²ÄÀÑʬΥ»¶ÈóÀþ·¿¥·¥å¥ì¡¼¥Ç¥£¥ó¥¬¡¼ÊýÄø¼°¤ÎÁ²¶á²òÀÏ Asymptotic analysis for the integrable discrete nonlinear Schrödinger equation
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1 Asymptotic analysis for the integrable discrete nonlinear Schrödinger equation ( ) ( )
2 1. Schrödinger focusing NLS iu t + u xx +2 u 2 u = 0 u(x, t) =2ηe 2iξx 4i(ξ2 η 2 )t+i(ψ 0 +π/2) sech(2ηx 8ξηt 2δ 0 ) (exp, ) (sech)
3 2. t - NLS: Fokas-Its ( 96), Kamvissis ( 95) : Krüger-Teschl ( 09) KdV: ( 75) Grunert-Teschl ( 09) (soliton resolution conjecture) Deift-Zhou ( 93) nonlinear steepest descent
4 3. NLS (IDNLS) 1 NLS (focusing) iu t + u xx +2 u 2 u = 0 Ablowitz-Ladik ( 75) focusing NLS (IDNLS) i d dt R n+(r n+1 2R n +R n 1 )+ R n 2 (R n+1 +R n 1 )=0 Lax (exp, ) (sech) IDNLS
5 4. NLS (IDNLS) 2 z 1, z 1 > 1, C 1 (0) (norming constant) ( ) bright soliton BS(n, t; z 1, C 1 (0)) = (exp carrier wave) (sech traveling wave) C 1 (0) PHASE SHIFT in exp and sech..
6 5. z 1 = exp(α 1 + iβ 1 ) α 1 > 0, BS(n, t; z 1, C 1 (0)) = C 1(0) C 1 (0) exp( i[2β 1 (n + 1) 2w 1 t] ) sinh(2α 1 )sech[2α 1 (n + 1) 2v 1 t θ 1 ]. v 1 = sinh(2α 1 ) sin(2β 1 ), w 1 = cosh(2α 1 ) cos(2β 1 ) 1, θ 1 = log C 1 (0) log sinh(2α 1 ). sech tw(z 1 ) = tw(exp(α 1 + iβ 1 )) = α 1 1 v 1 C 1 (0) PHASE SHIFT in exp and sech..
7 6. phase shift KdV
8 6. phase shift KdV t ± phase shift phase shift =
9 6. phase shift KdV t ± phase shift phase shift = 1- soliton resolution phase shift
10 7. IDNLS Lax i d dt R n+(r n+1 2R n +R n 1 )+ R n 2 (R n+1 +R n 1 ) = 0 (IDNLS) iu t + u xx + 2 u 2 u = Lax (AKNS) [ ] z Rn X n+1 = X n (n-part) d dt X n = [ R n z 1 a complicated matrix (IDNLS) d dt X n+1 }{{} n-part ] X n = ( d dt X m }{{} t-part z C \ {0} spectral parameter ) m=n+1 (t-part)
11 8. n-part R n 0 (rapidly) as n ± X n+1 [ ] z 0 0 z 1 X n.
12 8. n-part R n 0 (rapidly) as n ± X n+1 [ ] z 0 0 z 1 X n. ϕ n (z, t), ψ n (z, t): z = 1 ψn(z, t): z = 1 [ ] z n ϕ n (z, t) as n, 0 [ ] [ ] 0 z ψ n (z, t) z n, ψn(z, n t) as n
13 9. 1 z = 1, a(z), b(z) = b(z, t) ϕ n = bψ n + aψ n a(z) O( z > 1) C 0 ( z 1), b(z) C ( z = 1).
14 9. 1 z = 1, a(z), b(z) = b(z, t) ϕ n = bψ n + aψ n a(z) O( z > 1) C 0 ( z 1), b(z) C ( z = 1). a(z j ) = 0 a( z j ) = 0. {±z j, ± z 1 j }.. a(z).
15 10. z = 1, r(z) r(z) := b(z) a(z) [ [ 0 1 Recall: ψ n z n, ψn 1 z n 0 r ( ) [ ] z rψ n + ψn n const. (n ). 0 ] ] as n. r(z, t) = r(z) exp (it(z z 1 ) 2 ), where r(z) = r(z, 0).
