# Sine-Gordon方程

Sine-Gordon方程是十九世纪发现的一种偏微分方程：

${\displaystyle \varphi _{tt}-\varphi _{xx}=\sin \varphi }$

## 孤立子解

### 扭型孤立子

${\displaystyle p1:=-4*arctan((1/2)*(1.5*exp(-4*sqrt(2))-exp(2*x*sqrt(2)))*exp(t-2-x*sqrt(2)+2*sqrt(2))*sqrt(2)/(1.5*exp(-4)+exp(2*t)))}$

${\displaystyle p2:=-4*arctan((1/2)*(1.5*exp(2*x*sqrt(2))-exp(-4*sqrt(2)))*exp(t-2-x*sqrt(2)+2*sqrt(2))*sqrt(2)/(1.5*exp(2*t)+exp(-4)))}$

 Sine-Gordon kink soliton plot1 Sine-Gordon kink soliton plot2

### 钟型孤立子

Sine-Gordon方程有如下孤立子解：

${\displaystyle \varphi _{\text{soliton}}(x,t):=4\arctan e^{m\gamma (x-vt)+\delta }\,}$

${\displaystyle \gamma ^{2}={\frac {1}{1-v^{2}}}.}$
 顺时针孤立子 反时针孤立子

### 双孤立子解

${\displaystyle px1:=(8*(1.5*exp(-4)+exp(2*t)))*exp(t-2-x*sqrt(2)+2*sqrt(2))*(exp(2*x*sqrt(2))+1.5*exp(-4*sqrt(2)))/(4.50*exp(-8)+2*exp(4*t)+2.25*exp(2*t-4-2*x*sqrt(2)-4*sqrt(2))+3.0*exp(2*t-4)+exp(2*t-4+2*x*sqrt(2)+4*sqrt(2)))}$

${\displaystyle px2:=-(8*(1.5*exp(2*t)+exp(-4)))*exp(t-2-x*sqrt(2)+2*sqrt(2))*(1.5*exp(2*x*sqrt(2))+exp(-4*sqrt(2)))/(4.50*exp(4*t)+2*exp(-8)+2.25*exp(2*t-4+2*x*sqrt(2)+4*sqrt(2))+3.0*exp(2*t-4)+exp(2*t-4-2*x*sqrt(2)-4*sqrt(2)))}$

 Sine-Gordon colliding soltons plot1 Sine-Gordon colliding soltons plot2
 Sine-Gordon bright & dark solitons plot1 & dark solitons plot2
 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

### 三孤立子解

 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

## 呼吸子解

Sine-Gordon方程的呼吸子解
${\displaystyle u=4\arctan \left({\frac {{\sqrt {1-\omega ^{2}}}\;\cos(\omega t)}{\omega \;\cosh({\sqrt {1-\omega ^{2}}}\;x)}}\right),}$

${\displaystyle pz1:=4*arctan((28.460498941515413988*(exp(1.8973665961010275992*x)+sqrt(exp(3.7947331922020551984*x))))*sin(OmegaT)*csgn(1/cos(OmegaT))*exp(-.94868329805051379960*x)/(18.+5.*exp(1.8973665961010275992*x)))}$

${\displaystyle pz2:=4*arctan((28.460498941515413988*(exp(1.8973665961010275992*x)+sqrt(exp(3.7947331922020551984*x))))*sin(OmegaT)*csgn(1/cos(OmegaT))*exp(-.94868329805051379960*x)/(18.+5.*exp(1.8973665961010275992*x)))}$

 Sine-Gordon breather plot1 Sine-Gordon breather plot2

## 几何解释

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[2]

## 参考文献

1. ^ Inna Shingareva Carlos Lizarraga Celaya, Solving Nonlinear Partial Differential Equations with Maple and Mathematica, p86-94,Springer
2. ^ 陈省身 Geometrical interpretation of the sinh-Gordon equation。annals Polonici Mathematici XXXIX 1981
• Rajaraman, R. (1989). Solitons and Instantons: An Introduction to Solitons and Instantons in Quantum Field Theory. North-Holland Personal Library. 15. North-Holland. pp. 34–45. ISBN 978-0-444-87047-6.
• Polyanin, Andrei D.; Valentin F. Zaitsev (2004). Handbook of Nonlinear Partial Differential Equations. Chapman & Hall/CRC Press. pp. 470–492. ISBN 978-1-58488-355-5.
• 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.
• Georgiev DD, Papaioanou SN, Glazebrook JF (2004). "Neuronic system inside neurons: molecular biology and biophysics of neuronal microtubules". Biomedical Reviews 15: 67–75.
• Georgiev DD, Papaioanou SN, Glazebrook JF (2007). "Solitonic effects of the local electromagnetic field on neuronal microtubules". Neuroquantology 5 (3): 276–291.