Korteweg-de Vries 2011 03 29
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1 1 3 1.1 K-dV........................ 3 1.2.............................. 4 2 K-dV 5 2.1............................. 5 2.2.............................. 8 2.3 K-dV............. 11 2.4 1.............................. 13 3 2 15 3.1 2........................... 15 3.2 2....................... 17 3.3 K-dV......................... 20 3.4 2........................ 21 4 25
2 26 27
1 3 1 1.1 K-dV.,.,.,, (solitary wave) -on..,,. Korteweg-de Vries (, K-dV ), 1895 D. J. Korteweg G. de Vries,,. K-dV. u t 6u u x + 3 u = 0. (1.1.1) x3 x, t, u. 1965, N. J. Zabusky M. D. Kruskal K-dV *1,., K-dV., K-dV. *1 Zabusky and Kruskal (1965) K-dV u t + u u x + δ2 3 u x 3 = 0, (1.1.1),.
1 4 1.2. 2, K-dV, K-dV, K-dV., 1,. 3, K-dV 2., K-dV,,. 4.
2 K-dV 5 2 K-dV 2.1 K-dV,. u t 6u u x + 3 u x 3 = 0 (2.1.1). u/ t, K-dV (2.1.1) *1., (2.1.2). u t 6u u x = 0 (2.1.2), u t + c u x = 0 c = const. u = g(x ct). g x ct. (2.1.2) u,, (2.1.2) u = f(η) η = x + 6ut (2.1.3) *1 (2.1.2 ) Burgers
2 K-dV 6. (2.1.2), ( u u t 6u u x = 6t., t 6u u x f = du dη ) f = 0 (2.1.4). (2.1.4), (2.1.3) (2.1.2). (2.1.3), K-dV. (2.1.2) (2.1.3). η = x + 6ut, 6u.,,. u,. (2.1.2),,.. u t 6u u x = 0 (2.1.2),. 3.0, 128, 1.0 10 5.,, 32 *2. u, 100.0, 2. 2.1.1. (2.1.2) u = f(x + 6ut).,, 2.1.1,,., 2.1.1,,.,.,,. *2 u,.
2 K-dV 7 (a) (b) 16 (c) 32 2.1.1: (2.1.2). x, u.
2 K-dV 8 2.2 K-dV u t 6u u x + 3 u x 3 = 0 (2.1.1). (2.1.1),. u t + 3 u x = 0 (2.2.1) 3, (2.2.1). t = 0 u = ϕ(x). ϕ(x) 2l. ϕ(x) 2l/n (n = 1, 2, ), ϕ(x) = n=0 α n cos nπx l + n=1 β n sin nπx l (2.2.2). α n, β n. k n k n = nπ/l, (2.2.2) ϕ(x) = α n cos k n x + n=0 β n sin k n x (2.2.3). n ω n, u n u n exp(ik n x ω n t), (2.2.1), n n=1 ω n = k 3 n (2.2.4). t > 0 k n x ω n t, t > 0 (2.2.1) u = α n cos (k n x ω n t) + n=0. β n sin (k n x ω n t), n 1 (2.2.5) n=1 α 0 = 1 2l α n = 1 l β n = 1 l l l l l l l ϕ(x)dx, ϕ(x) cos k n x dx, n 1 ϕ(x) sin k n x dx, n 1 (2.2.6)
2 K-dV 9. (2.2.5), (2.2.1)., (2.2.4), k n, v gn v gn = 3k 2 n (2.2.7)., n 2.,,,.,., A n Fourier u = A n = α 2 n + β 2 n, n = 0, 1, 2.,,,.,,.. (2.2.1),. 3.0, 128, 1.0 10 6., 1000.,,, u = 10 3 sech 2 ( x 3.0/2.0 12.0/10 3 ) *3. 2.2.1.,, x..,,, (2.2.6),. *3 K-dV. 2.4 K-dV,.
2 K-dV 10 (a) (b) 400 (c) 800 2.2.1: (2.2.1). x, u.
2 K-dV 11 2.3 K-dV, K-dV. u t 6u u x + 3 u x 3 = 0 (2.1.1) 2.3.1.., (a) u., (b).,.,,,.,.,,,.,,,. (a) (b) 2.3.1:. x, u, u.,.
2 K-dV 12,,., K-dV,,.
2 K-dV 13 2.4 1, K-dV, K-dV (2.1.1), 1., c (c > 0) u = f(η), η = x ct. (2.1.1), K-dV. cf 6ff + f = 0. (2.4.1) f f η. η, f = f = f = = 0, (2.4.1) f = (c + 3f)f. f, (f ) 2 = (C f)f 2. c/2 = C. c > 0, C < 0., > 0 C f < 0. g = C f, ln g i C g + i C = i C(η x 0 ). i, x 0,. g g = i C tanh{i C/2(η x 0 )}. u = f(η) = C + g 2, C = A { } A u = Asech 2 2 (x 2At x 0) (2.4.2). 1. C < 0 A > 0.
