional subalgebras up to conjugacy is derived and used to construct distinct exact solutions. These solutions describe the dynamics of nonlinear waves in isothermal multicomponent magnetized plasmas.

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