Symmetry reduction, exact solutions and conservation laws of the Bogoyavlenskii equation

2019 ◽  
Vol 35 (01) ◽  
pp. 1950339
Author(s):  
Zhenli Wang ◽  
Chuan Zhong Li ◽  
Lihua Zhang
Keyword(s):  
Conservation Laws ◽  
Exact Solutions ◽  
Hyperbolic Function ◽  
Group Invariant ◽  
New Exact Solutions ◽  
Power Series Solutions ◽  
Hyperbolic Function Solutions ◽  
Trigonometric Function Solutions ◽  
Group Invariant Solutions ◽  
Symmetry Method

In this paper, by applying the direct symmetry method, we obtain the symmetry reductions, group invariant solutions and some new exact solutions of the Bogoyavlenskii equation, which include hyperbolic function solutions, trigonometric function solutions and power series solutions. We also give the conservation laws of the Bogoyavlenskii equation.

Exact Solutions for Coupled mKdV Equations by a New Symbolic Computation Method

2010 ◽  
Vol 20-23 ◽  
pp. 184-189 ◽  
Author(s):  
Bang Qing Li ◽  
Yu Lan Ma
Keyword(s):  
Exact Solutions ◽  
Symbolic Computation ◽  
Evolution Equations ◽  
Expansion Method ◽  
Nonlinear Evolution Equations ◽  
Hyperbolic Function ◽  
Computation Method ◽  
New Exact Solutions ◽  
Hyperbolic Function Solutions ◽  
Trigonometric Function Solutions

By introducing (G′/G)-expansion method and symbolic computation software MAPLE, two types of new exact solutions are constructed for coupled mKdV equations. The solutions included trigonometric function solutions and hyperbolic function solutions. The procedure is concise and straightforward, and the method is also helpful to find exact solutions for other nonlinear evolution equations.

The Multiple Exact Solutions for the Variable Coefficient KdV Equation

2014 ◽  
Vol 513-517 ◽  
pp. 4474-4477
Author(s):  
Lin Tian ◽  
Jia Qing Miao
Keyword(s):  
Exact Solutions ◽  
Variable Coefficient ◽  
Kdv Equation ◽  
Hyperbolic Function ◽  
New Exact Solutions ◽  
Kdv Equations ◽  
Hyperbolic Function Solutions ◽  
Trigonometric Function Solutions ◽  
Auxiliary Differential Equation Method ◽  
Differential Equation Method

The auxiliary differential equation method has recently been proposed ,It is introduced to construct more new exact solutions for the variable coefficient KdV equations. As a result , hyperbolic function solutions, trigonometric function solutions, and elliptic function solutions rational function solutions with parameters are obtained.

scholarly journals Exact solutions with arbitrary functions of the (4+1)-dimensional fokas equation

2019 ◽  
Vol 23 (4) ◽  
pp. 2403-2411 ◽  
Author(s):  
Bo Xu ◽  
Sheng Zhang
Keyword(s):  
Exact Solutions ◽  
Elliptic Function ◽  
Expansion Method ◽  
Trigonometric Function ◽  
High Dimensional ◽  
Hyperbolic Function ◽  
Jacobi Elliptic Function ◽  
New Exact Solutions ◽  
Hyperbolic Function Solutions ◽  
Trigonometric Function Solutions

In this paper, the (4+1)-dimensional Fokas equation is solved by the generalized F-expansion method, and new exact solutions with arbitrary functions are obtained. The obtained solutions include Jacobi elliptic function solutions, hyperbolic function solutions and trigonometric function solutions. It is shown that the generalized F-expansion method can be used for constructing exact solutions with arbitrary functions of some other high dimensional partial differential equations in fluids.

Analytic study of solutions for a two-component Novikov system

2020 ◽  
pp. 2150025
Author(s):  
Hui Gao ◽  
Gangwei Wang
Keyword(s):  
Exact Solutions ◽  
Stationary Solutions ◽  
Invariant Solutions ◽  
Group Invariant ◽  
New Exact Solutions ◽  
Two Component ◽  
Group Invariant Solutions ◽  
Symmetry Transformations ◽  
Symmetry Method ◽  
Similarity Reductions

Under investigation in this paper is a two-component Novikov system (also called Geng-Xue equation), which was proposed by Geng and Xue in 2009. Firstly, via the Lie symmetry method, infinitesimal generators, commutator table of Lie algebra and symmetry groups of the two-component Novikov system are presented. At the same time, some group invariant solutions are computed through similarity reductions. In particular, we construct peakon solution by applying the distribution theory. In addition, based on obtained group invariant solutions and symmetry transformations, we derive some new exact solutions, which include stationary solutions, smooth solutions, and a weak solution. The analytical properties to some of group invariant solutions and new exact solutions are discussed, such as decay, asymptotic behavior, and boundedness.

scholarly journals Bifurcation and exact solutions for the ($2+1$)-dimensional conformable time-fractional Zoomeron equation

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Zhao Li ◽  
Tianyong Han
Keyword(s):  
Exact Solutions ◽  
Traveling Wave Solutions ◽  
Expansion Method ◽  
Phase Portraits ◽  
Hyperbolic Function ◽  
Bifurcation Method ◽  
New Exact Solutions ◽  
Hyperbolic Function Solutions ◽  
Trigonometric Function Solutions ◽  
Reliable Methods

AbstractIn this paper, the bifurcation and new exact solutions for the ($2+1$ 2 + 1 )-dimensional conformable time-fractional Zoomeron equation are investigated by utilizing two reliable methods, which are generalized $(G'/G)$ ( G ′ / G ) -expansion method and the integral bifurcation method. The exact solutions of the ($2+1$ 2 + 1 )-dimensional conformable time-fractional Zoomeron equation are obtained by utilizing the generalized $(G'/G)$ ( G ′ / G ) -expansion method, these solutions are classified as hyperbolic function solutions, trigonometric function solutions, and rational function solutions. Giving different parameter conditions, many integral bifurcations, phase portraits, and traveling wave solutions for the equation are obtained via the integral bifurcation method. Graphical representations of different kinds of the exact solutions reveal that the two methods are of significance for constructing the exact solutions of fractional partial differential equation.

