JOURNAL OF MATHEMATICAL PHYSICS 51, 103522 共2010兲
Multidimensional partial differential equation systems:
Nonlocal symmetries, nonlocal conservation laws,
exact solutions
Alexei F. Cheviakov1,a兲 and George W. Bluman2,b兲
1
Department of Mathematics and Statistics, University of Saskatchewan,
Saskatoon S7N 5E6, Canada
2
Department of Mathematics, University of British Columbia, Vancouver V6T 1Z2,
Canada
共Received 11 June 2010; accepted 11 September 2010; published online 25 October 2010兲
For systems of partial differential equations 共PDEs兲 with n ⱖ 3 independent variables, construction of nonlocally related PDE systems is substantially more complicated than is the situation for PDE systems with two independent variables. In
particular, in the multidimensional situation, nonlocally related PDE systems can
arise as nonlocally related subsystems as well as potential systems that follow from
divergence-type or lower-degree conservation laws. The theory and a systematic
procedure for the construction of such nonlocally related PDE systems is presented
in Part I 关A. F. Cheviakov and G. W. Bluman, J. Math. Phys. 51, 103521 共2010兲兴.
This paper provides many new examples of applications of nonlocally related systems in three and more dimensions, including new nonlocal symmetries, new nonlocal conservation laws, and exact solutions for various nonlinear PDE systems of
physical interest. © 2010 American Institute of Physics. 关doi:10.1063/1.3496383兴
I. INTRODUCTION
In the situation of two independent variables, nonlocally related systems of partial differential
equations 共PDEs兲 have proven to be useful for many given nonlinear and linear PDE systems of
physical interest. For a given PDE system, one can systematically construct nonlocally related
potential systems and subsystems2,3 having the same solution set as the given system. Due to
nonlocal relations between solution sets, analysis of such nonlocally related systems can yield new
results for the given system.
Examples include results for nonlinear wave and diffusion equations, gas dynamics equations,
continuum mechanics, electromagnetism, plasma equilibria, as well as other nonlinear and linear
PDE systems.2–13 New results for such physical systems include systematic computations of nonlocal symmetries and nonlocal conservation laws, systematic constructions of further invariant and
nonclassical solutions, and the systematic construction of noninvertible linearizations.
This paper follows from Ref. 1 and is concerned with the construction and use of nonlocally
related PDE systems with three or more independent variables for specific examples. As shown in
Ref. 1, the situation for obtaining and using nonlocally related PDE systems is considerably more
complex than in the two-dimensional case. In particular, the usual 共divergence-type兲 conservation
laws give rise to vector potential variables subject to gauge freedom, i.e., defined to within
arbitrary functions of the independent variables, making the corresponding potential system underdetermined.
Another important difference between two-dimensional and multidimensional PDE systems is
that in higher dimensions, there can exist several types of conservation laws 共divergence-type and
a兲
Author to whom correspondence should be addressed. Electronic mail: chevaikov@math.usask.ca.
Electronic mail: bluman@math.ubc.ca.
b兲
0022-2488/2010/51共10兲/103522/26/$30.00
51, 103522-1
© 2010 American Institute of Physics
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103522-2
A. F. Cheviakov and G. W. Bluman
J. Math. Phys. 51, 103522 共2010兲
lower-degree conservation laws兲. For example, in the case of n = 3 independent variables, one can
have a vanishing divergence or a vanishing curl; for n ⬎ 3, n − 1 types of conservation laws exist.
关It is important to note that in many physical examples, the most commonly arising conservation
laws are of divergence-type 共degree r = n − 1兲 conservation laws, which yield underdetermined
potential systems.兴
Due to such complexity and, furthermore, the difficulty of performing computations for PDE
systems involving many dependent and independent variables, very few results have been obtained so far for multidimensional systems. In this paper, building on the framework presented in
Ref. 1, we present new results for important examples as well as discuss and synthesize some
previously known results. The symbolic software package GeM for MAPLE 共Ref. 14兲 was used for
the symbolic computations.
An important use of nonlocally related systems is the computation of nonlocal symmetries of
a given PDE system. A nonlocal symmetry is a symmetry for which the components of its
infinitesimal generator, corresponding to the variables of the given system, have an essential
dependence on nonlocal variables. Only determined nonlocally related systems can yield nonlocal
symmetries of a given system.11 Consequently, one seeks nonlocal symmetries of a given PDE
system 共with n ⱖ 3 independent variables兲 through seeking local symmetries of the following types
of nonlocally related PDE systems.1
• Nonlocally related subsystems 共always determined兲.
• Potential systems of degree one 共always determined兲. 共In R3, such potential systems arise
from curl-type conservation laws.兲
• Potential systems of degree r : 1 ⬍ r ⱕ n − 1, appended with an appropriate number of gauge
constraints.
Examples of nonlocal symmetries arising from all three of the above types are given in this
paper.
Another important use of nonlocally related systems is the computation of nonlocal conservation laws of a given PDE system. A nonlocal conservation law is a conservation law whose
fluxes depend on nonlocal variables, and which is not equivalent to any local conservation law of
the given system.1,15 Unlike nonlocal symmetries, nonlocal conservation laws can arise from both
determined and underdetermined potential systems, as illustrated by examples in this paper.
The sections of the paper below pertain to particular examples of nonlocally related PDE
systems and their applications to construction of nonlocal symmetries, nonlocal conservation laws,
and exact solutions of PDE systems in n ⱖ 3 dimensions. Examples of results for multidimensional
PDE systems in this paper include the following 共new results are marked by an asterisk兲.
• A nonlocal symmetryⴱ arising from a nonlocally related subsystem of a nonlinear PDE
system in 共2 + 1兲 dimensions 共Sec. II兲.
• Nonlocal symmetriesⴱ and nonlocal conservation lawsⴱ of a nonlinear “generalized plasma
equilibrium” PDE system in three space dimensions 共Sec. III兲. 关These nonlocal symmetries
and nonlocal conservation laws arise from local symmetries and local conservation laws of a
potential system following from a lower-degree 共curl-type兲 conservation law.兴
• Nonlocal symmetries of the linear wave equation in 共2 + 1兲 dimensions,11 arising from local
symmetries of an underdetermined potential system of degree of 2, appended with a Lorentz
gauge 共Sec. IV兲. 共Nonlocal conservation laws of this equation were also obtained in Ref. 11.兲
• Nonlocal symmetriesⴱ of dynamic Euler equations of incompressible fluid dynamics arising
from axially and helically symmetric reductions 共Sec. V兲.
• Nonlocal symmetries and nonlocal conservation laws of Maxwell’s equations in
共2 + 1兲-dimensional Minkowski space, arising from local symmetries and local conservation
laws of a determined potential system of degree 1 and an underdetermined potential system
of degree of 2, appended with a Lorentz gauge.11 Additional nonlocal conservation laws arise
from local conservation laws of a potential system appended with algebraicⴱ and divergenceⴱ
gauges 共Sec. VI兲.
• Nonlocal symmetries and nonlocal conservation laws of Maxwell’s equations in
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103522-3
J. Math. Phys. 51, 103522 共2010兲
Applications of nonlocal PDE systems in multi-D
共3 + 1兲-dimensional Minkowski space, arising from local symmetries and local conservation
laws of an underdetermined potential system of degree 2, appended with a Lorentz gauge.12
Additional nonlocal conservation laws arise from local conservation laws of a potential
system appended with algebraicⴱ and divergenceⴱ gauges. 共Sec. VII兲.
• Nonlocal symmetries 共following from a curl-type conservation law兲 and exact solutions of
the nonlinear three-dimensional magnetohydrodynamics 共MHD兲 equilibrium equations16,17
共Sec. VIII兲.
Finally, in Sec. IX, some open problems are discussed.
II. A NONLOCAL SYMMETRY ARISING FROM A NONLOCALLY RELATED SUBSYSTEM
IN THREE DIMENSIONS
The first example illustrates the use of nonlocally related subsystems to obtain nonlocal
symmetries of PDE systems in higher dimensions.
Consider the PDE system UV兵t , x , y ; u , v1 , v2其 in one time and two space dimensions, given
by
vt = grad u,
共2.1兲
ut = K共兩v兩兲div v.
In 共2.1兲, v = 共v1 , v2兲 is a vector function and K共兩v兩兲 is a constitutive function of the indicated scalar
argument. In 共2.1兲 and throughout this paper, subscripts are used to denote the corresponding
partial derivatives.
PDE system 共2.1兲 has the nonlocally related subsystem V兵t , x , y ; v1 , v2其, given by
共2.2兲
vtt = grad关K共兩v兩兲div v兴.
Consider the one-parameter class of constitutive functions given by
K共兩v兩兲 = 兩v兩2m = 共共v1兲2 + 共v2兲2兲m .
共2.3兲
It is interesting to compare the symmetry classifications of systems 共2.1兲 and 共2.2兲 with respect to
the constitutive parameter m ⫽ 0.
For arbitrary m in 共2.3兲, one can show that the point symmetries of given PDE system 共2.1兲
are given by the seven infinitesimal generators,
X1 =
⳵
,
⳵t
X2 =
⳵
,
⳵x
X5 = t
⳵
⳵
⳵
+x +y ,
⳵t
⳵x
⳵y
X6 = − y
冉
X7 = m x
X3 =
⳵
,
⳵y
X4 =
⳵
,
⳵u
⳵
⳵
⳵
⳵
+ x − v2 1 + v1 2 ,
⳵x
⳵y
⳵v
⳵v
冊
⳵
⳵
⳵
⳵
⳵
+y
+ 共m + 1兲u + v1 1 + v2 2 .
⳵x
⳵y
⳵u
⳵v
⳵v
共2.4兲
In contrast, subsystem 共2.2兲 has the point symmetries given by the six infinitesimal generators,
Y 1 = X 1,
Y 2 = X 2,
Y 3 = X 3,
Y 4 = X 5,
Y5 = X6 ,
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103522-4
J. Math. Phys. 51, 103522 共2010兲
A. F. Cheviakov and G. W. Bluman
冉
Y6 = m x
冊
⳵
⳵
⳵
⳵
+y
+ v1 1 + v2 2 .
