相关论文: Integrability of planar polynomial differential sy…
Given a planar differential system with a first integral, we show how to find a normalizer. For systems with a center, we give an integral formula for the derivative of its period function.
We investigate which polynomials can possibly occur as factors in the denominators of rational solutions of a given partial linear difference equation (PLDE). Two kinds of polynomials are to be distinguished, we call them /periodic/ and…
The Painleve test is very useful to construct not only the Laurent-series solutions but also the elliptic and trigonometric ones. Such single-valued functions are solutions of some polynomial first order differential equations. To find the…
We study the projective systems in both continuous and discrete settings. These systems are linearizable by construction and thus, obviously, integrable. We show that in the continuous case it is possible to eliminate all variables but one…
We show that for a natural polynomial Hamiltonian system the existence of a single Darboux polynomial (a partial polynomial first integral) is equivalent to the existence of an additional first integral functionally independent with the…
This paper is devoted to searching for Riemannian metrics on 2-surfaces whose geodesic flows admit a rational in momenta first integral with a linear numerator and denominator. The explicit examples of metrics and such integrals are…
We present a new technique for constructing solutions of quasilinear systems of first-order partial differential equations, in particular inhomogeneous ones. A generalization of the Riemann invariants method to the case of inhomogeneous…
Three symbolic algorithms for testing the integrability of polynomial systems of partial differential and differential-difference equations are presented. The first algorithm is the well-known Painlev\'e test, which is applicable to…
Let $\bbK$ be an ordinary differential field with derivation $\partial$. Let $\cP$ be a system of $n$ linear differential polynomial parametric equations in $n-1$ differential parameters with implicit ideal $\id$. Given a nonzero linear…
An effective method to obtain exact analytical solutions of equations describing the coherent dynamics of multilevel systems is presented. The method is based on the usage of orthogonal polynomials, integral transforms and their discrete…
Given a single (differential-algebraic) input-output equation, we present a method for finding different representations of the associated system in the form of rational realizations; these are dynamical systems with rational right-hand…
New families of fourth-order composition methods for the numerical integration of initial value problems defined by ordinary differential equations are proposed. They are designed when the problem can be separated into three parts in such a…
In this article, firstly we develop a method for a type of difference equations, applicable to solve approximately a class of first order ordinary differential equation systems. In a second step, we apply the results obtained to solve a…
In this paper, a matrix representation for the differential resultant of two generic ordinary differential polynomials $f_1$ and $f_2$ in the differential indeterminate $y$ with order one and arbitrary degree is given. That is, a…
We use homotopy theory to define certain rational coefficients characteristic numbers with integral values, depending on a given prime number q and positive integer t. We prove the first nontrivial degree formula and use it to show that…
Consider a planar polynomial vector field $X$, and assume it admits a symbolic first integral $\mathcal{F}$, i.e. of the $4$ classes, in growing complexity: Rational, Darbouxian, Liouvillian and Riccati. If $\mathcal{F}$ is not rational, it…
In this paper we construct the general solutions of two families of quad-equations, namely the trapezoidal $H^{4}$ equations and the $H^{6}$ equations. These solutions are obtained exploiting the properties of the first integrals in the…
Variational integrators are well-suited for simulation of mechanical systems because they preserve mechanical quantities about a system such as momentum, or its change if external forcing is involved, and holonomic constraints. While they…
In this research, the Bernoulli polynomials are introduced. The properties of these polynomials are employed to construct the operational matrices of integration together with the derivative and product. These properties are then utilized…
We derive a five-dimensional nonlinear first order matrix PDE which is a generalization of the completely integrable (2+1)-dimensional $N$-wave equation. Similar to the $\bar\partial$-problem, our algorithm is based on the linear integral…