玻恩-英费尔德方程

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

${\displaystyle \displaystyle (1-u_{t}^{2})u_{xx}+2u_{x}u_{t}u_{xt}-(1+u_{x}^{2})u_{tt}=0}$

行波解

玻恩-英费尔德方程有许多行波解^[2]

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

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

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

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

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

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

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

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

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

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

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参考文献

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

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