Related papers: Darboux polynomials for Lotka-Volterra systems in …
Given a constant skew-symmetric matrix A, it is a difficult open problem whether the associated Lotka-Volterra system is integrable or not. We solve this problem in the special case when A is a Toepliz matrix where all off-diagonal entries…
Integrable systems in low dimensions, constructed through the symmetry reduction method, are studied using phase portrait and variable separation techniques. In particular, invariant quantities and explicit periodic solutions are…
We classify all the possible $delta$-vectors of d-dimensional integral convex polytopes whose volumes are less than or equal to 3/(d!).
Simple derivation is presented of the four families of infinitely many shape invariant Hamiltonians corresponding to the exceptional Laguerre and Jacobi polynomials. Darboux-Crum transformations are applied to connect the well-known shape…
In this paper we review two concepts directly related to the Lax representations: Darboux transformations and Recursion operators for integrable systems. We then present an extensive list of integrable differential-difference equations…
We show that some hard to detect properties of quadratic ODEs (eg certain preserved integrals and measures) can be deduced more or less algorithmically from their Kahan discretization, using Darboux Polynomials (DPs). Somewhat similar…
We prove a Darboux-Jouanolou type theorem on the algebraic integrability of polynomial differential $r$-forms over arbitrary fields ($r\geq 1$). We also investigate the Darboux's method for producing integrating factors.
In this paper we utilize the covariance of Ricatti equation with respect to linear fractional transformations to define classes of conformally equivalent second order differential equations. This motivates then the introduction of…
In the article the problem of the integrable classification of nonlinear lattices depending on one discrete and two continuous variables is studied. By integrability we mean the presence of reductions of a chain to a system of hyperbolic…
An algebraic classification is given for spaces of holomorphic vector-valued modular forms of arbitrary real weight and multiplier system, associated to irreducible, T-unitarizable representations of the full modular group, of dimension…
We derive formulae for the calculation of Taylor coefficients of solutions to systems of Volterra integral equations, both linear and nonlinear, either without singularities or with singularities of Abel type and logarithmic type. We also…
In this paper we investigate factorizations of polynomials over the ring of dual quaternions into linear factors. While earlier results assume that the norm polynomial is real ("motion polynomials"), we only require the absence of real…
We classify integrable third order equations in 2+1 dimensions which generalize the examples of Kadomtsev-Petviashvili, Veselov-Novikov and Harry Dym equations. Our approach is based on the observation that dispersionless limits of…
The concept of extended Hamiltonian systems allows the geometrical interpretation of several integrable and superintegrable systems with polynomial first integrals of degree depending on a rational parameter. Until now, the procedure of…
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 plane wave solutions of the three-wave resonant interaction in the plane are considered. It is shown that rank-one constraints over the right derivatives of invertible operators on an arbitrary linear space gives solutions of the…
We prove a singular Darboux type theorem for homogeneous polynomial closed $2$-forms of degree one on $\mathbb{C}^n$. As application, we classify non-integrable codimension one distributions, of degree one, and arbitrary classes on…
The multi-indexed Laguerre and Jacobi polynomials form a complete set of orthogonal polynomials. They satisfy second-order differential equations but not three term recurrence relations, because of the 'holes' in their degrees. The…
We introduce a new type of reduction of inversive difference polynomials that is associated with a partition of the basic set of automorphisms $\sigma$ and uses a generalization of the concept of effective order of a difference polynomial.…
We study pseudo-Abelian integrals associated with polynomial deformations of slow-fast Darboux integrable systems. Under some assumptions we prove local boundedness of the number of their zeros.