⳵x
⳵y
⳵v
⳵v
共2.5兲
Additional point symmetries arise for 共2.1兲 if and only if m = −1 and for 共2.2兲 if and only if m =
−1 , −2. In the case m = −1, one can show that both systems have an infinite number of point
symmetries. In the case m = −2, subsystem 共2.2兲 has an additional point symmetry,
Y7 = t2
⳵
⳵
⳵
+ tv1 1 + tv2 2 ,
⳵t
⳵v
⳵v
共2.6兲
whereas given PDE system 共2.1兲 still has same point symmetries 共2.4兲. It follows that 共2.6兲 yields
a nonlocal symmetry of given PDE system UV兵t , x , y ; u , v1 , v2其 共2.1兲.
III. NONLOCAL SYMMETRIES AND NONLOCAL CONSERVATION LAWS OF A
NONLINEAR PDE SYSTEM IN THREE DIMENSIONS
As a second example, consider the time-independent “generalized plasma equilibrium” PDE
system H兵x , y , z ; h1 , h2 , h3其 in three space dimensions, given by
curl共K共兩h兩兲共curl h兲 ⫻ h兲 = 0,
div h = 0.
共3.1兲
In 共3.1兲, h = 共h1 , h2 , h3兲 is a vector of dependent variables. The first equation in PDE system 共3.1兲
is a conservation law of degree one 共curl-type conservation law兲. The corresponding potential
system HW兵x , y , z ; h1 , h2 , h3 , w其 is given by
K共兩h兩兲共curl h兲 ⫻ h = grad w,
div h = 0,
共3.2兲
where w共x , y , z兲 is a scalar potential variable. Potential system 共3.2兲 is determined and hence needs
no gauge constraints.
A. Nonlocal symmetries of PDE system „3.1…
First, a comparison is made of the classifications of point symmetries of the PDE systems
H兵x , y , z ; h1 , h2 , h3其 and HW兵x , y , z ; h1 , h2 , h3 , w其 for the one-parameter family of constitutive
functions K共兩h兩兲 given by
K共兩h兩兲 = 兩h兩2m ⬅ 共共h1兲2 + 共h2兲2 + 共h3兲2兲m ,
共3.3兲
where m is a parameter.
For an arbitrary m, given system H兵x , y , z ; h1 , h2 , h3其 共3.1兲 has eight point symmetries, given
by
X1 =
X5 = − z
⳵
,
⳵x
X2 =
⳵
,
⳵y
X3 =
⳵
,
⳵z
⳵
⳵
⳵
⳵
+ x − h3 1 + h1 3 ,
⳵x
⳵z
⳵h
⳵h
X7 = z
⳵
⳵
⳵
⳵
− y + h3 2 − h2 3 ,
⳵y
⳵z
⳵h
⳵h
X4 = x
X6 = y
⳵
⳵
⳵
+z +y ,
⳵x
⳵z
⳵y
⳵
⳵
⳵
⳵
− x + h2 1 − h1 2 ,
⳵x
⳵y
⳵h
⳵h
X8 = h1
⳵
2 ⳵
3 ⳵
,
1 +h
2 +h
⳵h
⳵h
⳵ h3
共3.4兲
corresponding to invariance, respectively, under three spatial translations, one dilation, three rotations, and one scaling.
For m ⫽ −1, potential system HW兵x , y , z ; h1 , h2 , h3 , w其 共3.2兲 has nine point symmetries, eight
of them corresponding to symmetries 共3.4兲, plus an extra translational symmetry in the potential
variable,
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103522-5
J. Math. Phys. 51, 103522 共2010兲
Applications of nonlocal PDE systems in multi-D
Y i = X i,
i = 1, . . . ,7;
Y8 = X8 + 2共m + 1兲w
⳵
,
⳵w
Y9 =
⳵
.
⳵w
共3.5兲
For m = −1, the point symmetries of H兵x , y , z ; h1 , h2 , h3其 remain the same, whereas the potential
system HW兵x , y , z ; h1 , h2 , h3 , w其 has an additional infinite number of point symmetries given by
Y⬁ = F共w兲
冉
冊
⳵
⳵
⳵
⳵
+ h1 1 + h2 2 + h3 3 ,
⳵w
⳵h
⳵h
⳵h
共3.6兲
depending on an arbitrary smooth function F共w兲. Symmetries 共3.6兲 are nonlocal symmetries of
given PDE system H兵x , y , z ; h1 , h2 , h3其 共3.1兲.
Note that symmetries 共3.6兲 cannot be used for the construction of invariant solutions since
they do not involve spatial components. However, one can use symmetries 共3.6兲 to map any
known solution of PDE system 共3.1兲 共with a corresponding potential variable w兲 to an infinite
family of solutions of 共3.1兲.
B. Nonlocal conservation laws arising from potential system „3.2…
We now seek divergence-type conservation laws of PDE system H兵x , y , z ; h1 , h2 , h3其 共3.1兲,
using the direct method, applied first to given system 共3.1兲 itself, and then to potential system
HW兵x , y , z ; h1 , h2 , h3 , w其 共3.2兲 for the one-parameter family of constitutive functions K共兩h兩兲 given
by 共3.3兲 for an arbitrary m. 共For the details on the direct method of construction of conservation
laws, see Ref. 1.兲
First, we seek local divergence-type conservation laws of PDE system H兵x , y , z ; h1 , h2 , h3其
共3.1兲, using multipliers of the form ⌳␴ = ⌳␴共x , y , z , H1 , H2 , H3兲, ␴ = 1 , . . . , 4. 关Here and below, to
underline the fact that multipliers are sought off of the solution space of a given PDE system, the
arbitrary functions corresponding to dependent variables are denoted by capitals. Then if a linear
combination of equations of the system with a set of multipliers gives a divergence expression,
one obtains a conservation law on solutions of the system. 共For details and notation, see Ref. 1 or
Ref. 15, Chap. 1.兲兴
From solving the corresponding set of multiplier determining equations, one finds the nontrivial conservation law multipliers given by
⌳1 = AH1,
⌳2 = AH2,
⌳3 = AH3,
⌳4 = B,
where A, B are arbitrary constants. 共In particular, the conservation law corresponding to the
constant B is simply the fourth PDE div h = 0.兲
Second, we apply the direct method to potential system HW兵x , y , z ; h1 , h2 , h3 , w其 共3.2兲, to seek
additional conservation laws of given PDE system H兵x , y , z ; h1 , h2 , h3其 共3.1兲. As shown in Theorem 6.3 of Ref. 1 共see also Ref. 15, Chap. 3兲, in order to obtain nonlocal divergence-type conservation laws, one must seek multipliers that essentially depend on potential variables. For four
ˆ 共H1 , H2 , H3 , W兲, ␴ = 1 , . . . , 4. In terms of
ˆ =⌳
equations 共3.2兲, we seek multipliers of the form ⌳
␴
␴
an arbitrary function G共W兲, one finds an infinite family of such multipliers, given by
ˆ = HiG⬘共W兲,
⌳
i
i = 1,2,3,
⌳4 = G共W兲,
with the corresponding divergence-type conservation laws given by
3
⳵
i
兺
i 关共G共w兲 + 2共m + 1兲wG⬘共w兲兲h 兴 = 0.
⳵
x
i=1
共3.7兲
In 共3.7兲, 共x1 , x2 , x3兲 = 共x , y , z兲. Conservation laws 共3.7兲 have an evident geometrical meaning. From
the vector equation in 共3.2兲, it follows that grad w is orthogonal to h, i.e., the vector field h is
tangent to level surfaces w = const. Expression 共3.7兲 can be rewritten as div共M共w兲h兲
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103522-6
J. Math. Phys. 51, 103522 共2010兲
A. F. Cheviakov and G. W. Bluman
⬅ M ⬘共w兲grad共w兲 · h + M共w兲div h = 0, where M共w兲 = G共w兲 + 2共m + 1兲wG⬘共w兲, and hence is equivalent to grad共w兲 · h = 0, provided that M ⬘共w兲 ⫽ 0.
By a similar argument, it follows that PDE system H兵x , y , z ; h1 , h2 , h3其 共3.1兲 has another
family of nonlocal conservation laws given by
共3.8兲
div关Q共w兲curl h兴 = 0
for an arbitrary Q共w兲.
IV. NONLOCAL SYMMETRIES OF THE TWO-DIMENSIONAL LINEAR WAVE EQUATION
Consider the linear wave equation U兵t , x , y ; u其 given by
共4.1兲
utt = uxx + uyy .
Equation 共4.1兲 is a divergence-type conservation law as it stands. Following Ref. 11, we introduce
a vector potential v = 共v0 , v1 , v2兲. The resulting potential equations are underdetermined, therefore
in order to seek nonlocal symmetries, a gauge constraint is needed. A Lorentz gauge is chosen
since it complies with the geometrical symmetries of given PDE 共4.1兲.11 The resulting determined
potential system UV兵t , x , y ; u , v其 is given by
ut = v2x − v1y ,
− ux = v0y − v2t ,
− uy = v1t − v0x ,
共4.2兲
v0t − v1x − v2y = 0.
A comparison is now made of the point symmetries of PDE systems U兵t , x , y ; u其 共4.1兲 and
UV兵t , x , y ; u , v其 共4.2兲. Modulo the infinite number of point symmetries of any linear PDE system,
linear wave equation 共4.1兲 has ten point symmetries:
• three translations X1 , X2 , X3 given by
X1 =
⳵
,
⳵t
X2 =
⳵
,
⳵x
X3 =
⳵
;
⳵y
• one dilation given by
X4 = t
⳵
⳵
⳵
+x +y ;
⳵t
⳵x
⳵y
• one rotation and two space-time rotations 共boosts兲 given by
X5 = x
⳵
⳵
−y ,
⳵y
⳵x
X6 = t
⳵
⳵
+x ,
⳵x
⳵t
X7 = t
⳵
⳵
+y ;
⳵y
⳵t
• three additional conformal transformations given by
X8 = 共t2 + x2 + y 2兲
X9 = 2tx
⳵
⳵
⳵
⳵
+ 2tx + 2ty − tu ,
⳵x
⳵y
⳵u
⳵t
⳵
⳵
⳵
⳵
+ 共t2 + x2 − y 2兲 + 2xy − xu ,
⳵t
⳵y
⳵u
⳵x
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103522-7
Applications of nonlocal PDE systems in multi-D
X10 = 2ty
J. Math. Phys. 51, 103522 共2010兲
⳵
⳵
⳵
⳵
+ 2xy + 共t2 − x2 + y 2兲 − yu .
