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.

54 pages; v2-v6 : minor corrections

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Cited by in corpus (56)

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• The four-dimensional Martinez Alonso--Shabat equation: reductions and nonlocal symmetries
• From Higher Spins to Strings: A Primer
• Parent formulations, frame-like Lagrangians, and generalized auxiliary fields
• Pseudo-local Theories: A Functional Class Proposal
• Gauge PDE and AKSZ-type Sigma Models
• Boundary values of mixed-symmetry massless fields in AdS space
• The four-dimensional Martinez Alonso - Shabat equation: differential coverings and recursion operators
• A unified approach to computation of integrable structures
• Symmetries and conservation laws for the Karczewska--Rozmej--Rutkowski--Infeld equation
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• Presymplectic structures and intrinsic Lagrangians for massive fields
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• Presymplectic AKSZ formulation of Einstein gravity
• Computing with Hamiltonian operators
• Deformations of infinite-dimensional Lie algebras, exotic cohomology, and integrable nonlinear partial differential equations
• Presymplectic BV-AKSZ formulation of Conformal Gravity
• Infinitely many nonlocal conservation laws for the $ABC$ equation with $A+B+C\ne 0$
• Higher symmetries of cotangent coverings for Lax-integrable multi-dimensional partial differential equations and Lagrangian deformations
• On integrable structures for a generalized Monge-Ampere equation
• Lax representations with non-removable parameters and integrable hierarchies of PDEs via exotic cohomology of symmetry algebras
• Deformations of non semisimple Poisson pencils of hydrodynamic type
• Presymplectic gauge PDEs and Lagrangian BV formalism beyond jet-bundles
• Integrability in differential coverings
• Evolutionary Hirota Type (2+1)-Dimensional Equations: Lax Pairs, Recursion Operators and Bi-Hamiltonian Structures
• Weakly nonlocal Poisson brackets, Schouten brackets and supermanifolds
• Recursion Operators for Multidimensional Integrable PDEs
• Symplectic structures related with higher order variational problems
• Integrable partial differential equations and Lie--Rinehart algebras
• On the variational noncommutative Poisson geometry
• Meta-Symplectic Geometry of $3^{\mathrm{rd}}$ Order Monge-Ampère Equations and their Characteristics
• Nonlocal symmetries of a class of scalar and coupled nonlinear ordinary differential equations of any order
• Involutive distributions of operator-valued evolutionary vector fields and their affine geometry. II
• Integrability properties of some symmetry reductions
• Lax pairs, recursion operators and bi-Hamiltonian representations of (3+1)-dimensional Hirota type equations
• On Koszul-Tate resolutions and Sullivan models
• Integrability of S-deformable surfaces: conservation laws, Hamiltonian structures and more
• Field Theory via Higher Geometry I: Smooth Sets of Fields
• Integration of quadratic Lie algebroids to Riemannian Cartan-Lie groupoids
• Remarks on non-maximal integral elements of the Cartan plane in jet spaces
• Homological evolutionary vector fields in Korteweg-de Vries, Liouville, Maxwell, and several other models
• A cohomological approach to immersed submanifolds via integrable systems
• A comparison of definitions for the Schouten bracket on jet spaces
• Consistent deformations in the presymplectic BV-AKSZ approach
• Internal Lagrangians of PDEs as variational principles
• Gardner's deformation of the Krasil'shchik-Kersten system
• On the geometry of the Batalin-Vilkovisky Laplacian
• A convenient criterion under which Z_2-graded operators are Hamiltonian
• Lax representations with non-removable parameter and exotic cohomology of symmetry algebras of PDEs
• Recursion operators and bi-Hamiltonian representations of cubic evolutionary (2+1)-dimensional systems
• On the (non)removability of spectral parameters in $Z_2$-graded zero-curvature representations and its applications
• Recursion operators in the cotangent covering of the rdDym equation
• The geometry of the space of Cauchy data of nonlinear PDEs