Geometry of jet spaces and integrable systems

TL;DR

This paper reviews recent geometric approaches to partial differential equations, focusing on integrable systems, and introduces new structures like tangent bundles and the variational Schouten bracket to analyze their properties.

Contribution

It presents novel geometric constructions such as analogs of tangent and cotangent bundles and the variational Schouten bracket for PDEs, enhancing the theoretical framework for integrable systems.

Findings

Introduction of tangent and cotangent bundle analogs for PDEs

Definition of the variational Schouten bracket

Illustration of theoretical constructions through examples

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.