We study nonlocal symmetries and their similarity reductions of Riccati and Abel chains. Our results show that all the equations in Riccati chain share the same form of nonlocal symmetry. The similarity reduced Nth order ordinary differential equation (ODE), N = 2, 3, 4, …, in this chain yields (N − 1)th order ODE in the same chain. All the equations in the Abel chain also share the same form of nonlocal symmetry (which is different from the one that exist in Riccati chain) but the similarity reduced Nth order ODE, N = 2, 3, 4, …, in the Abel chain always ends at the (N − 1)th order ODE in the Riccati chain. We describe the method of finding general solution of all the equations that appear in these chains from the nonlocal symmetry.
REFERENCES
1.
2.
H. T.
Davis
, Introduction to Nonlinear Differential and Integral Equations
(Dover
, New York
, 1962
).3.
M. J.
Ablowitz
and P. A.
Clarkson
, Solitons: Nonlinear Evolution Equations and Inverse Scattering
(Cambridge University Press
, Cambridge, England
, 1991
).4.
J. F.
Cariñena
, P.
Guha
, and M. F.
Rañada
, Nonlinearity
22
, 2953
(2009
).5.
J. F.
Cariñena
and A.
Ramos
, Int. J. Mod. Phys. A
14
, 1935
(1999
).6.
N.
Euler
and P. G. L
Leach
, Theor. Math. Phys.
159
, 474
(2009
).7.
V. K.
Chandrasekar
, M.
Senthilvelan
, A.
Kundu
, and M.
Lakshmanan
, J. Phys. A
39
, 9743
(2006
);V. K.
Chandrasekar
, M.
Senthilvelan
, A.
Kundu
, and M.
Lakshmanan
, J. Phys. A
, 39
10945
(2006
).8.
R. Gladwin
Pradeep
, V. K.
Chandrasekar
, M.
Senthilvelan
, and M.
Lakshmanan
, J. Math. Phys.
51
, 103513
(2010
).9.
V. K.
Chandrasekar
, M.
Senthilvelan
, and M.
Lakshmanan
, J. Phys. A
40
, 4717
(2007
);V. K.
Chandrasekar
, S. N.
Pandey
, M.
Senthilvelan
, and M.
Lakshmanan
, Chaos, Solitons Fractals
26
, 1399
(2005
).10.
V. K.
Chandrasekar
, M.
Senthilvelan
, and M.
Lakshmanan
, Proc. R. Soc. London, Ser. A
462
, 1831
(2006
).11.
V. K.
Chandrasekar
, M.
Senthilvelan
, and M.
Lakshmanan
, J. Phys. A
37
, 4527
(2004
).12.
A.
Gonzalez-Lopez
, Phys. Lett. A
4
, 190
(1988
).13.
M. C.
Nucci
, J. Nonlinear Math. Phys.
15
, 205
(2008
).14.
A. A.
Adam
and F. M.
Mahomed
, Nonlinear. Dyn.
30
, 267
(2002
).15.
B.
Abraham-Shrauner
, A.
Govinder
, and P. G. L.
Leach
, Phys. Lett. A
203
, 169
(1995
).16.
B.
Abraham-Shrauner
, J. Nonlinear Math. Phys.
9
, 1
(2002
).17.
C.
Muriel
and J. L.
Romero
, IMA J. Appl. Math.
66
, 111
(2001
);C.
Muriel
and J. L.
Romero
, Theor. Math. Phys.
133
, 1565
(2002
);P. J.
Olver
, J. Differ. Equations
222
, 164
(2006
);P. J.
Olver
, J. Nonlinear Math. Phys.
15
, 300
(2008
).18.
E.
Pucci
and G.
Saccomandi
, J. Phys. A
35
, 6145
(2002
).19.
B.
Abraham-Shrauner
, IMA J. Appl. Math.
56
, 235
(1996
);M. C.
Nucci
and P. G. L.
Leach
, J. Math. Anal. Appl.
251
, 871
(2000
);B.
Abraham-Shrauner
and K. S.
Govinder
, J. Nonlinear Math. Phys.
13
, 612
(2006
).20.
G. W.
Bluman
and S. C.
Anco
, Symmetries and Integration Methods for Differential Equations
(Springer-Verlag
, New York
, 2002
).21.
P. J.
Olver
, Applications of Lie Groups to Differential Equations
(Springer-Verlag
, New York
, 1986
).22.
M.
Lakshmanan
and M. Senthil
Velan
, J. Math. Phys.
33
, 4068
(1992
).23.
M. L.
Gandarias
, Theor. Math. Phys.
159
, 778
(2009
).24.
M. L.
Gandarias
and M. S.
Bruzon
, J. Nonlinear Math. Phys.
18
, 123
(2011
).25.
G. W.
Bluman
and S.
Kumei
, Symmetries and Differential Equations
(Springer
, Berlin
, 1989
).26.
P. E.
Hydon
, Symmetry Methods and Differential Equations
(Cambridge University Press
, Cambridge, England
, 2000
).27.
G. W.
Bluman
and G. R.
Reid
, IMA J. App. Math.
40
, 87
(1988
).28.
M. L.
Gandarias
, E.
Medina
, and C.
Muriel
, Nonlinear Anal.: Theory, Methods Appl.
47
, 5167
(2001
);M. L.
Gandarias
, E.
Medina
, and C.
Muriel
, J. Nonlinear Math. Phys.
9
, 47
(2002
).29.
E. S.
Cheb-Terrab
, L. G. S.
Duarte
, and L. A. C. P.
da Mota
, Comput. Phys. Commun.
108
, 90
(1998
).© 2012 American Institute of Physics.
2012
American Institute of Physics
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