Motions of curves in similarity geometries and Burgers-mKdV hierarchies

Communicated by Prof. M. Wadati
https://doi.org/10.1016/S0960-0779(03)00060-2Get rights and content

Abstract

Integrable systems satisfied by the curvatures of curves under inextensible motions in similarity geometries are identified. It is shown that motions of curves in two-, three- and n-dimensional (n>3) similarity geometries are described respectively by the Burgers hierarchy, Burgers-mKdV hierarchy and a multi-component generalizations of these hierarchies.

Introduction

Many 1 + 1-dimensional integrable equations have been shown to be related to motions of inextensible curves, such relationship provides new insight and geometric explanations to properties and structure of integrable equations. In an intriguing paper of Goldstein and Petrich [1], they showed that the mKdV hierarchy arises naturally (“naturally” means that the recursion operator of the mKdV equation is appeared in the equation for the curvature) from a local motion of non-stretching plane curves in Euclidean space R2. After that, Nakayama et al. [2] obtained the sine-Gordon equation by considering a non-local motion of curves in R2, they also pointed out that the Frenet–Serret equations for curves in R2 and R3 formulas are equivalent to the AKNS spectral problem without spectral parameter [2], [3]. In the earlier works of Hasimoto [4], he showed that the Schrödinger equation arises from motion of inextensible curves in R3, the Schrödinger hierarchy was also obtained by Langer and Perline [7], they provided a geometrical explanation to the recursion operator of the Schrödinger equation. Using the Hasimoto transformation, Lamb [5] obtained the mKdV and sine-Gordon equations from the motion of curves in R3. The Heisenberg spin chain model which is gauge equivalent to the Schrödinger equation was derived by Lakshmanan [6]. Recently, Schief and Rogers [8] obtained an extended Harry–Dym equation and sine-Gordon equation from binormal motions of curves with constant curvature or torsion. An important trend on this topic is to study motions of curves in classical geometries. Nakayama [9] showed that the defocusing non-linear Schrödinger equation, the Regge–Lund equation, a coupled of system of KdV equations and their hyperbolic type arise from motions of curves in hyperboloids in the Minkowski space. In [10] he realized the full AKNS scheme in a hyperboloid in M3,1. Motion of plane curves in the Minkowski space M2,1 was also considered by Gürses [11]. Recently we found that many 1 + 1-dimensional integrable equations including KdV, Sawada–Kotera, Burgers, Harry–Dym hierarchies and Kaup–Kupershmidt, Camassa–Holm equations naturally arise from motions of plane curves in centro-affine, similarity, affine and fully affine geometries [12], [13]. Motions of curves in three-dimensional centro-affine and affine geometries were also considered [14]. In particular, the Burgers hierarchy describes motion of non-stretching plane curves in similarity geometry [12]. We point out here that no Burgers hierarchy found from motions of curves in other geometries.

In this paper, we are mainly concerned with motions of curves in similarity geometries Pn, this problem has been involved in Sapiro and Tannenbaum [15]. The isometry group of similarity geometry is a composition of Euclidean motion and the dilatation. For instance, the corresponding Lie algebras of the isometry groups are generated by {∂x,∂u,xuux,xx+uu} for n=2 [16] and {∂x,∂y,∂u,xuux,xyyx,yuuy,xx+yy+uu} for n=3. All geometric quantities are invariant under the isometry groups. A nice fact is that we can define curvatures and arc-length of the similarity geometry [12], they are characterized by differential invariants and invariant one form of the isometry group. More precisely, let κ1, κ2,…,κn−1 be the curvatures in Euclidean space Rn, then they are differential invariants of n-dimensional Euclidean motion. One can readily verify that α1=κ1,s/k12, αi=ki/k1, i=2,…,n−1, are differential invariants under the similarity motion, we define them to be curvatures in n-dimensional similarity geometry, where s is the arc-length of a curve in Euclidean space. Also, dθ=k1ds is an invariant one-form, where θ is the angle between the tangent and a fixed direction, we define it to be the arc-length of a curve in Pn. After that we can define its frame vectors τi in terms of Euclidean’s ti: τi1ti, i=1,2,…,n. Using them we can represent a geometric motion in the formγt=∑i=1nAiτi,where Ai are the velocities along τi and depend on the curvatures αi and their derivatives with respect to the arc-length θ. By differentiating (1) with respect to time t, we obtain time evolution for vector fields τi, i=1,2,…,n,τ1τ2τnt=Uτ1τ2τn.By the Frenet–Serret formulas in Euclidean geometry,t1t2tns=0κ10−κ10κ20−κ200κn−10−κn−10t1t2tn,one obtains the Frenet–Serret formulas in the similarity geometry Pnτ1τ2τnθ=Vτ1τ2τn,with V an n×n matrix given as follows. The zero curvature conditionVt−Uθ+[V,U]=0,gives the equations for the curvatures αi, i=1,2,…,n−1.

