Related papers: Darboux-Jouanolou Integrability for Arbitrary Fiel…
The problem of discretization of Darboux integrable equations is considered. Given a Darboux integrable continuous equation, one can obtain a Darboux integrable differential-discrete equation, using the integrals of the continuous equation.…
We define four different kinds of multiplicity of an invariant algebraic curve for a given polynomial vector field and investigate their relationships. After taking a closer look at the singularities and at the line of infinity, we improve…
The article contributes to the theory of infinitesimal bendings of smooth surfaces in Euclidean 3-space. We derive a linear differential equation of the first order, which previously did not appear in the literature and which is satisfied…
We discuss an integrable discretization of the principal chiral field models equations and its involutive reduction. We present a Darboux transformation and general construction of solution solutions for these discrete equations.
We prove an existential finiteness Varchenko-Khovanskii type result for integrals of rational 1-forms over the level curves of Darbouxian integrals.
This paper details the lesser known conditions on ${\mathbb {R}}^{n}$ for the integrability of pfaffian forms, or 1-forms. Emphasis is given to locality of these conditions, and proofs in some additional detail are provided for theorems due…
M.F. Singer [Liouvillian first integrals of differential equations, Trans. Amer. Math. Soc. 333 (1992), 673--688] proved the equivalence between Liouvillian integrability and Darboux integrability for two dimensional polynomial differential…
We study differential-difference equation of the form $$ \frac{d}{dx}t(n+1,x)=f(t(n,x),t(n+1,x),\frac{d}{dx}t(n,x)) $$ with unknown $t(n,x)$ depending on continuous and discrete variables $x$ and $n$. Equation of such kind is called Darboux…
We establish a Liouville type theorem for some conformally invariant fully nonlinear equations
Some ideas and remarks are presented concerning a possible Lagrangian approach to the study of internal boundary conditions relating integrable fields at the junction of two domains. The main example given in the article concerns single…
We consider a discrete equation, defined on the two-dimensional square lattice, which is linearizable, namely, of the Burgers type and depends on a parameter $\alpha$. For any natural number $N$ we choose $\alpha$ so that the equation…
In this paper we present a far-reaching generalization of E. Vessiot's analysis of the Darboux integrable partial differential equations in one dependent and two independent variables. Our approach provides new insights into this classical…
In this article we bound the number of remarkable values of a polynomial vector field. The proof is short and based on Jouanolou's theorem about rational first integrals of planar polynomial derivations. Our bound is given in term of the…
We define two isomorphic algebras of differential operators: the first algebra consists of ordinary differential operators and contains the hypergeometric differential operator, while the second one consists of partial differential…
In this paper we present a short material concerning to some results in Morales-Ramis theory, which relates two different notions of integrability: Integrability of Hamiltonian Systems through Liouville Arnold Theorem and Integrability of…
Nonlinear semi-discrete equations of the form t_x(n+1)=f(t(n), t(n+1), t_x(n)) are studied. An adequate algebraic formulation of the Darboux integrability is discussed and the attempt to adopt this notion to the classification of Darboux…
In this paper we derive new two-component integrable differential difference and partial difference systems by applying a Lax-Darboux scheme to an operator formed from an ${\mathfrak{sl}}_3({\mathbb{C}})$-based automorphic Lie algebra. The…
We establish a general Liouville type theorem for conformally invariant fully nonlinear equations.
In this paper we analyze the tangential symmetries of Darboux integrable decomposable exterior differential systems. The decomposable systems generalize the notion of a hyperbolic exterior differential system and include the classic notion…
The number of singularities, counted with multiplicity, of a generic codimension one holomorphic distribution on a compact toric orbifold is determined. As a consequence, we give the classification of regular distributions on rational…