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.

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 SOLUTION AND CONSERVATION LAWS OF FRACTIONAL COUPLED WAVE INTERACTION EQUATION

Fractals ◽  
2019 ◽  
Vol 27 (02) ◽  
pp. 1950019
Author(s):  
HEMANTA MANDAL ◽  
B. BIRA
Keyword(s):  
Differential Equations ◽  
Conservation Laws ◽  
Invariant Solution ◽  
Group Analysis ◽  
Wave Interaction ◽  
Power Series Solution ◽  
Group Invariant ◽  
Coupled Equations ◽  
Kdv Equations ◽  
The Given

In this paper, we consider a time-fractional coupled KdV equations describing the interaction of equatorial and midlatitude Rossby waves. From the application of Lie group analysis, the governing system of fractional partial differential equations (FPDEs) is reduced to a system of fractional ordinary differential equations (FODEs). Further, we construct the group-invariant solution as well as the power series solution for the given coupled equations. Next, the evolutionary behavior of the waves under the influence of the fractional order derivative [Formula: see text] is studied graphically through the group-invariant solution. Finally, the conservation laws for the given system of FPDEs are obtained.

On the invariant analysis, symmetry reduction with group-invariant solution and the conservation laws for (2 + 1)-dimensional modified Heisenberg ferromagnetic system

2020 ◽  
Vol 34 (31) ◽  
pp. 2050305
Author(s):  
Vinita ◽  
S. Saha Ray
Keyword(s):  
Differential Equations ◽  
Conservation Laws ◽  
Invariant Solution ◽  
Magnetization Vector ◽  
Physical Structure ◽  
Similarity Solutions ◽  
Lie Symmetry ◽  
Group Invariant ◽  
Symmetry Approach ◽  
Ferromagnetic System

In this paper, a [Formula: see text]-dimensional modified Heisenberg ferromagnetic system, which appears in the biological pattern formation and in the motion of magnetization vector of the isotropic ferromagnet, is being investigated with the aim of exploring its similarity solutions. With the aid of Lie symmetry analysis, this system of partial differential equations has been reduced to a new system of ordinary differential equations, which brings an analytical solution of the main system. Infinitesimal generators, commutator table, and the group-invariant solutions have been carried out by using Lie symmetry approach. Moreover, conservation laws of the above mentioned system have been obtained by utilizing the new conservation theorem proposed by Ibragimov. By applying this analysis, the obtained results might be helpful to understand the physical structure of this model and show the authenticity and effectiveness of the proposed method.

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

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.

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