查看“︁正弦-戈尔登方程”︁的源代码

[[File:Bellsoliton2.gif|200px|thumb|right|钟形孤立子]]
'''正弦-戈尔登方程'''是十九世纪发现的一种偏微分方程：

<math>\varphi_{tt}- \varphi_{xx} = \sin\varphi </math>

來自下面的[[拉格朗日量|拉量]]：

<math>\mathcal{L} = \frac{1}{2} (\varphi_{t}^2- \varphi_{x}^2) + \cos\varphi </math>

由于正弦-戈尔登方程有多种[[孤立子]]解而倍受瞩目。

名字是物理家熟悉的[[克莱因-戈尔登方程]]（Klein-Gordon）的雙關語。<ref>{{Cite book|chapter=Solitons and instantons : an introduction to solitons and instantons in quantum field theory|url=https://www.worldcat.org/oclc/17480018|date=(1987 [printing])|location=Amsterdam|isbn=0-444-87047-4|oclc=17480018|last=Rajaraman, R.}}</ref>

==孤立子解==
利用[[分离变数法]]可得正弦-戈尔登方程的多种孤立子解。<ref>Inna Shingareva  Carlos Lizarraga Celaya, Solving Nonlinear  Partial Differential Equations with Maple and Mathematica, p86-94,Springer</ref>
===扭型孤立子===

<math>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)))</math>

<math>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)))</math>
{|
|[[File:Sine-Gordon kink soliton plot1.gif|frame|Sine-Gordon kink soliton plot1]]
|[[File:Sine-Gordon kink soliton plot2.gif|frame|Sine-Gordon kink soliton plot2]]
|}

===钟型孤立子===
正弦-戈尔登方程有如下孤立子解：

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

其中

: <math>\gamma^2 = \frac{1}{1 - v^2}.</math>

{|
|-
| [[Image:Sine gordon 1.gif|frame|顺时针孤立子]]
 
| [[Image:Sine gordon 2.gif|frame|反时针孤立子]]
|}

===双孤立子解===
<math>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)))</math>

<math>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)))</math>
{|
|-
|[[File:Sine-Gordon colliding soltons plot1.gif|frame|Sine-Gordon colliding soltons plot1]]
|[[File:Sine-Gordon colliding soltons plot2.gif|frame|Sine-Gordon colliding soltons plot2]]
|}
{|
|[[File:Sine-Gordon bright and dark solitons plot1.gif|frame|Sine-Gordon bright & dark solitons plot1]]
|[[File:Sine-Gordon bright and dark solitons plot2.gif|frame|& dark solitons plot2]]
|}
{|
|-
| [[File:Sine gordon 3.gif|frame|''扭型与反扭型碰撞'']]
| [[Image:Sine gordon 4.gif|frame|''扭型-扭型碰撞'' ]]
|}

{|
|-
| [[Image:Sine gordon 5.gif|frame|''驻波[[呼吸子]]'']]
| [[File:Sine gordon 6.gif|frame|''大振幅行波呼吸子]]
 [[File:Sine gordon 7.gif|frame|小振幅呼吸子]]

|}

===三孤立子解===
{|
|-
| [[Image:Sine gordon 5.gif|frame|''扭型行波呼吸子与驻波呼吸子碰撞'']]
| [[File:Sine gordon 6.gif|frame|''反扭型行波呼吸子与驻波波呼吸子碰撞]]
|}

==呼吸子解==
[[File:Breather.gif|thumb|right|250px|正弦-戈尔登方程的呼吸子解]]
:<math>u = 4 \arctan\left(\frac{\sqrt{1-\omega^2}\;\cos(\omega t)}{\omega\;\cosh(\sqrt{1-\omega^2}\; x)}\right),</math>

<math>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)))</math>

<math>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)))</math>
{|
|[[File:Sine-Gordon breather plot1.gif|frame|Sine-Gordon breather plot1]]
|[[File:Sine-Gordon breather plot2.gif|frame|Sine-Gordon breather plot2]]
|}

==几何解释==
[[File:Pseudosphere.png|thumb|right|300px|三维欧几里德空间的负常曲率曲面]]
根據[[陳省身]]的研究，正弦-戈尔登方程有一个几何解释：三维[[欧几里德空间]]的负常[[曲率]]曲面（[[偽球面]]）。<ref>[[陈省身]] Geometrical interpretation of the sinh-Gordon equation。annals Polonici Mathematici XXXIX 1981</ref>

正弦-戈尔登方程是：<ref>{{Cite book|edition=2nd ed|chapter=Handbook of nonlinear partial differential equations|url=https://www.worldcat.org/oclc/751520047|publisher=CRC Press|date=2012|location=Boca Raton, FL|isbn=978-1-4200-8723-9|oclc=751520047|last=Poli︠a︡nin, A. D. (Andreĭ Dmitrievich)}}</ref>

<math>\varphi_{tt}- \varphi_{xx} = \sinh\varphi </math>

跟{{Le|戶田場論|Toda field theory}}有關。<ref>{{Cite journal|title=A unified method for solving sinh-Gordon-type equations|url=http://doi.org/10.1393/ncb/i2005-10164-6|last=Xie|first=Yuanxi|last2=Tang|first2=Jiashi|date=2006-05-10|journal=Il Nuovo Cimento B|issue=2|doi=10.1393/ncb/i2005-10164-6|volume=121|pages=115–120|issn=0369-3554}}</ref>

== 量子場論 ==
正弦-戈尔登是{{Le|Thirring模特|Thirring model}}的[[S對偶]]。

半經典量子化：<ref>{{Cite journal|title=Quantum theory of solitons|url=https://linkinghub.elsevier.com/retrieve/pii/0370157378900583|last=Faddeev|first=L.D.|last2=Korepin|first2=V.E.|date=1978-06|journal=Physics Reports|issue=1|doi=10.1016/0370-1573(78)90058-3|volume=42|pages=1–87|language=en|access-date=2020-02-03|archive-date=2021-03-08|archive-url=https://web.archive.org/web/20210308055324/https://linkinghub.elsevier.com/retrieve/pii/0370157378900583|dead-url=no}}</ref>

== 參見 ==

* [[瞬子]]
* [[孤子]]
*[[統計場論]]的Coulomb氣體

==参考文献==
<references/>

== 閱讀 ==

*Bour E (1862). [https://gallica.bnf.fr/ark:/12148/bpt6k433694t/f5.item "Théorie de la déformation des surfaces"] {{Wayback|url=https://gallica.bnf.fr/ark:/12148/bpt6k433694t/f5.item |date=20200812035856 }}. J. Ecole Imperiale Polytechnique. 19: 1–48.
*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.

{{非线性偏微分方程}}
{{量子场论}}
{{弦理论}}
[[category:非线性偏微分方程]]