New Exact Solutions of Date Jimbo Kashiwara Miwa Equation Using Lie Symmetry Groups

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Khudija Bibi ◽  
Khalil Ahmad
Keyword(s):  
Exact Solutions ◽  
Linear Partial Differential Equation ◽  
Lie Symmetry ◽  
Lie Symmetry Method ◽  
New Exact Solutions ◽  
Independent Variable ◽  
Symmetry Generators ◽  
Logarithmic Functions ◽  
Djkm Equation ◽  
Symmetry Method

In this article, new exact solutions of 2 + 1 -dimensional Date Jimbo Kashiwara Miwa (DJKM) equation are constructed by applying the Lie symmetry method. By considering similarity variables obtained through Lie symmetry generators, considered 2 + 1 -dimensional DJKM equation is transformed into a linear partial differential equation with reduction of one independent variable. Afterwards by using Lie symmetry generators of this linear PDE, different invariant solutions involving exponential and logarithmic functions are explored which lead to the new exact solutions of the DJKM equation. Graphical representations of the obtained solutions are also presented to show the significance of the current work.

scholarly journals Analysis of the Symmetries and Conservation Laws of the Nonlinear Jaulent-Miodek Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Mehdi Nadjafikhah ◽  
Mostafa Hesamiarshad
Keyword(s):  
Conservation Laws ◽  
Symmetry Group ◽  
Symbolic Computation ◽  
Lie Symmetry ◽  
Lie Symmetry Method ◽  
Reduced Equations ◽  
Symmetry Generators ◽  
Symmetry Method

Lie symmetry method is performed for the nonlinear Jaulent-Miodek equation. We will find the symmetry group and optimal systems of Lie subalgebras. The Lie invariants associated with the symmetry generators as well as the corresponding similarity reduced equations are also pointed out. And conservation laws of the J-M equation are presented with two steps: firstly, finding multipliers for computation of conservation laws and, secondly, symbolic computation of conservation laws will be applied.

scholarly journals Time fractional Kupershmidt equation: symmetry analysis and explicit series solution with convergence analysis

2019 ◽  
Vol 27 (2) ◽  
pp. 171-185 ◽  
Author(s):  
Astha Chauhan ◽  
Rajan Arora
Keyword(s):  
Differential Equation ◽  
Power Series ◽  
Analytic Solutions ◽  
Lie Symmetry ◽  
Symmetry Analysis ◽  
Power Series Method ◽  
Lie Symmetry Method ◽  
Liouville Fractional Derivative ◽  
Symmetry Generators ◽  
Symmetry Method

AbstractIn this work, the fractional Lie symmetry method is applied for symmetry analysis of time fractional Kupershmidt equation. Using the Lie symmetry method, the symmetry generators for time fractional Kupershmidt equation are obtained with Riemann-Liouville fractional derivative. With the help of symmetry generators, the fractional partial differential equation is reduced into the fractional ordinary differential equation using Erdélyi-Kober fractional differential operator. The conservation laws are determined for the time fractional Kupershmidt equation with the help of new conservation theorem and fractional Noether operators. The explicit analytic solutions of fractional Kupershmidt equation are obtained using the power series method. Also, the convergence of the power series solutions is discussed by using the implicit function theorem.

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.

scholarly journals Parametric conditions and exact solution for the Duffing-Van der Pol class of equations

2018 ◽  
Vol 40 (3) ◽  
pp. 251-264
Author(s):  
Dao Huy Bich ◽  
Nguyen Dang Bich
Keyword(s):  
Exact Solution ◽  
Exact Solutions ◽  
Lower Order ◽  
Main Idea ◽  
Lie Symmetry ◽  
First Integrals ◽  
Original Equation ◽  
Van Der Pol ◽  
Lie Symmetry Method ◽  
Symmetry Method

