1202.4593 (M. S. Bruzon et al.)

Nonlocal symmetries of Riccati and Abel chains and their similarity
reductions    [PDF]

M. S. Bruzon, M. L. Gandarias, M. Senthilvelan

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.

View original: http://arxiv.org/abs/1202.4593