Hamiltonian structures for integrable models of field theory

Theoretical and Mathematical Physics -     

geometry of differential equations

This paper contains a survey of papers on the geometry of differential equations, which appeared no earlier than 1972, continuing the general survey (RZhMat, 1974, 11A800), and considers in more detail a special cycle of investigations of the geometry of systems of partial differential equations, distinguished by the presence of practical applications. Then we continue the survey of new results on the geometry of an ordinary differential equation of arbitrary order, started in (RZhMat, 1978, 1A645). There is constructed a general theory of invariants of equations, and classes of equations admitting a simplified coordinate representation are invariantly distinguished.Translated from Itogi Nauki i Tekhniki, Seriya Problemy Geometrii, Vol. 12, pp. 127–164, 1981.     

Oscillatory solutions of some integrable differential equations

We present a class of second order linear differential equations that are integrable and also possess a vanishing-at-infinity oscillatory solution. This feature shows that integrability might play a significant role in the theory of perturbed oscillations.     

On the integrable rational Abel differential equations

In Cheb-Terrab and Roche (Comput Phys Commun 130(1–2):204–231, 2000) a classification of the Abel equations known as solvable in the literature was presented. In this paper, we show that all the integrable rational Abel differential equations that appear in Cheb-Terrab and Roche (Comput Phys Commun 130(1–2):204–231, 2000) and consequently in Cheb-Terrab and Roche (Eur J Appl Math 14(2):217–229, 2003) can be reduced to a Riccati differential equation or to a first-order linear differential equation through a change with a rational map. The change is given explicitly for each class. Moreover, we have found a unified way to find the rational map from the knowledge of the explicitly first integral.     

Some models in differential geometry

Some variants of the axiomatics of the algebras of “vector fields” in models of noncommutative differential geometry are considered. In the case of a commutative model (the de Rham complex) a matrix analogue of the Kadomtsev-Petviashvili hierarchy is constructed. The corresponding Sato system is presented. The method of deformations of D-modules is used. Bibliography: 14 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 205, 1993, pp. 122–153. Translated by A. M. Nikitin.     

Bifurcations of completely integrable first-order ordinary differential equations

We consider an implicit first-order ordinary differential equation with complete integral. In [3], the authors give a generic classifications of first-order ordinary differential equations with complete integral with respect to the equivalence relation which is given by the group of point transformations. The classification problem is reduced to the classification of a certain class of divergent diagrams of mapping germs. In this paper, we give a generic classifications of bifurcations of such differential equations as an application of the Legendrian singularity theory. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 33, Suzdal Conference-2004, Part 1, 2005.     

Deformations of nonassociative algebras and integrable differential equations

A new class of nonassociative algebras related to integrable PDE's and ODE's is introduced. These algebras can be regarded as a noncommutative generalization of Jordan algebras. Their deformations are investigated. Relationships between such algebras and graded Lie algebras are established.     

Methods of oscillation theory of half-linear second order differential equations

In this paper we investigate oscillatory properties of the second order half-linear equation

Quantum integrable systems and differential Galois theory

This paper is devoted to a systematic study of quantum completely integrable systems (i.e., complete systems of commuting differential operators) from the point of view of algebraic geometry. We investigate the eigenvalue problem for such systems and the correspondingD-module when the eigenvalues are in generic position. In particular, we show that the differential Galois group of this eigenvalue problem is reductive at generic eigenvalues. This implies that a system is algebraically integrable (i.e., its eigenvalue problem is explicitly solvable in quadratures) if and only if the differential Galois group is commutative for generic eigenvalues. We apply this criterion of algebraic integrability to two examples: finite-zone potentials and the elliptic Calogero-Moser system. In the second example, we obtain a proof of the Chalyh-Veselov conjecture that the Calogero-Moser system with integer parameter is algebraically integrable, using the results of Felder and Varchenko.     

The geometry of differential difference equations

To each maximal commuting subalgebra h of gl_m( )is associated a system of differential difference equations, generalizing several known systems. Starting from a Grassmann manifold, solutions are constructed, their properties are discussed and the relation with other systems is given. Finally it is shown how to express these solutions in τ-functions.     

Differential-geometric structures in the theory of two-dimensional integrable equations

The gauge representations of the integrable generalization of the Heisenberg magnet in (2+1)-dimensional space-time are interpreted in terms of topological charge. Restrictions on the class of solutions to the equation for a two-dimensional magnet are described for which it becomes gauge-equivalent to the Davy-Stuartson equation.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 189, pp. 75–81, 1991.The author is grateful to P. P. Kulish for stimulating discussions.     

Catastrophe theory and differential geometry

In the study of maps the partial derivatives may be replaced by covariants, which gives the possibility of talking about vector forms. We establish a connection between these vector forms and the well-known objects of differential geometry in the sense of A. P. Norden and G. F. Laptev and Porteous derivatives. We explain the origin of these vector forms under a morphism of the multiple bundles in which the connections are defined.Translated from Itogi Nauki i Tekhniki, Seriya Problemy Geometrii, Vol. 16, pp. 35–80, 1984.