scholarly journals Lie Symmetry Analysis of Kudryashov-Sinelshchikov Equation

2011 ◽  
Vol 2011 ◽  
pp. 1-9 ◽  
Author(s):  
Mehdi Nadjafikhah ◽  
Vahid Shirvani-Sh
Keyword(s):  
Nonlinear Evolution ◽  
Lie Symmetry ◽  
Lie Symmetry Analysis ◽  
Group Invariant ◽  
Lie Symmetry Method ◽  
Group Invariant Solutions ◽  
Fifth Order ◽  
Symmetry Method ◽  
Preliminary Classification

The Lie symmetry method is performed for the fifth-order nonlinear evolution Kudryashov-Sinelshchikov equation. We will find ones and two-dimensional optimal systems of Lie subalgebras. Furthermore, preliminary classification of its group-invariant solutions is investigated.

scholarly journals Symmetry Reductions for a Generalized Fifth Order KdV Equation

2017 ◽  
Vol 2 (2) ◽  
pp. 485-494 ◽  
Author(s):  
M.S. Bruzón ◽  
T.M. Garrido ◽  
R. de la Rosa
Keyword(s):  
Kdv Equation ◽  
Nonlinear Problems ◽  
Mathematical Physics ◽  
Lie Symmetry ◽  
Shallow Waters ◽  
Lie Symmetry Analysis ◽  
Group Invariant ◽  
Group Invariant Solutions ◽  
Fifth Order ◽  
Wave Phenomena

AbstractIn this work, Lie symmetry analysis is performed on a generalized fifth-order KdV equation. This equation describes many nonlinear problems with great physical interest in mathematical physics, nonlinear dynamics and plasma physics, among them it is a useful model for the description of wave phenomena in plasma and solid state and internal solitary waves in shallow waters. Group invariant solutions are obtained which allow us to transform the equation into ordinary differential equations. Furthermore, taking into account the conservation laws that the ordinary differential equation admits we reduce the order of the equations. Finally, we obtain some exact solutions.

Similarity Reduction and Exact Solutions of a Boussinesq-like Equation

2018 ◽  
Vol 73 (4) ◽  
pp. 357-362 ◽  
Author(s):  
Bo Zhang ◽  
Hengchun Hu
Keyword(s):  
Exact Solutions ◽  
Direct Method ◽  
Similarity Solutions ◽  
Lie Symmetry ◽  
Invariant Solutions ◽  
Similarity Reduction ◽  
Group Invariant ◽  
Lie Symmetry Method ◽  
Group Invariant Solutions ◽  
Symmetry Method

AbstractThe similarity reduction and similarity solutions of a Boussinesq-like equation are obtained by means of Clarkson and Kruskal (CK) direct method. By using Lie symmetry method, we also obtain the similarity reduction and group invariant solutions of the model. Further, we compare the results obtained by the CK direct method and Lie symmetry method, and we demonstrate the connection of the two methods.

Group Invariant Solutions and Conservation Laws of the Fornberg– Whitham Equation

2014 ◽  
Vol 69 (8-9) ◽  
pp. 489-496 ◽  
Author(s):  
Mir Sajjad Hashemi ◽  
Ali Haji-Badali ◽  
Parisa Vafadar
Keyword(s):  
Exact Solutions ◽  
Reduction Method ◽  
Lie Symmetry ◽  
First Integrals ◽  
Invariant Solutions ◽  
Analysis Method ◽  
Lie Symmetry Analysis ◽  
Group Invariant ◽  
Group Invariant Solutions ◽  
Whitham Equation

In this paper, we utilize the Lie symmetry analysis method to calculate new solutions for the Fornberg-Whitham equation (FWE). Applying a reduction method introduced by M. C. Nucci, exact solutions and first integrals of reduced ordinary differential equations (ODEs) are considered. Nonlinear self-adjointness of the FWE is proved and conserved vectors are computed

scholarly journals Lie Symmetry Analysis of Burgers Equation and the Euler Equation on a Time Scale

Symmetry ◽  
2019 ◽  
Vol 12 (1) ◽  
pp. 10 ◽  
Author(s):  
Mingshuo Liu ◽  
Huanhe Dong ◽  
Yong Fang ◽  
Yong Zhang
Keyword(s):  
Euler Equation ◽  
Time Scale ◽  
Flow Model ◽  
Burgers Equation ◽  
Lie Symmetry ◽  
Symmetry Analysis ◽  
Lie Symmetry Analysis ◽  
Group Invariant ◽  
Discrete Equations ◽  
Group Invariant Solutions

As a powerful tool that can be used to solve both continuous and discrete equations, the Lie symmetry analysis of dynamical systems on a time scale is investigated. Applying the method to the Burgers equation and Euler equation, we get the symmetry of the equation and single parameter groups on a time scale. Some group invariant solutions in explicit form for the traffic flow model simulated by a Burgers equation and Euler equation with a Coriolis force on a time scale are studied.

