The Riccati and Ermakov-Pinney hierarchies

Abstract

The concept and use of recursion operators is well-established in the study of evolution, in particular nonlinear, equations. We demonstrate the application of the idea of recursion operators to ordinary differential equations. For the purposes of our demonstration we use two equations, one chosen from the class of linearisable hierarchies of evolution equations studied by Euler et al (Stud Appl Math 111 (2003) 315-337) and the other from the class of integrable but nonlinearisible equations studied by Petersson et al (Stud Appl Math 112 (2004) 201-225). We construct the hierarchies for each equation. The symmetry properties of the first hierarchy are considered in some detail. For both hierarchies we apply the singularity analysis. For both we observe intersting behaviour of the resonances for the different possible leading order behaviours. In particular we note the proliferation of subsidiary solutions as one ascends the hierarchy.

Rota–Baxter Operators on Pre-Lie Superalgebras

“…The construction of associative, Lie, pre-Lie and L-dendriform superalgebras is extended to the corresponding categories of bimodules. See [9,29,[31][32][33]43,46] about further results and [10,11,41,48,49] about relationships with Yang-Baxter equation.…”

Rota–Baxter coalgebras and Rota–Baxter bialgebras

“…Rota-Baxter algebras were introduced in [21] in the context of differential operators on commutative Banach algebras and since [4] intensively studied in probability and combinatorics, and more recently in mathematical physics, such as dendriform algebras * Corresponding author 1 (see [3,6,7,11]), free Rota-Baxter algebras (see [5,8,9,13,14]), Lie algebras (see [2,18]), multiple zeta values (see [8,15]) and Connes-Kreimer renormalization theory in quantum field theory (see [10]), etc. One can refer to the book [12] for the detailed theory of Rota-Baxter algebras.…”

Construction of Rota-Baxter algebras via Hopf module algebras

“…During the past four decades, this algebraic object has been investigated extensively by many mathematicians with various motivations. At present, Rota-Baxter algebras have become a useful tool in many branches of mathematics, such as combinatorics (see [9]), Loday type algebras (see [8,10]), pre-Lie and pre-Poisson algebras (see [1,2,20]), multiple zeta values (see [11,16]), and so on. Besides their own interest in mathematics, Rota-Baxter algebras also have many important applications in mathematical physics.…”