16 11. a(z j ) = 0 (order 1). ±z j ± z j 1. ϕ n (z j ) = b j ψ n (z j ) for some b j. C j := b j d a(z dz j) {(±z j, ± z j 1, C j )} J j=1, r(z)
17 11. a(z j ) = 0 (order 1). ±z j ± z j 1. ϕ n (z j ) = b j ψ n (z j ) for some b j. C j := b j d a(z dz j) {(±z j, ± z j 1, C j )} J j=1, r(z) R n. Riemann-Hilbert.
18 12. Riemann-Hilbert (RHP) Γ: ( + side). m(z): C \ Γ : 1. Γ = R, m(z) ±Im z > Γ = { z = 1}, m(z) z = 1.
19 12. Riemann-Hilbert (RHP) Γ: ( + side). m(z): C \ Γ : 1. Γ = R, m(z) ±Im z > Γ = { z = 1}, m(z) z = 1. m +, m : ± sides Γ RHP: m + = m v on Γ (v : given ) v = I m m(z) I as z.
20 13. RHP RHP: m + = m v on Γ
21 13. RHP RHP: m + = m v on Γ RHP RHP. v m ( ) 1. v = I ( ) on ˆΓ Γ, m[original] = m[with ˆΓ deleted] 2. v I on ˆΓ, m[original] m[with ˆΓ deleted]
22 14. Riemann-Hilbert (RHP) m(z):, C \ Γ RHP: m + = m v on Γ m(z)
23 14. Riemann-Hilbert (RHP) m(z):, C \ Γ RHP: m + = m v on Γ m(z), m(z) RHP m(z) ( )
24 Res(m(z); z j ) = lim z zj m(z) 15. blowup [ ] 0 0 z 2n j C j (t) 0 RHP D(z j, ε),. m. ˆm. z j. C(z j, ε). 1-
25 16. IDNLS C j (t) = C j (0) exp ( it(z j z 1 j ) 2), r(z, t) = r(z) exp ( it(z z 1 ) 2) on z = 1, where r(z) := r(z, 0) R n (0) t = 0. t > 0 RHP RHP. R n (t) (t > 0)
26 17. IDNLS: ( =0) r(z) = r(z, 0) = 0 r(z, t) = 0 for all t R n (t) =. t 1- PHASE SHIFT (formal proof in Ablowitz et al. 04) BS(n, t, z j, p j T (z j ) 2 C j (0)) phase shift : p j := k>j z 2 k z 2 k, T (z j) := k>j z 2 k (z2 j z 2 k ) z 2 j z 2 k
27 17. IDNLS: ( =0) r(z) = r(z, 0) = 0 r(z, t) = 0 for all t R n (t) =. t 1- PHASE SHIFT (formal proof in Ablowitz et al. 04) BS(n, t, z j, p j T (z j ) 2 C j (0)) phase shift : p j := k>j z 2 k z 2 k, T (z j) := k>j z 2 k (z2 j z 2 k ) z 2 j z 2 k phase shift
28 18. : R n (0), generic 1- s n < 2t timelike : phase shift R n (t) = BS ( n, t; z s, δ(0)δ(z s ) 2 p s T (z s ) 2 C s (0) ) +O(t 1/2 ). δ(z) r(z). p s T (z s ) z k s (k s). n 2t R n (t) = BS ( n, t; z s, p s T (z s ) 2 C s (0) ) + O(t 1/3 ). n > 2t R n (t) = BS (n, t; z s, p s T (z s ) 2 C s (0)) + O(n k ), k.