scholarly journals Residual Symmetry Reduction and Consistent Riccati Expansion to a Nonlinear Evolution Equation

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-9 ◽  
Author(s):  
Lamine Thiam ◽  
Xi-zhong Liu
Keyword(s):  
Evolution Equation ◽  
Nonlinear Evolution Equation ◽  
Nonlinear Evolution ◽  
Symmetry Reduction ◽  
Lie Symmetry ◽  
Residual Symmetry ◽  
Interaction Solutions ◽  
Lie Symmetry Method ◽  
Consistent Riccati Expansion ◽  
Symmetry Method

The residual symmetry of a (1 + 1)-dimensional nonlinear evolution equation (NLEE) ut+uxxx−6u2ux+6λux=0 is obtained through Painlevé expansion. By introducing a new dependent variable, the residual symmetry is localized into Lie point symmetry in an enlarged system, and the related symmetry reduction solutions are obtained using the standard Lie symmetry method. Furthermore, the (1 + 1)-dimensional NLEE equation is proved to be integrable in the sense of having a consistent Riccati expansion (CRE), and some new Bäcklund transformations (BTs) are given. In addition, some explicitly expressed solutions including interaction solutions between soliton and cnoidal waves are derived from these BTs.

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.

Symmetry analysis and invariant solutions of (3 + 1)-dimensional Kadomtsev–Petviashvili equation

2018 ◽  
Vol 15 (08) ◽  
pp. 1850125 ◽  
Author(s):  
Vishakha Jadaun ◽  
Sachin Kumar
Keyword(s):  
Lie Algebra ◽  
Zeta Functions ◽  
Lie Symmetry ◽  
Symmetry Analysis ◽  
Lie Symmetry Analysis ◽  
Power Series Method ◽  
Lie Symmetry Method ◽  
Petviashvili Equation ◽  
Symmetry Method ◽  
Similarity Reductions

Based on Lie symmetry analysis, we study nonlinear waves in fluid mechanics with strong spatial dispersion. The similarity reductions and exact solutions are obtained based on the optimal system and power series method. We obtain the infinitesimal generators, commutator table of Lie algebra, symmetry group and similarity reductions for the [Formula: see text]-dimensional Kadomtsev–Petviashvili equation. For different Lie algebra, Lie symmetry method reduces Kadomtsev–Petviashvili equation into various ordinary differential equations (ODEs). Some of the solutions of [Formula: see text]-dimensional Kadomtsev–Petviashvili equation are of the forms — traveling waves, Weierstrass’s elliptic and Zeta functions and exponential functions.

Multiple invariant solutions of the 3D potential Yu–Toda–Sasa–Fukuyama equation via symmetry technique

2020 ◽  
Vol 34 (20) ◽  
pp. 2050188
Author(s):  
Rodica Cimpoiasu
Keyword(s):  
Invariant Solutions ◽  
Lie Symmetry Analysis ◽  
Dynamical Models ◽  
Nonlinear Dynamical ◽  
Group Invariant ◽  
Potential Form ◽  
Multiple Structures ◽  
Group Invariant Solutions ◽  
First Time ◽  
Symmetry Method

This paper studies the potential form of the 3D potential Yu–Toda–Sasa–Fukuyama equation through the perspective of Lie symmetry analysis. This technique combined with symbolic computations does prove that the general Lie operator depends on five parameters and six independent arbitrary functions that are variable in respect to time. The group invariant solutions associated to some 1D subalgebras are systematically construct and they do involve arbitrary functions. When these functions are expressed under several specific forms, the associated wave solutions possess multiple structures. Graphical representations of some particular solutions are as well provided. As far as we know such general solutions are presented here for the first time and do indicate the symmetry method to be applied in order to solve other multidimensional, integrable, or nonintegrable nonlinear dynamical models.

A (2+1)-dimensional Korteweg–de Vries type equation in water waves: Lie symmetry analysis; multiple exp-function method; conservation laws

2016 ◽  
Vol 30 (28n29) ◽  
pp. 1640001 ◽  
Author(s):  
Abdullahi Rashid Adem
Keyword(s):  
Conservation Laws ◽  
Water Waves ◽  
Function Method ◽  
Type Equation ◽  
Lie Symmetry ◽  
Lie Symmetry Analysis ◽  
Shallow Water Waves ◽  
Lie Symmetry Method ◽  
De Vries ◽  
Symmetry Method

We consider a (2+1)-dimensional Korteweg–de Vries type equation which models the shallow-water waves, surface and internal waves. In the analysis, we use the Lie symmetry method and the multiple exp-function method. Furthermore, conservation laws are computed using the multiplier method.

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