([28] Bao-Feng Feng (UTP-TX), ( ), [20], [16], [24]. 1 ([3], [17]) p t = 1 2 κ2 T + κ s N -259-

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1 5,..,. [8]..,,.,.., Bao-Feng Feng UTP-TX,, UTP-TX,,. [0], [6], [4].. ps ps, t. t ps, 0 = ps. s 970 [0] []. [3], [7] p t = κ T + κ s N -59-

2 , κs, t κ t + 3 κ κ s + κ sss = 0. T s, t, Ns, t., - mkdv. mkdv. [] Musso []. [9] mkdv,,. R M p TpM = {at + bn a bκ = 0}. M presymplectic ω ωpat + bn, T + N = at + bn, αt + βn ds = b + aκβds S S M H Hp = S κs ds H X H X H = κ T + κ N mkdv KdV [3], [5], [].,.,,. [5], [6], [8] mkdv 977 [4]. mkdv -60-

3 .,., R, Z n p n. p n+ p n =. p n..,. p n+ p n =. p n+ p n ϵ = ϕ n. p n, p n = p n+ p n p n. p p n ϕ n. p n p n+ S n. p n. T n = p n + p,. Tim Hoffmann.,. ps. rs. κs = /rs κs., +, /rs κs.. p n p, p n+ 3 C v n p n. 3-6-

4 . p n, T n C v n. r v n r v n = sin ϕ n, κ v n = ±/r v n = ± sinϕ n / S, S n, S n+. p p n p, p n. p n, p n+, O e n., 3 S, S n, S n+, p n. r e n r e n = tan ϕ n + tan ϕ n+. /rn e, 3,. Hoffmann [8]. p n,, S S n. r n = tan ϕ n. /r n κ n,. [8]. 4, O, Musin[] S. Tabachnikov[6],. 3 Hoffmann Kutz[9] mkdv.. p n+ p n ϵ = -6-

5 d dt p n = ϵ p n+ p n p n+ p n d dt κ n = + ϵ ϵ 4 κ n κ n+ κ n mkdv [7] ϕ n,. [9]. n, m p m n p m n+ p m n =, a n, p m n+ p m n a n p m+ n p m n b m = RK m n pm n p m a, Frenet, = RW m n pm n+ p m n a n, RK m n K m n p m n+ p m n a n = t cos Ψ m n, sin Ψ m n K m n = Ψ m n Ψ m. 3 [9] p n+ m+ = p m+ n+. Θ m n tan Θ m+ n+ Θm n = b m + a n Θ m+ n Θ m n+a tan 4 b m a n 4. W m n n Θ m n+, Kn m = Θm n+ Θ m = Θm+ mkdv [5]. mkdv ps = xs, ys T s = x s, y s = cos θs, sin θs -63-

6 θ s = κs mkdv θs, t mkdv. θ t + θ s 3 + θ sss = 0 mkdv. Hoffmann-Kutz κ m n = tan Km n a n [4], a n = a, b m = b t := n + m, l := n m, δ := a + b, ϵ := a b δ 0 Hoffmann-Kutz 4 mkdv bilinear method, 3 fs, y, t gs, y, t D i sd j yd k t f g := s s i y y j t t k fx, y, tgx, y, t s=s,y=y,t=t. D s. n D n s f g = n r=0 n r s n r f s r g 4 [3] τ m n s, t; y D sd y τ m n τ m n = τ m n, 3 D sτ m n τ m n = 0, 4 D 3 s + D t τ m n τ m n = 0, 5 D y τ m n+ τ m n = a n τ m n+ τ m n, 6 D y τn m+ τn m = b m τ n+ τ m n m, 7 b m τ n m+ τn+ m a n τ n+τ m n m+ + a n b m τ n+ m+ n m =

7 p m n s, t; y := Θ m n s, t; y := log τ m n τ m n y logτ m n τ m n logτ m n / τ m n m, n Z, y R s, t p m n s, t; y mkdv s, t, y R n, m p m n s, t; y mkdv τ m n. breather solution [3]. N Z 0, τ m n u = τ m n s, t; y, z; u τ m n u = exp f i u [ s + n a n + m b m = f u i s, t; y, z; m, n i =,..., N y ] det f i u+j, i,j=,...,n f i u s = f i u+, f u i z = f i u+, f u i t = 4f i u+3, f u i y = f i u, f u i m, n f u m, n = f i u+ a m, n, f u i m, n f u m, n = f i u+ m, n n b m. N = 0, detf i u+j i,j=...,n =. f u i f u i = e η i + e µ i, 9 e η i = α i p u i a n p i b m p i e pis+p i z 4p3 i t+ p y i, n m 0 e µ j = β i qi u a n q i b m q i e qis+q i z 4q3 i t+ q y i, p i, q i, α i, β i. n m -65-

