Related papers: Rational recursion operators for integrable differ…
In this paper we are developing a theory of rational (pseudo) difference Hamiltonian operators, focusing in particular on its algebraic aspects. We show that a pseudo--difference Hamiltonian operator can be represented as a ratio $AB^{-1}$…
In this paper we discuss the concept of cosymmetries and co--recursion operators for difference equations and present a co--recursion operator for the Viallet equation. We also discover a new type of factorisation for the recursion…
First we use a new approach to give a graded Lie algebra whose Maurer-Cartan elements characterize pre-Lie algebra structures. Then using this graded Lie bracket we define the notion of a Nijenhuis operator on a pre-Lie algebra which…
In this paper we study the algebraic properties of a new integrable differential-difference equation. This equation can be seen as a deformation of the modified Narita-Itoh-Bogoyavlensky equation and has the Kaup-Kupershmidt equation in its…
In this paper, we first introduce the notion of a hyper relative differential operator on a Lie algebra, in which Nijenhuis operators are used to characterize the relative differential operators and their inverse. We then introduce the…
Led by the key example of the Korteweg-de Vries equation, we study pairs of Hamiltonian operators which are non-homogeneous and are given by the sum of a first-order operator and an ultralocal structure. We present a complete classification…
We introduce the idea of constructing recursion operators for full-fledged nonlocal symmetries and apply it to the reduced quasi-classical self-dual Yang-Mills equation. It turns out that the discovered recursion operators can be…
In this paper we continue studying of matrix $n\times n$ linear differential intertwining operators. The problems of minimization and of reducibility of matrix intertwining operators are considered and criterions of weak minimizability and…
Hamiltonian operators are used in the theory of integrable partial differential equations to prove the existence of infinite sequences of commuting symmetries or integrals. In this paper it is illustrated the new Reduce package \cde for…
A recursion operator is an integro-differential operator which maps a generalized symmetry of a nonlinear PDE to a new symmetry. Therefore, the existence of a recursion operator guarantees that the PDE has infinitely many higher-order…
The core object of this paper is a pair $(L, e)$, where $L$ is a Nijenhuis operator and $e$ is a vector field satisfying a specific Lie derivative condition, i.e., $Lie_{e}L=\operatorname{Id}$. Our research unfolds in two parts. In the…
Two aspects on the important notion of pre-Lie algebras are pre-Lie bialgebras (or left-symmetric bialgebras) with motivation from para-K\"ahler Lie algebras, and Nijenhuis operators on pre-Lie algebras arising from their deformation…
An explicit formula to find symmetry recursion operators for partial differential equations (PDEs) is obtained from new results connecting variational integrating factors and non-variational symmetries. The formula is special case of a…
Some results on (pre-)Jacobi-Jordan algebras and their representations are proved. Moreover, the notion of matched pairs and relative Rota-Baxter operators on these algebras are introduced and studied. The cohomology theory of relative…
The importance of the theory of pseudo-differential operators in the study of non linear integrable systems is point out. Principally, the algebra $\Xi $ of nonlinear (local and nonlocal) differential operators, acting on the ring of…
We present first heavenly equation of Pleba\'nski in a two-component evolutionary form and obtain Lagrangian and Hamiltonian representations of this system. We study all point symmetries of the two-component system and, using the inverse…
In this paper we investigate the algebraic structure related to a new type of correlator associated to the moduli spaces of $S^1$-parametrized curves in contact homology and rational symplectic field theory. Such correlators are the natural…
Noetherian operators are differential operators that encode primary components of a polynomial ideal. We develop a framework, as well as algorithms, for computing Noetherian operators with local dual spaces, both symbolically and…
Formally symmetric differential operators on weighted Hardy-Hilbert spaces are analyzed, along with adjoint pairs of differential operators. Eigenvalue problems for such operators are rather special, but include many of the classical…
A local resolution of the Problem of Time has recently been given, alongside reformulation as a local theory of Background Independence. The classical part of this can be viewed as requiring just Lie's Mathematics, albeit entrenched in…