Geometry of jet spaces and integrable systems

Abstract

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.Comment: 54 pages; v2-v6 : minor correction

The four-dimensional Martínez Alonso–Shabat equation: Differential coverings and recursion operators

“…On the other hand, symmetries ofẼ are called [16,18,5,17,15] nonlocal symmetries of E associated with the coveringẼ.…”

Integrability of S-deformable surfaces: Conservation laws, Hamiltonian structures and more

“…System (2) admits (in a generalized sense of [9,11,13]) a pair of compatible local Hamiltonian structures P i of the form…”

The Prolongation Structures for the System of the Reaction-Diffusion Type

“…We constructed ( ) ( ) ( ) 2, sl R R t ρ × prolongation structure and gave the corresponding Ablowitz-Kaup-Newell-Segur (AKNS-) type equations and Bäcklund transformation. In 2012 Krasil'shchik and Verbovetsky [8] gave an overview of the recent results on the geometry of partial differential equations (PDEs) in application to integrable systems. These results are essentially based on the geometrical approach to partial differential equations developed since 1970s by A.M. Vinogradov and his school [9] whose sections play the role of unknown functions (fields).…”

“…This attitude allows applying to PDEs powerful techniques of differential geometry and homological algebra. Readers can refer [8] for more information. It is noticed that the WE prolongation structures are an essentially special type of coverings [13] [15].…”