Nonlocal symmetries of Riccati and Abel chains and their similarity reductions

Abstract

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.

Accepted for publication in J. Math. Phys

References in corpus (2)

• New aspects of integrability of force-free Duffing-van der Pol oscillator and related nonlinear systems
• A nonlocal connection between certain linear and nonlinear ordinary differential equations/oscillators

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• Finding nonlocal Lie symmetries algorithmically
• A generalization of the S-function method applied to a Duffing-Van der Pol forced oscillator
• Order preserving contact transformations and dynamical symmetries of scalar and coupled Riccati and Abel chains
• Solving 1ODEs with functions