DP 方程

DP 方程(Degasperis-Procesi equation)是一個模擬彌散介質中非線性波動非線性偏微分方程：

${\displaystyle \displaystyle u_{t}-u_{xxt}+2\kappa u_{x}+4uu_{x}=3u_{x}u_{xx}+uu_{xxx}}$

Degasperis-Procesi equation plot

Degasperis-Procesi equation animation

解析解

DP方程有解析解^[1]

${\displaystyle A(x):=b^{3}+3*b^{3}*{\frac {({\frac {(1-(1/4)*p^{2})}{(1-p^{2})}}-1)*({\frac {(4*(1-(1/4)*p^{2})}{(1-p^{2})}}-1)}{((2*{\frac {(1-(1/4)*p^{2}))}{(1-p^{2})}}-1+{\sqrt {\frac {(1-(1/4)*p^{2}}{(1-p^{2}))}}}*cosh(x))}}}$

${\displaystyle B(x,t):={\frac {x}{p}}+4*a^{2}*t+log({\frac {(\alpha +1+exp(x*(\alpha -1)))}{(alpha-1+(e)^{(}x*(\alpha +1)))}})}$

其中

${\displaystyle alpha:={\frac {((2*a-1)*(a+1)}{((2*a+1)*(a-1)))^{(}1/2)}}}$

代入，

取參數p = .3, b = 1.1

${\displaystyle A(x)=1.331+{\frac {.9764}{(1.148+1.036*cosh(x))}}}$

${\displaystyle B(x,t)=3.33*x+4.2967*t+ln({\frac {(5.418+exp(3.418*x))}{(3.418+exp(5.4181*x))}})}$

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