# 玻恩-英费尔德方程

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

## 行波解

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

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

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

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

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

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

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

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

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

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

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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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