⳵t
⳵x
⳵u
⳵y
Potential system UV兵t , x , y ; u , v0 , v1 , v2其 共4.2兲 has seven point symmetries Y1 , . . . , Y7 that
project onto the point symmetries X1 , . . . , X7 of wave equation 共4.1兲. However, the three additional conformal symmetries of potential system 共4.2兲 given by
Y8 = X8 + 共y v1 − xv2 − tu兲
− 共xv0 + 2tv1 − yu兲
⳵
⳵
− 共2tv0 + xv1 + y v2兲 0
⳵u
⳵v
⳵
⳵
0
2
,
1 − 共y v + 2tv + xu兲
⳵v
⳵ v2
Y9 = X9 − 共y v0 + tv2 + xu兲
− 共tv0 + 2xv1 + y v2兲
⳵
⳵
− 共2xv0 + tv1 − yu兲 0
⳵u
⳵v
⳵
⳵
1
2
,
1 + 共y v − 2xv − tu兲
⳵v
⳵ v2
Y10 = X10 + 共xv0 + tv1 − yu兲
− 共2y v1 − xv2 − tu兲
⳵
⳵
− 共2y v0 + tv2 + xu兲 0
⳵u
⳵v
⳵
⳵
− 共tv0 + xv1 + 2y v2兲 2
⳵ v1
⳵v
共4.3兲
clearly yield nonlocal symmetries of wave equation 共4.1兲. Moreover, potential system 共4.2兲 has
three duality-type point symmetries given by
Y11 = v0
⳵
⳵
⳵
⳵
− u 0 − v2 1 + v1 2 ,
⳵v
⳵u
⳵v
⳵v
Y12 = v1
⳵
⳵
⳵
⳵
+ v2 0 + u 1 + v0 2 ,
⳵v
⳵u
⳵v
⳵v
Y13 = v2
⳵
⳵
⳵
⳵
− v1 0 − v0 1 + u 2 ,
⳵v
⳵u
⳵v
⳵v
共4.4兲
that also yield nonlocal symmetries of wave equation U兵t , x , y ; u其 共4.1兲. In summary, potential
system UV兵t , x , y ; u , v0 , v1 , v2其 共4.2兲 with the Lorentz gauge yields six nonlocal symmetries of
linear wave equation 共4.1兲.11
One can show that no nonlocal symmetries of the wave equation arise from the potential
system UV兵t , x , y ; u , v0 , v1 , v2其 if the Lorentz gauge is replaced by any one of the algebraic gauges
15
vk = 0 for k 苸 兵0 , 1 , 2其, the divergence gauge, the Poincaré gauge, or the Cronstrom gauge.
In Ref. 11, potential system UV兵t , x , y ; u , v0 , v1 , v2其 共4.2兲 was used to obtain additional 共nonlocal兲 conservation laws of wave equation U兵t , x , y ; u其 共4.1兲.
V. NONLOCAL SYMMETRIES OF THE EULER EQUATIONS
Consider the Euler equations describing the motion for an incompressible inviscid fluid in R3,
which in Cartesian coordinates are given by
div u = 0,
共5.1a兲
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103522-8
J. Math. Phys. 51, 103522 共2010兲
A. F. Cheviakov and G. W. Bluman
ut + 共u · ⵜ兲u + grad p = 0,
共5.1b兲
where the fluid velocity vector u = u1ex + u2ey + u3ez and fluid pressure p are functions of x , y , z , t.
The fluid vorticity is a local vector variable defined by
␻ = curl u.
共5.2兲
Using the vector calculus identity
共u · ⵜ兲u = grad
兩u兩2
+ 共curl u兲 ⫻ u,
2
vector momentum equation 共5.1b兲 can be rewritten as
冉
ut + ␻ ⫻ u + grad p +
冊
兩u兩2
= 0.
2
共5.3兲
One can construct a vorticity subsystem of PDE system 共5.1兲 by taking the curl of Eq. 共5.3兲,
div u = 0,
共5.4a兲
␻t + curl共␻ ⫻ u兲 = 0,
共5.4b兲
␻ = curl u.
共5.4c兲
PDE system 共5.4兲 is nonlocally related to Euler equations 共5.1兲. By definition, Euler system 共5.1兲
is a potential system of PDE system 共5.4兲 following from curl-type 共degree one兲 conservation law
共5.4b兲.
Below we compare point symmetries of Euler equations and the vorticity subsystem in two
symmetric settings.26
A. Axially symmetric case
Rewriting Euler equations 共5.1兲 in cylindrical coordinates 共r , z , ␸兲 with
u = uer + ve␸ + wez ,
we restrict the dependence of each of u , v , w , p to the coordinates t , r , z only due to the invariance
of the Euler equations under 共azimuthal兲 rotations in ␸. Consequently, one obtains the reduced
axially symmetric Euler system AE兵t , r , z ; u , v , w , p其 given by
1
1
ur + u + v␸ + wz = 0,
r
r
共5.5a兲
1
ut + uur + wuz − v2 + pr = 0,
r
共5.5b兲
1
vt + uvr + wvz + uv = 0,
r
共5.5c兲
1
wt + uwr + wwz + pz = 0.
r
共5.5d兲
In terms of cylindrical coordinates, the vorticity is represented in the form
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Applications of nonlocal PDE systems in multi-D
␻ = mer + ne␸ + qez .
Using the invariance of 共5.2兲 under the same azimuthal rotations, we again assume axial symmetry
and rewrite three scalar equations 共5.2兲 as
m + vz = 0,
n − uz + wr = 0,
q−
1 ⳵
共rv兲 = 0,
r ⳵r
共5.6兲
where m , n , q are functions of t , r , z.
Combining PDE systems 共5.5兲 and 共5.6兲, we obtain the PDE system
AEW兵t , r , z ; u , v , w , p , m , n , q其, which is obviously locally related to the axially symmetric Euler
system AE兵t , r , z ; u , v , w , p其 共5.5兲, since vorticity components are local variables in terms of
u , v , w.
The point symmetries of the system AEW兵t , r , z ; u , v , w , p , m , n , q其 are given by
X1 =
X2 = t
X3 = r
⳵
,
⳵t
⳵
⳵
⳵
⳵
⳵
⳵
+r +z −m
−n −q ,
⳵t
⳵r
⳵z
⳵m
⳵n
⳵q
⳵
⳵
⳵
⳵
⳵
⳵
+z +u +v +w
+ 2p ,
⳵r
⳵z
⳵u
⳵v
⳵w
⳵p
X4 = F共t兲
⳵
⳵
⳵
+ F⬘共t兲
− zF⬙共t兲 ,
⳵z
⳵w
⳵p
X5 = G共t兲
X6 =
1
rv
2 2
冉
−v
⳵
,
⳵p
冊
⳵
⳵
⳵
⳵
+ v2 + q + m
,
⳵v
⳵q
⳵m
⳵p
共5.7兲
in terms of arbitrary functions F共t兲 and G共t兲. Point symmetries 共5.7兲 correspond to the invariance
of reduced system AE兵t , r , z ; u , v , w , p其 共5.5兲 under time translations, two scalings, Galilean invariance in z, pressure invariance, and the additional symmetry X6 which corresponds to the an
introduction of a vortex at the origin given by
共 v ⬘兲 2 = v 2 +
2C
,
r2
p⬘ = p −
C
,
r2
C = const.
Now consider vorticity subsystem 共5.4兲. Under the assumption of axial symmetry, it is denoted by
AW兵t , r , z ; u , v , w , m , n , q其 and given by
1
1
ur + u + v␸ + wz = 0,
r
r
共5.8a兲
⳵
共wm − uq兲 = 0,
⳵z
共5.8b兲
mt +
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A. F. Cheviakov and G. W. Bluman
nt +
⳵
⳵
共un − vm兲 + 共nw − vq兲 = 0,
⳵r
⳵z
共5.8c兲
1 ⳵
共r共uq − wm兲兲 = 0,
r ⳵r
共5.8d兲
qt +
m + vz = 0,
n − uz + wr = 0,
q−
1 ⳵
共rv兲 = 0.
r ⳵r
共5.8e兲
PDE system AW兵t , r , z ; u , v , w , m , n , q其 共5.8兲 is a nonlocally related subsystem of the Euler reduced system with vorticity AEW兵t , r , z ; u , v , w , p , m , n , q其 共5.5兲 and 共5.6兲, and hence is nonlocally related to Euler reduced system AE兵t , r , z ; u , v , w , p其 共5.5兲.
One can show that the point symmetries of system AW兵t , r , z ; u , v , w , m , n , q其 共5.8兲 are given
by
Y 1 = X 1,
Y 2 = X 2,
Y3 = r
Y4 = F共t兲
⳵
⳵
⳵
⳵
⳵
+z +u +v +w
⬃ X3 ,
⳵r
⳵z
⳵u
⳵v
⳵w
⳵
⳵
+ F⬘共t兲
⬃ X4 ,
⳵z
⳵w
共5.9兲
in terms of an arbitrary function F共t兲. It follows that the symmetry X6 in 共5.7兲, which is a point
symmetry of PDE systems AE兵t , r , z ; u , v , w , p其 and AEW兵t , r , z ; u , v , w , p , m , n , q其, yields a nonlocal symmetry of the vorticity subsystem AW兵t , r , z ; u , v , w , m , n , q其..
B. Helically symmetric case
Now consider helical coordinates 共r , ␩ , ␰兲 in R3,
␰ = az + b␸,
␩ = a␸ − bz/r2,
a,b = const,
a2 + b2 ⬎ 0.
In helical coordinates, r is the cylindrical radius; each helix is defined by r = const, ␰ = const; ␩ is
a variable along a helix.
