Sine-Gordon方程

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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]

利用分离变数法可得Sine-Gordon方程的多种孤立子解[1]

扭型孤立子[编辑]

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)))

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[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

双孤立子解[编辑]

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)))

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[编辑]

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[编辑]

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

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)))

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[编辑]

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[2]

参考文献[编辑]

  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.

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