Sine-Gordon方程

钟形孤立子

Sine-Gordon方程是十九世纪发现的一种偏微分方程：

$\varphi_{tt}- \varphi_{xx} = \sin\varphi$

由于Sine-Gordon方程有多种孤立子解而倍受瞩目。

孤立子解 [编辑]

Sine-Gordon方程有如下孤立子解：

$\varphi_\text{soliton}(x, t) := 4 \arctan e^{m \gamma (x - v t) + \delta}\,$

其中

$\gamma^2 = \frac{1}{1 - v^2}.$

顺时针孤立子

反时针孤立子

扭型与反扭型碰撞

扭型-扭型碰撞

驻波呼吸子

大振幅行波呼吸子

小振幅呼吸子

扭型行波呼吸子与驻波呼吸子碰撞

反扭型行波呼吸子与驻波波呼吸子碰撞

三维欧几里德空间的负常曲率曲面

sin-Gordon 方程有一个几何解释：三维欧几里德空间的负常曲率曲面^[1]

^ 陈省身 Geometrical interpretation of the sinh-Gordon equation。annals Polonici Mathematici XXXIX 1981

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Dodd, Roger K.; J. C. Eilbeck, J. D. Gibbon, H. C. Morris (1982). Solitons and Nonlinear Wave Equations. London: Academic Press. ISBN 978-0-12-219122-0.
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