Straightening soliton curves

A canonical straightening process is described for soliton curves associated with the localized induction hierarchy. Following computer animated examples, the present topic is placed in the context of a larger theme: the soliton class is a natural setting for representation of diverse topological and geometrical behavior of curves and their motions. The constructions we describe here pertain to the soliton class, Γ, of unit speed curves γ in Euclidean 3-space: Γ = ⋃∞ n=0 Γn, where Γ1 = {lines}, Γ2 = {helices}, Γ3 = {elastic rods}, Γ4 ⊃ {buckled rings under pressure}, . . . For concise definition, consider X = ∑∞ n=0 λ Xn, a formal series of vectorfields along γ(s) satisfying X0 = − ∂s = −T and JXn = ∂Xn−1, n = 1, 2, . . . , i.e., JX = λ∂X (1) Here, J = T× (cross product with unit tangent), and ∂ = ∂ ∂s = ∇T (covariant derivative). Let Γn = {γ : 0 = Xn}; as will be seen, Γn is defined by an n-order ODE for T = γs, depending on n constants. Since J = −Id on normal vectorfields, Xn = fnT − J∂Xn−1, where ∂fn = ∂〈T,Xn〉 = 〈∂T,Xn〉 + 〈T, ∂Xn〉 = 〈−JX1, Xn〉 determines fn up to a constant of integration. Alternatively, (1) implies λ∂〈X,X〉 = 2〈JX,X〉 = 0, so for some constants Cn, 〈X,X〉 = p(λ) = 1 + ∞ ∑