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.

scholarly journals A New Variable-Coefficient Riccati Subequation Method for Solving Nonlinear Lattice Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Fanwei Meng
Keyword(s):  
Variable Coefficient ◽  
Toda Lattice ◽  
Traveling Wave Solutions ◽  
Trigonometric Function ◽  
Hyperbolic Function ◽  
Lattice Equation ◽  
New Exact Solutions ◽  
Hyperbolic Function Solutions ◽  
Trigonometric Function Solutions ◽  
Toda Lattice Equation

We propose a new variable-coefficient Riccati subequation method to establish new exact solutions for nonlinear differential-difference equations. For illustrating the validity of this method, we apply it to the discrete (2 + 1)-dimensional Toda lattice equation. As a result, some new and generalized traveling wave solutions including hyperbolic function solutions, trigonometric function solutions, and rational function solutions are obtained.

scholarly journals The Improvedexp⁡-Φξ-Expansion Method and New Exact Solutions of Nonlinear Evolution Equations in Mathematical Physics

2019 ◽  
Vol 2019 ◽  
pp. 1-8 ◽  
Author(s):  
Guiying Chen ◽  
Xiangpeng Xin ◽  
Hanze Liu
Keyword(s):  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Traveling Wave Solutions ◽  
Expansion Method ◽  
Nonlinear Evolution Equations ◽  
Hyperbolic Function ◽  
Jacobi Elliptic Function ◽  
New Exact Solutions ◽  
Hyperbolic Function Solutions ◽  
Trigonometric Function Solutions

Theexp(-Φ(ξ))-expansion method is improved by presenting a new auxiliary ordinary differential equation forΦ(ξ). By using this method, new exact traveling wave solutions of two important nonlinear evolution equations, i.e., the ill-posed Boussinesq equation and the unstable nonlinear Schrödinger equation, are constructed. The obtained solutions contain Jacobi elliptic function solutions which can be degenerated to the hyperbolic function solutions and the trigonometric function solutions. The present method is very concise and effective and can be applied to other types of nonlinear evolution equations.

scholarly journals Applications of an Extended -Expansion Method to Find Exact Solutions of Nonlinear PDEs in Mathematical Physics

2010 ◽  
Vol 2010 ◽  
pp. 1-19 ◽  
Author(s):  
E. M. E. Zayed ◽  
Shorog Al-Joudi
Keyword(s):  
Exact Solutions ◽  
Traveling Wave Solutions ◽  
Boussinesq Equations ◽  
Expansion Method ◽  
Nonlinear Pdes ◽  
Hyperbolic Function ◽  
Soliton Equation ◽  
New Exact Solutions ◽  
Wave Solutions ◽  
Hyperbolic Function Solutions

We construct the traveling wave solutions of the (1+1)-dimensional modified Benjamin-Bona-Mahony equation, the (2+1)-dimensional typical breaking soliton equation, the (1+1)-dimensional classical Boussinesq equations, and the (2+1)-dimensional Broer-Kaup-Kuperschmidt equations by using an extended -expansion method, whereGsatisfies the second-order linear ordinary differential equation. By using this method, new exact solutions involving parameters, expressed by three types of functions which are hyperbolic, trigonometric and rational function solutions, are obtained. When the parameters are taken as special values, some solitary wave solutions are derived from the hyperbolic function solutions.

Time-fractional inhomogeneous nonlinear diffusion equation: Symmetries, conservation laws, invariant subspaces, and exact solutions

2018 ◽  
Vol 32 (32) ◽  
pp. 1850401 ◽  
Author(s):  
Wei Feng ◽  
Songlin Zhao
Keyword(s):  
Diffusion Equation ◽  
Conservation Laws ◽  
Exact Solutions ◽  
Nonlinear Diffusion ◽  
Fractional Derivatives ◽  
Nonlinear Diffusion Equation ◽  
Group Invariant ◽  
Liouville Fractional Derivative ◽  
Group Invariant Solutions ◽  
Dimensional Dynamical System

In this paper, a class of time-fractional inhomogeneous nonlinear diffusion equation (tFINDE) with Riemann–Liouville fractional derivative is studied. All point symmetries admitted by this equation are derived. The optimal system of one-dimensional subalgebras is classified to perform the symmetry reductions. It is shown that the tFINDE can be reduced to fractional ordinary differential equations (FODEs), including Erdélyi–Kober fractional derivatives. As the results, some explicit group-invariant solutions are obtained. Through nonlinear self-adjointness, all conservation laws admitted by tFINDE arising from these point symmetry groups are listed. The method of invariant subspace is also applied to reduce the tFINDE to a two-dimensional dynamical system (DS). The admitted point symmetries of DS are used to derive the exact solutions of DS, which determine the exact solutions of the original tFINDE.

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