2 K-dV 14 K-dV 1 (2.4.2)., A > 0,.. 2A,.,., sech 1, 1/ A/2.,., K-dV,. (2.4.2) 1,.,,.
3 2 15 3 2 3.1 2 1,,., 2. K-dV u t 6u u x + 3 u x 3 = 0 (2.1.1).,. K-dV (2.1.1). u t 6uu x + u xxx = 0 (3.1.1), u φ u = 2 2 ln φ (3.1.2) x2. x, (3.1.1) ( ) 2 2 2 t x ln φ + 6 x ln φ + 4 2 x ln φ = 0 4., K-dV. φ x φ t φφ xt = 3(φ xx ) 2 4φ x φ xxx + φφ xxxx (3.1.3) φ = 1 + e 2θ, θ = kx ωt (3.1.4)
3 2 16, (3.1.3) ω = 4k 3 (3.1.5). k, ω (3.1.4). (3.1.5), (3.1.4) K-dV., (3.1.4) (3.1.2), u, u = 2k 2 sech 2 (kx ωt) (3.1.6). (3.1.6) 1, (3.1.4) K-dV., (3.1.4) A,, (3.1.2) A = e 2δ, φ = 1 + Ae 2θ (3.1.7) u = 2k 2 sech 2 (kx 4k 3 t δ), A. (3.1.7),,, i, φ = 1 + A i e 2θ i, θ i = k i x ω i t, i = 1, 2. K-dV.,, A 3 φ = 1 + A 1 e 2θ 1 + A 2 e 2θ 2 + A 3 e 2(θ 1+θ 2 ) (3.1.8), ω i = 4ki 4, i = 1, 2 (3.1.9) ( ) 2 k1 k 2 A 3 = A 1 A 2 (3.1.10) k 1 + k 2, K-dV *1. (3.1.8) (3.1.2), K-dV 2,., φ (3.1.8), 2. *1 (3.1.8) K-dV (3.1.3), e 2θ i, e 2(θ 1+θ 2 ), e 4θ 1+2θ 2, e 2θ 2+4θ 2,, 0 (3.1.9), (3.1.10).
3 2 17 3.2 2, φ 2 φ = 1 + A 1 e 2θ 1 + A 2 e 2θ 2 + A 3 e 2(θ 1+θ 2 ) (3.1.8).., θ i = k i x ω i t ω i = 4ki 4, i = 1, 2 (3.1.9) ( ) 2 k1 k 2 A 3 = A 1 A 2 (3.1.10) k 1 + k 2.,. (3.1.9), (3.1.10), A 1, A 2, k 1, k 2., (3.1.6), k i, (3.1.7) (3.1.2) A i., t ± u *2., φ. 1 0, k 1 x ω 1 t 0. 2 θ 2, θ 2 θ 1, θ 2 = k 2 x ω 2 t = k 2 (θ 1 ) + k ( 2 1 ω 2 k 1 k 1 ω 1 k 1 = k 2 k 1 (θ 1 ) + k 2 k 1 ( 1 k2 2 k 2 1 k 2 ) ω 1 t ) ω 1 t (3.2.1). k 1 > k 2 > 0. 1, 2,. t, θ 2 0, k 1 > k 2 (3.2.1) θ 2 k ( ) 2 1 k2 2 ω k 1 k1 2 1 t t., e 2θ 2 0, φ φ = 1 + A 1 e 2θ 2 (3.2.2) *2.