In a helically symmetric setting, the velocity and vorticity vectors are given by
u = u re r + u ␩e ␩ + u ␰e ␰,
␻ = ␻ re r + ␻ ␩e ␩ + ␻ ␰e ␰ ,
where the vector components as well as the pressure p are functions of t , r , ␰. 共Note that in the
limit a = 1, b = 0, helical coordinates become cylindrical coordinates with ␩ = ␸, ␰ = z.兲
Rewriting Euler equations 共5.1兲 in helical coordinates and imposing helical symmetry 共independence of ␩兲,18 one obtains the reduced helically symmetric PDE system
HE兵t , r , ␰ ; ur , u␩ , u␰ , p其, given by
1 ⳵ u␰
ur ⳵ ur
+
+
= 0,
r
⳵ r B共r兲 ⳵␰
共ur兲t + ur共ur兲r +
冉
1 ␰ r
B2共r兲 b ␰
u 共u 兲␰ −
u + au␩
B共r兲
r
r
共u␩兲t + ur共u␩兲r +
共5.10a兲
冊
2
+ pr = 0,
1 ␰ ␩
a2B2共r兲 r ␩
u 共u 兲␰ +
u u = 0,
B共r兲
r
共5.10b兲
共5.10c兲
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103522-11
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共u␰兲t + ur共u␰兲r +
1 ␰ ␰
2abB2共r兲 r ␩ b2B2共r兲 r ␰
1
u 共u 兲␰ +
p␰ = 0.
uu +
uu +
2
3
B共r兲
r
r
B共r兲
共5.10d兲
In 共5.10兲,
B共r兲 =
r
冑a r
2 2
+ b2
.
The helically symmetric version of 共5.2兲 is given by the three scalar equations,
共u␩兲␰
,
B共r兲
共5.11a兲
1 ⳵
1
abB2共r兲 ␩ a2B2共r兲 ␰
共ru␰兲 − 2
u +
u +
共ur兲␰ ,
2
⳵
r
r
r
r
B共r兲
共5.11b兲
␻r = −
␻␩ = −
␻␰ =
a2B2共r兲 ␩
u + 共u␩兲r .
r
共5.11c兲
One can consider the system HEW兵t , r , ␰ ; ur , u␩ , u␰ , ␻r , ␻␩ , ␻␰其, given by the combination of PDE
systems 共5.10兲 and 共5.11兲. This PDE system is locally related to helically symmetric Euler system
HE兵t , r , ␰ ; ur , u␩ , u␰ , p其 共5.10兲, since vorticity components are local functions of velocity components, their derivatives, and independent variables.
The point symmetries of system HEW兵t , r , ␰ ; ur , u␩ , u␰ , ␻r , ␻␩ , ␻␰其 共5.10兲 and 共5.11兲 are given
by
X1 =
X3 = t
⳵
,
⳵t
X2 =
⳵
,
⳵␰
⳵
⳵
⳵
⳵
⳵
⳵
⳵
⳵
− ur r − u␩ ␩ − u␰ ␰ − 2p − ␻r r − ␻␩ ␩ − ␻␰ ␰ ,
⳵t
⳵u
⳵u
⳵u
⳵p
⳵␻
⳵␻
⳵␻
X4 = t
⳵ bB共r兲 ⳵
⳵
−
,
␩ + B共r兲
⳵␰
ar ⳵ u
⳵ u␰
X5 = F共t兲
⳵
,
⳵p
共5.12兲
in terms of an arbitrary function F共t兲. Due to the local relation, point symmetries of helically
symmetric Euler system HE兵t , r , ␰ ; ur , u␩ , u␰ , p其 共5.10兲 are given by projections of symmetries
共5.12兲 onto the space of variables t , r , ␰, ur , u␩ , u␰ , p.
The corresponding helically symmetric version of vorticity subsystem 共5.4兲, where pressure
has been excluded through the application of a curl, is denoted by HW兵t , r , ␰ ; ur , u␩ , u␰ , ␻r , ␻␩ , ␻␰其
and given by
1 ⳵ u␰
ur ⳵ ur
+
+
= 0,
r
⳵ r B共r兲 ⳵␰
共 ␻ r兲 t +
1 ⳵ ␰ r
共u ␻ − ur␻␰兲 = 0,
B共r兲 ⳵␰
共5.13a兲
共5.13b兲
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103522-12
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1 ⳵
1
a2B2共r兲 r ␩
共r共ur␻␩ − u␩␻r兲兲 −
共u ␻ − u␩␻r兲 +
共u␰␻␩ − u␩␻␰兲
r ⳵r
r
B共r兲
共 ␻ ␩兲 t +
+
2abB2共r兲 ␰ r
共u ␻ − ur␻␰兲 = 0,
r2
⳵ r ␰ ␰ r a2B2共r兲 r ␰ ␰ r
共u ␻ − u ␻ 兲 +
共u ␻ − u ␻ 兲 = 0,
⳵r
r
共5.13d兲
共u␩兲␰
,
B共r兲
共5.13e兲
1 ⳵
1
abB2共r兲 ␩ a2B2共r兲 ␰
共ru␰兲 − 2
u +
u +
共ur兲␰ ,
2
r ⳵r
r
r
B共r兲
共5.13f兲
共 ␻ ␰兲 t +
␻r = −
␻␩ = −
共5.13c兲
␻␰ =
a2B2共r兲 ␩
u + 共u␩兲r .
r
共5.13g兲
Its point symmetries are given by
Y2 = X3 − ␻r
Y 1 = X 1,
Y3 = G共t兲
⳵
␩ ⳵
␰ ⳵
,
r −␻
␩ −␻
⳵␻
⳵␻
⳵␻␰
⳵ bB共r兲
⳵
⳵
G⬘共t兲 ␩ + B共r兲G⬘共t兲 ␰ .
−
⳵␰
ar
⳵u
⳵u
共5.14兲
in terms of an arbitrary function G共t兲. 关Note that symmetries X2 , X4 共5.12兲 are special cases of the
infinite family of symmetries Y3.兴
Comparing symmetry classifications 共5.12兲 and 共5.14兲, one observes that the full Galilei group
in the direction of ␰ only occurs as a point symmetry of reduced vorticity subsystem
HW兵t , r , ␰ ; ur , u␩ , u␰ , ␻r , ␻␩ , ␻␰其 共5.13兲, and thus yields a nonlocal symmetry of helically symmetry reduced Euler system HE兵t , r , ␰ ; ur , u␩ , u␰ , p其 共5.10兲.
VI. NONLOCAL SYMMETRIES AND NONLOCAL CONSERVATION LAWS OF MAXWELL’S
EQUATIONS IN „2 + 1… DIMENSIONS
The linear system of Maxwell’s equations in a vacuum in three space dimensions is given by
div B = 0,
Et = curl B,
div E = 0,
共6.1兲
Bt = − curl E,
where B = B ex + B ey + B ez is a magnetic field, E = E ex + E ey + E ez is an electric field, 共x , y , z兲
are Cartesian coordinates, and t is time.
Following Ref. 11, we consider PDE system 共6.1兲 in three-dimensional Minkowski space
共t , x , y兲. It is assumed that B = B共x , y兲ez, E = E1共x , y兲ex + E2共x , y兲ey. Then Maxwell’s equations 共6.1兲
can be written as the PDE system M兵t , x , y ; B , E1 , E2其 in terms of the four equations given by
1
2
3
R1关e1,e2,b兴 = e1x + e2y = 0,
1
2
3
R2关e1,e2,b兴 = e1t − by = 0,
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R3关e1,e2,b兴 = e2t + bx = 0,
R4关e1,e2,b兴 = bt + e2x − e1y = 0.
共6.2兲
We now seek nonlocal symmetries and nonlocal conservation laws of PDE system 共6.2兲. Following the systematic procedure described in Ref. 1, we first construct potential systems for PDE
system 共6.2兲. Note that each of the four equations in 共6.2兲 is a divergence expression as it stands.
Hence for each equation in 共6.2兲, one can introduce a three-component vector potential. This
yields 12 potential variables. From Theorem 6.1 in Ref. 1, it follows that in order to obtain
nonlocal symmetries of Maxwell’s equations 共6.2兲, gauge constraints are required. Since the form
of gauge constraints that could yield nonlocal symmetries is not known a priori, a different
approach is chosen. In particular, the system of Maxwell’s equations 共6.2兲 is equivalent to the
union of a divergence-type conservation law and a curl-type lower-degree conservation law, with
the latter requiring no gauge constraints.1,11 In particular, considering the electromagnetic field
tensors,
冢
0 − e1 − e2
Fij = e1 0
e2 − b
b
0
冣 冢
,
0
e1
e2
冣
Fij = − e1 0 b ,
− e2 − b 0
共6.3兲
and the dual tensor of Fij, given by ⴱFk = 21 ␧ijkFij, where ␧ijk is the Levi–Civita symbol, one can
rewrite Maxwell’s equations 共6.2兲 as
d ⴱ F = 0,
dF = 0,
共6.4兲
where the differential forms are given, respectively, by
F = − e1dt ∧ dx − e2dt ∧ dy + bdx ∧ dy,
ⴱ F = bdt − e2dx + e1dy.
If the three-dimensional Minkowski space 共t , x , y兲 is treated as R3, Eqs. 共6.4兲 can be written in the
conserved form M兵t , x , y ; e1 , e2 , b其,
div共t,x,y兲关b,e2,− e1兴 = 0,
curl共t,x,y兲关b,− e2,e1兴 = 0.
共6.5兲
Using the curl-type conservation law in 共6.5兲, one obtains a determined singlet potential system
MW兵t , x , y ; b , e1 , e2 , w其 given by
− e2 = wx ,
b = w t,
e1 = w y,
bt + e2x − e1y = 0.
共6.6兲
Using the divergence-type conservation law in 共6.5兲, one introduces a vector potential variable
a = 共a0 , a1 , a2兲 to obtain the underdetermined singlet potential system MA兵t , x , y ; b , e1 , e2 , a其 given
by
b = a2x − a1y ,
− e1 = a1t − a0x ,
e1t − by = 0,
e2 = a0y − a2t ,
e1x + e2y = 0,
e2t + bx = 0,
a0t − a1x − a2y = 0,
共6.7兲
appended by a Lorentz gauge for determinedness.
