玻恩-英费尔德方程

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玻恩-英费尔德方程(Born-Infeld equation)是马克斯·玻恩利奥波德·英费尔德创建的非线性偏微分方程[1]

\displaystyle (1-u_t^2)u_{xx} +2u_xu_tu_{xt}-(1+u_x^2)u_{tt}=0

行波解[编辑]

玻恩-英费尔德方程有许多行波解[2]

u[1] := 2*atan(\sqrt((a0+b1)/(a0-b1))*tan((1/2)*\sqrt(a0^2-b1^2)*\eta))+(1/2)*\pi

u[2] := 2*atan(\sqrt((a0+b1)/(a0-b1))*cot((1/2)*\sqrt(a0^2-b1^2)*\eta))+(1/2)*\pi

u[3] := 2*atan(\sqrt(a1^2+b1^2)*tan((1/2)*\sqrt(a1^2+b1^2)*\eta)/a1+b1/a1)

u[4] := 2*atan(\sqrt(a1^2+b1^2)*cot((1/2)*\sqrt(a1^2+b1^2)*\eta)/a1+b1/a1)

u[5] := 2*atan(\sqrt(a1^2+b1^2)*(coth(\sqrt(a1^2+b1^2)*eta)+csc(\sqrt(a1^2+b1^2)*eta))/a1+b1/a1)

u[6] := 2*atan(\sqrt(a1^2+b1^2)*(cosh(\sqrt(a1^2+b1^2)*\eta)+1)/(a1*sinh(\sqrt(a1^2+b1^2)*\eta))+b1/a1)

u[7] := 2*atan(\sqrt(a1^2+b1^2)*sinh(\sqrt(a1^2+b1^2)*\eta)/(a1*(cosh(\sqrt(a1^2+b1^2)*\eta)+1))+b1/a1)

u[8] := 2*atan(\sqrt((a0+b1)/(b1-a0))*cot(\sqrt(b1^2-a0^2)*\eta+sec(h*\sqrt(b1^2-a0^2)*\eta)))+(1/2)*\pi

u[9] := 2*atan(\sqrt((a0+b1)/(b1-a0))*sinh(\sqrt(b1^2-a0^2)*\eta)/(cosh(\sqrt(b1^2-a0^2)*eta)+1))

2*atan(\sqrt((a0+b1)/(b1-a0))*(cosh(\sqrt(b1^2-a0^2)*\eta)+1)/sinh(\sqrt(b1^2-a0^2)*\eta))

Born Infeld equation animation1 
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参考文献[编辑]

  1. ^ D.B. Fairlie and J.A. Mulvey Integrable Generalisations of the 2-dimensional Born Infeld Equation,University of Durham, 1993
  2. ^ Yuanxi Xie and Jiashi Tang,New Explicit Exact Solutions of the Born Infeld Equation,2005
  1. *谷超豪 《孤立子理论中的达布变换及其几何应用》 上海科学技术出版社
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