九州大学学術情報リポジトリ Kyushu University Institutional Repository ソリトンの二次元相互作用について 及川, 正行九州大学応用力学研究所 辻, 英一九州大学応用力学研究所 Oikawa, Masayuki Research Institute for A
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1 九州大学学術情報リポジトリ Kyushu University Institutional Repository ソリトンの二次元相互作用について 及川, 正行九州大学応用力学研究所 辻, 英一九州大学応用力学研究所 Oikawa, Masayuki Research Institute for Applied Mechanics, Kyushu University Tsuji, Hidekazu Research Institute for Applied Mechanics, Kyushu University 出版情報 : 応用力学研究所研究集会報告. 1ME-S7 1), 1-3. 九州大学応用力学研究所バージョン : 権利関係 :
2 No.1ME-S7 1 COE Reports of RIAM Symposium No.1ME-S7 Current and Future Research on Nonlinear Waves Perspectives for the Next Decade Proceedings of a symposium held at Chikushi Campus, Kyushu Universiy, Kasuga, Fukuoka, Japan, November 19-1, 9 Co-organized by Kyushu University Global COE Program Education and Research Hub for Mathematics - for - Industry Article No. 1 pp ) OIKAWA Masayuki), TSUJI Hidekazu) Received February 16, 1 Research Institute for Applied Mechanics Kyushu University March, 1
3 Masayuki OIKAWA) Hidekazu TSUJI) Research Institute for Applied Mechanics, Kyushu Univ.) Kadomtsev-Petviashvili KP) KP Benjamin-Ono, modified KP, extended KP, intermediate long wave KP V 1 Zabusky & Kruskal[1] Bullough[] Russell[3] Benney & Luke[4] 1964 KP KP Kadomtsev & Petviashvili[5] KP Zakharov & Shabat[6] Dryuma[7] KP Lax pair Zakharov & Shabat[6] Satsuma[8] N- Miles[9] KP Satsuma - singular Miles[1] - regular singular Mach ψ i ψ c 1 Mach Russell[3] [] 1
4 C B C ψ r ψ i B 1 ψ r ψ i O P PB PC ψ i ψ r Mach Mach stem PA P P A Miles[9] ψ i KP - sin ψ i < 3α sin ψ i ψ i ψ i < 3α 1) singular α h ψ i < 3α 1 P Mach [1] ψ i < 3α stem ψ i = 3α ajima et al.[11] Boussinesq Kako & ajima[1, 13] [11] Boussinesq Funakoshi[14] Boussinesq Miles Melville[15] Mach Miles Folkes et al.[16], Nishida & Nagasawa[17], Nagasawa & Nishida[18] KP [19] τ KP [, 1,, 3, 4, 5] N- M N, N)-M N, N)- y N y M N -,)-
5 Miles - singular regular,)- 314)-type,)- Mach [5] KdV KP Benjamin-OnoBO) modified KdV KdV KP [6] [7] Benjamin-Ono [8] extended KPEKP) BO [9] u T + u u + H u )) + u =, ) Hf) = 1 π P f ) d f) Hilbert P u,, T ) = a 1 + a [ + Ω a/4 + Ω )T ] /16. 3) a Ω, ) V ψ i B Q P R B PBPB QR PB QR ψ i ψ i QR 3
6 BO ) 4) a) Ω =, T = 1.56, b) Ω = 1, T = 5.9, c) Ω =.5, T = u,, ) = a 1 + a 7L /8 Ω L / ) /16, < < L 4) a =, Ω = tan ψ i > L L L = 819.L Ω = L / [9] 3 Ω =, 1,.5 T = 1.56, 5.9, u,, T ) u = 765 Ω Ω Mach Ω = Ω = 1 Ω =.5 stem stem stem 3b),c) stem 4a) u = 765 u u max Ω Ω = 1.33, 1.43 Ω Ω = 1.33 u max 9 4 4b) stem L s L s stem 4
7 Ω = 1.33 Ω = 1.43 Mach ) u = u/a, = a, = a 3/, T = a T 5) Ω Ω := Ω/ a 5 Ω 15 Ls a) u Ω = 765 u Ω =.5,.67, 1, 1.33, 1.43, 1.67, b) stem + : Ω =.5, : Ω = 1, : Ω = 1.43, : Ω = 1.67, : Ω =. 3 MKP KP MKP [3]. u u + 6u T + 3 u 3 ) + u =. 6) u,, T ) = ±A sech[a + Ω A + Ω )T )] 7) u,, ) = A sech[a Ω )], for < L / 8) 6) u = u/a, = a, = a, T = a 3 T 9) L Ω := Ω/A 5 a) Ω/A b) Ω/A Mach Mach stem stem 5
