A direct method for finding similarity reductions of partial differential equations is applied to a specific case of the Grad–Shafranov equation. As an illustration of the method, the frequently used Solov’ev equilibrium is derived. The method is employed to obtain a new family of exact analytical solutions, which contain both the classical and nonclassical group-invariant solutions of the Grad–Shafranov equation and thus greatly extends the range of the available analytical solutions. All the group-invariant solutions based on the classical Lie symmetries are shown to be particular cases in the new family of solutions.
REFERENCES
1.
N. H.
Ibragimov
, A Practical Course in Differential Equations and Mathematical Modelling: Classical and New Methods. Nonlinear Mathematical Models. Symmetry and Invariance Principles
(World Scientific
, Singapore
, 2009
).2.
G. I.
Barenblatt
, Scaling, Self-Similarity, and Intermediate Asymptotics: Dimensional Analysis and Intermediate Asymptotics
(Cambridge University Press
, Cambridge
, 1996
).3.
P.
Olver
, Applications of Lie Groups to Differential Equations
(Springer
, New York
, 1993
).4.
R. L.
White
and R. D.
Hazeltine
, Phys. Plasmas
16
, 123101
(2009
).5.
H.
Grad
and J.
Rubin
, Proceedings of the Second United Nations International Conference on the Peaceful Uses of Atomic Energy
, United Nations, Geneva, 1958
, Vol. 31
, p. 190
.6.
7.
8.
9.
P. J.
Olver
and P.
Rosenau
, Phys. Lett. A
114
, 107
(1986
).10.
P. A.
Clarkson
and M. D.
Kruskal
, J. Math. Phys.
30
, 2201
(1989
).11.
D.
Levi
and P.
Winternitz
, J. Phys. A
22
, 2915
(1989
).12.
E. K.
Maschke
, Plasma Phys.
15
, 535
(1973
).13.
P. J.
Mc Carthy
, Phys. Plasmas
6
, 3554
(1999
).14.
C. V.
Atanasiu
, S.
Günter
, K.
Lackner
, and I. G.
Miron
, Phys. Plasmas
11
, 3510
(2004
).15.
L.
Guazzotto
and J. P.
Freidberg
, Phys. Plasmas
14
, 112508
(2007
).16.
S. B.
Zheng
, A. J.
Wootton
, and E. R.
Solano
, Phys. Plasmas
3
, 1176
(1996
).17.
A. J.
Cerfon
and J. P.
Freidberg
, Phys. Plasmas
17
, 032502
(2010
).18.
O. I.
Bogoyavlenskij
, Phys. Lett. A
276
, 257
(2000
).19.
O. I.
Bogoyavlenskij
, Phys. Rev. Lett.
84
, 1914
(2000
).20.
© 2010 American Institute of Physics.
2010
American Institute of Physics
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