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.

Order preserving contact transformations and dynamical symmetries of scalar and coupled Riccati and Abel chains

“…The above two chains possess several interesting geometric structures and certain common mathematical properties, see for example [20,25]. The results which we present here will further illuminate the shared mathematical properties of these two chains.…”

“…One can again straightforwardly verify that the contact transformation (25) linearizes the nonlinear ODE (21) toü = 0, which is of the same order as (21). Using the contact transformation (25) we can find the general solution of the second order Abel equation as well.…”

“…One can again straightforwardly verify that the contact transformation (25) linearizes the nonlinear ODE (21) toü = 0, which is of the same order as (21). Using the contact transformation (25) we can find the general solution of the second order Abel equation as well. Substituting the solution u of the free particle equation into (25) and rewriting it, we get the following first order ODE, namely (ẋ + x 3 )(t + I 1 ) − x = 0.…”

Dealing with rational second order ordinary differential equations where both Darboux and Lie find it difficult: The S-function method

“…Leach, F.M. Mahomed, A.A. Adam, M. L. Gandarias, M. S. Bruzón, M. Senthilvelan and others worked with the concept of hiden and non local symetries [11,12,13,14,15,16,17,18]. C. Muriel and J.L.…”

A generalization of the S-function method applied to a Duffing-Van der Pol forced oscillator

“…Therefore, there is a constant search for generalizations of these methods, which, in turn, leads to the definition of a series of concepts and alternative procedures to attack these more complicated cases. Among these, we highlight: [5,6,7,8,9,10,11,12,13,14] in the 'symmetry side' and [15,16,17,18,19,20,21,22,23,24] in the 'Darbouxian side'.…”