3 2 18. A 1 = exp{ 2δ ( ) 1 }, (3.1.2) u, u = 2k 2 1sech 2 {k 1 x ω 1 t δ ( ) 1 } (3.2.3)., t θ 1 = 0. (3.2.3), k 2 1., t +, θ 2 k ( ) 2 1 k2 2 ω k 1 k1 2 1 t + t +, e 2θ 2 e 2θ 1. φ, ( φ = A 2 1 + A ) 3 e 2θ 1 e 2θ 2 (3.2.4) A 2. t, u, u = 2k 2 1sech 2 {k 1 x ω 1 t δ (+) 1 } (3.2.5)., δ (+) 1 = 1 ( ) 2 ln A3 A 2., θ 2 0 t ± u, t, t +. u = 2k2sech 2 2 {k 2 x ω 2 t δ ( ) 2 }, (3.2.6) δ ( ) 2 = 1 ( ) 2 ln A3, A 1 u = 2k 2 2sech 2 {k 2 x ω 2 t δ (+) 2 }, (3.2.7) δ (+) 2 = 1 2 ln A 2
3 2 19, t 2 [ ] u = 2ki 2 sech 2 {k i x ki 3 t δ ( ) i }, (3.2.8) t + i=1 δ ( ) 1 = 1 2 ln A 1, δ ( ) 2 = 1 2 ln ( A3 A 1 ), u = 2 [ 2ki 2 sech 2 {k i x ki 3 t δ (+) i } ], (3.2.9) i=1 δ (+) 1 = 1 ( ) 2 ln A3, A 2 δ (+) 2 = 1 2 ln A 2. (3.2.8) (3.2.9), δ (±) i ± i x i, x i 4k 2 i t, i = 1, 2, t. t x 1 < x 2, t +, x 1 > x 2. +,., 2 2k 2 1, 2k 2 2., t t +., 2k 2 1 1, 1 = 1 k 1 (δ (+) 1 δ ( ) 1 ), = 1 2k 1 ln A 1A 2 A 3., 2k2 2 2 2 = 1 ln A 3 2k 2 A 1 A 2. (3.1.10) k 1 > k 2, ( ) 2 A 1 A 2 k1 + k 2 = > 1 A 3 k 1 k 2,, 1 < 0, 2 > 0., 1 < 2,.
3 2 20 3.3 K-dV 2,, 2,,, K-dV. K-dV u t 6uu x + u xxx = 0 (3.1.1) u t = x (3u2 u xx ) (3.3.1). x u, u x, u xx, 0, (3.3.1) x +,, d dt + udx = 0. I 1 = + udx (3.3.2) (3.1.1) u uu t = 6u 2 u x uu xxx ( ) u 2 = (2u 3 uu xx + (u ) x) 2 t 2 x 2. x +,, + ( ) 1 I 2 = 2 u2 dx (3.3.3)., (3.1.1) u 2, u 3, u 4,. K-dV u,, u x, u xx,.
3 2 21 3.4 2 2. K-dV. u t uu x + u xxx = 0 (3.4.1), 1 2,. 2,,. (3.4.1). t, x, j, i, K-dV u j+1 i u j 1 i 2 t = u j i u j i+1 uj i 1 2 x uj i+2 2uj i+1 + 2uj i 1 uj i 2 2( x) 3 (3.4.2)., u, x + 0., u. (3.4.2),,,. (3.4.2) u., u 2. u, u 2., u 2, (3.4.2) u x,. K-dV (3.4.1) u, t ( u 2 2 ) (u j 2) 2 uj i+1 uj i 1 u j u j i+2 2uj i+1 + 2uj i 1 uj i 2 i (3.4.3) 2 x 2( x) 3, u, 0., 0., (3.4.1) u 2. (3.4.1) ) uu x = αuu x + β x α + β = 1 ( 1 2 u2, (3.4.4)
3 2 22., uu x αu i u i 1 u i+1 2 x + β 1 2 u2 i+1 1 2 u2 i 1 2 x (3.4.5). (3.4.5), u α, β. u 2. (3.4.5) u, u u i,, 1 3 x (u3 ) u i u i+1 ( αu i + β 2 u i+1 ) u i u i 1 ( αu i + β 2 u i 1 x,.,, ). α = 1 3, β = 2 3 (3.4.6),,., (3.4.6) (3.4.5), (3.4.2)., u j+1 i u j 1 i 2 t = u j i (u j i+1 uj i 1 ) 6 x + (uj i+1 )2 (u j i 1 )2 6 x uj i+2 2uj i+1 + 2uj i 1 uj i 2 2( x) 3.. 3.0, 256, 2.0 10 7., 25000. ( ) ( ) x X1 x X2 u = U1sech 2 U2sech 2 2α/U1 2α/U2.. α = 6.0, U1 = 1440.0, U2 = 720.0, X1 = 3.0/4.0, X2 = 3.0/2.0. xu 3.4.1. (a) 2
3 2 23 (a) (b) 6000 (c) 12000 (d) 24000 3.4.1: K-dV 2., (b) x. 3.2, A 2A. u,.,. (c),.,.,,,,.. (d).., 3.2
3 2 24 (a) (b) 3.4.2: a: xtu. z u.,.. b: xt u x.,,., 3.4.2. (a). x 2.0, t 2.5,. xt 3.4.2(b). (b),,,.,., t t +.
4 25 4 K-dV. K-dV,, K-dV,. 2,,.
4 26,.,..,.,.,,., DCL.
4 27 [1], 1995: 30 3 30,, 219pp. [2],, 1984: 6,, 183pp. [3], 1985:,, 249pp. [4], 2007: - KdV -, [5] Zabusky, J. N. and Kruskal, M. D., 1965: Interatction of solitons in a collisionless plasma and the recurrence of initial states, P hys.rev.lett., 15, 240-243.