42 research outputs found
Equivalence of conservation laws and equivalence of potential systems
We study conservation laws and potential symmetries of (systems of)
differential equations applying equivalence relations generated by point
transformations between the equations. A Fokker-Planck equation and the Burgers
equation are considered as examples. Using reducibility of them to the
one-dimensional linear heat equation, we construct complete hierarchies of
local and potential conservation laws for them and describe, in some sense, all
their potential symmetries. Known results on the subject are interpreted in the
proposed framework. This paper is an extended comment on the paper of J.-q. Mei
and H.-q. Zhang [Internat. J. Theoret. Phys., 2006, in press].Comment: 10 page
Conservation Laws and Symmetries of Semilinear Radial Wave Equations
Classifications of symmetries and conservation laws are presented for a
variety of physically and analytically interesting wave equations with power
onlinearities in n spatial dimensions: a radial hyperbolic equation, a radial
Schrodinger equation and its derivative variant, and two proposed radial
generalizations of modified Korteweg--de Vries equations, as well as
Hamiltonian variants. The mains results classify all admitted local point
symmetries and all admitted local conserved densities depending on up to first
order spatial derivatives, including any that exist only for special powers or
dimensions. All such cases for which these wave equations admit, in particular,
dilational energies or conformal energies and inversion symmetries are
determined. In addition, potential systems arising from the classified
conservation laws are used to determine nonlocal symmetries and nonlocal
conserved quantities admitted by these equations. As illustrative applications,
a discussion is given of energy norms, conserved H^s norms, critical powers for
blow-up solutions, and one-dimensional optimal symmetry groups for invariant
solutions.Comment: 16 pages. Final version with minor revision
Connections Between Symmetries and Conservation Laws
This paper presents recent work on connections between symmetries and
conservation laws. After reviewing Noether's theorem and its limitations, we
present the Direct Construction Method to show how to find directly the
conservation laws for any given system of differential equations. This method
yields the multipliers for conservation laws as well as an integral formula for
corresponding conserved densities. The action of a symmetry (discrete or
continuous) on a conservation law yields conservation laws. Conservation laws
yield non-locally related systems that, in turn, can yield nonlocal symmetries
and in addition be useful for the application of other mathematical methods.
From its admitted symmetries or multipliers for conservation laws, one can
determine whether or not a given system of differential equations can be
linearized by an invertible transformation.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and
Applications) at http://www.emis.de/journals/SIGMA