8 τ τ m n = exp [ s + n a n + m b m y ] det f i j, i,j=,...,n e η i e µ j n f i u = e η i + e µ i, = α i p u i a n p i b m p i e pis 4p n m 3 i t+ p i y, = β i p i u + a n p i + b m p i e pis+4p m 3 i t p i y. p i, α i R, β i R i =,..., N, τ m n mkdv mkdv N- N = M, τ m n p i, α i, β i C i =,..., M, p k = p k k =,..., M, α k = α k, β k = β k k =,..., M mkdv mkdv M- 5 mkdv ps, t px, t = x, yx, t. mkdv u x u t = + u 3/, u = y x xx WKI rs, t := exp θs, t, zs, t := s 0 rs, t ds Dym r t = r 3 r zzz []. -66-

9 [] B.-F. Feng, J. Inoguchi, K. Kajiwara, K. Maruno and Y. Ohta, Discrete integrable systems and hodograph transformations arising from motion of deiscrete plane curves, preprint, arxiv:07.48, 0. [] A. Fujioka and T. Kurose, Hamiltonian formalism for the higher KdV flows on the space of closed complex equicentroaffine curves, Int. J. Geom. Methods Modern Phys. 700, no., [3] R. E. Goldstein and D. M. Petrich, The Korteweg-de Vries hierarchy as dynamics of closed curves in the plane, Phys. Rev. Lett. 6799, no. 3, [4], Nonlinear partial difference equations. I. A difference analogue of the Kortewegde Vries equation, J. Phys. Soc. Japan 43977, no. 4, [5], Discretization of the potential modified KdV equation, J. Phys. Soc. Japan 67998, no. 4, [6],,, 003. [7] M. Hisakado, K. Nakayama, and M. Wadati, Motion of discrete curves in the plane, J. Phys. Soc. Japan 64995, no. 7, [8] T. Hoffmann, Discrete Differential Geometry of Curves and Surfaces, COE, Vol. 8, 009 [8]. [9] T. Hoffmann and N. Kutz, Discrete curves in CP and the Toda lattice, Stud. Appl. Math [0],,, 007. [],,, 00. [],,, 00 0, 6 3. [3] J. Inoguchi, K. Kajiwara, N. Matsuura, Y. Ohta, Motion and Bäcklund transformations of discrete plane curves, Kyushu J. Math., to appear arxiv: [4] J. Inoguchi, K. Kajiwara, N. Matsuura, Y. Ohta, Explicit solutions to semi-discrete modified KdV equation and motion of discrete plane curves, preprint. [5],, [8]. [6] W. Rossman,, 48 00, 9 5. [7] G. L. Lamb Jr., Solitons and the motion of helical curves, Phys. Rev. Lett , no. 5, [8],,,, 000, pp [9] N. Matsuura, Discrete KdV and discrete modified KdV equations arising from motions of planar discrete curves, Int. Math. Res. Notices, to appear doi:0.093/imrn/rnr

10 9, [0],,, A0-580,, pp [] O. R. Musin, Curvature extrema and four-vertex theorems for polygons and polyhedra, J. Math. Sci , no. 0, [] E. Musso, An experimental study of Goldstein-Petrich curves, Rend. Sem. Mat. Univ. Pol. Torino 67009, no. 4, [3] U. Pinkall, Hamiltonian flows on the space of star-shaped curves, Results Math.7995, no. 3-4, [4] W. Rossman, Discrete Constant Mean Curvature Surfaces via Conserved Quantities, COE 5, 00. [5] G. Segal, The geometry of the KdV equation, Int. J. Modern Phys. A 6 99, [6] S. Tanachnikov, A four vertex theorem for polygons, Amer. Math. Monthly 07000, no. 9, [7] M. Umehara, A unified approach to the four vertex theorems. I.,in: Differential and Symplectic Topology of Knots and Curves, 85 8, Amer. Math. Soc. Transl. Ser., 90999, pp [8],,, 00. inoguchi@sci.kj.yamagata-u.ac.jp -68-

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