From singlet potential systems 共6.6兲 and 共6.7兲, one obtains the couplet potential system
MAW兵t , x , y ; a , w其 given by
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103522-14
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A. F. Cheviakov and G. W. Bluman
wt = a2x − a1y ,
− wx = a0y − a2t ,
− wy = a1t − a0x ,
a0t − a1x − a2y = 0,
共6.8兲
where the components of the electric and magnetic fields have been excluded through appropriate
substitutions.
The corresponding tree of nonlocally related PDE systems for given PDE system
M兵t , x , y ; e1 , e2 , b其 共6.5兲 was presented in Fig. 1 in Ref. 1.
A. Nonlocal symmetries of Maxwell’s equations „6.2…
Maxwell’s equations 共6.2兲 have eight point symmetries: three translations, one rotation, two
space-time rotations 共boosts兲, one dilation, and one scaling, given by the infinitesimal generators,
X1 =
⳵
,
⳵t
X2 =
X5 = x
⳵
,
⳵x
X3 =
⳵
,
⳵y
X4 = − y
⳵
⳵
⳵
⳵
+ t + b 2 + e2 ,
⳵t ⳵x
⳵e
⳵b
X7 = t
⳵
⳵
⳵
+x +y ,
⳵t
⳵x
⳵y
X6 = y
X8 = e1
⳵
⳵
⳵
⳵
+ x − e2 1 + e1 2 ,
⳵x
⳵y
⳵e
⳵e
⳵
⳵
⳵
⳵
+ t − b 1 − e1 ,
⳵t ⳵x
⳵e
⳵b
⳵
⳵
2 ⳵
.
1 +e
2 +b
⳵b
⳵e
⳵e
共6.9兲
We now seek nonlocal symmetries of PDE system 共6.2兲 that arise as point symmetries of its
potential systems. As discussed in Ref. 1, nonlocal symmetries can only arise from a potential
system if the latter is determined. The point symmetries of determined singlet potential systems
MW兵t , x , y ; b , e1 , e2 , w其 共6.6兲, MA兵t , x , y ; b , e1 , e2 , a其 共6.7兲, and determined couplet potential system MAW兵t , x , y ; a , w其 共6.8兲 are as follows.
Potential system MW兵t , x , y ; b , e1 , e2 , w其 共6.6兲 has eight point symmetries that project onto
point symmetries 共6.9兲 of PDE system 共6.2兲, plus three additional conformal-type point symmetries given by
W1 = 共t2 + x2 + y 2兲
⳵
⳵
⳵
⳵
+ 2tx + 2ty − 共3te1 + 2yb兲 1
⳵x
⳵y
⳵t
⳵e
− 共3te2 − 2xb兲
W2 = 2tx
⳵
⳵
⳵
1
2
− tw ,
2 − 共2ye − 2xe + 3tb + w兲
⳵w
⳵e
⳵b
⳵
⳵
⳵
⳵
+ 共t2 + x2 − y 2兲 + 2xy − 共3xe1 + 2ye2兲 1
⳵t
⳵y
⳵x
⳵e
+ 共2ye1 − 3xe2 + 2tb + w兲
W3 = 2ty
⳵
⳵
⳵
2
− xw ,
2 + 共2te − 3xb兲
⳵w
⳵e
⳵b
⳵
⳵
⳵
⳵
+ 2xy + 共t2 − x2 + y 2兲 − 共3ye1 − 2xe2 + 2tb + w兲 1
⳵t
⳵x
⳵y
⳵e
− 共2xe1 + 3ye2兲
⳵
⳵
⳵
1
− yw ,
2 − 共2te + 3yb兲
⳵w
⳵e
⳵b
共6.10兲
that yield nonlocal symmetries of Maxwell’s equations 共6.2兲.
Potential system MA兵t , x , y ; b , e1 , e2 , a其 共6.7兲 has five point symmetries. They project onto
point symmetries Xi, i = 1 , 2 , 3 , 7 , 8 共6.9兲 of Maxwell’s equations 共6.2兲.
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103522-15
Applications of nonlocal PDE systems in multi-D
J. Math. Phys. 51, 103522 共2010兲
Couplet potential system MAW兵t , x , y ; a , w其 共6.8兲 is potential system 共4.2兲 for the wave
equation 共with w = u, ai = vi兲. Hence it has duality-type symmetries 共4.4兲. In particular, one can
write them as first-order symmetries,
Z1 = a0t
⳵
⳵
⳵
⳵
⳵
⳵
⳵
+ a0y 1 − a0x 2 + a0
− w 0 − a2 1 + a1 2 ,
⳵a
⳵b
⳵e
⳵e
⳵w
⳵a
⳵a
Z2 = a1t
⳵
⳵
⳵
⳵
⳵
⳵
⳵
+ a1y 1 − a1x 2 + a1
+ a2 0 + w 1 + a0 2 ,
⳵a
⳵b
⳵e
⳵e
⳵w
⳵a
⳵a
Z3 = a2t
⳵
⳵
⳵
⳵
⳵
⳵
⳵
+ a2y 1 − a2x 2 + a2
− a1 0 − a0 1 + w 2 .
⳵a
⳵b
⳵e
⳵e
⳵w
⳵a
⳵a
共6.11兲
Symmetries 共6.11兲 yield three additional nonlocal symmetries of Maxwell’s equations 共6.2兲.11
B. Nonlocal conservation laws of Maxwell’s equations „6.2…
(A) The potential system MAW兵t , x , y ; a , w其 with the Lorentz gauge. Potential system
MAW兵t , x , y ; a , w其 共6.8兲 with the Lorentz gauge was used in Ref. 11 to obtain additional conservation laws with explicit dependence of the multipliers on potential variables. As an example,
consider a linear combination of the equations of 共6.8兲 with multipliers depending only on potential variables and their derivatives: ⌳␴共A , W , ⳵A , ⳵W兲, ␴ = 1 , . . . , 4. The solution of the corresponding determining equations1 yields eight sets of nontrivial multipliers given by
⌳1 = C1W + C2A1 + C3A2 + C4A0t + C5 ,
⌳2 = C1A2 + C2A0 + C3W + C4A1t + C6 ,
⌳3 = − C1A1 − C2W + C3A0 + C4A2t + C7 ,
⌳ 4 = C 1A 0 + C 2A 2 − C 3A 1 − C 4W t + C 8 ,
where C1 , . . . , C8 are arbitrary constants. The constants C5 , . . . , C8 simply yield four divergence
expressions 共6.8兲, whereas the constants C1 , . . . , C4 yield conservation laws,
⳵
⳵
1⳵ 2
共w + 共a0兲2 + 共a1兲2 + 共a2兲2兲 + 共− a0a1 − a2w兲 + 共a1w − a0a2兲 = 0,
2 ⳵t
⳵x
⳵y
1 ⳵
⳵ 2
⳵
共a w − a0a1兲 +
共− w2 + 共a0兲2 + 共a1兲2 − 共a2兲2兲 + 共a1a2 − a0w兲 = 0,
2 ⳵x
⳵t
⳵y
1 ⳵
⳵ 1
⳵
共a w + a0a2兲 + 共− a1a2 − a0w兲 +
共− w2 − 共a0兲2 + 共a1兲2 + 共a2兲2兲 = 0,
2 ⳵y
⳵t
⳵x
⳵
⳵
共wa0t − a0wt − a1a2t + a2a1t 兲 + 共a1wt − wa1t + a0a2t − a2a0t 兲
⳵t
⳵x
+
⳵ 2
共a wt − wa2t + a1a0t − a0a1t 兲 = 0.
⳵y
共6.12兲
Since the fluxes in conservation laws 共6.12兲 explicitly involve potential variables 共and not the
combinations of derivatives of potential variables which are identified with the given dependent
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103522-16
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A. F. Cheviakov and G. W. Bluman
variables b , e1 , e2 through potential equations兲, conservation laws 共6.12兲 yield four nonlocal conservation laws of Maxwell’s equations 共6.2兲.
(B) The potential system MAW兵t , x , y ; a , w其 with an algebraic gauge. Now consider the
potential system MAW兵t , x , y ; a , w其,
wt = a2x − a1y ,
− wx = a0y − a2t ,
− wy = a1t − a0x ,
共6.13兲
a2 = 0,
which has the algebraic 共spatial兲 gauge a2 = 0 instead of the Lorentz gauge. One can show that
choosing multipliers
⌳ 1 = A 1,
⌳ 2 = A 0,
⌳3 = − W,
⌳4 = 0,
one obtains an additional nonlocal conservation law of Maxwell’s equations 共6.2兲 given by
1 ⳵ 2
⳵ 1
⳵
共a w兲 + 共a0w兲 +
共w − 共a0兲2 + 共a1兲2兲 = 0.
2 ⳵y
⳵t
⳵x
共6.14兲
Using respectively the algebraic gauges a0 = 0 and a1 = 0, one obtains two further nonlocal conservation laws of Maxwell’s equations 共6.2兲.
(C) The potential system MAW兵t , x , y ; a , w其 with the divergence gauge. Now consider the
potential system MAW兵t , x , y ; a , w其 with the divergence gauge, given by
wt = a2x − a1y ,
− wy = a1t − a0x ,
− wx = a0y − a2t ,
a0t + a1x + a2y = 0.
共6.15兲
We again seek conservation law multipliers depending only on potential variables and their derivatives: ⌳␴共A , W , ⳵A , ⳵W兲, ␴ = 1 , . . . , 4. One can obtain an additional divergence-type conservation law
⳵
⳵
1⳵ 2
共w − 共a0兲2 + 共a1兲2 + 共a2兲2兲 + 共− a2w − a0a1兲 + 共a1w − a0a2兲 = 0
2 ⳵t
⳵x
⳵y
共6.16兲
following from the set of multipliers
⌳1 = W,
⌳ 2 = A 2,
⌳ 3 = − A 1,
⌳4 = A0 ,
which yields a nonlocal conservation law of Maxwell’s equations 共6.2兲.
