On an integrable multi-dimensionally consistent 2 n + 2 n -dimensional heavenly-type equation

Based on the commutativity of scalar vector fields, an algebraic scheme is developed which leads to a privileged multi-dimensionally consistent 2 n + 2 n -dimensional integrable partial differential equation with the associated eigenfunction constituting an infinitesimal symmetry. The ‘universal’ character of this novel equation of vanishing Pfaffian type is demonstrated by retrieving and generalizing to higher dimensions a great variety of well-known integrable equations such as the dispersionless Kadomtsev–Petviashvili and Hirota equations and various avatars of the heavenly equation governing self-dual Einstein spaces.

The gap phenomenon in parabolic geometries

The infinitesimal symmetry algebra of any Cartan geometry has maximum dimension realized by the flat model, but often this dimension drops significantly when considering non-flat geometries, so a gap phenomenon arises. For general (regular, normal) parabolic geometries of type

Symmetry properties of conservation laws

Symmetry properties of conservation laws of partial differential equations are developed by using the general method of conservation law multipliers. As main results, simple conditions are given for characterizing when a conservation law and its associated conserved quantity are invariant (and, more generally, homogeneous) under the action of a symmetry. These results are used to show that a recent conservation law formula (due to Ibragimov) is equivalent to a standard formula for the action of an infinitesimal symmetry on a conservation law multiplier.

NON-NOETHER CONSERVATION LAWS

In this paper we will give a formula for computing conservation laws for a Hamiltonian system that admits non-Noether infinitesimal symmetry. The formula involves the differential operator associated with the dual Lefschetz operator corresponding to a symplectic form.

TENSOR MODELS AND HIERARCHY OF n-ARY ALGEBRAS

Tensor models are generalization of matrix models, and are studied as models of quantum gravity. It is shown that the symmetry of the rank-three tensor models is generated by a hierarchy of n-ary algebras starting from the usual commutator, and the 3-ary algebra symmetry reported in the previous paper is just a single sector of the whole structure. The condition for the Leibnitz rules of the n-ary algebras is discussed from the perspective of the invariance of the underlying algebra under the n-ary transformations. It is shown that the n-ary transformations which keep the underlying algebraic structure invariant form closed finite n-ary Lie subalgebras. It is also shown that, in physical settings, the 3-ary transformation practically generates only local infinitesimal symmetry transformations, and the other more nonlocal infinitesimal symmetry transformations of the tensor models are generated by higher n-ary transformations.

Finding symmetries by incorporating initial conditions as side conditions

It is generally believed that in order to solve initial value problems using Lie symmetry methods, the initial condition needs to be left invariant by the infinitesimal symmetry generator that admits the invariant solution. This is not so. In this paper we incorporate the imposed initial value as a side condition to find ‘infinitesimals’ from which solutions satisfying the initial value can be recovered, along with the corresponding symmetry generator.