Tautochrone | Korteweg-de Vries方程与逆散射变换

作者介绍:牛津大学数学专业

目录与序言

目录:
  • Korteweg-de Vries方程
  • Schrödinger方程的散射
  • Lax算子对
  • 正散射问题
  • 逆散射问题
  • 散射资料的演化
  • 孤立子解
  • 总结与挖坑
Russell最早观察到水面上的孤立波现象,他在1844年记录到:
I was observing the motion of a boat which was rapidly drawn along a narrow channel by a pair of horses, when the boat suddenly stopped - not so the mass of water in the channel which it had put in motion; it accumulated round the prow of the vessel in a state of violent agitation, then suddenly leaving it behind, rolled forward with great velocity, assuming the form of 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. I followed it on horseback, and overtook it still rolling on at a rate of some eight or nine miles an hour, preserving its original figure some thirty feet long and a foot to a foot and a half in height. 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 which I have called the Wave of Translation’.
这种波能以稳定的形状传播,并且在与其它波进行非线性相互作用后,仍然能保持原来的形状,因此被称为孤立波(solitary wave),因其粒子性也称为孤立子(soliton)。

Korteweg-de Vries方程


Schrödinger方程的散射

接下来介绍一种被称为逆散射变换(inverse scattering transform, IST)的方法处理KdV方程的初值问题,这种方法也能被用于多个非线性PDE系统(“可积系统”)。
给定势能求散射资料称为正散射问题,而给定散射资料求势能称为逆散射问题。

Lax算子对

那么散射问题和KdV方程有什么关系呢?我们如何把KdV方程与Schrödinger方程联系在一起?
接下来分步具体求解。

正散射问题


逆散射问题

求解逆散射问题是非常复杂的。此处使用的逆散射方法由Gel'fand和Levitan提出。首先我们利用上面给出的散射资料,定义函数:

散射资料的演化


孤立子解


总结与挖坑

我们用逆散射变换完全求解了KdV方程的初值问题,因此KdV方程就成为了可积系统的一个例子。而这个方法也能用来求解别的可积系统。孤立子现象,逆散射变换,可积系统是一个超级大坑,资料浩如烟海。我个人水平有限,只能理解其中很小的一部分。关于逆散射变换,根据project需要我可能会继续写如下相关的课题:
  • 利用WKB近似处理非线性PDE的逆散射问题,并引入Hamilton力学的范
  • 通过求解对应的Riemann-Hilbert边值问题来处理逆散射问题

参考

[1] Ablowitz & Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, London Mathematical Society Lecture Note Series, 149.
[2] Dunajski, Cambridge Part II notes on Integrable Systems.
[3] Arnold, Mathematical Methods of Classical Mechanics (4th ed.)
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