12 research outputs found
On the variational interpretation of the discrete KP equation
We study the variational structure of the discrete Kadomtsev-Petviashvili
(dKP) equation by means of its pluri-Lagrangian formulation. We consider the
dKP equation and its variational formulation on the cubic lattice as well as on the root lattice . We prove that, on a lattice
of dimension at least four, the corresponding Euler-Lagrange equations are
equivalent to the dKP equation.Comment: 24 page
Post-Lie Algebras, Factorization Theorems and Isospectral-Flows
In these notes we review and further explore the Lie enveloping algebra of a
post-Lie algebra. From a Hopf algebra point of view, one of the central
results, which will be recalled in detail, is the existence of a second Hopf
algebra structure. By comparing group-like elements in suitable completions of
these two Hopf algebras, we derive a particular map which we dub post-Lie
Magnus expansion. These results are then considered in the case of
Semenov-Tian-Shansky's double Lie algebra, where a post-Lie algebra is defined
in terms of solutions of modified classical Yang-Baxter equation. In this
context, we prove a factorization theorem for group-like elements. An explicit
exponential solution of the corresponding Lie bracket flow is presented, which
is based on the aforementioned post-Lie Magnus expansion.Comment: 49 pages, no-figures, review articl