Related papers: Weierstrass integrability of differential equation…
The Li\'{e}nard equation is of a high importance from both mathematical and physical points of view. However a question about integrability of this equation has not been completely answered yet. Here we provide a new criterion for…
Here we present an efficient method to compute Darboux polynomials for polynomial vector fields in the plane. This approach is restricetd to polynomial vector fields presenting a Liouvillian first integral (or, equivalently, to rational…
Integrable systems with a linear periodic integral for the Lie algebra $\mathrm{e}(3)$ are considered. One investigates singulariries of the Liouville foliation, bifurcation diagram of the momentum mapping, transformations of Liouville…
Polynomial dynamical systems describing interacting particles in the plane are studied. A method replacing integration of a polynomial multi--particle dynamical system by finding polynomial solutions of a partial differential equations is…
The geometric theory of Lie systems will be used to establish integrability conditions for several systems of differential equations, in particular Riccati equations and Ermakov systems. Many different integrability criteria in the…
The concepts of differentiation and integration for matrices were introduced for studying zeros and critical points of complex polynomials. Any matrix is differentiable, however not all matrices are integrable. The purpose of this paper is…
An extension of the notion of solvable structure for involutive distributions of vector fields is introduced. The new structures are based on a generalization of the concept of symmetry of a distribution of vector fields, inspired in the…
The method of obtaining new integrable coupled equations through enlarging spectral problems of known integrable equations, which was recently proposed by W.-X. Ma, can produce nonintegrable systems as well. This phenomenon is demonstrated…
We consider the discrete Boussinesq integrable system and the compatible set of differential difference, and partial differential equations. The latter not only encode the complete hierarchy of the Boussisesq equation, but also incorporate…
Different definitions of integrability, as a rule, use linearization of initial equation and/or expansion on some basic functions which are themselves solutions of some linear differential equation. Important fact here is that linearization…
We establish an analogue of the first fundamental theorem of calculus for functions defined on the Wasserstein space of probability measures. Precisely, we show that if a function on the Wasserstein space is sufficiently regular in the…
The geometric theory of Lie systems is used to establish integrability conditions for several systems of differential equations, in particular some Riccati equations and Ermakov systems. Many different integrability criteria in the…
First order integrals of motion for Schr\"odinger equations with position dependent masses are classified. Seventeen classes of such equations with non-equivalent symmetries are specified. They include integrable, superintegrable and…
We use a recently found method to characterise all the invertible fourth-order difference equations linear in the extremal values based on the existence of a discrete Lagrangian. We also give some result on the integrability properties of…
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…
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…
We study differential-difference equation of the form $t_{x}(n+1)=f(t(n),t(n+1),t_x(n))$ with unknown $t=t(n,x)$ depending on $x$, $n$. The equation is called Darboux integrable, if there exist functions $F$ (called an $x$-integral) and $I$…
On the base of Lie algebraic and differential geometry methods, a wide class of multidimensional nonlinear integrable systems is obtained, and the integration scheme for such equations is proposed.
Every orthonomic system of partial differential equations is known to possess a finite number of integrability conditions sufficient to ensure the validity of all. Herewith we offer an efficient algorithm to construct a sufficient set of…
We investigate the first-order system `$s\,' = c^3, \, c\,' = - s^3; \, s(0) = 0, \, c(0) = 1$'. Its solutions have the property that $s \, c$, $s^2$ and $c^2$ extend to simply-poled elliptic functions, which we explicitly identify in terms…