Related papers: One-dimensional dynamical systems type delta over …
Starting from the notion of discriminantly separable polynomials of degree two in each of three variables, we construct a class of integrable dynamical systems. These systems can be integrated explicitly in genus two theta-functions in a…
All Darboux integrable difference equations on the quad-graph are described in the case of the equations that possess autonomous first-order integrals in one of the characteristics. A generalization of the discrete Liouville equation is…
By taking the Weyl equation with external electro-magnetic potentials as the simplest representative for a system of PDOs, we give a new method of treating non-commutativity of coefficients matrices. More precisely, we construct a Fourier…
This text is a revised version of the authors Habilitationsschrift which was submitted to the University of Augsburg, 1993. Fuchs type differential operators are used to model the analysis on manifolds with cone--like singularities, or more…
We study the first order structure of $L^2$-differential one-forms on the Sierpi\'{n}ski gasket. We consider piecewise energy finite functions related to one-forms, normal parts, and show self-similarity properties of one-forms as in the…
A deformed differential calculus is developed based on an associative star-product. In two dimensions the Hamiltonian vector fields model the algebra of pseudo-differential operator, as used in the theory of integrable systems. Thus one…
The theory of abstract Friedrichs operators, introduced by Ern, Guermond and Caplain (2007), proved to be a successful setting for studying positive symmetric systems of first order partial differential equations (Friedrichs, 1958),…
We consider here a new operator, called ``super nabla'', which is shown to be generic among operators for which the modified Macdonald polynomials are joint eigenfunctions. All previously known Macdonald eigenoperators can readily be…
This paper is a survey of our recent work on operator algebras associated to dynamical systems that lead to classification results for the systems in terms of algebraic invariants of the operator algebras.
We introduce the notion of discrete Baker-Akhiezer (DBA) modules, which are modules over the ring of difference operators, as a certain discretization of Baker-Akhiezer modules which are modules over the ring of differential operators. We…
The solvability for infinite dimensional differential algebraic equations possessing a resolvent index and a Weierstra{\ss} form is studied. In particular, the concept of integrated semigroups is used to determine a subset on which…
We address the problem of classification of integrable differential-difference equations in 2+1 dimensions with one/two discrete variables. Our approach is based on the method of hydrodynamic reductions and its generalisation to dispersive…
Integration of nonlinear partial differential equations with the help of the non-commutative integration over octonions is studied. An apparatus permitting to take into account symmetry properties of PDOs is developed. For this purpose…
We construct explicit differential operators on hermitian modular forms, extending methods developed for Siegel modular forms. These differential operators are closely related to the two-variable spherical pluriharmonic polynomials. We…
The dependence on the domain is studied for the Dirichlet eigenvalues of an elliptic operator considered in bounded domains. Their proximity is measured by a norm of the difference of two orthogonal projectors corresponding to the reference…
In this article, a new definition of fractional Hilfer difference operator is introduced. Definition based properties are developed and utilized to construct fixed point operator for fractional order Hilfer difference equations with initial…
We develop an approach analogous to classical Frobenius theory for the analysis of singularities of ODEs in the case of discrete dynamical systems. Our methodology is based on the Roman-Rota theory of finite operators and relies crucially…
In this work, we introduce the new class of functions which can use to solve the nonlinear/linear multi-dimensional differential equations. Based on these functions, a numerical method is provided which is called the Developed Lagrange…
A deformation of the standard prolongation operation, defined on sets of vector fields in involution rather than on single ones, was recently introduced and christened "\sigma-prolongation"; correspondingly one has "\sigma-symmetries" of…
In their work on differential operators in positive characteristic, Smith and Van den Bergh define and study the derived functors of differential operators; they arise naturally as obstructions to differential operators reducing to positive…