Computing with Hamiltonian operators

Abstract

Hamiltonian operators are used in the theory of integrable partial differential equations to prove the existence of infinite sequences of commuting symmetries or integrals. In this paper it is illustrated the new Reduce package \cde for computations on Hamiltonian operators. \cde can compute the Hamiltonian properties of skew-adjointness and vanishing Schouten bracket for a differential operator, as well as the compatibility property of two Hamiltonian operators and the Lie derivative of a Hamiltonian operator with respect to a vector field. It can also make computations on (variational) multivectors, or functions on supermanifolds. This can open the way to applications in other fields of Mathematical Physics.

35 pages, published version; software is available on the web page of the author http://poincare.unisalento.it/vitolo

References in corpus (23)

• A field-theory motivated approach to symbolic computer algebra
• Bihamiltonian Hierarchies in 2D Topological Field Theory At One-Loop Approximation
• Geometry of jet spaces and integrable systems
• Poisson vertex algebras in the theory of Hamiltonian equations
• Hamiltonian operators and l*-coverings
• A Symbolic Algorithm for Computing Recursion Operators of Nonlinear PDEs
• Flat pencils of metrics and Frobenius manifolds
• On deformation of Poisson manifolds of hydrodynamic type
• A simple way of making a Hamiltonian system into a bi-Hamiltonian one
• Symmetry reductions and exact solutions of Lax integrable $3$-dimensional systems
• Projective-geometric aspects of homogeneous third-order Hamiltonian operators
• On integrability of Weingarten surfaces: a forgotten class
• Alternative bi-Hamiltonian structures for WDVV equations of associativity
• Systems of conservation laws with third-order Hamiltonian structures
• Exact Poisson pencils, $\tau$-structures and topological hierarchies
• Coverings over Lax integrable equations and their nonlocal symmetries
• Towards the classification of homogeneous third-order Hamiltonian operators
• Hamiltonian structures for general PDEs
• On a class of third-order nonlocal Hamiltonian operators
• Compatible Dubrovin-Novikov Hamiltonian operators, Lie derivative and integrable systems of hydrodynamic type
• Remarks on the Lagrangian representation of bi-Hamiltonian equations
• MasterPVA and WAlg: Mathematica packages for Poisson vertex algebras and classical affine $\mathcal{W}$-algebras
• Symbolic Computing with Grassman Variables

Cited by in corpus (8)

• Applications of Nijenhuis geometry III: Frobenius pencils and compatible non-homogeneous Poisson structures
• Bi-Hamiltonian structure of the Oriented Associativity Equation
• Quasilinear systems of first order PDEs with nonlocal Hamiltonian structures
• Weakly nonlocal Poisson brackets: tools, examples, computations
• Weakly nonlocal Poisson brackets, Schouten brackets and supermanifolds
• Classification of degenerate non-homogeneous Hamiltonian operators
• WDVV equations and invariant bi-Hamiltonian formalism
• Bi-Hamiltonian structures of WDVV-type