贝克隆德变换 是两个非线性偏微分方程之间的一对变换关系[1]
两个非线性偏微分方程
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{\displaystyle F_{1}(u,x,t,u_{x},u_{t},u_{xx},u_{tt},u_{xt},u_{tt})=0,}
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{\displaystyle F_{2}(w,\xi ,\eta ,w_{\xi },w_{\eta },w_{\xi },w_{\xi \xi },w_{\xi \eta },w_{\eta \eta })=0}
之间的贝克隆德变换，指的是这样一对关系
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{\displaystyle \phi _{1}(u,x,t,u_{x},u_{t},w,\xi ,\eta ,w_{\xi },w_{\eta })=0,}
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{\displaystyle \phi _{2}(u,x,t,u_{x},u_{t},w,\xi ,\eta ,w_{\xi },w_{\eta })=0.}
贝克隆德变换是求非线性偏微分方程 精确解的一种重要的变换。
1876年瑞典数学家贝克隆德 发现正弦-戈尔登方程 的不同解u、v
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{\displaystyle u_{xt}=\sin u.\,}
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{\displaystyle v_{xt}=\sin v.\,}
之间有如下关系：[2]
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{\displaystyle {\begin{aligned}v_{x}&=u_{x}-2\beta \sin {\Bigl (}{\frac {u+v}{2)){\Bigr )}\\v_{t}&=-u_{t}+{\frac {2}{\beta ))\sin {\Bigl (}{\frac {v-u}{2)){\Bigr )}\end{aligned))\,\!}
这就是正弦-戈尔登方程的贝克隆德自变换。
将贝克隆德自变换第一式对t取微商，二式对x微商：
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{\displaystyle bt1:=(1/2)*u_{xt}-(1/2)*v_{xt}=\beta *cos((1/2)*u+(1/2)*v)*((1/2)*u_{t}+(1/2)*v_{t})}
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{\displaystyle bt2:=(1/2)*u_{xt}+(1/2)*v_{xt}=cos((1/2)*u-(1/2)*v)*((1/2)*u_{x}+(1/2)*v_{x})/\beta }
消除v即得
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{\displaystyle u_{xt}=\sin u.\,}
消除u项即得
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{\displaystyle v_{xt}=\sin v.\,}
贝克隆德变换常用于求正弦-戈尔登方程 、高维广义Burger I型方程、高维广义Burger II型方程的精确解：[3]
Sine-gordon kink2d Sine-gordon 3D animation1 Sine-gordon 3D animation2 利用正弦-戈尔登方程 的自贝克隆德变换解正弦-戈尔登方程：
由贝克隆德自变换
v
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{\displaystyle v_{x}=u_{x}-2\beta \sin({\frac {u+v}{2)))}
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{\displaystyle u_{x}=2\beta \sin {\Bigl (}{\frac {u}{2)){\Bigr )))
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{\displaystyle 2*\beta =u[x]/sin((1/2)*u)}
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{\displaystyle 2*\beta *x=2*ln(csc((1/2)*u)-cot((1/2)*u))}
对贝克隆德自变换第二式作同样运算得：
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{\displaystyle 2*t/\beta =2*ln(csc((1/2)*u)-cot((1/2)*u))}
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{\displaystyle 2\beta *x=2*ln(tan(u/4))}
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{\displaystyle 2t/\beta =2*ln(tan(u/4))}
二式相加得：
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{\displaystyle 2*beta*x+2*t/beta=4*ln(tan((1/4)*u))}
分离u得正弦-戈尔登方程 的一个解析解：
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{\displaystyle u(x,t)=4*arctan(e^{\frac {\beta ^{2}*x+t}{2\beta )))}
又从
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{\displaystyle 2*t/\beta =2*ln(csc((1/2)*u)-cot((1/2)*u))}
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{\displaystyle u(x,t)=2*arctan(2*exp((1/2)*(\beta ^{2}*x+t)/\beta )/(1+(exp((1/2)*(\beta ^{2}*x+t)/\beta ))^{2}))}
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{\displaystyle u(x,t)=-2*arctan(((exp((1/2)*(\beta ^{2}*x+t)/\beta ))^{2}-1)/(1+(exp((1/2)*(\beta ^{2}*x+t)/\beta ))^{2}))}
可积系统
KdV方程
^ Inna Shignareve and Carlos Lizarraga-Celaya, Solving Nonlinear Partial Differential Equations with Maple and Methematica, p46, Springer
^ 阎振亚著《复杂非线性波的构造性理论及其应用》6页科学出版社2007年
^ 阎振亚著《复杂非线性波的构造性理论及其应用》106-111页科学出版社2007年
![{\displaystyle 2*\beta =u[x]/sin((1/2)*u)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/559dec483508e6e3892dfd5a0cca09dbcc2b6432)
