arXiv subjects
Publications and source records attributed to Joseph Krasil'shchik.
In the framework of the theory of differential coverings \cite{KV}, we discuss a general geometric construction that serves the base for the so-called Lax pairs containing differentiation with respect to the spectral parameter \cite{OS}. Such kind of objects arise, for example, when studying integrability properties of equations like the Gibbons-Tsarev one \cite{GT}.
An overview of some recent results on the geometry of partial differential equations in application to integrable systems is given. Lagrangian and Hamiltonian formalism both in the free case (on the space of infinite jets) and with constraints (on a PDE) are discussed. Analogs of tangent and cotangent bundles to a differential equation are introduced and the variational Schouten bracket is defined. General theoretical constructions are illustrated by a series of examples.
In the context of the cohomological deformation theory, infinitesimal description of one-parametric families of Backlund transformations of special type including classical examples is given. It is shown that any family of such a kind evolves in the direction of a nonlocal symmetry shadow.
The coupled KdV-mKdV system arises as the classical part of one of superextensions of the KdV equation. For this system, we prove its complete integrability, i.e., existence of a recursion operator and of infinite series of symmetries.
These lecture notes are a systematic and self-contained exposition of the cohomological theories naturally related to partial differential equations: the Vinogradov C-spectral sequence and the C-cohomology, including the formulation in terms of the horizontal (characteristic) cohomology. Applications to computing invariants of differential equations are discussed. The lectures contain necessary introductory material on the geometric theory of differential equations and homological algebra.
Using covering theory approach (zero-curvature representations with the gauge group SL2), we insert the spectral parameter into the Gauss-Mainardi-Codazzi equations in Tchebycheff and geodesic coordinates. For each choice, four integrable systems are obtained.