孤立波：从连续到离散

1. 孤立波：从连续到离散 上 海大学 张大军 （静宜大学 2013年11月）

7. 孤立波的特征：波+粒子 Unlike normal waves they will never merge—so a small wave is overtaken by a large one, rather than the two combining.

8. KdV 2-soliton

9. 伟大的水波 • The Scott Russell Aqueduct on the Union Canal near Heriot-Watt University, 12 July 1995. For the technically minded, the aqueduct is 89.3 m long, 4.13m wide, and 1.52m deep.

10. 自然界中的孤立波

11. 实验室中的孤立波

12. 伟大的水波 • The Scott Russell Aqueduct on the Union Canal near Heriot-Watt University, 12 July 1995. For the technically minded, the aqueduct is 89.3 m long, 4.13m wide, and 1.52m deep.

13. John Scott Russell (9 May 1808-8 June 1882) Education: Edinburgh, St. Andrews, Glasgow • August, 1834 • z

14. Russell’s observation • A large solitary elevation, a rounded, smooth and well defined heap of water, which continued its course along the channel apparently without change of form or diminution of speed … Its height gradually diminished, and after a chase of one or two miles I lost it in the windings of the channel. Such, in the month of August 1834, was my first chance interview with that singular and beautiful phenomenon. (Russell, 1838)

15. Russell的实验

16. Russell的实验

18. 研 究 结 论 • The waves are stable, and can travel over very large distances (normal waves would tend to either flatten out, or steepen and topple over) • The speed depends on the size of the wave, and its width on the depth of water. • Unlike normal waves they will never merge—so a small wave is overtaken by a large one, rather than the two combining. • If a wave is too big for the depth of water, it splits into two, one big and one small.

19. The Great Wave Translation • Solitary waves --- J.S. Russell • Airy: “even in an uniform-canal of rectangular section, are no longer propagated without change of type.” Solitary waves of permanent form do not exist! • Russell: “completely the opposite of that to which we should be led on the same grounds.”

20. 非线性模型：波的坍塌 • 非线性方程： • 行波解： • 速度： 速度快 速度慢

21. 非线性模型：波的坍塌 t = 0 t > 0

22. Scott Russell 的其他 • 组建 the Royal Commission for the Exhibition of 1851 • 成立J Scott Russell & Co. shipbuilding company The Great Eastern

23. Scott Russell 的其他 • 组建 the Royal Commission for the Exhibition of 1851 • 成立J Scott Russell & Co. shipbuilding company • 评价：未提Solitary waves • a better scientist than a businessman

24. 1834 ~ 1895 J Scott Russell (1808-1882) Diederik Korteweg (1848-1941)

25. Korteweg-de Vries(KdV)方程 • Korteweg(1848-1941) Amsterdam大学教授 • Gustav de Vries : K的学生 流体力学基本模型 KdV方程： 行波解：

26. Russell’s Grate Wave---Solitary Wave Travelling wave

27. animation

28. 1895 ~ 1960s Diederik Korteweg (1848-1941) Martin D. Kruskal (1925-2006)

29. FPU问题 • Fermi-Pasta-Ulam problem （Los Alamos, 1950’s） • Study the thermalization process of a solid • Computer use (Maniac)

30. Birth of Solitons (孤立子) • Martin David Kruskal • 1925-2006 • 导师：Courant • 院士 • Father of “Soliton” • FPU问题 • Toda Lattice • KdV方程的数值解 Solitons(partical property, 1965)

31. 粒子特征（ Soliton）

33. Inverse Scattering Transform （反散射变换）

34. Exact solutions to the KdV 1-soliton solution

35. Exact solutions to the KdV 2-soliton solution

36. sine-Gordon方程

37. sine-Gordon方程 • 机械孤子： Kink Anti-Kink

38. animation

39. animation

40. animation

41. http://www.math.h.kyoto-u.ac.jp/~takasaki/soliton-lab/gallery/solitons/sg-e.htmlhttp://www.math.h.kyoto-u.ac.jp/~takasaki/soliton-lab/gallery/solitons/sg-e.html Breather

42. 方法举例 • 反散射变换/Riemann-Hilbert方法，基于Lax对 • Hirota方法 /双线性方法 • Royal Hirota 日本学者

43. Hirota双线性方法 变换: 双线性方程: KdV方程: 级数解: 1孤子解:

44. Hirota双线性方法 • 2孤子解：

45. Hirota双线性方法 • n孤子解：

46. Hirota双线性方法 • 反散射变换 • Hirota方法 • Sato理论 • 2小时/天 X 2周 • 2小时/天 X 1天 • 2小时/天 X 2月

47. 离 散 • 如何看待离散，为什么要离散？ • 可积离散 • 多维相容性 • 从离散到连续：连续极限

48. u (x, t) u (n, m) t = t0 + m q （x0, t0） x = x0 + n p 如何看待离散？ (×) 实数x 整数n: 映射 f(x) f(n)

49. 非局部特征

50. 例：非线性叠加公式与离散方程 递推关系=离散系统