Related papers: Internality of autonomous algebraic differential e…
Our purpose in this paper is to study when a planar differential system polynomial in one variable linearizes in the sense that it has an inverse integrating factor which can be constructed by means of the solutions of linear differential…
The problem of constructing semi-discrete integrable analogues of the Liouville type integrable PDE is discussed. We call the semi-discrete equation a discretization of the Liouville type PDE if these two equations have a common integral.…
In this paper we present an algebraic study concerning the general second order linear differential equation with polynomial coefficients. By means of Kovacic's algorithm and asymptotic iteration method we find a degree independent…
Even if it is nonintegrable, a differential equation may nevertheless admit particular solutions which are globally analytic. On the example of the dynamical system of Kuramoto and Sivashinsky, which is generically chaotic and presents a…
In this paper, we study the algebraic, rational and formal Puiseux series solutions of certain type of systems of autonomous ordinary differential equations. More precisely, we deal with systems which associated algebraic set is of…
The integrability problem consists in finding the class of functions a first integral of a given planar polynomial differential system must belong to. We recall the characterization of systems which admit an elementary or Liouvillian first…
The purpose of this article is to develop an algebraic approach to the problem of integrable classification of differential-difference equations with one continuous and two discrete variables. As a classification criterion, we put forward…
We make explicit certain results around the Galois correspondence in the context of definable automorphism groups, and point out the relation to some recent papers dealing with the Galois theory of algebraic differential equations when the…
A criterion in the spirit of Rosenlicht is given, on the rational function f(x), for when the planar vector field defined by x'=f(x) and y'=xy admits a pair of algebraically independent first integrals over some extension of the base field.…
We consider an ordinary nonlinear differential equation with generalized coefficients as an equation in differentials in algebra of new generalized functions. Then the solution of such equation will be a new generalized function. In the…
In this paper are examined general classes of linear and non-linear analytical systems of partial differential equations. Indeed the integrability conditions are found and if they are satisfied, the solutions are given as functional series…
There exists a well established differential topological theory of singularities of ordinary differential equations. It has mainly studied scalar equations of low order. We propose an extension of the key concepts to arbitrary systems of…
In this paper we address the problem of representing solutions of a system of scalar linear partial difference equations akin to state space equations of 1-D systems theory. We first obtain a representation formula for a special class of…
We present further developments on the Lagrangian 1-form description for one-dimensional integrable systems in both discrete and continuous levels. A key feature of integrability in this context called a closure relation will be derived…
An inverse problem to identify unknown coefficients of a partial differential equation by a single interior measurement is considered. The equation considered in this paper is a strongly elliptic second order scalar equation which can have…
It is shown that, by appropriately defining the eigenfunctions of a function defined on the extended phase space, the Liouville theorem on solutions of the Hamilton--Jacobi equation can be formulated as the problem of finding common…
We investigate the behavior of integral formulations of variable coefficient elliptic partial differential equations (PDEs) in the presence of steep internal layers. In one dimension, the equations that arise can be solved analytically and…
We consider entire solutions to $L u= f(u)$ in $\RR^2$, where $L$ is a general nonlocal operator with kernel $K(y)$. Under certain natural assumtions on the operator $L$, we show that any stable solution is a 1D solution. In particular, our…
An elementary example shows that the number of zeroes of a component of a solution of a system of linear ordinary differential equations cannot be estimated through the norm of coefficients of the system alone.
In dimension n isolated singularities -- at a finite point or at infinity -- for solutions of finite total mass to the n-Liouville equation are of logarithmic type. As a consequence, we simplify the classification argument in…