正弦-戈爾登方程

正弦-戈爾登方程是十九世紀發現的一種偏微分方程：

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

來自下面的拉量：

${\mathcal {L}}={\frac {1}{2}}(\varphi _{t}^{2}-\varphi _{x}^{2})+\cos \varphi$

由於正弦-戈爾登方程有多種孤立子解而倍受矚目。

名字是物理家熟悉的克萊因-戈爾登方程（Klein-Gordon）的雙關語。^[1]

孤立子解

利用分離變數法可得正弦-戈爾登方程的多種孤立子解。^[2]

扭型孤立子

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

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

Sine-Gordon kink soliton plot1

Sine-Gordon kink soliton plot2

鍾型孤立子

正弦-戈爾登方程有如下孤立子解：

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

其中

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

順時針孤立子

反時針孤立子

雙孤立子解

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

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

Sine-Gordon colliding soltons plot1

Sine-Gordon colliding soltons plot2

Sine-Gordon bright & dark solitons plot1

& dark solitons plot2

扭型與反扭型碰撞

扭型-扭型碰撞

駐波呼吸子

大振幅行波呼吸子

小振幅呼吸子

三孤立子解

扭型行波呼吸子與駐波呼吸子碰撞

反扭型行波呼吸子與駐波波呼吸子碰撞

呼吸子解

正弦-戈爾登方程的呼吸子解

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

幾何解釋

三維歐幾里德空間的負常曲率曲面

根據陳省身的研究，正弦-戈爾登方程有一個幾何解釋：三維歐幾里德空間的負常曲率曲面（偽球面）。^[3]

正弦-戈爾登方程是：^[4]

$\varphi _{tt}-\varphi _{xx}=\sinh \varphi$

跟戶田場論（英語：Toda field theory）有關。^[5]

量子場論

正弦-戈爾登是Thirring模特（英語：Thirring model）的S對偶。

半經典量子化：^[6]

參見

瞬子
孤子
統計場論的Coulomb氣體

參考文獻

^ Rajaraman, R. Solitons and instantons : an introduction to solitons and instantons in quantum field theory. Amsterdam https://www.worldcat.org/oclc/17480018. (1987 [printing]). ISBN 0-444-87047-4. OCLC 17480018. 缺少或|title=為空 (幫助)
^ Inna Shingareva Carlos Lizarraga Celaya, Solving Nonlinear Partial Differential Equations with Maple and Mathematica, p86-94,Springer
^ 陳省身 Geometrical interpretation of the sinh-Gordon equation。annals Polonici Mathematici XXXIX 1981
^ Poli︠a︡nin, A. D. (Andreĭ Dmitrievich). Handbook of nonlinear partial differential equations. 2nd ed. Boca Raton, FL: CRC Press https://www.worldcat.org/oclc/751520047. 2012. ISBN 978-1-4200-8723-9. OCLC 751520047. 缺少或|title=為空 (幫助)
^ Xie, Yuanxi; Tang, Jiashi. A unified method for solving sinh-Gordon-type equations. Il Nuovo Cimento B. 2006-05-10, 121 (2): 115–120. ISSN 0369-3554. doi:10.1393/ncb/i2005-10164-6.
^ Faddeev, L.D.; Korepin, V.E. Quantum theory of solitons. Physics Reports. 1978-06, 42 (1): 1–87 [2020-02-03]. doi:10.1016/0370-1573(78)90058-3. （原始內容存檔於2021-03-08）（英語）.

閱讀

Bour E (1862). "Théorie de la déformation des surfaces" （頁面存檔備份，存於網際網路檔案館）. 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.

[1] Rajaraman, R. Solitons and instantons : an introduction to solitons and instantons in quantum field theory. Amsterdam https://www.worldcat.org/oclc/17480018. (1987 [printing]). ISBN 0-444-87047-4. OCLC 17480018. 缺少或|title=為空 (幫助)

[2] Inna Shingareva Carlos Lizarraga Celaya, Solving Nonlinear Partial Differential Equations with Maple and Mathematica, p86-94,Springer

[3] 陳省身 Geometrical interpretation of the sinh-Gordon equation。annals Polonici Mathematici XXXIX 1981

[4] Poli︠a︡nin, A. D. (Andreĭ Dmitrievich). Handbook of nonlinear partial differential equations. 2nd ed. Boca Raton, FL: CRC Press https://www.worldcat.org/oclc/751520047. 2012. ISBN 978-1-4200-8723-9. OCLC 751520047. 缺少或|title=為空 (幫助)

[5] Xie, Yuanxi; Tang, Jiashi. A unified method for solving sinh-Gordon-type equations. Il Nuovo Cimento B. 2006-05-10, 121 (2): 115–120. ISSN 0369-3554. doi:10.1393/ncb/i2005-10164-6.

[6] Faddeev, L.D.; Korepin, V.E. Quantum theory of solitons. Physics Reports. 1978-06, 42 (1): 1–87 [2020-02-03]. doi:10.1016/0370-1573(78)90058-3. （原始內容存檔於2021-03-08）（英語）.

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