Nonlocal symmetries of Riccati and Abel chains and their similarity reductions

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 N th order ordinary differential equation (ODE), N = 2, 3, 4, …, in this chain yields (N...

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Bibliographic Details
Published in:Journal of mathematical physics 2012-02, Vol.53 (2), p.023512-023512-8
Main Authors: Bruzon, M. S., Gandarias, M. L., Senthilvelan, M.
Format: Article
Language:English
Subjects:
Exact sciences and technology
Mathematical methods in physics
Mathematics
Ordinary differential equations
Physics
Sciences and techniques of general use
Solutions
Symmetry
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Summary: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 N th 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 N th 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.
ISSN:0022-2488
1089-7658
DOI:10.1063/1.3682473