29 19. timelike δ(z) n < 2t (the ( timelike region) 2 ) A := n/t i 2 n/t. S 1 := e πi/4 A, S 2 := e πi/4 Ā, S 3 := S 1, S 4 := S 2. z = 1. ( [ 1 S2 δ(z) := exp + 2πi S 1 { z = 1}. δ(z). δ(z) 1 S4 S 3 ] ) (τ z) 1 log(1 + r(τ) 2 ) dτ
30 20. Riemann-Hilbert m + (z) = m (z)v(z) on z = 1 [ 1 + r(z) 2 e v(z) = 2φ r(z) ] e 2φ r(z) 1 φ = 1 2 it(z z 1 ) 2 n log z! m(z) R n (t) = d dz m(z) 21 z=0
31 21. φ = 1it(z 2 z 1 ) 2 n log z Re φ(z) = 0 z = 1 n < 2t n = 2t n > 2t Re φ(z) = 0 steepest descent path
32 References Ablowitz M.J., Prinari B., Trubatch A.D., Discrete and continuous nonlinear Schrödinger systems, Deift P., Zhou X., A steepest descent method for oscillatory Riemann Hilbert problems. Asymptotics for the MKdV equation, (1993) P. Deift, S. Kamvissis, T. Kriecherbauer and X. Zhou, The Toda rarefaction problem, (1996). K. Grunert and G. Teschl, Long-time asymptotics for the Korteweg-de Vries equation via nonlinear steepest descent, (2009). Yamane H., Long-time asymptotics for the defocusing integrable discrete nonlinear Schrödinger equation, (2014), II (2015)
33 . Yamane H., Long-Time Asymptotics for the integrable discrete nonlinear Schrödinger equation: the focusing case, arxiv:
34 24. 4
35 25. tw(z j ) j [ ] 0 0 Res(m(z); ±z j ) = lim m(z) z ±z j z 2n j C j (t) 0 z 2n j C j (t) = C j (0) exp [ 2α j t{tw(z j ) n/t} ]
36 25. tw(z j ) j [ ] 0 0 Res(m(z); ±z j ) = lim m(z) z ±z j z 2n j C j (t) 0 z 2n j C j (t) = C j (0) exp [ 2α j t{tw(z j ) n/t} ] s : n/t tw(z s )
37 25. tw(z j ) j [ ] 0 0 Res(m(z); ±z j ) = lim m(z) z ±z j z 2n j C j (t) 0 z 2n j C j (t) = C j (0) exp [ 2α j t{tw(z j ) n/t} ] s : n/t tw(z s ) tw(z j ) tw(z s ) < 0 z 2n j C j (t).
38 25. tw(z j ) j [ ] 0 0 Res(m(z); ±z j ) = lim m(z) z ±z j z 2n j C j (t) 0 z 2n j C j (t) = C j (0) exp [ 2α j t{tw(z j ) n/t} ] s : n/t tw(z s ) tw(z j ) tw(z s ) < 0 z 2n j C j (t). tw(z j ) tw(z s ) > 0 z 2n j C j (t),..
39 25. tw(z j ) j [ ] 0 0 Res(m(z); ±z j ) = lim m(z) z ±z j z 2n j C j (t) 0 z 2n j C j (t) = C j (0) exp [ 2α j t{tw(z j ) n/t} ] s : n/t tw(z s ) tw(z j ) tw(z s ) < 0 z 2n j C j (t). tw(z j ) tw(z s ) > 0 z 2n j C j (t),.. ±z s, ± z s 1. C s (t) p s T (z s ) 2. p s := j>s z 2 j z 2 j, T (z s ) := j>s z 2 j (z 2 s z 2 j ) z 2 s z 2 j
40 26. reduction P. Deift, S. Kamvissis, T. Kriecherbauer and X. Zhou, The Toda rarefaction problem, Comm. Pure Appl. Math. 49(1) (1996), KdV K. Grunert and G. Teschl, Long-time asymptotics for the Korteweg-de Vries equation via nonlinear steepest descent, Math. Phys. Anal. Geom. 12(3) (2009) 4 r(z) 0 blowup (4 ) ( ), r(z) 0 ( ), 4 r(z) 0 4, : 1- (BS)
41 Res(m(z); z j ) = lim z zj m(z) 27. blowup [ ] 0 0 z 2n j C j (t) 0. D(z j, ε),. m. ˆm. z j. C(z j, ε)
42 28. blowup 2 [ ] 0 0 Res(m(z); z j ) = lim m(z) z zj z 2n. j C j (t) 0 m(z) 1 z 2n j C j (t) (m(z) 2 ) m(z) 1 = z 2n j C j (t) m(z) 2 mod O zj z z j 1 0 D(z j, ε) ˆm(z) := m(z) z 2n j C j (t) 1 z z j z j C(z j, ε).
43 29. (blowup) 1 0 ˆm(z) := m(z) z 2n j C j (t) in D(±z j, ε)( ) 1 z z j blowup 1 0 ˆm + (z) = ˆm (z) z 2n j C j (t) on C(±z j, ε)( ) 1 z z j
44 29. (blowup) 1 0 ˆm(z) := m(z) z 2n j C j (t) in D(±z j, ε)( ) 1 z z j blowup 1 0 ˆm + (z) = ˆm (z) z 2n j C j (t) on C(±z j, ε)( ) 1 z z j t? z 2n j C j (t). ( ) z 2n j C j (t) ( ).