(D) Other gauges. One can directly show that for conservation law multipliers depending on
potential variables and their first derivatives, no additional conservation laws arise for the potential
system MAW兵t , x , y ; a , w其 with Cronstrom or Poincaré gauges. Other gauges have not been examined.
VII. NONLOCAL SYMMETRIES AND NONLOCAL CONSERVATION LAWS OF
MAXWELL’S EQUATIONS IN „3 + 1… DIMENSIONS
Now consider Maxwell’s equations M兵t , x , y , z ; e , b其 共6.1兲 in four-dimensional Minkowski
space-time 共x0 , x1 , x2 , x3兲 = 共t , x , y , z兲.
As it is written, each of the eight equations in 共6.1兲 is a divergence-type conservation law. As
per Table II in Ref. 1 in n = 4 dimensions, each divergence-type conservation law gives rise to
n共n − 1兲 / 2 = 6 potential variables, i.e, if one directly uses all equations of 共6.1兲 to introduce potentials, one obtains 48 scalar potential variables, and a highly underdetermined potential system.
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103522-17
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Applications of nonlocal PDE systems in multi-D
Instead of using divergence-type conservation laws, lower-degree conservation laws can be
effectively used, as follows.12
In four-dimensional Minkowski space-time, the 4 ⫻ 4 metric tensor is given by ␩␮␯ = ␩␮␯
= diag共−1 , 1 , 1 , 1兲; ␮ , ␯ = 0 , 1 , 2 , 3. The electromagnetic field tensor F␮␯ and its dual ⴱF␮␯ are
given by the matrices
F ␮␯ =
冢
0 − e1 − e2 − e3
e1
−b
e3
b2
e
− b2
b3
0
2
3
1
0
b
− b1
0
冣 冢
,
ⴱ F ␮␯ =
b1
0
− b1
0
2
−e
− b3
e2
−b
3
b2
b3
e3
− e2
0
e1
− e1
0
冣
,
where the dual is defined by ⴱF␮␯ = 21 ␧␮␯␣␤F␣␤ = 21 ␧␮␯␣␤␩␣␥␩␤␦F␥␦ and ␧␮␯␣␤ is the fourdimensional Levi–Civita symbol. 共In this section, Greek indices are assumed to take on the values
0,1,2,3, whereas Latin indices take on the values 1,2,3 and correspond to spatial coordinates.兲
Through use of the differential 2-forms F = F␮␯dx␮ ∧ dx␯, ⴱF = ⴱ F␮␯dx␮ ∧ dx␯, Maxwell’s equations 共6.1兲 can be written as two conservation laws of degree 2,
dF = 0,
d ⴱ F = 0.
共7.1兲
In particular, the equation dF = 0 is equivalent to the four scalar equations div B = 0, Bt =
−curl E, and the equation d ⴱ F = 0 is equivalent to the remaining four equations of 共6.1兲.
Using Poincaré’s lemma, one introduces the magnetic potential a and the electric potential c,
F = da,
ⴱ F = dc,
共7.2兲
where a and c are four-component one-forms a = a␮dx␮, c = c␮dx␮ 共a total of eight scalar potential
variables兲.
The corresponding determined singlet potential system MA兵t , x , y , z ; e , b , a其 is given by
div E = 0,
Et = curl B,
e1 = a0x − a1t ,
e2 = a0y − a2t ,
e3 = az0 − a3t ,
b1 = a3y − az2 ,
b2 = az1 − a3x ,
b3 = a2x − a1y ,
共7.3兲
the determined singlet potential system MC兵t , x , y , z ; e , b , c其 is given by
div B = 0,
Bt = − curl E,
e1 = c3y − cz2,
e2 = cz1 − c3x ,
e3 = c2x − c1y ,
b1 = c1t − c0x ,
b2 = c2t − c0y ,
b3 = c3t − cz0 ,
共7.4兲
and the determined couplet potential system AC兵t , x , y , z ; a , c其 is given by
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103522-18
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A. F. Cheviakov and G. W. Bluman
a3y − az2 = c1t − c0x ,
az1 − a3x = c2t − c0y ,
a2x − a1y = c3t − cz0,
a0x − a1t = c3y − cz2
a0y − a2t = cz1 − c3x ,
az0 − a3t = c2x − c1y ,
共7.5兲
where electric and magnetic field components have been excluded through substitutions.
The above potential systems are underdetermined. In particular, both a and c are defined to
within arbitrary four-dimensional gradients. It is natural to use Lorentz gauges for these potentials
due to the Minkowski geometry, as well as the symmetry and linearity of Maxwell’s equations
共6.2兲. In Sec. VII B, we will show that other gauges are also useful for finding nonlocal conservation laws.
A. Nonlocal symmetries
Consider the determined potential system which consists of six PDEs 共7.5兲 appended by
Lorentz gauges,
a0t − a1x − a2y − az3 = 0,
c0t − c1x − c2y − cz3 = 0.
共7.6兲
This appended potential system has 23 point symmetries including four space-time translations,
one dilation, six rotations/boosts, six internal rotations/boosts, one scaling, one duality-rotation,
and four conformal symmetries. In particular, the conformal symmetries,
X1 = − 共t2 + x2 + y 2 + z2兲
⳵
⳵
⳵
⳵
− 2tx − 2ty − 2tz
⳵x
⳵y
⳵z
⳵t
+ 共3ta0 + xa1 + ya2 + za3兲
+ 共ya0 + 3ta2 − zc1 + xc3兲
+ 共3tc0 + xc1 + yc2 + zc3兲
⳵
0
3
1
2 ⳵
2 + 共za + 3ta + yc − xc 兲
⳵a
⳵ a3
⳵
2
3
0
1 ⳵
0 + 共− za + ya + xc + 3tc 兲
⳵c
⳵ c1
+ 共za1 − xa3 + yc0 + 3tc2兲
X2 = 2tx
⳵
0
1
2
3 ⳵
0 + 共xa + 3ta + zc − yc 兲
⳵a
⳵ a1
⳵
1
2
0
3 ⳵
,
2 + 共− ya + xa + zc + 3tc 兲
⳵c
⳵ c3
共7.7兲
⳵
⳵
⳵
⳵
+ 共t2 + x2 − y 2 − z2兲 + 2xy + 2xz
⳵t
⳵y
⳵z
⳵x
+ 共− 3xa0 + ta1 + zc2 + yc3兲
⳵
0
1
2
3 ⳵
0 − 共ta + 3xa + ya + za 兲
⳵a
⳵ a1
+ 共ya1 + 3xa2 − zc0 − tc3兲
⳵
⳵
+ 共za1 − 3xa3 + yc0 + tc2兲 3
⳵ a2
⳵a
+ 共za2 − ya3 − 3xc0 − tc1兲
⳵
⳵
− 共tc0 + 3xc1 + yc2 + zc3兲 1
⳵ c0
⳵c
+ 共za0 + ta3 + yc1 − 3xc2兲
⳵
⳵
+ 共− ya0 − ta2 + zc1 − 3xc3兲 3 ,
⳵ c2
⳵c
共7.8兲
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103522-19
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Applications of nonlocal PDE systems in multi-D
X3 = 2ty
⳵
⳵
⳵
⳵
+ 2xy + 共t2 − x2 + y 2 − z2兲 + 2yz
⳵t
⳵x
⳵z
⳵y
+ 共− 3ya0 − ta2 + zc1 − xc3兲
⳵
2
3
0
1 ⳵
2 + 共za − 3ya − xc − tc 兲
⳵a
⳵ a3
− 共ta0 + xa1 + 3ta2 + za3兲
+ 共− za1 + xa3 − 3yc0 − tc2兲
− 共tc0 + xc1 + 3yc2 + zc3兲
X4 = 2tz
⳵
1
2
0
3 ⳵
0 + 共− 3ya + xa + zc + tc 兲
⳵a
⳵ a1
⳵
0
3
1
2 ⳵
0 + 共− za − ta − 3yc + xc 兲
⳵c
⳵ c1
⳵
⳵
+ 共xa0 + ta1 + zc2 − 3yc3兲 3 ,
⳵ c2
⳵c
共7.9兲
⳵
⳵
⳵
⳵
+ 2xz + 2yz + 共t2 − x2 − y 2 + z2兲
⳵t
⳵x
⳵y
⳵z
+ 共− 3za0 + ta3 + yc1 + xc2兲
⳵
⳵
+ 共− 3za1 + xa3 − yc0 − tc2兲 1
⳵ a0
⳵a
+ 共− 3za2 + ya3 + xc0 + tc1兲
⳵
⳵
− 共ta0 + xa1 + ya2 + 3za3兲 3
⳵ a2
⳵a
+ 共ya1 − xa2 − 3zc0 − tc3兲
⳵
⳵
+ 共ya0 + ta2 − 3zc1 + xc3兲 1
⳵ c0
⳵c
+ 共− xa0 − ta1 − 3zc2 + yc3兲
⳵
0
1
2
3 ⳵
,
2 − 共tc + xc + yc + 3zc 兲
⳵c
⳵ c3
共7.10兲
can be shown to correspond to four nonlocal symmetries of Maxwell system M兵t , x , y , z ; e , b其
共6.1兲. In particular, one can show that the symmetry components corresponding to the electric and
magnetic fields e , b essentially depend on symmetric combinations of derivatives of the potential
variables and are not expressible through local variables via potential equations 共7.3兲 and 共7.4兲.
Additional nonlocal symmetries of Maxwell’s equations 共6.1兲 in four-dimensional space-time
were obtained in Ref. 12 which arise as local 共first-order兲 symmetries of determined potential
system AC兵t , x , y , z ; a , c其 共7.5兲 appended by Lorentz gauges.
