Related papers: Linear Differential Systems with Small Coefficient…
Several recently discovered properties of multiple families of special polynomials (some orthogonal and some not) that satisfy certain differential, difference or q-difference equations are reviewed. A general method of construction of…
We study the relation between the Galois group $G$ of a linear difference-differential system and two classes $\mathcal{C}_1$ and $\mathcal{C}_2$ of groups that are the Galois groups of the specializations of the linear difference equation…
We investigate the Liouvillian integrability of Hamiltonian systems describing a universe filled with a scalar field (possibly complex). The tool used is the differential Galois group approach, as introduced by Morales-Ruiz and Ramis. The…
This paper is dedicated to the differential Galois theory in the complex analytic context for Lie-Vessiot systems. Those are the natural generaliza- tion of linear systems, and the more general class of differential equations adimitting…
In this paper we consider a reduction of a non-homogeneous linear system of first order operator equations to a totally reduced system. Obtained results are applied to Cauchy problem for linear differential systems with constant…
We obtain conditions for the differentiability of weak solutions for a second-order uniformly elliptic equation in divergence form with a homogeneous co-normal boundary condition. The modulus of continuity for the coefficients is assumed to…
To every Darboux integrable system there is an associated Lie group $G$ which is a fundamental invariant of the system and which we call the Vessiot group. This article shows that solving the Cauchy problem for a Darboux integrable partial…
We extend Kovacic's algorithm to compute the differential Galois group of some second order parameterized linear differential equation. In the case where no Liouvillian solutions could be found, we give a necessary and sufficient condition…
We study dissipative translationally invariant free-fermionic theories with quadratic Liouvillians. Using a Lie-algebraic approach, we solve the Lindblad equation and find the density matrix at all times for arbitrary time dependence of the…
Picard--Vessiot theorem (1910) provides a necessary and sufficient condition for solvability of linear differential equations of order $n$ by quadratures in terms of its Galois group. It is based on the differential Galois theory and is…
Liouville theorems for scaling invariant nonlinear parabolic equations and systems (saying that the equation or system does not possess nontrivial entire solutions) guarantee optimal universal estimates of solutions of related initial and…
A Dirichlet-type problem is studied for an equation of even order with variable coefficients. A criterion for the uniqueness of a solution is given. The solution is built in the form of a Fourier series. When justifying the convergence of…
Consider a third order linear differential equation $L(f)=0$, where $L\in\mathbb{Q}(z)[\partial_z]$. We design an algorithm computing the Liouvillian solutions of $L(f)=0$. The reducible cases devolve to the classical case of second order…
A categorical theory for the discretization of a large class of dynamical systems with variable coefficients is proposed. It is based on the existence of covariant functors between the Rota category of Galois differential algebras and…
The linearization of complex ordinary differential equations is studied by extending Lie's criteria for linearizability to complex functions of complex variables. It is shown that the linearization of complex ordinary differential equations…
Let $A=(a_{ij})_{n\times n}$ be a nonnegative, symmetric, irreducible and invertible matrix. We prove the existence and uniqueness of radial solutions to the following Liouville system with singularity: $$\{{array}{ll} \Delta…
We prove that the monodromy group of a reduced irreducible square system of general polynomial equations equals the symmetric group. This is a natural first step towards the Galois theory of general systems of polynomial equations, because…
This paper considers the problem of testing whether there exists a solution satisfying certain non-negativity constraints to a linear system of equations. Importantly and in contrast to some prior work, we allow all parameters in the system…
We revisit the results on admissible transformations between normal linear systems of second-order ordinary differential equations with an arbitrary number of dependent variables under several appropriate gauges of the arbitrary elements…
In the present paper we derive Liouville type results and existence of periodic solutions for $\chi^{(2)}$ type systems with non-homogeneous nonlinearities. Moreover, we prove both universal bounds as well as singularity and decay estimates…