From recursion operators to Hamiltonian structures. The factorization method

We describe a simple algorithmic method of constructing Hamiltonian structures for nonlinear PDE. Our approach is based on the geometrical theory of nonlinear differential equations and is in a sense inverse to the well-known Magri scheme. As an illustrative example, we take the KdV equation and the Boussinesq equation. Further applications, including construction of previously unknown Hamiltonian structures, are in preparation

The Monge-Ampère equation: Hamiltonian and symplectic structures, recursions, and hierarchies

Using methods of geometry and cohomology developed recently, we study the Monge-Ampère equation, arising as the first nontrivial equation in the associativity equations, or WDVV equations. We describe Hamiltonian and symplectic structures as well as recursion operators for this equation in its orginal form, thus treating the independent variables on an equal footing. Besides this we present nonlocal symmetries and generating functions (cosymmetries)

On symmetries and cohomological invariants of equations possessing flat representations

We study the equation E_fc of flat connections in a fiber bundle and discover a specific geometric structure on it, which we call a flat representation. We generalize this notion to arbitrary PDE and prove that flat representations of an equation E are in 1-1 correspondence with morphisms f: E\to E_fc, where E and E_fc are treated as submanifolds of infinite jet spaces. We show that flat representations include several known types of zero-curvature formulations of PDE. In particular, the Lax pairs of the self-dual Yang-Mills equations and their reductions are of this type. With each flat representation we associate a complex C_f of vector-valued differential forms such that its first cohomology describes infinitesimal deformations of the flat structure, which are responsible, in particular, for parameters in Backlund transformations. In addition, each higher infinitesimal symmetry S of E defines a 1-cocycle c_S of C_f. Symmetries with exact c_S form a subalgebra reflecting some geometric properties of E and f. We show that the complex corresponding to E_fc itself is 0-acyclic and 1-acyclic (independently of the bundle topology), which means that higher symmetries of E_fc are exhausted by generalized gauge ones, and compute the bracket on 0-cochains induced by commutation of symmetries.Comment: 30 page

The D-Boussinesq equation: Hamiltonian and symplectic structures; Noether and inverse Noether operators

Using new methods of analysis of integrable systems,based on a general geometric approach to nonlinear PDE,we discuss the Dispersionless Boussinesq Equation, which is equivalent to the Benney-Lax equation,being a system of equations of hydrodynamical type. The results include: a description of local and nonlocal Hamiltonian and symplectic structures, hierarchies of symmetries, hierarchies of conservation laws, recursion operators for symmetries and generating functions of conservation laws. Highly interesting are the appearences of the Noether and Inverse Noether operators ,leading to multiple infinite hierarchies of these operators as well as recursion operators

(Non)local Hamiltonian and symplectic structures, recursions, and hierarchies: a new approach and applications to the N=1 supersymmetric KdV equation

Using methods of math.DG/0304245 and [I.S.Krasil'shchik and P.H.M.Kersten, Symmetries and recursion operators for classical and supersymmetric differential equations, Kluwer, 2000], we accomplish an extensive study of the N=1 supersymmetric Korteweg-de Vries equation. The results include: a description of local and nonlocal Hamiltonian and symplectic structures, five hierarchies of symmetries, the corresponding hierarchies of conservation laws, recursion operators for symmetries and generating functions of conservation laws. We stress that the main point of the paper is not just the results on super-KdV equation itself, but merely exposition of the efficiency of the geometrical approach and of the computational algorithms based on it.Comment: 16 pages, AMS-LaTeX, Xy-pic, dvi-file to be processed by dvips. v2: nonessential improvements of exposition, title change

A convenient criterion under which Z_2-graded operators are Hamiltonian

We formulate a simple and convenient criterion under which skew-adjoint Z_2-graded total differential operators are Hamiltonian, provided that their images are closed under commutation in the Lie algebras of evolutionary vector fields on the infinite jet spaces for vector bundles over smooth manifolds.Comment: J.Phys.Conf.Ser.: Mathematical and Physical Aspects of Symmetry. Proc. 28th Int. colloq. on group-theoretical methods in Physics (July 26-30, 2010; Newcastle-upon-Tyne, UK), 6 pages (in press

High daily energy expenditure of incubating shorebirds on High Arctic tundra: a circumpolar study

1. Given the allometric scaling of thermoregulatory capacity in birds, and the cold and exposed Arctic environment, it was predicted that Arctic-breeding shorebirds should incur high costs during incubation. Using doubly labelled water (DLW), daily energy expenditure (DEE) during incubation was measured in eight shorebird species weighing between 29 and 142 g at various sites in the Eurasian and Canadian High Arctic. The results are compared with a compilation of similar data for birds at lower latitudes. 2. There was a significant positive correlation between species average DEE and body mass (DEE (kJ day−1) = 28·12 BM (g)^0·524, r^2 = 0·90). The slopes of the allometric regression lines for DEE on body mass of tundra-breeding birds and lower latitude species (a sample mostly of passerines but including several shorebirds) are similar (0·548 vs 0·545). DEE is about 50% higher in birds on the tundra than in temperate breeding areas. 3. Data for radiomarked Red Knots for which the time budgets during DLW measurements were known, indicated that foraging away from the nest on open tundra is almost twice as costly as incubating a four-egg clutch. 4. During the incubation phase in the High Arctic, tundra-breeding shorebirds appear to incur among the highest DEE levels of any time of the year. The rates of energy expenditure measured here are among the highest reported in the literature so far, reaching inferred ceilings of sustainable energy turnover rates.

Algebraic theories of brackets and related (co)homologies

A general theory of the Frolicher-Nijenhuis and Schouten-Nijenhuis brackets in the category of modules over a commutative algebra is described. Some related structures and (co)homology invariants are discussed, as well as applications to geometry.Comment: 14 pages; v2: minor correction