Riccati和Abel链的非局部对称性及其相似约化
Nonlocal symmetries of Riccati and Abel chains and their similarity reductions
摘要：
研究了Riccati链和Abel链的非局部对称性及其相似约化。结果表明，Riccati链中的所有方程都具有相同的非局部对称性。在这条链上的相似性约化了$N^{th}$阶常微分方程（ODE）$N=2, 3,4,...$，在同一条链上产生了$(N-1)^{th}$阶常微分方程。Abel链中的所有方程也具有相同形式的非局部对称性（这与Riccati链中存在的非局部对称性不同），但Abel链中的相似约化$N^{th}$阶ODE，$N=2, 3,4,$，总是在Riccati链中的$(N-1)^{th}$阶ODE处结束。我们描述了从非局部对称性出发求这些链中出现的所有方程的通解的方法。
英文摘要：
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.
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