8 a) b) 図 5 MKP 方程式 6) の初期値 8) に対する解の鳥瞰図 a) A =.5, Ω = 4Ω/A = 8), T =.95. b) A =.5, Ω = 1Ω/A = ), T = A= umax/a Ω/A 図6 漸近的最大振幅の倍率 umax /A の Ω/A 依存性 を持つ波が寄生している 図 6 は MKP 方程式 6) の初期値 8) に対する数値解の漸近的最大振 幅の倍率 umax /A が Ω/A にどのように依存するかを示している A =.5,.75, 1 に対する結果が 一つの曲線上に載っていることが示唆され scaling 則が成り立っていることが確かめられる ま た Ω/A が 3 と 4 の間で反射のタイプが変わることが推定できる さらに 漸近的最大振幅の最 大値は初期振幅の 4 倍より小さい可能性が高い MKP 方程式 6) では 正負のソリトン解が存在するので これらがどのような相互作用をする かは興味深い これを調べるために初期条件 { u,, ) = A sech[a + Ω )] for Ly / < <, A sech[a Ω )] for < < Ly /, 1) の下で MKP 方程式 6) を数値的に解いた A は正の部分と負の部分とのつなぎ目のところの不 連続性を滑らかにするための因子で 初期の解の過渡的な挙動を抑制する以外は本質的な役割はし ない 数値計算は A = 1 で行ったが いくつか行った A = 1.5 の場合の計算は 解の主要部分に ついては scaling 則が成り立つことを示した 図 7 は a) Ω = 3, T = 4, b) Ω = 1/3, T = 5 の場 合の鳥瞰図である この場合は = での境界条件が u = となる仮想的な境界での反射と同等 である 従って 等角反射の場合 反射波の符号が変わる他は通常の等角反射と同じである a) では確かにそうなっていて Ω/A が大きいときは等角反射が起こるようだ しかし Ω/A が小さ 6
9 stem b) MKP 6) 1) a) A = 1, Ω = 3Ω/A = 3), T = 4, b) A = 1, Ω = 1/3Ω/A = 1/3), T = 5. 4 EKP EKP [8] u T + 6u u EKdV u T ) u Qu + 3 u u =, Q > 11) + 6u u Qu u + 3 u 3 = 1) EKdV a sech η u, T ) = ), η = κ 4κ T ), κ = 1 a 1 tanh η a 1/Q a Qa 6 ) 13) a a < 6/Q u,, ) = 1 a sech κ Ω L / ) ) tanh κ Ω L / ) a 1/Q a 14) κ 13) 11) u = u/a, = a 1/, = a, T = a 3/ T, Q = aq 15) Q, Ω := Ω/ a a < 6/Q Q < 6 [31]Q KP Ω c Ω > Ω c Ω < Ω c Mach stem 7
10 Q Ω Ω = 4 Ω Ω = 1 stem stem EKP 11) 14) a) Q = 5, Ω = 4, T = 6, b) Q = 5, Ω = 1, T =. 9 EKP 11) 14) Ω Q Q = KP 16) Q 6 Ω Q = 5, Ω = 4 8a) Q = 5, Ω = 1 8b) Ω stem stem Q < 6 9 Q Ω 8
11 Q = KP ) u max = 1 + Ω ) 3 for Ω < Ω c = 3 4Ω Ω + Ω 1 for Ω > Ω c = 3 16) Ω c = 3 = Ω > Ω c Ω < Ω c Mach Ω = Ω c u max = 4 Q =.5 Ω c u max Q = Q 3 Ω c u max Q 4 u max Ω Ω u max 5 ILW intermediate long wave ILW) u T + u u + ) P G )u,, T )d G) := 1 [ coth χ π χ ) ] sgn) + u =, 17) [3] ILW u T + u u + P G )u, T )d = 19) l h h 1 18) χ χ := h 1 /l 17) u,, T ) = 18) K sinkχ), < Kχ < π) ) cosh[k + Ω ct )] + coskχ) c = 1 χ K cotkχ) + Ω 1) [33]K, Ω, K tankχ/) u,, ) = K sinkχ) cosh[k Ω )] + coskχ) ) Ω P G )u )d = χ 3 u + χ u 3 + Oχ5 ). 3) 9
12 χ 17) ) KP u T + u u + χ 3 ) u u =, 4) u,, ) = a sech [ a 4χ Ω ) ] 5) χ Kχ = π K/λ, λ > ) λ K [33]17) ) DBO ) u,, ) = a 1 + a Ω ) /16 6) a λ = a/4 DILW ) 6) u = u/a, = a, = a 3/, T = a T, χ = aχ 7) χ = aχ Ω := Ω/ a K = K/a χ T = 1 u χ =., Ω = a) DILW 17) 1 ), b) KP O-type,). χ =., 1, 5, 1 Ω χ =. KP 4) a = 1 5) [6] KP Ω > Ω c O-type,)- Ω < Ω c 314)-type,)-χ =.,)- 1 a) Ω =, T = 1 b) KP 4) a = 1 5) O-type,)- Ω T 11 a) Ω =.5, T = 4 b) KP 4) a = 1 5) 314)-type,)- Ω T 1
13 T = 4 u χ =., Ω =.5a) DILW 17) 1 ), b) KP 314)-type,) u a Ω u a Ω + : χ =., : χ = 1, : χ = 5, : χ = 1, : χ =. KP 4) a = 1 5) Mach 314)-type stem.5,.5 KP Mach Ω = Ω c = 1 u a 1 + Ω) for Ω < Ω c = 1 u a = 4Ω Ω + for Ω > Ω c = 1 Ω 1 1 χ Ω Ω Mach Mach stem χ stem 11 8)
14 3b), c) 1 Ω χ KP 8) χ Ω c 1 χ =. KP Ω χ u a DILW KP DBO 6 KP Mach KP stem KP [6] [7] KP V V KP 314)-type,)- [17, 18]. 314)-type,)- KP ψ c [1] Zabusky, N. J. and Kruskal, M. D. : Phys. Rev. Lett ) 4. [] Bullough, R. K. : The wave par excellence, the solitary progressive great wave of equilirium of the fluid: an early history of the solitary wave. in Solitons Introduction and Applications, ed. M. Lakshmanan, Springer-Verlag, Berlin, 1988) pp.7-4. [3] Russell, J. S. : Report on Waves, British Association Reports 1844). 1
