Received: 21 August 2017
DOI: 10.1002/mma.4665
RESEARCH ARTICLE
Lie symmetries of a system arising in plasma physics
K. Charalambous
1
C. Sophocleous
2
1
Department of Mathematics, University
of Nicosia, Nicosia, CY 1700, Cyprus
2
Department of Mathematics and
Statistics, University of Cyprus, Nicosia,
CY 1678, Cyprus
Correspondence
Christodoulos Sophocleous, Department
of Mathematics and Statistics, University
of Cyprus, Nicosia CY 1678, Cyprus.
Email: christod@ucy.ac.cy
Communicated by: F. Colombo
MOS Classification: 35A30; 35K55; 58J70
Lie group classification for a diffusion-type system that has applications in
plasma physics is derived. The classification depends on the values of 5 param-
eters that appear in the system. Similarity reductions are presented. Certain
initial value problems are reduced to problems with the governing equations
being ordinary differential equations. Examples of potential symmetries are also
presented.
KEYWORDS
diffusion-type systems, equivalence group, group classification, initial value problems, potential
symmetries, similarity reductions
1 INTRODUCTION
Rosenau and Hyman
1,2
introduced the system
u
t
=(u
m
v
l
u
x
)
x
, (uv)
t
= (u
n+1
v
p
v
x
)
x
+(u
m
v
l+1
u
x
)
x
(1)
to study the effect of nonlinearly coupled mass and heat diffusion in a plasma, which slowly diffuses in a strong magnetic
field. In this context, u and v denote the density and ionic temperature of the plasma, m and n are positive constants.
Here, we assume that m, n, l, p, and are real numbers. Bertsch and Kamin provided
3
a solution of the system (1) with
the initial conditions u(x, 0)= u
0
(x) and v(x, 0)= v
0
(x), and later, they described the large time behavior of this solution.
4
Here, we consider the problem of group classification of the system (1). We determine all values of the parameters
m, n, l, p, and such that the system (1) admits Lie symmetries. A Lie symmetry is a differential operator, which is the
generator of an infinitesimal transformation which leaves the differential equation or function upon which it operates
unchanged. The advantage of using an infinitesimal transformation is that the equations to be solved for the coefficients
of the differential operator are linear. When the symmetries have been determined, the corresponding infinitesimal trans-
formations can be exponentiated to obtain the group of finite transformations under the action of which the equation is
invariant. This method is easy to apply and has become well-established in the last decades, see, for example, the standard
texts.
5-11
Lie reduction method, which is based on Lie symmetries, is an efficient technique for solving nonlinear partial dif-
ferential equations. Basically, this method reduces the number of independent variables by one. For example, a partial
differential equation with two independent variables can be reduced into an ordinary differential equation. Lie symme-
tries can also be used to solve initial or/and boundary value problems (see, for example, the other works).
12,13
There is a
continuing interest in deriving Lie symmetries for systems of partial differential equations in the last few years.
14-22
When a partial differential equation admits conservation laws, then by an introduction of a new dependent variable,
it can be written as a system of partial differential equations. In many cases, Lie symmetries of this auxiliary system
leads to nonlocal (potential) symmetries of the original partial differential equation. The idea for deriving such nonlocal
symmetries was introduced by Bluman.
5,23,24
Math Meth Appl Sci. 2017;1–13. wileyonlinelibrary.com/journal/mma Copyright © 2017 John Wiley & Sons, Ltd. 1