Motions of curves in similarity geometries and Burgers-mKdV 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 . After that, Nakayama et al. [2] obtained the sine-Gordon equation by considering a non-local motion of curves in , they also pointed out that the Frenet–Serret equations for curves in and 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 , 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 . 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,x∂u−u∂x,x∂x+u∂u} for n=2 [16] and {∂x,∂y,∂u,x∂u−u∂x,x∂y−y∂x,y∂u−u∂y,x∂x+y∂y+u∂u} 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 , 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: , i=1,2,…,n. Using them we can represent a geometric motion in the formwhere 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,By the Frenet–Serret formulas in Euclidean geometry,one obtains the Frenet–Serret formulas in the similarity geometry Pnwith V an n×n matrix given as follows. The zero curvature conditiongives 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 respectively by t, n, s and κ. The Frenet–Serret formulas read
The curvature and arc-length in P2 are given respectively by and , tangent and normal vectors of a curve in P2 are τ1=γθ and . Via a direct computation we have the Frenet–Serret formulas in P2
Now motion of curves in P2 is specified by
We relate it to the Euclidean motionwhere f=B/κ, g=A/
Curves in P3 and Burgers-mKdV hierarchy
In the case of curves in P3, the Frenet–Serret formulas readwhere and are respectively the curvature and torsion in P3, they are related to the Euclidean curvature κ and torsion τ by
The motion is now described bywhere τi, i=1,2,3 are related to the Euclidean tangent , normal and binormal by
It is possible to relate (14) with the Euclidean motion in [2], [3]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 Pnwhere 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 . The curve motion in Pn is described by (1). The inextensibility condition is
The time evolution for the frame vectors τi is
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 , and (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)
Phys. Lett. A
(1998)- et al.
Physica D
(2002) - et al.
Phys. Rev. Lett.
(1991) - et al.
Phys. Rev. Lett.
(1992) - et al.
Phys. J. Phys. Soc. Jpn.
(1993) J. Fluid Mech.
(1972)J. Math. Phys.
(1977)J. Math. Phys.
(1979)- et al.
J. Nonlinear Sci.
(1991)