Bäcklund变换

Bäcklund变换是两个非线性偏微分方程之间的一对变换关系[1]

${\displaystyle phi[1](u,x,t,u_{x},u_{t},w,xi,eta,w_{x}i,w_{e}ta)=0}$

${\displaystyle phi[2](u,x,t,u_{x},u_{t},w,xi,eta,w_{x}i,w_{e}ta)=0}$

Bäcklund变换是求非线性偏微分方程精确解的一种重要的变换。

1876年瑞典数学家巴克隆德发现Sine-Gordon方程的不同解u、v

${\displaystyle u_{xt}=\sin u.\,}$
${\displaystyle v_{xt}=\sin v.\,}$

{\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}}\,\!}

${\displaystyle bt1:=(1/2)*u_{xt}-(1/2)*v_{xt}=\beta *cos((1/2)*u+(1/2)*v)*((1/2)*u_{t}+(1/2)*v_{t})}$

${\displaystyle bt2:=(1/2)*u_{xt}+(1/2)*v_{xt}=cos((1/2)*u-(1/2)*v)*((1/2)*u_{x}+(1/2)*v_{x})/\beta }$

${\displaystyle v_{xt}=\sin v.\,}$

Bäcklund变换常用于求Sine-Gordon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]

解Sine-Gordon方程

Sine-gordon kink2d

Sine-gordon 3D animation1

Sine-gordon 3D animation2

${\displaystyle u_{x}=2\beta \sin {\Bigl (}{\frac {u}{2}}{\Bigr )}}$ ，显然

${\displaystyle 2*\beta =u[x]/sin((1/2)*u)}$ ，两边对x积分，得：

${\displaystyle 2*\beta *x=2*ln(csc((1/2)*u)-cot((1/2)*u))}$

${\displaystyle 2*t/\beta =2*ln(csc((1/2)*u)-cot((1/2)*u))}$  经过三角函数运算，二式简化为

${\displaystyle 2\beta *x=2*ln(tan(u/4))}$

${\displaystyle 2t/\beta =2*ln(tan(u/4))}$

${\displaystyle 2*beta*x+2*t/beta=4*ln(tan((1/4)*u))}$

${\displaystyle u(x,t)=4*arctan(e^{\frac {\beta ^{2}*x+t}{2\beta }})}$

${\displaystyle u(x,t)=2*arctan(2*exp((1/2)*(\beta ^{2}*x+t)/\beta )/(1+(exp((1/2)*(\beta ^{2}*x+t)/\beta ))^{2}))}$

${\displaystyle u(x,t)=-2*arctan(((exp((1/2)*(\beta ^{2}*x+t)/\beta ))^{2}-1)/(1+(exp((1/2)*(\beta ^{2}*x+t)/\beta ))^{2}))}$

参考文献

1. ^ Inna Shignareve and Carlos Lizarraga-Celaya, Solving Nonlinear Partial Differential Equations with Maple and Methematica, p46, Springer
2. ^ 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
3. ^ 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