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On locally and nonlocally related potential systems

### Abstract

For any partial differential equation(PDE) system, a local conservation law yields potential equations in terms of some potential variable, which normally is a nonlocal variable. The current paper examines situations when such a potential variable is a local variable, i.e., is a function of the independent and dependent variables of a given PDE system, and their derivatives. In the case of two independent variables, a simple necessary and sufficient condition is presented for the locality of such a potential variable, and this is illustrated by several examples. As a particular example, two-dimensional reductions of equilibrium equations for fluid and plasma dynamics are considered. It is shown that such reductions with respect to helical, axial, and translational symmetries have conservation laws which yield local potential variables. This leads to showing that the well-known Johnson–Frieman–Kruskal–Oberman (JFKO) and Bragg–Hawthorne (Grad–Shafranov) equations are locally related to the corresponding helically and axially symmetric PDE systems of fluid/plasma dynamics. For the axially symmetric case, local symmetry classifications and arising invariant solutions are compared for the original PDE system and the Bragg–Hawthorne (potential) equation. The potential equation is shown to have additional symmetries, denoted as *restricted symmetries*. Restricted symmetries leave invariant a family of solutions of a given PDE system but not the whole solution manifold, and hence are not symmetries of the given PDE system. Corresponding reductions are shown to yield solutions, which are not obtained as invariant solutions from local symmetry reduction.

© 2010 American Institute of Physics

Received 24 October 2009
Accepted 28 April 2010
Published online 09 July 2010

Acknowledgments:
A.F.C. is grateful to NSERC and the University of Saskatchewan for research support. G.W.B. was supported by an NSERC grant.

Article outline:

I. INTRODUCTION
II. CONDITIONS FOR THE LOCALITY OF A POTENTIAL VARIABLE
A. Conservation laws and potential systems
B. The condition for locality of a potential variable
III. LOCAL POTENTIALS IN TWO-DIMENSIONAL REDUCTIONS OF FLUID AND PLASMA EQUILIBRIUMEQUATIONS
A. Equations of fluid and plasma equilibria in three dimensions
B. The three two-dimensional reductions
1. The general helically symmetric reduction
2. The axially symmetric reduction
3. The translationally symmetric reduction
IV. COMPARISON OF POINT SYMMETRIES OF LOCALLY RELATED PDE SYSTEMS (3.14)–(3.16)
A. Point symmetries of the potential system (3.15)
B. Point symmetries of the MHD equilibrium system (3.14)
C. Point symmetries of the scalar Bragg–Hawthorne equation(3.16)
1. Relations between the point symmetries
V. RELATIONS BETWEEN INVARIANT SOLUTIONS
A. Reductions of the nonlinear Bragg–Hawthorne equation(3.16)
B. Exact solutions of the linear Bragg-Hawthorne equation(3.16)
VI. DISCUSSION

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2010-07-09

2016-10-25

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