New connections between moving curves and soliton equations

Abstract

Lamb has identified a certain class of moving space curves with soliton equations. We show that there are two other classes of curve evolution that may be so identified. Hence three distinct classes of curve evolution are associated with a given integrable equation. The nonlinear Schrödinger equation is used to illustrate this explicitly.

New geometries associated with the nonlinear Schrödinger equation

“…We showed recently [3] that there are two other ways of making the association, which we call formulations II and III respectively, which lead to two other curve evolutions.…”

“…(2), and using Eq. (1), a short calculation [3] shows that the LL equation [7] S u = S × S ss ; S 2 = 1…”

“…Recently, we showed [3] that there are two other distinct ways of making such a connection. Thus three different space curve evolutions get associated with a given solution of the integrable equation.…”

“…In this paper, we use a different approach which obtains two more surfaces (or moving curves), in addition to the above surface. For the NLS, we first use the expressions [3] for the associated curve parameters to show that the three space curve evolutions all map to the integrable Landau-Lifshitz (LL) equation [7] for the time evolution of a spin vector S of a continuous one-dimensional Heisenberg ferromagnet. In other words, the tangent vector of the first moving curve, the binormal vector of the second, and the normal vector of the third, are shown to satisfy the LL equation.…”

Moving non-null curves according to Bishop frame in Minkowski 3-space

“…In the last years, the relationship between integrable evolution equations and the motion of curves was investigated by many authors [1][2][3][4][5][6][7][8][9][10][11][12][13][14].…”