Quaternionic Soliton Equations from Hamiltonian Curve Flows in HP^n

A bi-Hamiltonian hierarchy of quaternion soliton equations is derived from geometric non-stretching flows of curves in the quaternionic projective space $HP^n$. The derivation adapts the method and results in recent work by one of us on the Hamiltonian structure of non-stretching curve flows in Riemannian symmetric spaces $M=G/H$ by viewing $HP^n \simeq {\rm U}(n+1,H)/{\rm U}(1,H) \times {\rm U}(n,H)\simeq {\rm Sp}(n+1)/{\rm Sp}(1)\times {\rm Sp}(n)$ as a symmetric space in terms of compact real symplectic groups and quaternion unitary groups. As main results, scalar-vector (multi-component) versions of the sine-Gordon (SG) equation and the modified Korteveg-de Vries (mKdV) equation are obtained along with their bi-Hamiltonian integrability structure consisting of a shared hierarchy of quaternionic symmetries and conservation laws generated by a hereditary recursion operator. The corresponding geometric curve flows in $HP^n$ are shown to be described by a non-stretching wave map and a mKdV analog of a non-stretching Schrodinger map.Comment: 25 pages; typos correcte

Lagrangian Curves in a 4-dimensional affine symplectic space

Lagrangian curves in R4 entertain intriguing relationships with second order deformation of plane curves under the special affine group and null curves in a 3-dimensional Lorentzian space form. We provide a natural affine symplectic frame for Lagrangian curves. It allows us to classify La- grangrian curves with constant symplectic curvatures, to construct a class of Lagrangian tori in R4 and determine Lagrangian geodesic

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

TRANSVECTION AND DIFFERENTIAL INVARIANTS OF PARAMETRIZED CURVES

Abstract. In this paper we describe an sl2 representation in the space of differential invariants of parametrized curves in homogeneous spaces. The representation is described by three operators, one of them being the total derivative D. We use this representation to find a basis for the space of differential invariants of curves in a complement of the image of D, and so generated by transvection. These are natural representatives of first cohomology classes in the invariant bicomplex. We describe algorithms to find these basis and study most well-known geometries. 1

Morphometric study of the regeneration of individual rays in teleost tail fins

The results obtained using morphometric variables which describe fin ray regeneration patterns are reported for individual fin ray amputations in the goldfish (Carassius auratus) and zebrafish (Brachydanio rerio). Classical and updated experiments are compared to verify previous morphogenetic models of cell tractions (Oster et al. 1983) or epidermis-mesenchyme induction (Saunders et al. 1959) applied to the limb of other vertebrates. Position-dependent patterns within the fin of Carassius auratus are analysed under a comparative protocol using morphometric methods. Conditions in which the apical epidermis is separated from blastema may differentiate small fin rays, thus suggesting this epidermis is involved in blastemal formation. Blastemal cells differentiating as lepidotrichia forming cells (LFCs) may also be related to morphological changes in covering epidermis. Long-range interactions from neighbouring fin ray blastemas or short-range interactions within the blastema, may be postulated through the analysis of segmentation