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-
, 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-
.,., 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. 3 3. p n. 3-6-
. p n, T n C v n. r v n r v n = sin ϕ n, κ v n = ±/r v n = ± sinϕ n /.... 3 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-
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-
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 = 0. 8-64-
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-
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-
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