Related papers: On integrable by Euler planar differential systems
We consider the simplest gauge theories given by one- and two- matrix integrals and concentrate on their stringy and geometric properties. We remind general integrable structure behind the matrix integrals and turn to the geometric…
We determine a considerable class of nonlinear partial differential equation systems which have global regular solutions. Uniqueness is not a direct general consequence of this method. The scheme can be applied to the incompressible Navier…
A planar integral point set is a set of non-collinear points in plane such that for any pair of the points the Euclidean distance between the points is integral. We discuss the classification of planar integral point sets and provide…
In the paper, the author elementarily unifies and generalizes eight identities involving the functions $\frac{\pm1}{e^{\pm t}-1}$ and their derivatives. By one of these identities, the author establishes two explicit formulae for computing…
Euler equations are the basic system in fluid dynamics describing the motion of incompressible and inviscid ideal fluids. For a bounded smooth domain $\Omega$ in $\mathbb{R}^n$. The well-posedness of Euler equations is well-known in Sobolev…
Basic notions regarding classical integrable systems are reviewed. An algebraic description of the classical integrable models together with the zero curvature condition description is presented. The classical r-matrix approach for discrete…
We study problems of the calculus of variations and optimal control within the framework of time scales. Specifically, we obtain Euler-Lagrange type equations for both Lagrangians depending on higher order delta derivatives and…
It is a basic introduction to differential graded Lie algebras, Maurer-Cartan equation and associated deformation functors.
A survey of some recent and important results which have to do with integrable equations and their relationship with the theory of surfaces is given. Some new results are also presented. The concept of the moving frame is examined, and it…
The problem of inverting the total divergence operator is central to finding components of a given conservation law. This might not be taxing for a low-order conservation law of a scalar partial differential equation, but integrable systems…
At two examples dealt with in methodologically different ways it will be pointed out how the concept of an empirical theory (in the sense of the Structuralists) can be useful to specify contents relevant to maths didactics.
This work focuses on two notions of non-Hamiltonian integrable systems: B-integrability and Euler-Jacobi integrability. We first show that the first notion is stronger. We then investigate which possible "non-evident" properties one can add…
Relations among integrals of logarithms, polylogarithms and Euler sums are presented. A unifying element being the introduction of Nielsen's generalized polylogarithms.
We consider in this work planar polynomial differential systems having a polynomial first integral. We prove that these systems can be obtained from a linear system through a polynomial change of variables.
We present a proof given by Euler in his paper {\it ``De serierum determinatione seu nova methodus inveniendi terminos generales serierum"} \cite{E189} (E189:``On the determination of series or a new method of finding the general terms of…
We extend the theory of Euler integration from the class of constructible functions to that of "tame" real-valued functions (definable with respect to an o-minimal structure). The corresponding integral operator has some unusual defects (it…
This paper is devoted to a study of linear, differential and topological classifications for linear controlled systems governed by ordinary differential equations. The necessary and sufficient conditions for the linear and topological…
We announce a detailed numerical investigation for some class of difference schemes, which arises from Euler implicit scheme. Such schemes demonstrate unusual behavior and leads to origin of solitons. Applications to some nonlinear problems…
We derive the Euler equations from quantum dynamics for a class of fermionic many-body systems. We make two types of assumptions. The first type are physical assumptions on the solution of the Euler equations for the given initial data. The…
Two models of integral theory based on the concept of a differential as a certain infinitesimal quantity are considered. One theory treats an infinitesimal quantity as a zero-tending sequence. The second is as an infinitesimal Hyper-real.