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

References in corpus (10)

• KdV6: An Integrable System
• Conservation Laws and Potential Symmetries of Linear Parabolic Equations
• On integrability of Weingarten surfaces: a forgotten class
• Hamiltonian structures for general PDEs
• On integrability of the Camassa-Holm equation and its invariants. A geometrical approach
• Locality of symmetries generated by nonhereditary, inhomogeneous, and time-dependent recursion operators: a new application for formal symmetries
• On classification of Poisson vertex algebras
• Variational Poisson-Nijenhuis structures for partial differential equations
• (Non)local Hamiltonian and symplectic structures, recursions, and hierarchies: a new approach and applications to the N=1 supersymmetric KdV equation
• Recursion operators for vacuum Einstein equations with symmetries

Cited by in corpus (55)

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• A foundational approach to the Lie theory for fractional order partial differential equations
• From Higher Spins to Strings: A Primer
• The four-dimensional Martinez Alonso--Shabat equation: reductions and nonlocal symmetries
• Parent formulations, frame-like Lagrangians, and generalized auxiliary fields
• Pseudo-local Theories: A Functional Class Proposal
• Boundary values of mixed-symmetry massless fields in AdS space
• Gauge PDE and AKSZ-type Sigma Models
• 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
• Jacobi multipliers, non-local symmetries and nonlinear oscillators
• Presymplectic structures and intrinsic Lagrangians for massive fields
• Presymplectic structures and intrinsic Lagrangians
• Presymplectic AKSZ formulation of Einstein gravity
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• Deformations of infinite-dimensional Lie algebras, exotic cohomology, and integrable nonlinear partial differential equations
• Computing with Hamiltonian operators
• 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
• Lax representations with non-removable parameters and integrable hierarchies of PDEs via exotic cohomology of symmetry algebras
• On integrable structures for a generalized Monge-Ampere equation
• 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
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• Symplectic structures related with higher order variational problems
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• Nonlocal symmetries of a class of scalar and coupled nonlinear ordinary differential equations of any order
• Lax pairs, recursion operators and bi-Hamiltonian representations of (3+1)-dimensional Hirota type equations
• On Koszul-Tate resolutions and Sullivan models
• Integrability properties of some symmetry reductions
• Field Theory via Higher Geometry I: Smooth Sets of Fields
• Integration of quadratic Lie algebroids to Riemannian Cartan-Lie groupoids
• Integrability of S-deformable surfaces: conservation laws, Hamiltonian structures and more
• A cohomological approach to immersed submanifolds via integrable systems
• A comparison of definitions for the Schouten bracket on jet spaces
• 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
• Gardner's deformation of the Krasil'shchik-Kersten system
• Internal Lagrangians of PDEs as variational principles
• On the geometry of the Batalin-Vilkovisky Laplacian
• Consistent deformations in the presymplectic BV-AKSZ approach
• Recursion operators and bi-Hamiltonian representations of cubic evolutionary (2+1)-dimensional systems
• Recursion operators in the cotangent covering of the rdDym equation
• Lax representations with non-removable parameter and exotic cohomology of symmetry algebras of PDEs
• A convenient criterion under which Z_2-graded operators are Hamiltonian
• On the (non)removability of spectral parameters in $Z_2$-graded zero-curvature representations and its applications
• The geometry of the space of Cauchy data of nonlinear PDEs