22 research outputs found
On the integrability of the shift map on twisted pentagram spirals
In this paper we prove that the shift map defined on the moduli space of
twisted pentagram spirals of type possesses a non-standard Lax
representation with an associated monodromy whose conjugation class is
preserved by the map. We prove this by finding a coordinate system in the
moduli space of twisted spirals, writing the map in terms of the coordinates
and associating a natural parameter-free non-standard Lax representation. We
then show that the map is invariant under the action of a -parameter group
on the moduli space of twisted spirals, which allows us to construct
the Lax pair. We also show that the monodromy defines an associated Riemann
surface that is preserved by the map. We use this fact to generate invariants
of the shift map
On integrable generalizations of the pentagram map
In this paper we prove that the generalization to of the
pentagram map defined in \cite{KS} is invariant under certain scalings for any
. This property allows the definition of a Lax representation for the map,
to be used to establish its integrability
Discrete moving frames on lattice varieties and lattice based multispace
In this paper, we develop the theory of the discrete moving frame in two different ways. In the first half of the paper, we consider a discrete moving frame defined on a lattice variety and the equivalence classes of global syzygies that result from the first fundamental group of the variety. In the second half, we consider the continuum limit of discrete moving frames as a local lattice coalesces to a point. To achieve a well-defined limit of discrete frames, we construct multispace, a generalization of the jet bundle that also generalizes Olver’s one dimensional construction. Using interpolation to provide coordinates, we prove that it is a manifold containing the usual jet bundle as a submanifold. We show that continuity of a multispace moving frame ensures that the discrete moving frame converges to a continuous one as lattices coalesce. The smooth frame is, at the same time, the restriction of the multispace frame to the embedded jet bundle. We prove further that the discrete invariants and syzygies approximate their smooth counterparts. In effect, a frame on multispace allows smooth frames and their discretisations to be studied simultaneously. In our last chapter we discuss two important applications, one to the discrete variational calculus, and the second to discrete integrable systems. Finally, in an appendix, we discuss a more general result concerning equicontinuous families of discretisations of moving frames, which are consistent with a smooth frame