正弦-戈尔登方程

正弦-戈尔登方程是十九世纪发现的一种偏微分方程： φ t t − φ x x = sin ⁡ φ {\displaystyle \varphi _{tt}-\varphi _{xx}=\sin \varphi } 來自下面的拉量： L = 1 2 ( φ t 2 − φ x 2 ) + cos ⁡ φ {\displaystyle {\mathcal {L}}={\frac {1}{2}}(\varphi _{t}^{2}-\varphi _{x}^{2})+\cos \varphi } 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。 利用分离变数法可得正弦-戈尔登方程的多种孤立子解。 p 1 := − 4 ∗ a r c t a n ( ( 1 / 2 ) ∗ ( 1.5 ∗ e x p ( − 4 ∗ s q r t ( 2 ) ) − e x p ( 2 ∗ x ∗ s q r t ( 2 ) ) ) ∗ e x p ( t − 2 − x ∗ s q r t ( 2 ) + 2 ∗ s q r t ( 2 ) ) ∗ s q r t ( 2 ) / ( 1.5 ∗ e x p ( − 4 ) + e x p ( 2 ∗ t ) ) ) {\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)))} p 2 := − 4 ∗ a r c t a n ( ( 1 / 2 ) ∗ ( 1.5 ∗ e x p ( 2 ∗ x ∗ s q r t ( 2 ) ) − e x p ( − 4 ∗ s q r t ( 2 ) ) ) ∗ e x p ( t − 2 − x ∗ s q r t ( 2 ) + 2 ∗ s q r t ( 2 ) ) ∗ s q r t ( 2 ) / ( 1.5 ∗ e x p ( 2 ∗ t ) + e x p ( − 4 ) ) ) {\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)))} 正弦-戈尔登方程有如下孤立子解： φ soliton ( x , t ) := 4 arctan ⁡ e m γ ( x − v t ) + δ {\displaystyle \varphi _{\text{soliton}}(x,t):=4\arctan e^{m\gamma (x-vt)+\delta }\,} 其中 γ 2 = 1 1 − v 2 . {\displaystyle \gamma ^{2}={\frac {1}{1-v^{2}}}.} p x 1 := ( 8 ∗ ( 1.5 ∗ e x p ( − 4 ) + e x p ( 2 ∗ t ) ) ) ∗ e x p ( t − 2 − x ∗ s q r t ( 2 ) + 2 ∗ s q r t ( 2 ) ) ∗ ( e x p ( 2 ∗ x ∗ s q r t ( 2 ) ) + 1.5 ∗ e x p ( − 4 ∗ s q r t ( 2 ) ) ) / ( 4.50 ∗ e x p ( − 8 ) + 2 ∗ e x p ( 4 ∗ t ) + 2.25 ∗ e x p ( 2 ∗ t − 4 − 2 ∗ x ∗ s q r t ( 2 ) − 4 ∗ s q r t ( 2 ) ) + 3.0 ∗ e x p ( 2 ∗ t − 4 ) + e x p ( 2 ∗ t − 4 + 2 ∗ x ∗ s q r t ( 2 ) + 4 ∗ s q r t ( 2 ) ) ) {\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)))} p x 2 := − ( 8 ∗ ( 1.5 ∗ e x p ( 2 ∗ t ) + e x p ( − 4 ) ) ) ∗ e x p ( t − 2 − x ∗ s q r t ( 2 ) + 2 ∗ s q r t ( 2 ) ) ∗ ( 1.5 ∗ e x p ( 2 ∗ x ∗ s q r t ( 2 ) ) + e x p ( − 4 ∗ s q r t ( 2 ) ) ) / ( 4.50 ∗ e x p ( 4 ∗ t ) + 2 ∗ e x p ( − 8 ) + 2.25 ∗ e x p ( 2 ∗ t − 4 + 2 ∗ x ∗ s q r t ( 2 ) + 4 ∗ s q r t ( 2 ) ) + 3.0 ∗ e x p ( 2 ∗ t − 4 ) + e x p ( 2 ∗ t − 4 − 2 ∗ x ∗ s q r t ( 2 ) − 4 ∗ s q r t ( 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)))} u = 4 arctan ⁡ ( 1 − ω 2 cos ⁡ ( ω t ) ω cosh ⁡ ( 1 − ω 2 x ) ) , {\displaystyle u=4\arctan \left({\frac {{\sqrt {1-\omega ^{2}}}\;\cos(\omega t)}{\omega \;\cosh({\sqrt {1-\omega ^{2}}}\;x)}}\right),} p z 1 := 4 ∗ a r c t a n ( ( 28.460498941515413988 ∗ ( e x p ( 1.8973665961010275992 ∗ x ) + s q r t ( e x p ( 3.7947331922020551984 ∗ x ) ) ) ) ∗ s i n ( O m e g a T ) ∗ c s g n ( 1 / c o s ( O m e g a T ) ) ∗ e x p ( − .94868329805051379960 ∗ x ) / ( 18. + 5. ∗ e x p ( 1.8973665961010275992 ∗ x ) ) ) {\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)))} p z 2 := 4 ∗ a r c t a n ( ( 28.460498941515413988 ∗ ( e x p ( 1.8973665961010275992 ∗ x ) + s q r t ( e x p ( 3.7947331922020551984 ∗ x ) ) ) ) ∗ s i n ( O m e g a T ) ∗ c s g n ( 1 / c o s ( O m e g a T ) ) ∗ e x p ( − .94868329805051379960 ∗ x ) / ( 18. + 5. ∗ e x p ( 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）。 正弦-戈尔登方程是： φ t t − φ x x = sinh ⁡ φ {\displaystyle \varphi _{tt}-\varphi _{xx}=\sinh \varphi } 跟戶田場論（英语：Toda field theory）有關。 正弦-戈尔登是Thirring模特（英语：Thirring model）的S對偶。 半經典量子化： 瞬子 孤子 統計場論的Coulomb氣體 Rajaraman, R. 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