This paper presents a methodology to find the exact solution and respective parametric conditions to the Duffing-Van der Pol class of equations. The supposed method in this paper is different from the Prelle and Singer method and the Lie symmetry method. The main idea of the supposed method is implemented in finding the first integrals of the original equation and leading this equation to a solved equation of lower order to which the exact solution can be obtained. As results the parametric conditions and the exact solutions in parametric forms are indicated. The algorithm for determining integral constants and the investigation of solution characteristics are considered.

scholarly journals Exact Solutions of Generalized Boussinesq-Burgers Equations and (2+1)-Dimensional Davey-Stewartson Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-8 ◽  
Author(s):  
Isaiah Elvis Mhlanga ◽  
Chaudry Masood Khalique
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Nonlinear Partial Differential Equations ◽  
Travelling Wave ◽  
Lie Symmetry ◽  
Coupled Systems ◽  
Burgers Equations ◽  
Lie Symmetry Method ◽  
Symmetry Method

We study two coupled systems of nonlinear partial differential equations, namely, generalized Boussinesq-Burgers equations and (2+1)-dimensional Davey-Stewartson equations. The Lie symmetry method is utilized to obtain exact solutions of the generalized Boussinesq-Burgers equations. The travelling wave hypothesis approach is used to find exact solutions of the (2+1)-dimensional Davey-Stewartson equations.

A nonlocal Boussinesq equation: multiple-soliton solutions and symmetry analysis

2021 ◽  
Author(s):  
Xi-zhong Liu ◽  
Jun Yu
Keyword(s):  
Exact Solutions ◽  
Boussinesq Equation ◽  
Soliton Solutions ◽  
Symmetry Reduction ◽  
Lie Symmetry ◽  
Symmetry Analysis ◽  
Lie Symmetry Method ◽  
Multiple Soliton Solutions ◽  
Symmetry Method

Abstract A nonlocal Boussinesq equation is deduced from the local one by using consistent correlated bang method. To study various exact solutions of the nonlocal Boussinesq equation, it is converted into two local equations which contain the local Boussinesq equation. From the N-soliton solutions of the local Boussinesq equation, the N-soliton solutions of the nonlocal Boussinesq equation are obtained, among which the (N = 2, 3, 4)-soliton solutions are analyzed with graphs. Some periodic and traveling solutions of the nonlocal Boussinesq equation are derived directly from known solutions of the local Boussinesq equation. Symmetry reduction solutions of the nonlocal Boussinesq equation are also obtained by using the classical Lie symmetry method.

Exact solutions of Zabolotskaya–Khokhlov equation using optimal system

2018 ◽  
Vol 15 (07) ◽  
pp. 1850111
Author(s):  
Mohsin Umair ◽  
Tooba Feroze
Keyword(s):  
Exact Solutions ◽  
Lie Symmetry ◽  
Optimal System ◽  
Lie Symmetry Method ◽  
Symmetry Method

In this paper, using the Lie symmetry method, we obtain optimal system, group invariants and exact solutions of [Formula: see text] dimensional Zabolotskaya–Khokhlov equation.

scholarly journals Exact Solutions of a Mathematical Model Describing Competition and Co-Existence of Different Language Speakers

Entropy ◽  
2020 ◽  
Vol 22 (2) ◽  
pp. 154 ◽  
Author(s):  
Roman Cherniha ◽  
Vasyl’ Davydovych
Keyword(s):  
Mathematical Model ◽  
Exact Solutions ◽  
System Modeling ◽  
Language Shift ◽  
Reaction Diffusion ◽  
Lie Symmetry ◽  
Lie Symmetry Method ◽  
Traveling Fronts ◽  
Three Component Reaction ◽  
Symmetry Method

The known three-component reaction–diffusion system modeling competition and co-existence of different language speakers is under study. A modification of this system is proposed, which is examined by the Lie symmetry method; furthermore, exact solutions in the form of traveling fronts are constructed and their properties are identified. Plots of the traveling fronts are presented and the relevant interpretation describing the language shift that has occurred in Ukraine during the Soviet times is suggested.

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.

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