孤立波：从连续到离散 上 海大学 张大军 （静宜大学 2013年11月）
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.
KdV 2-soliton
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.
自然界中的孤立波
实验室中的孤立波
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.
John Scott Russell August, 1834 z (9 May 1808-8 June 1882) Education: Edinburgh, St. Andrews, Glasgow August, 1834 z
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)
Russell的实验
Russell的实验
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.
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.”
非线性模型：波的坍塌 非线性方程： 速度： 行波解： 速度快 速度慢
非线性模型：波的坍塌 t = 0 t > 0
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
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
1834 ~ 1895 J Scott Russell (1808-1882) Diederik Korteweg (1848-1941)
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：
Russell’s Grate Wave---Solitary Wave Travelling wave
animation
1895 ~ 1960s Diederik Korteweg (1848-1941) Martin D. Kruskal (1925-2006)
FPU问题 Fermi-Pasta-Ulam problem （Los Alamos, 1950’s） Study the thermalization process of a solid Computer use (Maniac)
Birth of Solitons (孤立子) Martin David Kruskal 1925-2006 导师：Courant 院士 Father of “Soliton” Solitons(partical property, 1965) FPU问题 Toda Lattice KdV方程的数值解
粒子特征（ Soliton）
Inverse Scattering Transform （反散射变换）
Exact solutions to the KdV 1-soliton solution
Exact solutions to the KdV 2-soliton solution
sine-Gordon方程
sine-Gordon方程 机械孤子： Kink Anti-Kink
animation
animation
animation
Breather http://www.math.h.kyoto-u.ac.jp/~takasaki/soliton-lab/gallery/solitons/sg-e.html
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
Hirota双线性方法 KdV方程: 双线性方程: 变换: 级数解: 1孤子解:
Hirota双线性方法 2孤子解：
Hirota双线性方法 n孤子解：
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
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
如何看待离散？ 映射 f(n) f(x) 实数x 整数n: (×) （x0, t0） x = x0 + n p t = t0 + m q u (x, t) u (n, m) 映射
非局部特征
例：非线性叠加公式与离散方程 递推关系=离散系统
例：特殊函数与离散方程 递推关系=离散系统
例：凸五边形映射
超离散系统
超离散系统 元胞自动机（Wolfram） 黑白格子
超离散可积系统 t=0 每次移动1球， 将最左边的球移到右边最近的空盒子， 重复此步骤直至所有的球都被移动一次。 t=1 2018/11/13 元胞自动机与孤子系统的离散化
[Tokihiro, Tokyo] t=0 t=1 t=2 t=3 t=4 2018/11/13 元胞自动机与孤子系统的离散化
基本运算 平移 差分 导数 Leibniz 公式
符 号
为什么要离散？ To be sure, it has been pointed out that the introduction of a space-time continuum may be considered as contrary to nature in view of the molecular structure of everything which happens on a small scale. It is manitained that perhaps the success of the Heisenberg method points to a purely algebraic description of nature, that is the elimination of continuum functions from physics. Then, however, we must also give up, by principle, the space-time continuum. It is not unimaginable that human ingenuity will some day find methods which will make possible to proceed along such a path. … … [A. Einstein, 1936]
为什么要离散？ Stanislav Smirnov 2010 ICM Fields Medal
Nijhoff: The study of integrability of discrete systems forms at the present time the most promising route towards a general theory of difference equations and discrete systems. Hietarinta: Continuum integrability is well established and all easy things have already been done; discrete integrability, on the other hand, is relatively new and in that domain there are still new things to be discovered.
Bobenko: The aim of discrete differential geometry is the discretization of classical differential geometry, that is, to find proper discrete analogs of differential geometric notions and to develop at the discrete level a corresponding theory.
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
Sato理论的离散化 反散射变换：Kruskal, 1967 双线性方法：Hirota, 1971 Sato’s Theory：Sato, 1980 (2003 Wolf) Kyoto Group: Data, Jimbo, Miwa, 1980’s Sato理论ABC Web: http://www.science.shu.edu.cn/siziduiwu/zdj/index.htm
指数函数 指数函数 离散指数 平面波因子 N次代数多项式的根
双线性等式
Miwa变换/映射 文献：Data, Jimbo, Miwa, JPSJ, 1981, 1982
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
多维相容性 KdV非线性叠加公式
多维相容性 KdV非线性叠加公式 相容性？ 相同的
Searching 多维相容系统 多维相容性 Frank Nijhoff(2011), Nijhoff(2012), Bobenko-Suris (2012) Adler-Bobenko-Suris (ABS) Classification (2003, 09) Hietarinta’s search 4D-相容：ABS(IMRN 2011) Consistency Around the Cube
ABS链方程(ABS, 2003, CMP) ……
多维相容性应用1：Bäcklund变换 Consistent Cube Bäcklund变换
多维相容性应用1：Lax Pair Bäcklund变换 Lax Pair
其他方法 双线性（理解离散的双线性） Cauchy矩阵（矩阵方程与离散系统） 直接线性化方法（特征曲线与离散系统） 。。。
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
连续极限 KdV的非线性叠加公式 链势KdV (lpKdV):
连续极限(I) lpKdV 半离散pKdV 变量关系
连续极限(II) 半离散pKdV pKdV 变量关系
问 题 离散的数学工具 量子几何 应用 离散化与数值计算
谢 谢 Web: www.science.shu.edu.cn/siziduiwu/zdj/index.htm Email: djzhang@staff.shu.edu.cn 主要合作者 曹策问老师 Jarmo Hietarinta (Turku, Finland) Frank W Nijhoff (Leeds, UK) KM Tamizhmani (Pondicherry, India)