B. Nonlocal divergence-type conservation laws
Consider system AC兵t , x , y , z ; a , c其 共7.5兲. We seek nonlocal conservation laws of Maxwell’s
equations 共6.1兲 arising as local conservation laws of its potential system 共7.5兲, using the direct
method, with multipliers depending only on potential variables and their derivatives:
⌳␴共A , C , ⳵A , ⳵C兲. 共In each subsequent case, only first derivatives that are not dependent through
the equations of the system are included in the dependence of the multipliers, in order to exclude
trivial conservation laws.兲
(A) Gauge-invariant nonlocal conservation laws. First, consider conservation laws arising
from underdetermined potential system 共7.5兲. It follows that such conservation laws will hold for
any gauge. Following the direct method, one obtains 2090 linear PDEs for the six unknown
multipliers. Its complete solution yields seven independent sets of multipliers. Six of these sets
correspond to conservation laws that are PDEs 共7.5兲 themselves. The other set is given by
⌳1 = C3y − Cz2 = E1,
⌳2 = Cz1 − C3x = E2,
⌳3 = C2x − C1y = E3 ,
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103522-20
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⌳4 = A3y − Az2 = B1,
⌳2 = Az1 − A3x = B2,
⌳6 = A2x − A1y = B3 .
共7.11兲
The corresponding conservation law given by the divergence expression 共exterior derivative兲,
d⌿ = 0,
⌿1 = a ∧ F + c ∧ ⴱ F,
共7.12兲
was found in Ref. 12 and is a gauge-invariant nonlocal conservation law of Maxwell’s equations
共6.1兲.
(B) Nonlocal conservation laws arising from algebraic gauges. As a specific example, consider potential system 共7.5兲 with the algebraic gauge a0 = c0 = 0. Here we seek local conservation
laws arising from multipliers of the form
⌳␴ = ⌳␴共Ã,C̃, ⳵ Ã, ⳵ C̃兲,
␴ = 1, . . . ,6,
à ⬅ 共A1,A1,A3兲,
C̃ ⬅ 共C1,C1,C3兲.
The complete solution of the corresponding determining equations yields seven sets of multipliers.
Six of these sets of multipliers correspond to PDEs 共7.5兲 as before, and the other set of multipliers
given by
⌳ i = C i,
⌳i+3 = Ai,
共7.13兲
i = 1,2,3,
yields the conservation law
⳵
1⳵
共amam + cmcm兲 − k ␧kij共aic j兲 = 0,
2 ⳵t
⳵x
共7.14兲
which is a nonlocal conservation law of Maxwell’s equations 共6.1兲.
(C) Nonlocal conservation laws for the divergence gauge. Now, consider potential system
共7.5兲 appended with two divergence gauges
a0t + a1x + a2y + az3 = 0,
共7.15兲
c0t + c1x + c2y + cz3 = 0.
We seek local conservation laws of the resulting determined potential system arising from multipliers of the form
⌳␴共A,C, ⳵ A, ⳵ C兲,
␴ = 1, . . . ,8.
The solution of the determining equations yields 11 sets of multipliers, corresponding to
• the eight obvious conservation laws 关PDEs 共7.5兲 and 共7.15兲兴;
• gauge-invariant conservation law 共7.12兲;
• two additional sets of multipliers,
⌳ i = C i,
⌳i = Ait,
⌳i+3 = Ai,
i = 1,2,3,
⌳ 7 = A 0,
⌳i+3 = − Cit,
i = 1,2,3,
⌳7 = − C0t ,
⌳8 = C0 ,
⌳8 = A0t .
共7.16兲
共7.17兲
The additional conservation law corresponding to multipliers 共7.16兲 is given by
⳵
1⳵
共␩␮␯a␮a␯兲 − k 共a0ak + c0ck + ␧kijaic j兲 = 0.
2 ⳵t
⳵x
The conservation law corresponding to multipliers 共7.17兲 is given by
冉
冊
冉
共7.18兲
冊
⳵
⳵
⳵ ␮␯ ␮ ⳵ 0 ␮ ⳵ 0
⳵
␩ a ␯ c − c ␯ a − k ␧kij ai j a0 + ci j c0 = 0.
⳵x
⳵x
⳵x
⳵x
⳵t
⳵x
共7.19兲
Conservation laws 共7.18兲 and 共7.19兲 are nonlocal conservation laws of Maxwell’s equations 共6.1兲.
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103522-21
Applications of nonlocal PDE systems in multi-D
J. Math. Phys. 51, 103522 共2010兲
(D) Nonlocal conservation laws for the Lorentz gauge. As a last example, for determined
potential system 共7.5兲 and 共7.6兲 with the Lorentz gauges, we obtain all local conservation laws
arising from zeroth-order multipliers,
⌳␴ = ⌳␴共A,C兲,
␴ = 1, . . . ,8.
关First-order conservation laws of potential system 共7.5兲 and 共7.6兲 are given in Ref. 12.兴
The solution of the multiplier determining equations yields 12 sets of multipliers. Eight of
them correspond to eight equations 共7.5兲 and 共7.6兲 which are divergence expressions as they stand.
The additional four sets of multipliers,
⌳ 1 = − p 0C 1 − p 1C 0 + p 2A 3 − p 3A 2,
⌳ 2 = − p 0C 2 − p 1A 3 − p 2C 0 + p 3A 1 ,
⌳ 3 = − p 0C 3 + p 1A 2 − p 2A 1 − p 3C 0,
⌳ 4 = − p 0A 1 − p 1A 0 − p 2C 3 + p 3C 2 ,
⌳ 5 = − p 0A 2 + p 1C 3 − p 2A 0 − p 3C 1,
⌳ 6 = − p 0A 3 − p 1C 2 + p 2C 1 − p 3A 0 ,
⌳ 7 = p 0A 0 + p 1A 1 + p 2A 2 + p 3A 3,
⌳ 8 = p 0C 0 + p 1C 1 + p 2C 2 + p 3C 3 ,
共7.20兲
involving arbitrary constants p1 , . . . , p4, respectively, yield four conservation laws which are nonlocal conservation laws of Maxwell’s equations 共6.1兲. In particular, the conservation law corresponding to p1 = 1, p2 = p3 = p4 = 0, is given by
⳵
1⳵
共兩a兩2 + 兩c兩2兲 − k 共a0ak + c0ck + ␧kijaic j兲 = 0,
2 ⳵t
⳵x
共7.21兲
and the other three are obtained from corresponding permutations of the indices.
VIII. NONLOCAL SYMMETRIES AND EXACT SOLUTIONS OF THE THREE-DIMENSIONAL
MHD EQUILIBRIUM EQUATIONS
Consider the PDE system of ideal MHD equilibrium equations in three space dimensions
given by
div共␳v兲 = 0,
共8.1a兲
div b = 0,
1
␳v ⫻ curl v − b ⫻ curl b − grad p − ␳ grad兩v兩2 = 0,
2
共8.1b兲
curl v ⫻ b = 0.
共8.1c兲
In 共8.1兲, the dependent variables are the plasma density ␳, the plasma velocity v = 共v , v , v 兲, the
pressure p, and the magnetic field b = 共b1 , b2 , b3兲; the independent variables are the spatial coordinates 共x , y , z兲. For closure, one must add an appropriate equation of state that relates pressure
and density to MHD equations 共8.1兲.
It has been shown16,17,20 that an infinite number of nonlocal symmetries exist for MHD
equations 共8.1兲 for two different equations of state. These symmetries have been used in the
literature for the construction of physical plasma equilibrium solutions. Moreover, the symmetries
for incompressible equilibria 共Sec. VIII A兲 preserve solution stability, i.e., map stable magnetohydrodynamic equilibria into stable magnetohydrodynamic equilibria. For additional details and
examples, see Refs. 16, 19, 21, and 22.
1
2
3
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103522-22
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A. F. Cheviakov and G. W. Bluman
A. Nonlocal symmetries for incompressible MHD equilibria
As a first simplified example, consider the incompressible MHD equilibrium system
I兵x , y , z ; b , v , p其 with constant density 共without loss of generality, ␳ = 1兲, given by
div v = 0,
共8.2a兲
div b = 0,
1
v ⫻ curl v − b ⫻ curl b − grad p − grad兩v兩2 = 0,
2
共8.2b兲
curl共v ⫻ b兲 = 0.
共8.2c兲
Using lower-degree conservation law 共8.2c兲, one introduces a potential variable ␺,
v ⫻ b = grad ␺ .
共8.3兲
共Note that ␺ has the direct physical meaning of a function enumerating magnetic surfaces, i.e.,
two-dimensional surfaces to which streamlines and magnetic field lines are tangent. In general,
every three-dimensional plasma domain is spanned by such surfaces.兲
The resulting determined potential system I⌿兵x , y , z ; b , v , p , ␺其 is given by
div v = 0,
div b = 0,
v ⫻ b = grad ␺ ,
1
v ⫻ curl v − b ⫻ curl b − grad p − grad兩v兩2 = 0.
2
共8.4a兲
共8.4b兲
Now a comparison is made between the point symmetries of PDE systems 共8.2兲 and 共8.4兲. Incompressible MHD equilibrium system 共8.2兲 has ten point symmetries: translations in pressure and
two scalings, given, respectively, by
Xp =
⳵
,
⳵p
XD = x
⳵
⳵
⳵
+y +z ,
⳵x
⳵y
⳵z
XS = bi
⳵
⳵
i ⳵
,
i +v
i + 2p
⳵b
⳵v
⳵p
the interchange symmetry given by
XI = vi
⳵
⳵
i ⳵
,
i +b
i − 共b · v兲
⳵b
⳵v
⳵p
and the Euclidean group 共three space translations and three rotations兲 given by
XE = ␨ ⳵
⳵
+ 共b · grad兲␨ ,
⳵x
⳵b
where the hook symbol denotes summation over vector components, x = 共x , y , z兲, ␨ = a + 共b ⫻ x兲,
and a, b are arbitrary constant vectors in R3.
The first nine symmetries of MHD system 共8.2兲 directly yield point symmetries of potential
system 共8.4兲. In addition, potential system 共8.4兲 has the obvious potential shift symmetry given by
X␺ =
⳵
,
⳵␺
as well as an infinite number of point symmetries given by
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103522-23
J. Math. Phys. 51, 103522 共2010兲
Applications of nonlocal PDE systems in multi-D
冉
X⬁ = M共␺兲 vi
冊
⳵
⳵
i ⳵
,
i +b
i − 共b · v兲
⳵b
⳵v
⳵p
共8.5兲
where M共␺兲 is an arbitrary smooth function of its argument. Point symmetries 共8.5兲 yield nonlocal
symmetries of incompressible MHD equilibrium system 共8.2兲. One can show that globally symmetries 共8.5兲 transform a given solution 共b , v , p兲 to a family of solutions 共b⬘ , v⬘ , p⬘兲 given by16,23
x⬘ = x,
y ⬘ = y,
z⬘ = z,
b⬘ = b cosh M共␺兲 + v sinh M共␺兲,
v⬘ = v cosh M共␺兲 + b sinh M共␺兲,
p⬘ = p + 共兩b兩2 − 兩b⬘兩2兲/2.