15 [4] Benney, D. J. and Luke, J. C. : J. Math. and Phys ) 39. [5] Kadomtsev, B. B. and Petviashvili, V. I. : Sov. Phys.- Doklady ) 539. [6] Zakharov, V. E. and Shabat, A. B. : Funct. Anal. Appl ) 6. [7] Dryuma, V. S. : Sov. Phys. JETP Lett ) 387. [8] Satsuma, J. : J. Phys. Soc. Jpn ) 86. [9] Miles, J. W. : J. Fluid Mech ) [1] Miles, J. W. : J. Fluid Mech ) [11] ajima, N., Oikawa, M. and Satsuma, J. : J. Phys. Soc. Jpn ) [1] Kako, F. and ajima, N. : J. Phys. Soc. Jpn ) 63. [13] Kako, F. and ajima, N. : J. Phys. Soc. Jpn ) 311. [14] Funakoshi, M. : J. Phys. Soc. Jpn ) 371. [15] Melville, W. K. : J. Fluid Mech ) 58. [16] Folkes, P. A., Ikezi, H. and Davis, R. : Phys. Rev. Lett ) 9. [17] Nishida,. and Nagasawa, T. : Phys. Rev. Lett ) 166. [18] Nagasawa, T. and Nishida,. : Phys, Rev. A ) 343. [19] Sato, M. : RIMS Kokyuroku Kyoto University) No ) 3. [] Biondini, G. and Kodama,. : J. Phys. A: Math. Gen. 36 3) [1] Kodama,. : J. Phys. A: Math. Gen. 37 4) [] Biondini, G. and Chakravarty, S. : J. Math. Phys. 47 6) [3] Biondini, G. and Chakravarty, S. : Math. Comput. Simul. 74 7) 37. [4] Chakravarty, S. and Kodama,. : J. Phys. A: Math. Theor. 41 8) 759. [5] Chakravarty, S. and Kodama,. : Studies Appl. Math. 13 9) 83. [6] Kodama,., Oikawa, M. and Tsuji, H. : J. Phys. A: Math. Theor. 4 9) 311. [7],,, : KPII 1). [8] Tsuji, H. and Oikawa, M. : J. Phys. Soc. Jpn ) [9] Tsuji, H. and Oikawa, M. : Fluid Dyn. Res. 9 1) 51. [3] Tsuji, H. and Oikawa, M. : J. Phys. Soc. Jpn. 73 4) 334. [31] Tsuji, H. and Oikawa, M. : J. Phys. Soc. Jpn. 76 7) [3] Tsuji, H. and Oikawa, M. : Preprint submitted in Fluid Dyn. Res. [33] Satsuma, J. and Ablowitz, M. J. : Solutions of an internal wave equation describing a stratified fluid with finite depth. in Nonlinear Partial Differential Equations in Engineering and Applied Science, ed. R. L. Sternberg et al., Marcel Dekker, New ork, 198) pp
163 KdV KP Lax pair L, B L L L 1/2 W 1 LW = ( / x W t 1, t 2, t 3, ψ t n ψ/ t n = B nψ (KdV B n = L n/2 KP B n = L n KdV KP Lax W Lax τ KP L ψ τ τ Cha
63 KdV KP Lax pair L, B L L L / W LW / x W t, t, t 3, ψ t n / B nψ KdV B n L n/ KP B n L n KdV KP Lax W Lax τ KP L ψ τ τ Chapter 7 An Introduction to the Sato Theory Masayui OIKAWA, Faculty of Engneering,
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Equlbrum dstrbuton of two-dmensonal pont vortces at postve and negatve absolute temperature ( < 0 ) Yuch YATSUYANAGI Faculty of Educaton, Shzuoka Unversty 1 Onsager 1949 2 [1] Onsager Boltzmann exp( βe)
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0 Newton Leibniz ( ) α1 ( ) αn (1) a α1,...,α n (x) u(x) = f(x) x 1 x n α 1 + +α n m 1957 Hans Lewy Lewy 1970 1 1.1 Example 1.1. (2) d 2 u dx 2 Q(x)u = f(x), u(0) = a, 1 du (0) = b. dx Q(x), f(x) x = 0