The outline of this paper is as follows: In Section 2, for completeness we study motion of plane curves in P2 by choosing tangent and normal vectors slightly different from [12]. Motions of curves in P3 and Pn (n>3) are discussed respectively in 3 Curves in, 4 Curves in. Section 5 is a concluding remarks on this work.

Section snippets

Curves in P2 and the Burgers hierarchy

We denote the tangent, normal, arc-length and curvature in Euclidean space R2 respectively by t, n, s and κ. The Frenet–Serret formulas readtns=0κ−κ0tn.

The curvature and arc-length in P2 are given respectively by κ̃s2 and dθ=κds, tangent and normal vectors of a curve in P2 are τ1=γθ and τ2=n. Via a direct computation we have the Frenet–Serret formulas in P2τ1τ2θ=κ̃1−1κ̃τ1τ2.

Now motion of curves in P2 is specified byγt=Aτ1+Bτ2.

We relate it to the Euclidean motionγt=fn+gt,where f=B/κ, g=A/

Curves in P3 and Burgers-mKdV hierarchy

In the case of curves in P3, the Frenet–Serret formulas readτ1τ2τ3θ=κ̃10−1κ̃τ̃0τ̃κ̃τ1τ2τ3,where κ̃ and τ̃ are respectively the curvature and torsion in P3, they are related to the Euclidean curvature κ and torsion τ byκ̃=κsκ2,τ̃=τκ.

The motion is now described byγt=Aτ1+Bτ2+Cτ3,where τi, i=1,2,3 are related to the Euclidean tangent t, normal n and binormal b byτ1=tκ,τ2=nκ,τ3=bκ.

It is possible to relate (14) with the Euclidean motion in R3 [2], [3]γt=Wt+Un+Vb,with W=A/κ, U=B/κ and V=C/κ. Using

Curves in Pn and a generalization of the Burgers-mKdV hierarchy

Similarly, we have the Frenet–Serret formulas in Pnτ1τ2τnθ=−α110−1−α1α20−α2−α10αn−10−αn−1−α1τ1τ2τn,where the curvatures αi of curves in Pn, i=2,…,n−1, are related to the Euclidean’s curvatures by α1=κ1,s/κ12, αi=κi/κ1, i=2,3,…,n−1 and the arc-length is given by dθ=∮κ1ds. The curve motion in Pn is described by (1). The inextensibility condition is(A2,θ−α1A2+A1−α2A3)θ2(A3,θ−α1A32A2−α3A4),∮α2(A3,θ−α1A32A2−α3A4)dθ=0.

The time evolution for the frame vectors τi isτi,t=(A1,θ−A2−α1A1i+∑

Concluding remarks

In this paper, we have carried out a study on motions of curves in similarity geometries Pn (n⩾2). It has been shown that the motions of inextensible curves in P2, P3 and Pn (n>3) yield Burgers hierarchy, Burgers-mKdV hierarchy and a multi-component generalization of these hierarchies respectively. This is compared with the motions of curves in R2, R3 and Rn (n>3) respectively yield the mKdV hierarchy, Schrödinger hierarchy and a multi-component generalization of mKdV-Schrödinger hierarchies.

In

Acknowledgements

This work was supported by an Earmarked Grant for Research, Hong Kong. The second author was also partially supported by the NSF (Grant No. 19901027) of China and Shaan Xi Province.

References (19)

  • M. Gürses

    Phys. Lett. A

    (1998)
  • K.S. Chou et al.

    Physica D

    (2002)
  • R.E. Goldstein et al.

    Phys. Rev. Lett.

    (1991)
  • K. Nakayama et al.

    Phys. Rev. Lett.

    (1992)
  • K. Nakayama et al.

    Phys. J. Phys. Soc. Jpn.

    (1993)
  • H. Hasimoto

    J. Fluid Mech.

    (1972)
  • G.L. Lamb

    J. Math. Phys.

    (1977)
  • M. Lakshmanan

    J. Math. Phys.

    (1979)
  • J. Langer et al.

    J. Nonlinear Sci.

    (1991)
There are more references available in the full text version of this article.

Cited by (0)

View full text