相关论文: On Rank Two Algebro--Geometric Solutions of an Int…

All exactly integrable systems connected with the semisimple algebras of the second rank with an arbitrary choice of the grading in them are presented in explicit form. General solution of such systems are expressed in terms of the matrix…

Second-order superintegrable systems in dimensions two and three are essentially classified. With increasing dimension, however, the non-linear partial differential equations employed in current methods become unmanageable. Here we propose…

Exactly integrable systems connected to semisimple algebras of second rank with an arbitrary choice of grading are presented in explicit form. General solutions of these systems are expressed in terms of matrix elements of two fundamental…

We consider a system of differential equations and obtain its solutions with exponential asymptotics and analyticity with respect to the spectral parameter. Solutions of such type have importance in studying spectral properties of…

The equivalence problem for linear differential operators of the second order, acting in vector bundles, is discussed. The field of rational invariants of symbols is described and connections, naturally accosiated with differential…

Several algebro-geometric properties of commutative rings of partial differential operators as well as several geometric constructions are investigated. In particular, we show how to associate a geometric data by a commutative ring of…

The $2n$ dimensional manifold with two mutually commutative operators of differentiation is introduced. Nontrivial multidimensional integrable systems connected with arbitrary graded (semisimple) algebras are constructed. The general…

In the paper a two-dimensional integro-differential system is considered. Using some variational methods we give sufficient conditions for the existence and uniqueness of a solution to the considered system. Moreover, we show that the…

The completeness of the group classification of systems of two linear second-order ordinary differential equations with constant coefficients is delineated in the paper. The new cases extend what has been done in the literature. These cases…

INTRODUCTION This papers deals with partial differential equations of second order, linear, with constant and not constant coefficients, in two variables, which admit real characteristics. I face the study of PDEs with the mentality of the…

We study a hypergeometric function in two variables and a system of hypergeometric differential equations associated with this function. This is a regular holonomic system of rank $9$. We give a fundamental system of solutions to this…

A deformed differential calculus is developed based on an associative star-product. In two dimensions the Hamiltonian vector fields model the algebra of pseudo-differential operator, as used in the theory of integrable systems. Thus one…

In this present paper, we study geometric structures of rank two prolongations of implicit second-order partial differential equations (PDEs) for two independent and one dependent variables and characterize the type of these PDEs by the…

The nature of so-called differential-algebraic operators and their approximations is constitutive for the direct treatment of higher-index differential-algebraic equations. We treat first-order differential-algebraic operators in detail and…

A class of two-dimensional systems of second-order ordinary differential equations is identified in which a system requires fewer Lie point symmetries than required to solve it. The procedure distinguishes among those which are…

We construct a 2-category of differential graded schemes. The local affine models in this theory are differential graded algebras, which are graded commutative with unit over a field of characteristic zero, are concentrated in non-positive…

We derive integrable discrete systems which are contiguity relations of two equations in the Painlev\'e-Gambier classification depending on some parameter. These studies extend earlier work where the contiguity relations for the six…

We reconsider non-degenerate second order superintegrable systems in dimension two as geometric structures on conformal surfaces. This extends a formalism developed by the authors, initially introduced for (pseudo-)Riemannian manifolds of…

Geometric problems are usually formulated by means of (exterior) differential systems. In this theory, one enriches the system by adding algebraic and differential constraints, and then looks for regular solutions. Here we adopt a dual…

Although this article can be read independently, it is a continuation of the introduction to integrable systems aspects of quantum cohomology given in part 1 (math.DG/0104274). In the same elementary style, i.e. assuming basic properties of…