Hamiltonian operators and ℓ*-coverings

Abstract

An efficient method to construct Hamiltonian structures for nonlinear evolution equations is described. It is based on the notions of variational Schouten bracket and ℓ * -covering. The latter serves the role of the cotangent bundle in the category of nonlinear evolution PDEs. We first consider two illustrative examples (the KdV equation and the Boussinesq system) and reconstruct for them the known Hamiltonian structures by our methods. For the coupled KdV-mKdV system, a new Hamiltonian structure is found and its uniqueness (in the class of polynomial (x, t)-independent structures) is proved. We also construct a nonlocal Hamiltonian structure for this system and prove its compatibility with the local one.

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

“…while a variational bivector is a Poisson structure (and the corresponding operator is Hamiltonian) if and only if [8] …”

Classification of integrable super-systems using the SsTools environment

“…First, the very algorithm [5,10] for finding recursions of symmetry algebras of equations E suggests to treat the recursions as symmetries of the linearized systems Lin E, taking into account the original equations (that is, replacing their left-hand sides with their right-hand sides) and differential consequences of them. The linearized system, which involves time derivatives of Cartan's forms, is included in the list pdes in form of relations df(f(i),t)=.…”

“…Also, in section 1.2 we review a geometric (coordinate-free) algorithm for constructing recursion operators. However, we refer to [5,7] for basic notions and concepts in the geometry of (super)PDE, see also [8,9,10,16] and references therein. The principal result of this paper is that there exist only four nonlinear coupled boson-fermion systems that satisfy the axioms and the weighting |f | = |b| = |D t | = 1 2 .…”

A Geometric Study of the Dispersionless Boussinesq Type Equation

“…Using the methods developed in [6], we rediscover the above mentioned biHamiltonian structure and show that it is only a part of the infinite-dimensional space of operators that take conservation laws of (1) (their generating functions, more precisely) to symmetries. These operators, in a standard way, generate an infinite associative (but not commutative) algebra of recursion operators for symmetries.…”

“…As it was mentioned in the Introduction, our computations are based on the results of paper [6] (see also [7]). For the general theoretical background we also refer the reader to books [1,10,12].…”