Related papers: Liouvillian integrability of three dimensional vec…

We study vector fields of the plane preserving the form of Liouville. We present their local models up to the natural equivalence relation, and describe local bifurcations of low codimension. To achieve that, a classification of univariate…

Liouville conformal field theory is a prototypical example of an exactly solvable quantum field theory, in the sense that the correlation functions in an arbitrary background can be determined exactly using only the constraints of unitarity…

In this paper, we study the properties of the first global term in the polyhomogeneous expansions for Liouville's equation. We obtain rigidity and gap results for the boundary integral of the global coefficient. We prove that such a…

We develop a method, based on Darboux' and Liouville's works, to find first integrals and/or invariant manifolds for a physically relevant class of dynamical systems, without making any assumption on these elements' form. We apply it to…

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…

There are many research available on the study of real-valued fractal interpolation function and fractal dimension of its graph. In this paper, our main focus is to study the dimensional results for vector-valued fractal interpolation…

Singular complex analytic vector fields on the Riemann surfaces enjoy several geometric properties (singular means that poles and essential singularities are admissible). We describe relations between singular complex analytic vector fields…

We give an algorithm for deciding whether a planar polynomial differential system has a first integral which factorizes as a product of defining polynomials of curves with only one place at infinity. In the affirmative case, our algorithm…

The notion of integrability is discussed for classical nonautonomous systems with one degree of freedom. The analysis is focused on models which are linearly spanned by finite Lie algebras. By constructing the autonomous extension of the…

Let $k$ be a differential field of characteristic zero and $E$ be a liouvillian extension of $k$. For any differential subfield $K$ intermediate to $E$ and $k$, we prove that there is an element in the set $K-k$ satisfying a linear…

We show how all the quantal systems related to the exceptional Laguerre and Jacobi polynomials can be constructed in a direct and systematic way, without the need of shape invariance and Darboux-Crum transformation. Furthermore, the…

We develop a general technique for computing functional integrals with fixed area and boundary length constraints. The correct quantum dimensions for the vertex functions are recovered by properly regularizing the Green function. Explicit…

Differential Galois theory has played important roles in the theory of integrability of linear differential equation. In this paper we will extend the theory to nonlinear case and study the integrability of the first order nonlinear…

An infinite set of operator-valued relations in Liouville field theory is established. These relations are enumerated by a pair of positive integers $(m,n)$, the first $(1,1)$ representative being the usual Liouville equation of motion. The…

In this study we work on a novel Hamiltonian system which is Liouville integrable. In the integrable Hamiltonian model, conserved currents can be represented as Binomial polynomials in which each order corresponds to the integral of motion…

We consider an integrable polynomial system with generalized Darboux first integral H_0. We assume that it defines a family of real cycles in a region bounded by a polycycle. To any polynomial form \eta one can associate the pseudo-abelian…

We consider the Hardy-H\'enon system $-\Delta u =|x|^a v^p$, $-\Delta v =|x|^b u^q$ with $p,q>0$ and $a,b\in {\mathbb R}$ and we are concerned in particular with the Liouville property, i.e. the nonexistence of positive solutions in the…

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…

In this work we compute the families of classical Hamiltonians in two degrees of freedom in which the Normal Variational Equation around an invariant plane falls in Schroedinger type with polynomial or trigonometrical potential. We analyze…

We consider Hamiltonians associated with 3 dimensional conformally flat spaces, possessing 2, 3 and 4 dimensional isometry algebras. We use the conformal algebra to build additional {\em quadratic} first integrals, thus constructing a large…