正弦-戈尔登方程

利用分离变数法可得正弦-戈尔登方程的多种孤立子解。^[2]

扭型孤立子

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

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

钟型孤立子

正弦-戈尔登方程有如下孤立子解：

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

其中

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

双孤立子解

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

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

三孤立子解

${\displaystyle u=4\arctan \left({\frac {{\sqrt {1-\omega ^{2}}}\;\cos(\omega t)}{\omega \;\cosh({\sqrt {1-\omega ^{2}}}\;x)}}\right),}$ 

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

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

根据陈省身的研究，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（伪球面）。^[3]

正弦-戈尔登方程是：^[4]

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

跟户田场论（英语：Toda field theory）有关。^[5]

正弦-戈尔登是Thirring模特（英语：Thirring model）的S对偶。

半经典量子化：^[6]

• 瞬子
• 孤子
• 统计场论的Coulomb气体

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.  请检查|date=中的日期值 (帮助); 缺少或|title=为空 (帮助)
2. ^ Inna Shingareva Carlos Lizarraga Celaya, Solving Nonlinear Partial Differential Equations with Maple and Mathematica, p86-94,Springer
3. ^ 陈省身 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=为空 (帮助) 引文格式1维护：冗余文本 (link)
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. （原始内容存档于2021-03-08） （英语）. 

• Bour E (1862). "Théorie de la déformation des surfaces" （页面存档备份，存于互联网档案馆）. J. Ecole Imperiale Polytechnique. 19: 1–48.
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