共8.6兲
Since transformations 共8.6兲 depend on an arbitrary function M共␺兲, that is, constant on magnetic
surfaces, they can be used to obtain families of physically interesting solutions from a known
MHD equilibrium solution. Transformations 共8.6兲 preserve magnetic surfaces: b⬘ ⫻ v⬘ is parallel
to b ⫻ v.
As a simple example, consider the well-known simple “transverse flow” solution of MHD
equilibrium system 共8.2兲 given by
b = H共r兲ez,
v = ␻共r兲共− yex + xey兲,
p共r兲 = F共r兲 − H2共r兲/2,
F共r兲 =
冕
r
q␻2共q兲dq
共8.7兲
0
depending on two arbitrary functions H共r兲, ␻共r兲. This solution describes the differential rotation of
a constant-density ideal gas plasma around the z-axis, for the vertical magnetic field; r = 冑x2 + y 2 is
a cylindrical radius. The magnetic surfaces ␺ = const are cylinders r = const around the z-axis. In
Fig. 1共a兲, field lines of solution 共8.7兲 tangent to the cylinder r = 1 are shown for H共r兲 = e−r, ␻共r兲
= 2e−2r. Using transformations 共8.6兲 with an arbitrary function M共␺兲 = f共r兲, one obtains an infinite
family of solutions 共8.7兲 for a noncollinear magnetic field and velocity given by
b = H共r兲cosh共f共r兲兲ez + v sinh共f共r兲兲,
v = cosh共f共r兲兲v + H共r兲sinh共f共r兲兲ez .
共8.8兲
Here the magnetic field lines and plasma streamlines are helices that are tangent to cylindrical
2
magnetic surfaces r = const, with slopes depending on r. For f共r兲 = e−r , original and transformed
magnetic field lines and streamlines tangent to the cylinder r = 1 are shown in Fig. 1.
One can show16,20 that for incompressible plasma equilibria with nonconstant plasma density,
there exist infinite sets of transformations that generalize 共8.6兲, as follows. If the density ␳ is
constant on magnetic surfaces, i.e.,
grad ␳ · B = grad ␳ · V = 0,
then the infinite set of transformations,
x⬘ = x,
B⬘ = b共⌿兲B + c冑␳V,
y ⬘ = y,
V⬘ =
z⬘ = z,
c共⌿兲
b共⌿兲
V,
B+
a共⌿兲
a共⌿兲冑␳
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103522-24
J. Math. Phys. 51, 103522 共2010兲
A. F. Cheviakov and G. W. Bluman
2
1.5
1
0.5
0
–2
–2
–1
–1
0
0
1
1
(a)
2
2
10
8
6
4
2
0
–2
–2
–1
–1
0
0
1
1
(b)
2
2
FIG. 1. Magnetic field lines and streamlines of “transverse flow” MHD equilibrium solution 共8.7兲 共a兲 and its transformed
version 共8.8兲 共b兲. Magnetic field lines are shown with thick lines and plasma streamlines with thin lines.
␳⬘ = a2共⌿兲␳,
1
P⬘ = CP + 共C兩B兩2 − 兩B⬘兩2兲,
2
共8.9兲
maps a given solution 共B , V , P , ␳兲 of PDE system 共8.1兲 into a family of solutions 共B⬘ , V⬘ , P⬘ , ␳⬘兲
with the same set of magnetic field lines. In 共8.9兲, a共⌿兲 and b共⌿兲 are arbitrary functions constant
on magnetic surfaces ⌿ = const and b2共⌿兲 − c2共⌿兲 = C = const.
B. Nonlocal symmetries for compressible adiabatic MHD equilibria
Now consider the system of compressible MHD equilibrium equations C兵x , y , z ; b , v , p , ␳其
given by
div共␳v兲 = 0,
div b = 0,
v · grad p + ␥ p div v = 0,
共8.10a兲
共8.10b兲
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103522-25
J. Math. Phys. 51, 103522 共2010兲
Applications of nonlocal PDE systems in multi-D
1
␳v ⫻ curl v − b ⫻ curl b − grad p − ␳ grad兩v兩2 = 0,
2
共8.10c兲
curl共v ⫻ b兲 = 0.
共8.10d兲
PDE system 共8.10兲 describes plasmas corresponding to the ideal gas equation of state and undergoing an adiabatic process. Here the entropy S = p / ␳␥ is constant throughout the plasma domain.
A determined potential system C⌿兵x , y , z ; b , v , p , ␳ , ␺其 is obtained, as before, through replacing conservation law 共8.10d兲 by potential equations 共8.3兲. The resulting potential system
C⌿兵x , y , z ; b , v , p , ␳ , ␺其 has an infinite number of point symmetries given by the infinitesimal
generator,
冉
XC = N共␺兲 vi
冊 冉冕
⳵
⳵
− 2␳
+
⳵ vi
⳵␳
N共␺兲d␺
冊
⳵
,
⳵␺
共8.11兲
where N共␺兲 is an arbitrary smooth function.17 Point symmetries 共8.11兲 yield nonlocal symmetries
of compressible MHD equilibrium system 共8.10兲. The finite form of the transformations of physical variables is readily found to be given by
x⬘ = x,
y ⬘ = y,
z⬘ = z,
v⬘ = f共␺兲v,
b⬘ = b,
␳⬘ = ␳/f 2共␺兲.
p⬘ = p,
共8.12兲
Some generalizations of symmetry transformations 共8.12兲 are considered in Refs. 16 and 20.
IX. DISCUSSION AND OPEN PROBLEMS
In this paper, the systematic framework for obtaining nonlocally related PDE systems in
multidimensions 共n ⱖ 3 independent variables兲, including procedures for obtaining determined
nonlocally related PDE systems, as presented in Ref. 1 has been illustrated with examples. Nonlocal symmetries and nonlocal conservation laws have been used as a measure of “usefulness” of
nonlocally related PDE systems due to their straightforward computation and, often, transparent
physical meaning. In particular, new examples of nonlocal symmetries and nonlocal conservation
laws have been found for the following situations.
• A nonlocal symmetry arising from a nonlocally related subsystem in 共2 + 1兲 dimensions 共Sec.
II兲.
• Nonlocal symmetries and nonlocal conservation laws of a nonlinear “generalized plasma
equilibrium” PDE system in three space dimensions. 关These nonlocal symmetries and nonlocal conservation laws arise as local symmetries and conservation laws of a potential system
following from a lower-degree 共curl-type兲 conservation law 共Sec. III兲.兴
• Nonlocal symmetries of dynamic Euler equations of incompressible fluid dynamics arising
from reduced systems for axial as well as helical symmetries 共Sec. V兲.
• Nonlocal conservation laws of Maxwell’s equations in 共2 + 1兲-dimensional Minkowski space,
arising from a potential system appended with algebraic and divergence gauges 共Sec. VI兲.
• Nonlocal symmetries and nonlocal conservation laws of Maxwell’s equations in
共3 + 1兲-dimensional Minkowski space, arising from a potential system of degree 2, appended
with algebraic and divergence gauges 共Sec. VII兲.
Moreover, known examples from the existing literature were discussed and synthesized within the
framework presented in Ref. 1.
The well-known Geroch group24,25 of nonlocal 共potential兲 symmetries of Einstein’s equations
with a metric that admits a Killing vector has not been considered in this paper. This example is
a natural generalization of ideas discussed above on the calculus on manifolds. It uses a conservation law of degree 1 to introduce a scalar potential variable. The symmetries are used in Refs.
24 and 25 to generate new exact solutions of Einstein’s equations. The Geroch group example
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103522-26
A. F. Cheviakov and G. W. Bluman
J. Math. Phys. 51, 103522 共2010兲
reinforces the understanding that a given PDE system has to have an internal geometric structure
共in this case, a Killing vector兲 in order to have lower-degree conservation laws.
Although this paper has substantially synthesized and extended known results for multidimensional nonlocally related PDE systems, many more examples are needed to arrive at a better
understanding of interconnections between nonlocally related PDE systems with n ⱖ 3 independent variables. The principal difficulty in performing computations lies in the complexity in
solving determining equations for symmetries and conservation laws in multidimensions. Open
problems include the following.
共1兲
共2兲
共3兲
Find examples of nonlinear PDE systems with n ⱖ 3 independent variables, for which nonlocal symmetries arise as local symmetries of a potential system following from a
divergence-type conservation law共s兲, appended with some gauge constraint共s兲.
Find efficient procedures to obtain “useful” gauge constraints 共e.g., yielding nonlocal symmetries and/or nonlocal conservation laws兲 for potential systems arising from divergencetype conservation laws 共as well as for underdetermined potential systems arising from lowerdegree conservation laws兲. In particular, do there exist further refinements of Theorems 6.1
and 6.3 of Ref. 1 that can rule out consideration of specific families of gauges for particular
classes of potential systems?
Find further examples of lower-degree conservation laws for PDE systems of physical importance. 关Conservation laws of degree one 共curl-type in R3兲 would be of particular interest,
since corresponding potential systems are determined.兴 Examples suggest that lower-degree
conservation laws are rather rare and are only expected to exist when a given PDE system
has a special geometrical structure. On the other hand, divergence-type conservation laws are
rather common.
ACKNOWLEDGMENTS
The authors are grateful to NSERC for research support.
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26
Note that system 共5.3兲 contains only first-order PDEs. Normally, in the point symmetry analysis procedure, if all
differential equations are of the same order, no differential consequences are used in symmetry determining equations.
However, in PDE system 共5.3兲, an important differential consequence of PDEs 共5.4c兲 is div ␻ = 0. Without explicitly
using this constraint, one misses infinite symmetries Y4 in 共5.9兲 and Y3 in 共5.14兲.
1
2
3
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