Narrow Results

We study the higher KdV flows on the space of closed complex equicentroaffine curves as Hamiltonian systems. Using a suitable presymplectic structure of the space, we give the Hamiltonian flows associated with the higher KdV equations and a map between the space and the space of closed curves in the complex plane, which induces the Miura transformation between the higher KdV equations and the higher mKdV equations.

In this paper, we investigate the flow of curve and its equiform geometry in 4-dimensional Galilean space. We obtain that the Frenet equations and curvatures of inextensible flows of curves and its equiformly invariant vector fields and intrinsic quantities are independent of time. We find that the motions of curves and its equiform geometry can be defined by the inviscid and stochastic Burgers’ equations in 4-dimensional Galilean space.