45 29. (blowup) 1 0 ˆm(z) := m(z) z 2n j C j (t) in D(±z j, ε)( ) 1 z z j blowup 1 0 ˆm + (z) = ˆm (z) z 2n j C j (t) on C(±z j, ε)( ) 1 z z j t? z 2n j C j (t). ( ) z 2n j C j (t) ( ). ±z j ± z 1 j
46 30. : Γ, z 2n j C(t) A. A M + (z) = M (z)v (z) on Γ, 1 0 M + (z) = M (z) A on C(±z 1 0, ε), z z 0 M + (z) = M (z) 1 z 2 0 Ā z z 0 1 on C(± z 0 1, ε), 0 1 M(z) I as z.
47 30. : Γ, z 2n j C(t) A. A M + (z) = M (z)v (z) on Γ, 1 0 M + (z) = M (z) A on C(±z 1 0, ε), z z 0 M + (z) = M (z) 1 z 2 0 Ā z z 0 1 on C(± z 0 1, ε), 0 1 M(z) I as z. A ( ) 1/A ( )?
48 31. 2 R(z, z 0 ) := z2 0(z 2 z 0 2 ). phase shift formula z 2 z0 2
49 31. 2 R(z, z 0 ) := z2 0(z 2 z 0 2 ). phase shift formula z 2 z0 2 RHP (by ) M + (z) = M (z)d(z) 1 V (z)d(z) on Γ, [ ] R(z, z0 ) D(z) := 1 0, on Γ, 0 R(z, z 0 ) [ M + (z) = M 1 R(z, z (z) 0 ) 2 z z ] 0 A on C(±z 0, ε), M + (z) = M (z) 2 z z 1 0 R(z, z 0 ) z 0 2 Ā 1 on C(± z 0 1, ε), M(z) I as z. A!
50 32. 3 M(z) = diag(z 2 0, z 2 0 )M(z)D(z) 1 z z 0 A A 0 z z 0 z 0 2 Ā D(z)= 0 z z 1 0 z z 1 0 z [ Ā ] R(z, z 0 ) R(z, z 0 ) [ R(z, z 0 ) R(z, z 0 ) [ R(z, z 0 ) 1 0 ±z 0 ± z 1 0. ] 0 R(z, z 0 ) ] in D(±z 0, ε), in D(± z 1 0, ε), elsewhere.
51 33. 4 blowup? [ 0 0 Res(M(z); ±p) = lim z ±p M(z) p 2n C 0 ± p 1. ]
52 33. 4 blowup? [ 0 0 Res(M(z); ±p) = lim z ±p M(z) p 2n C 0 ± p 1. M(z) [ ] Res( M(z); 0 0 ±p) = lim M(z) z ±p p 2n, τc 0 ( ) p τ = R(±p, z 0 ) 2 2 z0 2 2 =, z0(p 2 2 z 2 0 ) C τ. phase shift formula. ]
53 34. j s with n/t tw(z j ) 0, blowup. n/t tw(z s ) 0 ±z s, ± z s 1 z =1 (nonlinear steepest descent) n < 2t 4 z =
54 35.? φ = 1 2 it(z z 1 ) 2 n log z Re φ(z) = 0 original RHP z = 1 n < 2t( ) n = 2t n > 2t( ) δ(z).
55 36. Reφ(z) = 0 Re φ(z) = 0 RHP ( ) n < 2t ( ) n = 2t n > 2t ( ) Re φ(z) = 0 (4 )
56 References Ablowitz M.J., Prinari B., Trubatch A.D., Discrete and continuous nonlinear Schrödinger systems, Deift P., Zhou X., A steepest descent method for oscillatory Riemann Hilbert problems. Asymptotics for the MKdV equation, (1993) P. Deift, S. Kamvissis, T. Kriecherbauer and X. Zhou, The Toda rarefaction problem, (1996). K. Grunert and G. Teschl, Long-time asymptotics for the Korteweg-de Vries equation via nonlinear steepest descent, (2009). Yamane H., Long-time asymptotics for the defocusing integrable discrete nonlinear Schrödinger equation, (2014), II (2015)
57 . Yamane H., Long-Time Asymptotics for the integrable discrete nonlinear Schrödinger equation: the focusing case, arxiv:
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