Classification of homogeneous quadratic conservation laws with viscous terms

2007 ◽  
Vol 26 (2) ◽  
Author(s):  
Jane Hurley Wenstrom ◽  
Bradley J. Plohr
Keyword(s):  
Conservation Laws

scholarly journals Integrability of Riemann-Type Hydrodynamical Systems and Dubrovin’s Integrability Classification of Perturbed KdV-Type Equations

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 1077
Author(s):  
Yarema A. Prykarpatskyy
Keyword(s):  
Conservation Laws ◽  
Hamiltonian Structure ◽  
Necessary Condition ◽  
Type Representation ◽  
Infinite Hierarchy ◽  
Polynomial Coefficients ◽  
Kdv Type Equations ◽  
Special Case

Dubrovin’s work on the classification of perturbed KdV-type equations is reanalyzed in detail via the gradient-holonomic integrability scheme, which was devised and developed jointly with Maxim Pavlov and collaborators some time ago. As a consequence of the reanalysis, one can show that Dubrovin’s criterion inherits important parts of the gradient-holonomic scheme properties, especially the necessary condition of suitably ordered reduction expansions with certain types of polynomial coefficients. In addition, we also analyze a special case of a new infinite hierarchy of Riemann-type hydrodynamical systems using a gradient-holonomic approach that was suggested jointly with M. Pavlov and collaborators. An infinite hierarchy of conservation laws, bi-Hamiltonian structure and the corresponding Lax-type representation are constructed for these systems.

On the symmetry and conservation law classification of the de Sitter–Schwarzschild metric and the corresponding wave and Klein–Gordon equations

2020 ◽  
Vol 17 (11) ◽  
pp. 2050172
Author(s):  
Ashfaque H. Bokhari ◽  
A. H. Kara ◽  
F. D. Zaman ◽  
B. B. I. Gadjagboui
Keyword(s):  
Conservation Laws ◽  
Conservation Law ◽  
De Sitter ◽  
Killing Vectors ◽  
Schwarzschild Geometry ◽  
Klein Gordon ◽  
Spacetime Metric ◽  
Variational Symmetries ◽  
Symmetry And Conservation Law

The main purpose of this work is to focus on a discussion of Lie symmetries admitted by de Sitter–Schwarzschild spacetime metric, and the corresponding wave or Klein–Gordon equations constructed in the de Sitter–Schwarzschild geometry. The obtained symmetries are classified and the variational (Noether) conservation laws associated with these symmetries via the natural Lagrangians are obtained. In the case of the metric, we obtain additional variational ones when compared with the Killing vectors leading to additional conservation laws and for the wave and Klein–Gordon equations, the variational symmetries involve less tedious calculations as far as invariance studies are concerned.

Classification of Variational Conservation Laws of General Plane Symmetric Spacetimes

2017 ◽  
Vol 68 (3) ◽  
pp. 335 ◽  
Author(s):  
Usamah S. Al-Ali ◽  
Ashfaque H. Bokhari ◽  
A. H. Kara ◽  
Ghulam Shabbir
Keyword(s):  
Conservation Laws ◽  
Plane Symmetric ◽  
Symmetric Spacetimes ◽  
General Plane

Conservation laws and null divergences

1983 ◽  
Vol 94 (3) ◽  
pp. 529-540 ◽  
Author(s):  
Peter J. Olver
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Conservation Laws ◽  
Existence Of Solutions ◽  
Complete Classification ◽  
Complete Analysis ◽  
Partial Differential ◽  
Physical Systems

For a system of partial differential equations, the existence of appropriate conservation laws is often a key ingredient in the investigation of its solutions and their properties. Conservation laws can be used in proving existence of solutions, decay and scattering properties, investigation of singularities, analysis of integrability properties of the system and so on. Representative applications, and more complete bibliographies on conservation laws, can be found in references [7], [8], [12], [19]. The more conservation laws known for a given system, the more tools available for the above investigations. Thus a complete classification of all conservation laws of a given system is of great interest. Not many physical systems have been subjected to such a complete analysis, but two examples can be found in [11] and [14]. The present paper arose from investigations ([15], [16]) into the conservation laws of the equations of elasticity.

scholarly journals Group Classification and Conservation Laws of a Class of Hyperbolic Equations

2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
J. C. Ndogmo
Keyword(s):  
Differential Equations ◽  
Conservation Laws ◽  
Linear Dependence ◽  
Hyperbolic Equations ◽  
Group Classification ◽  
The Family ◽  
The Given ◽  
Low Order

Abstracts. A method for the group classification of differential equations is proposed. It is based on the determination of all possible cases of linear dependence of certain indeterminates appearing in the determining equations of symmetries of the equation. The method is simple and systematic and applied to a family of hyperbolic equations. Moreover, as the given family contains several known equations with important physical applications, low-order conservation laws of some relevant equations from the family are computed, and the results obtained are discussed with regard to the symmetry integrability of a particular class from the underlying family of hyperbolic equations.

Lie Symmetries and Conservation Laws of the Generalised Foam-Drainage Equation

2017 ◽  
Vol 72 (3) ◽  
pp. 261-267 ◽  
Author(s):  
Zhi-Yong Zhang ◽  
Kai-Hua Ma
Keyword(s):  
Power Series ◽  
Conservation Laws ◽  
Series Solution ◽  
Power Series Solution ◽  
One Dimensional ◽  
Symmetry Classification ◽  
Symmetry Operators ◽  
Foam Drainage Equation ◽  
Foam Drainage

AbstractWe perform a complete Lie point symmetry classification of the generalised foam-drainage equation and then construct an optimal system of one-dimensional subalgebra of the admitted symmetry operators and use them to reduce the equations under study. A power series solution of the reduced equation is constructed. Moreover, we find all multipliers of the equations and apply them to construct conservation laws.

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