For systems of partial differential equations (PDEs) with n≥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 [A. F. Cheviakov and G. W. Bluman, J. Math. Phys. 51, 103521 (2010)]. 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.

Topics
Einstein field equations, Einstein Maxwell equations, Gauge fixing, Algebraic number theory, Partial differential equations, Vector fields, Minkowski spacetime, Gauge field theory, Fluid dynamics, Vortex dynamics
1.
A. F.
 
Cheviakov
 and
G. W.
 
Bluman
,
J. Math. Phys.
 
51
,
103521
(
2010
).
Google Scholar
Crossref
Search ADS
 
2.
G. W.
 
Bluman
 and
A. F.
 
Cheviakov
,
J. Math. Phys.
 
46
,
123506
(
2005
).
https://doi.org/10.1063/1.2142834
Google Scholar
Crossref
Search ADS
 
3.
G. W.
 
Bluman
,
A. F.
 
Cheviakov
, and
N. M.
 
Ivanova
,
J. Math. Phys.
 
47
,
113505
(
2006
).
https://doi.org/10.1063/1.2349488
Google Scholar
Crossref
Search ADS
 
4.
G. W.
 
Bluman
 and
S.
 
Kumei
,
J. Math. Phys.
 
28
,
307
(
1987
).
https://doi.org/10.1063/1.527659
Google Scholar
Crossref
Search ADS
 
5.
G. W.
 
Bluman
 and
P.
 
Doran-Wu
,
Acta Appl. Math.
 
2
,
79
(
1995
).
Google Scholar
 
6.
A.
 
Ma
, “
Extended group analysis of the wave equation
,” M.Sc. thesis,
University of British Columbia
,
1991
.
Google Scholar
 
7.
I. S.
 
Akhatov
,
R. K.
 
Gazizov
, and
N. H.
 
Ibragimov
,
Sov. Math. Dokl.
 
35
,
384
(
1987
).
Google Scholar
 
8.
A.
 
Sjöberg
 and
F. M.
 
Mahomed
,
Appl. Math. Comput.
 
150
,
379
(
2004
).
https://doi.org/10.1016/S0096-3003(03)00259-5
Google Scholar
Crossref
Search ADS
 
9.
G. W.
 
Bluman
,
A. F.
 
Cheviakov
, and
J. -F.
 
Ganghoffer
,
J. Eng. Math.
 
62
,
203
(
2008
).
https://doi.org/10.1007/s10665-008-9221-7
Google Scholar
Crossref
Search ADS
 
10.
G. W.
 
Bluman
 and
A. F.
 
Cheviakov
,
J. Math. Anal. Appl.
 
333
,
93
(
2007
).
https://doi.org/10.1016/j.jmaa.2006.10.091
Google Scholar
Crossref
Search ADS
 
11.
S. C.
 
Anco
 and
G. W.
 
Bluman
,
J. Math. Phys.
 
38
,
3508
(
1997
).
https://doi.org/10.1063/1.531866
Google Scholar
Crossref
Search ADS
 
12.
S. C.
 
Anco
 and
D.
 
The
,
Acta Appl. Math.
 
89
,
1
(
2005
).
https://doi.org/10.1007/s10440-005-9001-6
Google Scholar
Crossref
Search ADS
 
13.
A. F.
 
Cheviakov
 and
S. C.
 
Anco
,
Phys. Lett. A
 
372
,
1363
(
2008
).
https://doi.org/10.1016/j.physleta.2007.09.065
Google Scholar
Crossref
Search ADS
 
14.
A. F.
 
Cheviakov
,
Comput. Phys. Commun.
 
176
,
48
(
2007
).
https://doi.org/10.1016/j.cpc.2006.08.001
Google Scholar
Crossref
Search ADS
 
15.
G. W.
 
Bluman
,
A. F.
 
Cheviakov
, and
S. C.
 
Anco
,
Applications of Symmetry Methods to Partial Differential Equations
,
Applied Mathematical Sciences
Vol.
168
(
Springer
,
New York
,
2010
).
Google Scholar
Crossref
Search ADS
 
16.
O. I.
 
Bogoyavlenskij
,
Phys. Lett. A
 
291
,
256
(
2001
).
https://doi.org/10.1016/S0375-9601(01)00650-8
Google Scholar
Crossref
Search ADS
 
18.
O.
 
Kelbin
,
A. F.
 
Cheviakov
, and
M.
 
Oberlack
 (unpublished).
19.
O. I.
 
Bogoyavlenskij
,
Phys. Rev. E
 
62
,
8616
(
2000
).
https://doi.org/10.1103/PhysRevE.62.8616
Google Scholar
Crossref
Search ADS
 
20.
O. I.
 
Bogoyavlenskij
,
Phys. Rev. E
 
66
,
056410
(
2002
).
https://doi.org/10.1103/PhysRevE.66.056410
Google Scholar
Crossref
Search ADS
 
21.
A. F.
 
Cheviakov
,
Phys. Rev. Lett.
 
94
,
165001
(
2005
).
https://doi.org/10.1103/PhysRevLett.94.165001
Google Scholar
Crossref
Search ADS
PubMed
 
22.
K. I.
 
Ilin
 and
V. A.
 
Vladimirov
,
Phys. Plasmas
 
11
,
3586
(
2004
).
https://doi.org/10.1063/1.1759337
Google Scholar
Crossref
Search ADS
 
23.
A. F.
 
Cheviakov
,
Phys. Lett. A
 
321
,
34
(
2004
).
https://doi.org/10.1016/j.physleta.2003.12.006
Google Scholar
Crossref
Search ADS
 
24.
R.
 
Geroch
,
J. Math. Phys.
 
12
,
918
(
1971
).
https://doi.org/10.1063/1.1665681
Google Scholar
Crossref
Search ADS
 
25.
R.
 
Geroch
,
J. Math. Phys.
 
13
,
394
(
1972
).
https://doi.org/10.1063/1.1665990
Google Scholar
Crossref
Search ADS
 
26.
Note that system (5.3) 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 (5.3), an important differential consequence of PDEs (5.4c) is div ω=0⁠. Without explicitly using this constraint, one misses infinite symmetries Y4 in (5.9) and Y3 in (5.14).
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2010
American Institute of Physics
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